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PAGES MISSING, WITHIN THE, , BOOK ONLY, DRENCHED BOOK
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1, , 64598
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00 P, , 750-28-4-8 - -U),UOO., 1, , OSMANIA UNIVERSITY LIBRARY, Call, , No., , ^ryJu 4, , Accession No., , Author, , Title, .*3t, , This book should be returned on or before the, , <**, , marked
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ELEMENTS OF, , PROPERTIES OF MATTER, WITH TYPICAL NUMERICALS SOLVED, I, , FOR DEGREE CLASSES, , ], , by, , D., , S, , ., , DELHI, , S,, , MATHUR, , C H AW-D, , &, , C, , NEW DELHI, JULLUNDUR, LTJCKNOW - BOMBAY
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SPECIAL FEATURES, 1., , Detailed, , and simple treatment, with each, , step fully explain-, , ed., 2., , 336 illustrative diagrams given., , 3, , A, , large, , number, , of typical numerical problems solved,, , cluding those set in the various University, covering 150 pages or more, of the book,, 4, , 5., , (in-, , Examinations),, , Illustrative solutions, with the use of logarithms,, the margin to the left in the first two chapters., , shown on, , Useful appendices, on Differential and Integral Calculus,, together with those on important Trigonometrical Relations, and the use of Logarithms, included, as also Logarithmic, Tables and Tables of Important Constants., , Published by, , S., , for, , CHAND &, , Shyam Lai, , 16B/4, Asaf, , (, , All profits, , from, , CO., , Charitable Trust,, , AH Road, New, , this, , Delhi, , book are spent on, , CHAND, , Ram, , & CO., NEW DBLKI, , Lamington Road, , DBLHI, JULLUNDUB, LUOKNQW, BOMBAY, , S., , Nagar, Fountain, M*i Hiran Gate, Hazrat Ganj, , First published, , Seventh Edition, , Price, , :, , October,, July,, , 1949, , J962, , Rs, 9-00, , 0. S. Sharma, for a. unana <s uo. t, Printed a* Rajendra Ravindra Printer*, (P) Ltd.,, , PublMed by, , charities-), , Kam, Ram, , Wagar,, , Nagar,, , New, New, , Delhi and, Delhi- 1.
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CONTENTS, PA<, Chapter, , I, , Units and Dimensions., , 1-, , Fundamental and Derived Units Principal Systems of Units, Dimensions Dimensional Formulae and Equations Uses of DimenLimitations of Dimensional Analysis Solved Exsional Equations, Units, , amples Exercise I., 20Chapter II Motion along a Curve The Projectile., Acceleration Couple Work, Rotation Angular, Velocity Angular, done by a Couple Relation between Couple and Angular Acceleration, in the, Hodograph Uniform Circular, Centrifugal Force Practical Applications, of Centripetal and Centrifugal Forces Other Effects and Applications, of Centrifugal Force The Projectile Motion of a Projectile in a nonHorizontal Range of a Projectile Maximum Height, resisting medium, attained by a Projectile Angle of Projection for Maximum Range, Range on an Inclined Plane Resultant Velocity of a Projectile at a, given instant Solved Examples Exercise II., , The Hodograph Velocity, Motion Centripetal Force, , 48, Moment of Inertia Energy of Rotation., of Inertia and its Physical Significance Radius of Gyration, Etpression for Moment of Inertia Torque General Theorems on, Moment of Inertia Calculation of the Moment of Inertia of a Body, Its Units etc., Particular Cases of Moments of Inertia Table of Moments of Inertia Routh's Rule Practical Methods for the Determination of Moments of Inertia Angular Moment and Angular ImpulseLaw of Conservation of Angular Momentum Laws of Rotation Kinetic Energy of RoUtiQri, Acceleration of a body rolling down an inclined" riline, uraphical Representation of Plane Vectors Precession, The Gyrostat Gyroscope The Gyrostatic Pendulum Case of a, Rolling Disc or Hoop Gyrostatic and, Gyroscopic Applications, Solved Examples Exercise III., , Chapter, , III, , Moment, , Chapter IV, , Ill, Simple Harmonic Motion., Characteristics, Linear S.H.M. Equation of Simple, oj#a, Harmohic Motion ComposKrfcm of Two Simple Harmonic Motions, (Graphicat^Qd Analytical JtXEftoas) Composition of two equal circular, motions in oppis^ite directing Energy of a Particle in simple Harmonic Motion, A vehkge Kinetv and Potential Energies of a Particle in, S. H.M., Solved ExaSlpl^s Exercise IV., Definition, , V Measurement of Mass The Balance., 146, Mass and Weight The Common Balance Essentials or Requisites of a, Good Balance Faults in a Balance Determination of True Weight, Correction for Buoyancy Solved Examples Exercise V., 160, Chapter VI Acceleration due to Gravity., Acceleration due to Gravity The Simple Pendulum Borda's Pendulum-^ Compound Pendulum -fnterchangeability of the Centres of Suspension and Oscillation Centre of Percussion Other points, collinear, with the e.g., about which the time-period is the same Conditions for, Maximum and Minimum Time -periods Bar Pendulum Owen's modification of the bar pendulum Kater *s Reversible Pendulum Kater' s, Method of Coincidences Computed Time BesseVs Contribution Errors, in the Compound Pendulunfand their Remedies, Other Improvements, due to Bessel Conical Pendulum Steam Eogine Governor Other, methods for the determination of # Variation of the value of g~~, Determination of the value of g at Sea Local and Temporal Changes in, the value of g Gravity SurveyGeophysical, Prospecting Solved, , Chapter, , -, , Examples, , Exercise VI.
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(v/), , 1'AOES, Chapter VII, , Gravitation., , 224273, , Kepler's Laws Note on Newton's deductions from Kepler's, Newton's Law of Gravitation Determination of the Gravitational Corstant, Density of the Earth Qualities of Gravitation Law, of Gravitation and the Theory of Relativity Gravitational Field, InGravitational Potential Potential Energy Gratensity of the Field, vitational potential at a point distant r from a body of mass m, VeloPotential at a point Outside, city of Escape Equipotential Surface, and Inside a Spherical Shell Gravitational Field Inside a Spherical, Shell or a Hollow Sphere, Potential and Field Intensity due to a Solid, Sphere at a point (p Inside the Sphere and (//) Outside the Sphere, Intensity and Potential of the Gravitational Field at a Point due to a, Circular Disc Intensity and Potential of the Gravitational Field at a, point due to an Infinite Plane Inertial and Gravitational Mass, Earthquakes Seismic Waves and Seismographs Seismology Seismographs GG litzin's Seismograph Determination of the Epicentre and, the Focus, Modern Applica ions of Seismology Solved Examples, , Historical, , laws, , Exercise VII., , Chapter VIII, , Elasticity., , 274-341, , Introductory Stress and Strain Hook 's Law Three Types of Elasticity, Equivalence of a shear to a Compression and an Extension at, right angles to each other, Shearing stress equivalent to an equal, linear, tensile stress, and an equal compression stress at right, angles to each other Work done per Unit Volume in a Strain, Deformation of a Cube Bulk Modulus Modulus of Rigidity Young's, Modulus Relation connecting the Elastic Constants Poisson's Ratio, Determination of Young's Modulus Determination of Poisson's, Ratio for Rubber Resilience Effect of a suddenly applied loadTwisting Couple on a Cylinder (or wire) Variation of stress in a, twisted cylinder (or wire) strain energy in a twisted cylinder (or wire), Alternative expression for strain energy in terms of stress Torsional, PendulumDetermination of the Coefficient of Rigidity (r\) for a Wire, Determination of Moment of Inertia with the help of a Torsional, Pendulum Bending of Beams Bending Moment The Cantilever (/), Loaded at the free end (/*) Loaded uniformly Limitations of the Simple, Theory of Bending Strongly bent beams Transverse vibrations of a, loaded cantilever, Depression of a beam supported at the ends (/) when, the beam is loaded at the centre 07) when the beam is loaded uniformly, for the comparison of Young's Modulus and coefficient, for a given, material. Strain energy in a bent beamResilience of bent beams, Columns, Pillars and Struts Critical load, for long columns (/') When the two ends of the column are rounded or, hinged (//) When the two ends of the column are fixed (///) When one, end of the column is fixed and the other loaded. Elastic waves (/) ComImpact coefficient of Restitution loss of kinetic, pressional waves, bo dies Solved, Energy on Impact Relative masses of colliding, Examples. Exercise VIII., , Searle's, , Method, , of Rigidity, , 342366, Chapter IX Hydrostatics., Fluids Liquids and Gases, Hydrostatic Pressure Hyprostatic Pressure due to a liquid Column The Hydrostatic Paradox A liquid, transmits Pressure equally in all directions Pascal's Law Thrust on, an Immersed Plane Centre of Pressure Particular Cases of Centre of, Pressure Change of Depth of Centre of Pressure Principle of ArchimedesEquilibrium of Floating Bodies Stability of EquilibriumRoll ing and Pitching of a Ship Determination of Metacentric Height, Pressure due to a Compressible Fluid or a Gas Measurement of, Atmospheric Pressure Correction of Barometric Reading Change of, Pressure with Altitude Solved Examples Exercise IX., , 367393, X Flying machines Jet planes, Rockets and Satellites, Flying machinesThe kite The Airplane- Different parts of an Airplane and thiif functions Jet propulsion -Thrust supplied by the jet, , Chapter
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(wV), , PAGES, jetEffect of smaller cross-section of the jet Rocket, fuel, Shape of the Rocket The, Specific impluse, planes Rocket, Multi-stage Rocket Take off of ttie rocket Salvaging of the various, stage rockets Satellites Conditions for a satellite to be placed in, orbit, Launching of the satelliteStability of the rocket during flightForm of the satellite Weight and size of the satellite Material of the, frame of the satellite Duration of satellite's existence Other essentials, Return of Artificial satellite uses of an artificial satellite Exercise X., Efficiency of the, , Friction and Lubrication, Principle of Virtual Work and its, 394417, Simple Applications., Laws of Friction Sliding Friction Angle of FricStatic Friction, Cone of Friction Acceleration down an Inclined PUne Rolling, tion, Friction Friction and Stability Friction, a necessity Simple Practical Applications of Friction, Rope Machines (/) The Prony Brake-'Hi) The Band Brakes Mechanism of Friction*(//) The Rope Brake, Lubricants Principle of Virtual Work (f> Case of a body in equilibrium on a smooth Inclined Plane undet the action of a force (ii) Cast, of equilibrium of a body on a rough Inclined Plane (Hi) Case of equiliirium of a system of two or mare connected bodies (/v) Relation between, Equilibrium and potential energy (v) Tension in a Fhwheel Solved, Examples Exercise XL, , Chapter XI, , &^, , *&S 453, Chapter XII Flow of Liquids Yi|S&i&, Rate of Flow of a liquid Lines and Tubes of Flow Energy of tnlP, Liquid -Bernoulli's Theorem and its important Anjpiications^-Important Applications of Bernoulli's Equation, Viscosity Coefficient of, Viscosity, Fugitive Elasticity Critical ^VclochyPoiseuille's Equation, for flow of liquid through a tube Experimental Determination of rj for, a liquid, Poheuillfs method Motion in a Viscous Medium Determination of Coefficient of Viscosity of a Liquid Stoics' Method, Rotation Viscomster Variation of Viscosityxrf a Liquid with TemperatureComparison of Viscosities Ostwald Viscometer Determination, Rankine's Method for the determination of the, of Viscosity of Gases, Viscosity of a Gas Solved Examples Exercise XII., 454 474, Chapter XIII Diffusion and Osmosis., Diffusion Pick's law Relation between Time of Diffusion and Length, of Column Experimental Measurement of Diffusivity Graham's Law, for Diffusion of Gaie s Effusion Transpiration and Transfusion, Osmosis and Osmotic Pressure La^s of Osmotic Pressure -Kinetic, Theory of Solutions Osmosis and Vapour Pressure of a Solution, Osmosis and Boiling Point of a Solution Osmosis and Freezing Point, of a Solution Determination of Percentage of Dissociation of an, Determination of Molecular Weight of a Substance from, Electrolyte, Elevation of Boiling point or Depression of Freezing-point of a Solution, of the substance Solved Examples Exercise XIII., <, , -, , ^tapter XIV Surface Tension Capillarity., 475, Molecular Force Molecular Range Sphere of Influence, Tension Explanation of Surface Tension Surface-Film and Surface, Energy Free Energy of a Surface and Surface Tension Pressure"", umerence across a Liquid Surface Drops and Bubbles Excess Pressure inside a Liquid Drop Excess Pressure inside a Soap Bubble, Determination of the Surface" tension -pf "g BubbleWork done in, blowing a Bubble -Curvature, Pressure and Surface Tension Layer of, Liquid between two plates Shape of Liquid Meniscus in a Capillary, Tube Angle of Contact Measurement of the Angle of Gontact Rise of, i uoe of insufficien t, Liquid in a Capillary Tube Rise of liquid in, Length Rise of liquid in a Conical Capillary Tube Energy required, to raised liquid in a Capillary Tube Rise of a liquid between two, Force between Bodies Partly" Immersed in a Liquid, Parallel Plates, Shape of Liquid Drop on a Horizontal Plate Experimental Determination of Surface Tension, (Different Methods) - Surface Tension of, , \
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(vm), , PAGSB, Liquid Interfaces Factors affecting Surface TensionExperimental, Study of the Variation of Surface Tension with Temperature Surface, Tension and Vapour Pressure over a Liquid Surface Effect on Evaporation and Condensation, Solved Examples Exercise XIV., , 532575, Chapter XV -Gases Kinetic Theory., The Kinetic Theory Introduction Kinetic Theory of Gases Pressure, Exerted by a Perfect Gas Value of c Relation between c and I, Deduction of Gas Laws on the basis of the Kinetic Theory Kinetic, Energy of a Molecule Value of the Gas Constant Van der Waal's, Equation Mean Free Path of a Molecule Viscosity of Gases Production of Low Pressure Exhaust Pumps Exhaust Pumps and their characDifferent Types of Pumps -The Common Air Pump Rotary, teristics, Oil Pumps (Gaede and Hyvac types), Molecular Pumps Diffusion Condensation Pumps (Gaede and Waran types) - Other methods of Producing, Vacua Measurement of Low Pressures- Manometers and Gauges, Common Mercury Manometers The Bourdon Gauge -Mcleod Vacuum, 1, , Improved modifications of Mcleod Gauge The Pirani ResisGauge Thermocouple Gauge- lonisation Gauges a-ray lonisa, Gauge The Knudsen Gauge Solved Examples Exercise XV., , Gauge, tance, tion, , APPENDICES, Appendix, , I, , Appendix, , 11, , Appendix, , Important Trigonometrical Relations, , 576577, , Logarithms, , 578, , III -Differential, , Appendix IV, , Calculus, , Integral Calculus, , Constant Tables, /., , //., , ///., / V., , V., , r/,, F/7., , VIII.", , Densities of, , 580, , 581-588, , 589-596, 597- 600, , Common, , Substances, , 597, , Elastic Constants, , 598, , Coefficients of Restitution, , 598, , Coefficients of Viscosity, , 598, , Molecular Elevation of Boiling Points of Solvents, , 599, , Molecular Depression of Freezing Point of Solvents, , 599, , Surface_Tensions of Important liquids, , 599, , Molecular Constants, , Logarithmic and Antilogarithmic Tables, Index, , (.00, , 602-605, 606
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CHAPTER, , I, , UNITS AND DIMENSIONS, The Physicist always seeks to reduce his physical, limits., 1., concepts and conclusions to measurable quantities, in the spirit of, Lord Kelvin's very shrewd and apt remark that 'when you can measure what you are speaking about and express it in numbers, you know, something about it, but when you cannot measure it in numbers, your, 9, knowledge is meagre and unsatisfactory , a remark which is at once a, challenge and an inspiration to men of science to sift and clarify, their ideas and notions until they become quite precise and clear-cut., , Now, measurement inevitably involves comparison with a, chosen standard or unit of a similar kind so that, the first essential, step to be taken is the selection of a suitable standard or unit in, .accordance with the nature of the physical quantity to be measured,, und the second, to determine its value in terms of the chosen unit., In other words, to form an exact idea of the magnitude of a physical, quantity, it is neq^sary to express (/) the standard or unit in which, the quantity is metipured and (ii) the number of ti^s the quantity, ;, , ., , *, , contains that unit., , Thus, for example, when we speak of a distance as being equal, to 5 miles, we mean that the standard or unit in which it is measured, is the mile, and that the distance in question is five times this unit., If we choose the yard (which is I/ 1760 of 1 mile) or thfe foot (which, is 1/1760x3 of 1 mile) as our unit, the same distance will be equal, to 8800 yards or 26,400 feet respectively, i.e., its numerical value, will be 1760 times or 1760x3 times 5. Thus, the larger the unit, the, smaller the numerical value of the quantity and the smaller the unit, the, larger its value. Or, the numerical value of a quantity is* inversely proportional to the magnitude of the unit selected as the standard. It, follows, therefore, that the product of the numerical value of the, quantity and the magnitude of the unit in which it is expressed is a, constant., Thus,, ;, , 5, , X, , I^, , Or, in general, if n t and /I 8 be the numerical values of a given, physical quantity, corresponding to the units xt and x, respectively,, ,, , we have, , Derived Units. For measuring different, 2., Fundamental aijtf, kinds of qu^$Jrefes, ^^itmst obyiously have different kinds of units. Ij, these be selected in any arbitrary manners they will be quite unrelated, to each other, and their use will create difficulties and complication), in actual practice., They are, therefore, all based on some funda, mental units, so as to be interdependent and properly related t<, each other, the guiding principle in their choice being to D, (a}, , they are well-defined and of a suitable, , size,
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PBOPBBTIES, , 2, (b), , Off, , MATTER, , they are easily reproducible at, , all places,, , to changes, (c) they are not subject to any secular changes (ie^, with time),, (d) they do not readily or appreciably vary with varying physica*, do r, conditions, like temperature, pressure etc., and, if they, their manner of variation is perfectly correctly known., , The fundamental units chosen, and internationally employed f, are those of mass, length and time which C. F. Gauss, in 1832, termed, as absolute units*. The reason why these alone are chosen as the, 'fundamental' units, and not any others, would seem to be that they, derived, represent our elementary scientific notions and cannot be, from one another nor can they be resolved into anything more basic, or fundamental. All other units in Mechanics can be derived from, them and are, therefore, called 'derived units, Thus, the units of, ;, , 9, , ., , area and volume are derived units, for they can both be derived from, the unit of length, the former being the area of a square, and the, Similarly, the, latter, the volume of a cube, each of unit length., unit of velocity is a derived unit and is the velocity of a body, which covers unit distance, or length, in unit time, and so en., , There are three principal, 3., Principal Systems of Units., systems of units in vogue, viz.,, the C G.S. system,, (/) the Centimetre-Gramme-Second system or, or the F. P. S. system and (/w) the, (ii) the Foot-Pound-Second system, Metre-Kilogramme-Second system or the M. K. S. system., In this system, the unit of length is, (i) The C. G. S. System., the centimetre, that of mass, the gramme and that of time, the, second., , The Centimetre, , is one-hundredth part of a metre, 'which is the, a temperature ofOC, between two lines on a platinum-iridium, bar, preserved at the International Bureau of Metric Weights and, Measures at Sevres, near Paris. Originally intended to be onethousand millionth part of the longitude of the earth from the north, pole to the equator, passing through Paris, it is found, however, to, be slightly smaller., The International Bureau of Weights and Measures has constructed a line standard metre, known as the Prototype Metre, copies, or replicas of which have been supplied to various Governments., The Gramme is one-thousandth part of a lump of platinum-indium,, called a Kilogramme, made by Borda, in accordance wjth a decree of, the French Republic, and also preserved at Sevres. It is equal to the, mass of water, whose volume is one cubic centimetre, at 4C, when it has, , distance, at, , its, , maximum, , density,, , The Second,, , (viz., I, , or the, , gm./c.c.), , mean, , solar second, as it is called, ig, 1/24 x 00 X 60/A, or 1/86400//? part of the mean solar day, which is tht, average value, for one year, of the solar day, or the time which elapse*, between two consecutive transits of the Sun across the meridian, at any, place on the Earth's surface., *In connection with the measurements of the earth'scarried out by him at Gottingen., , maenetk, , field
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TJUTTS ATffD, , 3, , DIMENSIONS, , Another unit of time, used in Astronomy, is the mean siderial, second, which is 1/86400*// part of the siderial day, or the true period, of revolution of the Earth on its axis, i.e., the interval which elapses, between two consecutive passages of a fixed star across the meridian., (ft*) The F. P. S. System., Here, the unit of length is the foot,, the unit of mass, the pound and the unit of time, the second., , The Foot w one- third of the distance between two transverse lines,, a temperature of62F, on two goldplygs in a bronze bar*, kept at the, Standards Office of the Board of Trade, London., , at, , The Pound (avoirdupois) is the mass of a platinum-indium cylinder,, marked "P S., 1844, I Ib." also kept at the Standards Office of the, Board of Trade, London., %, And, the Second, or the mean solar second,, , is, , the same as defined:, , above., , Other units, derived from those given above, are called the, units or the B. O. T. units., ,, , Board of Trade, , well be mentioned here that we generally choose our, smt the quantity to be measured. Thus, for example, for the, measurement of very small lengths or distances, we have successively, smaller units of length, v/z., the micron (//) = 10~ 3 mm., the millimicron (m ) = 10 6 mm. and the Angstrom unit (A. U. or, simply, A}, = U)- 7 mm. and, for the measurement of very large distances, like, those of interbteller space, we have correspondingly larger units, like, the light year, or the distance covered by light in vacuo, (with a, 10, cm. /sec.) in one full year. Similar being the case, velocity of 2 9.) x 10, with the units of mass and time., , It, , may as, , units to, , fl, , ;, , This is a comparatively new system,, (iii) The M.K.S. System., much akin to the C.G.S. system, in which the units of length,, mass and time are the Metre, the Kilogramme and the Second, very, , respectively., , The fir.^t system is the one invariably used in scientific work all, over, the second is more or less confined in its use to ojily Great, Britain and the third is now being increasingly adopted, electrical, engineering etc., where it is found to be more convenient and useful, , m, , 4., , Dimensional Formulae and Equations., , Dimensions., , The units of mass, length and tiine are, (a) Dimensions., denoted by the capioal letters, [M], [t] and [T}\, which merely iri&h, cate their nature and not their magnitude. And, since the unit of are*a, is. the product of two unit length*, we have the unit of area represented by [L] x fJL] or [L 2 ] and, similarly, the unit of volume, being, the product of three unit lengths, is represented by [L] x [L] x [L] or, 8, [L ]. We express this by saying that the unit of area is of two dimenin length., sionsjf. in length, and the unit of volume, of three dimensions, ;, , This bar has now also been replaced by a platinum-indium one., tThe square brackets merely indicate 'dimension of\ Once this is under-, , may as well be orrutted, as we shall quite often do., *, J Which is the abbreviated form of exponent of dimension', but, used, and, well, understood., commonly, stood, they, , lt, , is, , -now
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PROPERTIES OF MATTER, , 4, , Since neither the unit of area nor that of volume depends upon mass, and time, their dimensions are said to be zero in both mass and time, L*, L1 T* and, and we may, therefore, represent these units as, Z* respectively., , M, , M, , f, , "The dimensions of a derived unit may thus be defined as the, powers to which the fundamental units of mass, length and time must be, raised to represent it" Thus, if a derived unit depends upon the wth, power of a fundamental unit, it is said to be of n dimensions in that, fundamental unit. For example,, ., , ,, , velocity, , ^, , and hence the dimensions of the, in time., , zero,, , unit, , it is, , Again, since deceleration, f, , of acceleration are, , -, , M, , L 7- 1, ^, , ~, , =, ==, , M, , , ,, , - LT, , of velocity are, , independent of mass,, may, therefore, represent it by, , Since, , and we, , fL 1, \-T J, , distance or length, ------------, , =, , its, , 1 in length, , dimension, , MLT~, , in, , and, mass, , 1, is, , l, ., , the dimensions, ., , ., , of the, , unit, , ,, , r rr, LI-*, and, , so on., , It will thus be seen that the dimensions of a physical quantity, by simply defining it in terms of those physical quanti-, , are obtained, , the value, ties whose dimensions in mass, length and time are known,, of a derived unit depending upon the values of the fundamental units, from which it is derived. Thus, if we take a yard as our unit of length,*, 2, instead of a/oof, the units of area and volume will respectively be 3, and 3* times as big as their uptits itt the ordinary system. So that,, the dimensions of a physical quantity show how its nature and the value, , of its unit depend upon, , the fundamental units chosen., , and Equations. A dimensional formula, (b) Dimensional Formulae, an expression, showing how and which of the fundamental units enter, into the unit of a physical quantity. Thus, all the expressions in the, Table opposite, indicating the relation between the derived and, fundamental units, are dimensional formulae. For example, the, dimensional formula for work is ML*T~*. But when we put it in the, 2, =, r~ f it is called a dimensional equation for work., form,, This idea of dimensional formulae for physical quantities, as, we know it today, was first clearly given by Fourier, in the year, 1822, although it originated initially with Newton, who refers to the, principle of similitude in his famous and well celebrated Principia,, is, , W, , ML, , ,, , Proposition 32)., The student is no doubt aware that in Physics we come acrooo, two types of quantities, viz., variables and constants, which may, both be dimensional or non-dimensional (i.e., dimensionless). Thus,, (II,, , we have, Dimensional Variables. These are quantities like acceleration, velocity, force and most of the others which the Physicist has to, deal with, at every step. These are, so to speak, his 'current coin'., (/), , which have a constant, (0^ Dimensional ConstantsQuantities, value jmd yet have dimensions are called dimensional constants. As
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TOITS AKD DIMENSIONS, , 5, , examples of these may be cited G, the Gravitational Constant, and c,, the Velocity of Light in vacuo, whose dimensions are M~ 1 L*T~** and, , MLT~, , l, , respectively., , Non-dimensional Variables. These are quantities which are, and yet have no dimensions as, for example, specific, gravity, strain or an angle, (see Table below)., Here, we also meet with groups of dimensional variables (with, (Hi), , variables, , ;, , or without dimensisnal constants) such that their dimension, , is, , zero in, , each of the fundamental quantities, i.e., in length, mass and time. Thus,, for example, the, quantity t\/ gjl has no dimensions and so also the, quantity up//?, called Reynold's number, can be shown to have zero, dimensions in mass, length and time. Such quantities were given, the name 'numerics' by James Thomson., ;, , (to), , These are mere numbers, , Non-Dimensional Constants., , like, , -, , 3, 2, TT etc., , Thus, numerics, pure numbers and quantities like heat, electritemperature and dielectric constant have no dimensions in MLT., The following Table shows at a glance the dimensional, formulae for some important physical quantities., city,, , Dimensional formula, , Physical quantity, , 1., , Area, , =, , 2., , Volume, , - (length) 8, = length/time, = velocity/ time, , 3., , 4., , Velocity, Acceleration, , (length), , 5, , M*L*T,, , ML*T,, r,, , j, , 5., , MLT-\, , or, , ^, , 8, , or simply [L ], or simply [*], or [LT~ l\, , *, , or, , M*LT, , *, , or [LT~*], , Momentum, , 6., , Force, , 7., , Work*, , = (mass x velocity), = (mass x acceleration), = rate of change of, , MxL/T,, , or, , [MLT- 1], , momentum, (force, , x distance, or length), , 8., , 9., , Couple* = (force x length), Kinetic Energy*, 2, (i mass x velocity ), , MX (LIT XL, MX IL*IT*] =, 2, , ), , 11., , Potential Energy*, (mass x acceleration, due to gravity x, distance), Power, (or rate of doing work), , 12., , Density, , 13., , Specific gravity=a mere ratio., , 1 4., , Pressure = force/area, , MLT~*IL*, or, , 15., , Stress, , = force /area, , MLT-*IL*. or, , 10., , = work/time, = mass /volume, , [ML*T~*], , ', , [AfL*/T*] or [ML*T~*], , MIL*, or [ML~*T] or [ML~], , No, , dimensions, , *See Solved Example 1 (6), page 13., be noted that the demensions of couple, kinetic energy and, potential energy are the same as those for work, because they arc mutually, Same is the case witn pressure and s/re$5., convertible and energy is just work., *It will
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PROPERTIES OF MATTER, , 6, , Dimensional formula, , Physical quantity, , Strain, , 16., , **, , change of length or volume, original lengttTor, , length, , =a mere, , volume, , No, , volume, volume, , ^length, , dimensions., , number., , 1*1., , Coefficient of Elasticity, , 18., , Coefficient of Viscosity, fo rce, velocity, , =, , stress/strain, , MLT-* LT-*, L, , ', , L*, , =~~, , area "distance, Surface tension =force/length,, , 19., , =, , or,, , 20., , Frequency, Angle, , 21., , = [M LT~*], , energy /area, , -, , T- 1, , 1/T, , I/time, ', , or, , MLT~*IL,, , length /length, a number., , No, , ,, , or [MT-*], , [AfLT- 1 ], , or, , dimensions., , Uses of Dimensional Equations. A careful examination, 5., of the dimensional equations of the various physical quantities, involved in a relation, i.e., an analysis of their dimensions, is of great, help to us in more ways than one, the process beim* known as, distnensionai analysis., Its three chief uses are the following, :, , (a) conversion, , of one system of units, , into another,, , (b) checking the results arrived at,, , and, , (c) deriving, , a correct relationship, , between, , different, , physical, , quantities., , Let us consider these in some, , detail., , It is seen, another., (a) Conversion of one system of units into, that a physical quantity is expressed in terms of an appropriate unit, of the same nature, its value being equal to the product of a number, and that particular unit. Further, as shown in 1, its value remains, the same on all systems of units. This affords us an easy method, of changing over from one system of units to another., , Thus, suppose there, , is, , a physical, , mass length and time, a b c, L T, formula is, Then, if its, , and, , c in, , M, , ., , quantity of dimensions, , a,, , b, , respectively, /.e., whose dimensional, numerical* value be HJ in one lystem, , in vvhich the fundamental units are, , M, , Lt, , L x and T19, , it is, , clearly equal, , tonAM'LfTf]., Also,, , if its, , mental units, , M, , 2,, , numerical value be w a * n Another system of fundaL 2 and T2 it is equal to n^MJLfTJ] in this, ,, , ystem., , So, , that,, , whence,, , n&, l, , ~M
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AKB DIMENSIONS, , -=, So thad;, knowing the fundamental units in the two systems, ind the numerical value of the quantity in one of them, its numerical, value in the other system can be easily determined., Care must,, be taken to apply relation (i) above, after expressing, the given quantity in absolute units. Let us consider an example, or two., , however,, , (1) To convert, into dynes, (the unit, , We know, i Ib. =453-6, , a poundal, (the unit of force in the F.P.S. system),, of force in the C.G.S. system)., /, , MLT~* and that, x 2-54 <w.=30-48 cms. So that,, , that force has dimensions, , gms., and, , 1 ft., , =, , 12, , M units in F.P.S. system = 453 6 M units in C.G.S. system,, , L units in F.P.S. system = 30'48 L units in C.G.S. system,, T units in C.G.S. system,, and T units in F.P.S. system=, unit, of time being the same, viz., the second, in the, the fundamental, two systems,, MLT-* poundals, , .-., , ~, .r-*^, , log 453*6, log 30-48, , =, -, , <-,, , ., , 1, , Ur,, , tct-i, , ', , 2-6567, , ,, , 7, , poundal =*, ^, , 4840, 4- 1407, Antilog, = l'382xl0 4, , =, , (453-6M)(30-48 L)T~*., , 453-6MX 30-48, --,>, MxLx, -r, , 1, , s=s, , x, , 1-382, , 10*, , units, , in, , the, , C.G.S. system., , =, , l-382x 10* dynes. </, Thus, 1 poundal, (2) To convert one Horse Power, (F. P. S. system), into Watt*, (C.G.S. system). We know that, 1, , H. P. == 550 ft., , = 550 X 32-2 ft. poundals sec., =, 32-2 /*. /sec. y, g, , Ibs.jsec., , I, , 2, , and, Again, as shown in Ex., , (1),, , M units in F.P.S. system = 453-6 M units in C.G.S. system,, , and, , L units, T units, , ., , =, =, , 30-48, , L, , T, , -, , Since the dimensional formula for power, , is, , AfL2 r~8 we have, ,, , = 550x32-2(453-6M)x(30-48L) xr-., 453 6M X 30 48L x r-F P - 550 x 32, 32-2, H.P., 2x, , -, , H.P., , 2, , ', , M, , *This ratio, , MJM^, , if, , ^ be, , " the, iv, that, at in, F.P.S. system,, conYcrsion, ion factor,, factor., t,, , i.e.,, , -, , 8, , ', , (, , ), , ., , the unit of mass in the C.Q.S. system, the ratio 'gram to the pound" is called
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PROPERTIES OF MATTER, , 8, , Checking the results arrived at. This depends upon what, the principle of homogeneity of dimensions, again due to, Fourier, according to which the dimensions of all the terms on the two, sides of an equation must be the same., This follows at once from the, fact that it is not possible to compare twa physical quantities of, different natures, and that only quantities of the same nature can be, added up together, their resultant being also of the same nature., If, therefore, in a given relation the terms on either side have the same, dimensions, the relation is a correct one, but if they have not, there isa flaw somewhere, which must be diligently sought out., Let us again take a couple of examples, (b), , 10, , called, , :, , (1), , To check, , the accuracy of the relation,, , t, , =, , 2ir^i]if9 for, , a, , simple pendulum., , Here, the term / on the lef hand side has only one dimension in, or the dimension of t is [ T], its dimensions in both mass and, fc, , time,, , length being zero., , And, on the right hand side, 2ir has na dimensions, being just a., number / has one dimension in length, or its dimension is [L], those, in mass and time being zero and the dimensions of g, the acceleration due to gravity, are LT~ 2 that in mass being zero., Hence the, ;, , ;, , ,, , =, , =, , dimensions of the term, 2n y7//, [T],, ^HUlF* or \/~f*, it has, only one dimension in time, the same as the term on the, hand side. The relation /, 2n\/l/g is, therefore, a correct one., , i.e.,, , left, , =, , in t, , (2) To check the relation S, seconds by a body, having an, , =, , ut+\, , at*,, , for the distance covered, u and an acceleration a., , initial velocity, , Here, the dimension of the term S on the left hand side is one, or [L], and taking the terms on the right hand side, we have, LT* 1, (/) dimensions of u (velocity), , in length,, , (U) dimensions oft (time), (Hi) dimensions of J (a number), (iv), , and, , (v), .-,, , dimensions of a (acceleration), dimensions of f 2 (time 2 ), , dimensions of the term, ut -f \at*, , =, =, =, =, =, =, =, , T, Nil, , LT~*, I* 2 ., , LT~* x T+LT-* x T*r, , L+L,, , i.e., the dimension of each term on the right hand side is the same as, that oj the term on the left hand side ; hence the given relation i, , correct., , A similar dimensional homogeneity will be observed in the case, of any other relation, representing a physical phenomenon. The, method of dimensions has thus a very definite mnemonical value*, and enables the beginner to resolve his confusion between two alternative possibilities occurring to him regarding a particular half forgotten formula, as, for example^whether the time-period of a simple, 2n^/l/g, or whether the, 2?r\/^ or by f, pendulum is1given by t, formula iirr gives the surface area or the volume of a sphere etc., etc., , =, , *, i.e.,, , value at an aid to memory.
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9, , UNITS AND DIMXHIONS, , different pftysical, (c) Deriving a correct relationship between, The principle of homogeneity of dimensions also enables, quantities., us to deduce a relationship between different physical quantities, or,, , at any rate, a preliminary form of such a relationship, For, knowing, the factors on which a physical quantity may possibly depend*, and, this requires a little physical insight and a certain amount of 'horse, an expression for it can be obtained in terms of these factors,, sense', such that the dimensions of the terms on the two sides of the, expression are the same, the only acceptable form of the relationshipbeing the one which remains true irrespective of the system of units, employed. A few examples will illustrate the point., , To deduce an expression for, , (1), , the time-period, , of a simple pendu-, , lum., , The factors on which the time-period, are the following, (i) the mass of the bob (m),, , may, , (/), , possibly depend^, , :, , (ii), , the length of the pendulum, , (Hi) acceleration, (iv), , Let, , /, , due to gravity, , the angle of swing of the, , be proportional to, , t=K.m a l*y, , ma, , c, ,, , /*,, , g, , (/),, , (g), , and, , pendulum (6)., So that,, 6d, , and, , ., , where Kis a constant of proportionality., of the terms on either side of the sign of, dimensions, Taking, equality, we have, [T], , 6d ,, , = [Ma ][L*][LT~*Y =, , T = M*Ld + c T- 2e, , Or,, , M*L*Le T-*<. r* and, , ., , having no, dimensions., , 1, , Since the dimensions of the terms on the two sides must be the, L and T,, we have, equating the indices of, , M, , same,, , whence,, , ,, , and, 2c = 1,, a = 0, b+c =, =, b \ =0, or b =, c, | and, = K.I*. g~~*., , ., , ., , Therefore,, , t, , t^KVlfg., , Or/, , +J, , The value of K can be found out experimentally f, and comes to, 2ir, , ;, , so that, the required relation is, , t, , =, , 2?r, , \fTfg., , It will easily be noted, from the above, that, (/) the time-period of the pendulum is independent, , a fact we know to be true by actual experience, and (ii) the expression t^/gjl has no dimensions, as, dimensionless constant, and is thus a numeric., , of, , its, , mass,, , ;, , 9, , K, , An important, that, , if, , factors, , it is, , equal to the, , 9, , deduction emerges from this latter point,, , two pendulums having, , different lengths, (^, , and, , /t ),, , viz.,, , oscillating, , * It is, absolutely necessary to take into account all possible major, on which our result may reasonably be expected to depend, though one, , or more of these factors may get eliminated later. The method, however, ceases, to give any worthwhile result if the number of variables included is more than, six., , K, , tThe value of can be determined easily by substituting in the relation, obtained, the observed value of /, for known values of / and #.
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10, , PROPERTIES OF MATTER, , at two different places, where the values of the acceleration due to, gravity are ol and g+ respegtively, take time TA and T a to describe, equal arcs, they may have, , i.e.,, , the value of the non-dimensional expression or the numeric, be the same for both., , may, , And,, , if this, , be, , means that the two pendulums pass, , so, it, , through exactly the same phase for the same value of r\/g]T. This is a, case of what is called dynamical similarity, and all, moving systems of, this type are said to be, dynamically similar., , A very interesting and a classic example of this principle is the comparison of the speeds of fully grown animals with those of their young., Very reasonably, taking the density of the two animals to be the same and, muscular strengths directly proportional to the cross-section of their limbs,, we have the ratio between their densities equal to one and similarly that between, their strengths per unit area of cross-section of their limbs, also equal to one ; so, that, if subscripts 1 and 2 refer to the adult animal and to its young respectivetheir, , ly,, , we have, , ratio, , of their, , densities, i.e.,, , ^, , L>i, , /^ =1, f, , I, , L, z, , md also ratio between their muscular strengths per unit, From these two, , relations then,, , we, , area,, , i.e.,, , easily get, , X = 17', L"i, , L-\, , where L^IT^, , is, , the speed of the, , grown animal and L 2 /Tt9 that of its young., , full, , The speeds of the two animals are thus the same, a result which, at first, sight, appears to.be simply ridiculous. And yet it is an actual fact, the shorter, strides of the young being taken faster than the longer ones of the adult., (2) To deduce a relationship for the velocity of sound in a material, medium, the temperature of the medium remaining constant., , The, , E and, , (ii), , velocity K may depend upon, the density of the medium, p, , V = K.Ea ^ b, , >, , where, , (/), , the elasticity, , of the medium, , so that,, , ;, , K is a constant., , Again, taking dimensions of the terms on both sides,, >, , MOLT-* =|, m, j^j., , clear, , r*r*, ro, , ., , I, , 1, , ii, , /"., rn, , (", , v elasticity, , J, , \, , stress/ strain, , force /area, , I, , ^-rilrJzl, a ratio, mass/ volume., l^and density, , ', , Since the dimensions on the two sides must be the same ,, 2a, 1,, that, 1, and, a+b, , whence,, , Hence,, Or,, , I, , we have, , =, =, a, V=, , 'V^, , a3b =, b = a = - J., and, \, ;, , =, , it
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UNITS AND DIMENSIONS, , The value of, found to be, , K, , is, , in this case, , 1, , again determined by experiment, and, ;, , so that,, , V, , i*, , 5= \/~Ejp, , So far only simple cases have been considered. In other cases, the method used above may not always be applicable. Let us consider one such typical example by way of illustration of the method, adopted in such cases, (3), , in, , time, , t,, , To obtain a, , relation between the distance travelled, be u and acceleration a., , by a bod), , if its initial velocity, , Let the distance covered by the body in time t be represented, K.ua .ab .t c, Then, taking dimensions, we have, , =, , by S, , Or,, , [L], , ^, T =, , J, , =, , La T~* x Lb T~*b x, , c, , Since the dimensions on the two sides must be the same, we, , have, , a+b, of, , a,, , ==, , l...(i), , a2b+c =, , ;, , a+2b~~~c, , or,, , ;, , =, , ...(&'], , These two equations alone are not enough to give us the values, b and c. Hence we proceed as follows, :, , Suppose the body has no acceleration. Then,, S, K'u a t c where K' is another constant., , =, , ,, , Taking dimensions, we have, , L, a, , *, , whence,, , =, =, , La T~*Tc, 1, , ;, , and, , =, , L a Tc ~ a, , ca =, , 0,, , ,, , or c, , =, , a, , =, , 1,, , =, , S, , K' . ut., Now, suppose the body has no initial velocity. Then,, S = K" ab t c where K" is yet another constant,, , ...(A), , ,, , Again, taking dimensions,, , L, b, , =, , 1, , ==, ;, , Hence, , L Tb, , and, , we have, , T = Lb Tc -*b, c, 2b = 0, or, S = K".at*., , 26, , c, , ., , c, , =, , 26 =>, , 2., , ...(B), , therefore, a body has both, initial velocity as well as, acceleration, its equation of motion contains both the expressions,, so that, we have, (A) as well as (B), If,, , ;, , K' and K" can be determined experimentally,, and are found to be equal to 1 and \ respectively. Thus, the re2, t//-f-|a/, quired relation comes to be S, svhere the constants, , =, , ., , In addition to the three chief uses of dimensional analysis, discussed above, mention may also be made here of a couple of others., Thus,, (iv) it is helpful in selecting experiments likely to give some useful, information and avoiding others. In this connection, Lord Rayleigtfs, remark is worth quoting. Says he, 'I have often been impressed by, the scanty attention paid even by original workers in Physics to the, great principle of similitude. It happens not infrequently that, results in the form of 'laws' are put forward as novelties on th*
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PROPERTIES OF MATTER, , 12, , basis of elaborate experiments which might, minutes' consideration.', , have been predicted, , How true,, , a priori after a few, , indeed, , !, , it is a powerful aid to mathematical analysis,, .happens to be a, complex one and when no experiments to solve it are possible. Even if the number of variables, involved in the problem be a large one, dimensional analysis does, help obtain at least a partial solution of it., , Then, again,, , (v), , when the problem, , ^Limitations of Dimensional Analysis. It will be readily seen, from the examples, given above, that the method of dimensional, analysis is after all not quite so simple or straight in its application,, except in obviously easy cases. Very helpful, as far as it goes, it has, also its own limitations., Thus, for example, :, , (i), , Its, , tion about, , one obvious drawback, , pure numerics, , K, , (like, , is, , that, , it, , little or no informaconnon-dimensional, and, , gives, , t^/yjlin Ex. 1), , stants (like, in Ex. 2), involved in various physical relations, and, which, therefore, have to be determined by separate calculation or, , experiment., , can be ob(ii) Then, again, since at best only three equations, tained by equating the dimensions of [Af], [L] and [T}\ the method is, of no avail in deducing the exact form of a physical relation which, happens to depend upon more than three quantities. For, clearly, of a, given number of quantities involved, the indices of only three can be, expressed in terms of the rest, thus leaving us with a relation between?, the remaining number* of non-dimensional groups of terras so that,, what we may ultimately succeed in obtaining is just an equation in, terms of an undetermined function., ;, , It will thus be clear that, while the method of dimensional, analysis remains unrivalled and almost unique, in so far as conversion, from one system of units into another and checking the correctness, of physical relations are concerned, its use is not quite so safe or, certain when it comes to establishing a definite or exact relationship, , between a given set of physical quantities and, particularly, hands of beginners., , so,, , in the, , More often than not, the success of the method depends upon, the proper choice of dimensional constants (like G or c), which have to, be introduced as additional variables. And, it needs a trained,, subtle and intuitive mind, with the solid background of a mature, and a comprehensive knowledge of the subject, to decide, on the basis, of analysis or experience or perhaps just on that of some sort of, inspiration of the moment, what particular variables to select, and, how, when and where to introduce them. A very apt illustration in, support of these remarks is perhaps Raleigh's explanation, by the, method of dimensions, as to why the sky is blue., That the colour of the sky is due to the scattering of light by, suspended drops of moisture and dust particles etc. (of molecular, From this basic fact,, size) in the atmosphere is fairly well known., Raleigh proceeds as follows, *v/z.,, , :, , the given number of quantities minus three.
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UNITS AND DIMENSIONS, Let A t be the amplitude of the scattered wave., ble factors on which it may depend are, (/), , (it), , (Hi), , And, , (iv), , 13, , Then, the possi-, , Af the amplitude of the incident wave of light,, the linear dimension of the scattering particle,, r, the distance from the particle,, , I,, , \, the wave-length of light., A s in terms of all these variables,, , So that, expressing, , we have, , A g **K.Afl*r*K,, , K is a, , where, , constant of proportionality., , Or, taking dimensions,, , L, , L*, , the dimensions of, 0,, L, and those of, , for, obviously,, viz.,, , we have, , =, , We,, , ., , U, , all, , K=, , therefore,, , ., , If, , ., , U,, , these quantities are the, , ame., , have, , =, , ;, , a+b+c+f., , Now, we know that the, , scattered light is, araflfjtude of the, proportional to that of t$e incident light and (//) inversely, proportional to its distance from the scattering particle. This at, 1 and c, 1., once gives us a, And, therefore,, </) directly, , =, , =, i+bl+d, whence, d = 16., "* = K., A, = K A, V r* A, 1 ._, , So, , 1, , that,, , ., , Now, as Rayleigh remarks, 'from what we know of the dynamics of the situation** / varies directly as the volume of the scattering particle., , A, , Hence, , And, , 6=3., , And, therefore,, s, , =K, , ', ., , 2, , ., , Or,, , A, , s, , oc I/ A, , 2, ., , since intensity oc (amplitude)*, we have, 4, intensity of scattered light, Is oc I/ A ., , It thus follows, as a natural consequence, that the wave-length of blue, being roughly half that of red light, the in tensity of scattered blue light, is sixteen times that of scattered red light and that the sky, therefore, appears to, us to be blue., The student will appreciate how, in capable hands, the method of dimensional analysis can be made to yield results beyond the pale of elementary, , light, , analysis., , SOLVED EXAMPLES, Deduce the dimensions of (a) the, , the, Coefficient of Viscosity, and (, Constant of Gravitation (G)., Obtain a formula for the time of swing af a simple pendulum from a knowLedge of the dimensions of the physical quantfp Involved., (Punjab), (a) We know that the coefficient of viscosity (17) of a liquid is given by, w jpr/8v/,, the relation,, *?, 1., , =, , vhere P, , the pressure difference between the two ends of the capillary tube ;, % its radius ; /, its length and v, the rate of flow of the liquid through it, or the, >olume of liquid flowing out per second, is, , *v/z.,, , ight., , the ratio of the respective amplitudes of the incident and reflected
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n, , PEOPEETIES OF MATTER, Therefore, taking dimensions of all these quantities, we have, ML" 1 !"*, dimensions of P, [see Table on page, , M, , =, =, -, , r*, , v, /, , ^, , IT", , 1, , [v, , L, , and, Bare z^ro., Hence, dimensions of coefficient of viscosity, , and, , (b), , We know, , rate, , of flow, , =, , volume/time., , [Both being numbers., are, , TJ,, , that the value of G, the Garvitational Constant,, , G^CQd*IM.m.l., , the relation,, , Si, , is, , given by, , ,, , C, , is the restoring couple per unit twist of the wire; B, the angle of twist oj, where, the wire ; d, the distance between the centres of the near large and small balls ;, andm, the masses of the large and small balls respectively and /, the length of the, torsion rod, (Cavendish's experiment)., , M, , Therefore, taking the dimensions of the quantities involved, we have, 2, T~*, dimensions of C (couple) =, [See Table on page 5-, , =, =, =, =, , 9 (angle), , d, , 1, , M, , m, , ML, L2, , M, M, , Hence dimensions of G are, , or, , For answer to the second part of the question, see page, , 9, (Ex. 1)., , Find the unit of length if one minute be the unit of time ; one stone,, 32'2 ft per sec 2 )., the unit of mass, and one pound-weight, the unit of force., (g, 2., , We know, , that, , 1, , M units, , Now,, , T, and, , =, , 14, , =, , Jog, , 1, , in the, , ,,, , ,,, , Then, 32'2, , (14M) units, , 5-0643, , Or,, , 1J461, , di-, , ,,, , in the, , ordinary system,, , 1 stone *= 14 Ibs., (607) units in the ordinary system,, (xL) units in the ordinary system., , MLT~ =14MxxLx[6QT]~* units, = 14M.*L.60~ r- units, Z, , 2, , 3-5564, , and that the, , [v, , =, =, , ,,, , ,,, , ,,, , 5079, , 32'2 poundals,, , ., , new system, , ,,, , L, , let, , log 32*2, 60, 2 log, , = g poun dais, , Ib. wt., , 2, mensions of force are MLT~~, , 32-2-, , JJf., , x=, , And/., , in this, , system, , 2, , ^8283., , 14, , 3*9182", , the unit of length in the new system would be xL, Or,, 8283, =8283xL, i.e., equal to 8283/h, [since [I] = l ft., If the acceleration due to gravity be represented by unity and one, 3., second be the unit of time, what must be the unit of length ?, 1 cm., and that, In the ordinary system, in which the unit of length [L], , Antilog, , -, , =, , of time [71, , =, , sec, , 1, , ,, , we have, , =, , 2, , 1 cm. /.sec. , and acceleration!, unit of acceleration, [dimensions LT~*], to 981 cm./sec*.=9B\ LT~*., If the unit of length, in the new system, be LI, we have, ', L acceleration due to gravity, on this system, == 1 xL x sec 2, L{T~* 9, the unit of time being the same, i.e., 1 second, in this system also., , due to gravity equal, , ., , I^T-, , 2, , 981, , IT- 2, , =, , [1., , ., , 1 cm., 981 cms., since L, 981 L ; that is L l, Or,, LI, Thus, the unit of length in the new system is equal to 981 cms., Given that the unit of power is one million ergs per minute, the unit of, 4., force is 1000 dynes and the unit of time, 1/10 sec., what are the units of ma**-, , and length?, , Here (a),, , unit of power, [dimensions, , of force, [dimensions, unit of time [dimension T], , (b) unit, , and, , (c), , ML*T~*], , MLT~*], , =, , 1000,000 Srgslmt., 1000,000 16Q ergs per $rc~, IQQQ dynes,, 1/10 sec.
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UNITS AND DIMENSIONS, multiplying (a) by (c), we have, unit of power x unit of time, , /., , =, , unit of work., , - [ML T-*][T] = ML*T~*., _ 1000,000 x J_ _ 10*, 2, , ', , Dividing, unit, , this, , (b), , Or.iniro/mai*-, , x, , ,, , we have, , = ~~-cms., , ,, , JQQQ, , we have, , MIT', , of mass, , -, , ., , ~, , of distance or length, , Now, from, unit, , 10"""" 6, 60, unit of work by the unit of force,, , 1000x(l/10), , 2, , 2, , -f, , = MLT~*, , ^, , 1000x3, , F, , ^, , rv, r, , r, 6, , lOQxS*, , 5/3, , x, , ^!, , iii7, , v, , maw, , of, ' force =1000 dyne&, *,,, , and im/^/arc.-, , Therefore, the units of mass and length, in the given system, are 6 ,gms. and", 5/3 cms., respectively,, 5. If the fundamental units are the velocity of light in air, the acceleration, of gravity at Greenwich, and the density of mercury at 0C, find the units or, mass, length and time. (Velocity of light - 3 x i0 10 cm \see ; acceleration of gravity, 2, 2, at Greenwich = 9 81 x 10 cm.isec13*6 gm. per c.c.)., density of mercury, , =, , ;, , Here,, , (a) unit, (b), (c), , ,., , *=0'4771, , log 3, 8 log 10, , /., j, , acceleration,, , (, , density,, , (, , dividing (a) by, , =8000p|, , f time, , >, , j, , log 9*81, , -09917, , ML~ 3 = 13*6 gm.jcm*, ), , (, , we have, 3xl0 10, , (b),, , t, , mlt, , 8 477 1, , LT~ l )=3 x 10 10 cm.fsec., Lr- 2 )=9*81 x 10 2 cm.jsec.*, , (dimensions, , of velocity,, , 7), , =, , 9 X1 x, , 1, , 2, , 1, , !!L, , 7*4854, 7, 3'058 x 10, , Antilog, , ., , ^ 3xl0, , 8, ', , 9" 8"P, , _!., , Substituting this value in (a), we have, mit Of length, (L) ^LT~ l .T., =3 x 10 10 x 3-058 x 10 7 =9 174x 10" cms., , j, , |, , j, , 13-6= 1-1 335, 17, log(9'174x 10, =53 8878, , log, , j, , unit, , ;, , 1, , 051, , 8, xL, =13'6x(9-174x 10 17, ---, , 3, ), , ., , Thus, the required units of mass, length and time are10 7 seconds,, , Antilog 55-0213, , rosixio 65, , (c), we have, 3, of mass, (M) = ML-, , And, from, , ), , x 10 65 gms., 9'174x 10 17 cms. and 3'058x, , respectively., 6., , of length and force he each increased four times, show that, increased sixteen times., have unit of energy = unit of force x unit of distance., , If the units, , the unit of energy, , We, , is, , If now, the units of force and distance be made four times each, they, would be 4[MLT~*] and 4L respectively, and, therefore, the new unit of energy, a, 2, would be 4Afr- x4L=16AfL T- which is sixteen times ML 2 r~ 2 , the ordinary, J, , 1, , ,, , unit., , the unit, , Thus, we see that by increasing the unit of force ^md length four times each,, of energy is increased sixteen times., , Show by the method of dimensions that the relation, C=nnr 4 j2l for, 1., the couple per unit twist of a wire of length /, radius r and cefficient of rigidity, correct one., , is a, Let us take the dimensions of the terms, equality and see if they are the same. Thus,, , dimensions of, , and., , " and, , 2,, , C (couple), , =*, , =, , on the two, , ML, , 2, , sides of the sign, , of, , r~*, , ML^ 1 T" Z [same as for elasticity ~, , n, , (rigidity}, , r*, , (radius?, , L*, , /, , (length), , L, , being numbers, have no dimensions.
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PROPERTIES OF MATTER, , 16, , Therefore, the dimensions of the term, \AT IT-*, v f*, 1, XL, , ML, , are, , wirr*/2J, , _, , JL, , *he same as for C, on the, , hancTside., , left, , Hence, the relation C, wrr*/2/ is a correct one., Test by the method of dimensions the accuracy of the relation, , 8., tf, , =, , \Afc* +?*)/# f r tne time-period of a, , 2, , compound pendulum., , be -correct, the dimensions of the terms on either side of, the sign of equality must be tne same., If the relation, , Let, , -us, , put the relation as, , =, , t, , Now, the dimensions of /, K*, I, , 2A /, V, , _, , 4., , g, , lg, , = [T], = [L K being the radius of gyration., - [L], 2, , ],, , Therefore, the dimensions of the term on the right hand side are, , -V2^", Thus,, viz., [T]., , vri+5 *", , we see, , The, , that the dimensions of the terms on either side are the same,, relation is, therefore, a correct one., , 9. Find the dimensions of velocity and acceleration., Assuming that, -when a body falls from rest under gravity the velocity v is given by Kg*W 9 where, h is the distance fallen through, g, the acceleration of gravity and K, p and q are, Constants., Show, by a consideration of the dimensions involved, that v^K\/gh., (London Higher School Certificate), , one of the question, see Table on page, , 'For answer to part, , We, , are given that v, , =, , Taking dimensions, therefore, we have, , Kg*hP., , dimensions of v, , 5., , =, , =*, , ^-, , LT~ l, , ., , dimensions of h9, ,,, , K = 0, for it is a constant or a mere number., , Therefore, the dimensions of the term,, , Since dimensions on both sides of the sign cf equality must be the tame,, AVC have, , p+q, , Or,, , 1, , and -2p, , =, , KgW, v, , Or,, , 1,, , Kg*, , =, , whence, p, ., , A*, , i and, , 0=, , i., , JSTV^-, , K^/giT, , 10. The frequency of vibration (n) of a stretched string is a function, of the tension (T), the length (/) and the mass per unit length (p). Prove that, , "~, JLet, , n DC, , Ta/V., , dimensions of, , /, , Then, taking dimensions, we have, !// ** M*L*T~*- 9 or, T~\, >
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AND DIMENSIONS, , TTKITS, , J, , Dimensions of, , (force)*, , ,, , (length)*, , ,t, , /*,, , ", , pe >, , r, , dimensions of the term, , 17, , = [MLT~*]*, - /A, , a, , rami*, , M*L*T-**, , rM"\ c =JV/a, ., fr " cf, , U^/J -LrJ, , M, , T L*P =, a, , c, , y-i, , Or,, , a, , l*T-, , Za, , .L b, , .M Lr c, , 9, , -, , t, , +, , ., , jv/f, , Since dimensions of n must be the same on both sides, we have a 4 c, = Oand 2a = 1, or 2a = 1 i.e a = i, and hence i-f c=0, or c=, , c, , ,, , Also |4-A-(~i) =, , Ta, , Therefore,, , l, , 0, or, , b, , =, , f-h&~h4, , 0, i.e., l-t-6, , _if =, , i, , T1, , c, , ., , /- 1, , p, , ., , =, , 0,, , or 6, , ;, , |., , 1., , V "p-F, , A /, , 1, /, , _, , And hence, , n oc, , 11., The time of oscillation (n of a small drop of liquid under surface tension depends only on the density (/>), the radius (a), and the surface, , 1, tension (T)., Is a numeric., , Let, , t, , Show, , =, , K?, , a, , that the period of oscillation, , a b T*., , So, , that, taking dimensions,, , dimensions of, , --=, , ., , a 2 .T, , _l, , *,, , where, , K, , we have, , 2, , [/V/r- ]^, , M, , C, , T~ ZC, , [See page 5., , ,, , being a numeric., , dimensions of the term, , =, , Or., , = ZA, =, , b, , \L\, , 7^ =, , K has wo dimensions,, , KP^, , T,, , t, , a*, , and, , is, , K?ab T c, , M, , ML-* aU>McT-* c, , ~=, , ., , a + cL-* a +*T- 2 c., , Since the dimensions of the terms on both sides must be the same,, , we, , have, , /j-hc^O;, c, , Or,, , Hence, , //iff, , 12., , =, , =, =, , -3a+Z>, , J and, , f/m^ c/ oscillation of the drop,, , a, , /., , =, , t, , X"p2, , 0,, , and, , i, , and, , ., , a*, , ., , -2c -1., ^, , T, , f, , ., , *, , Explain the Principle of Homogeneity of dimensions in a physical, , equation., , M, , of the largest stone that can be moved by a, Assuming that the mass, flowing river depends on K, the velocity, p, the densitv of water and on g, show, varies with the sixth power of the velocity, of flow in the river., that, (Punjab), , M, , Let, , So, , M depend upon K, , a, ,, , P* and g c, , ., , M = KV &&g, a, , that,, , Taking dimensions,, , dimensions of, , 1, , *T has, , [K being a constant*, , M = [MJ, [, , and, , c., , we have, , L, T, , T, , =, , I, , no dimensions, being a mere number.
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PROPERTIES OF MATTER, , 18, , r~ w, , La :r-aAf*Zr'&, , Af, , Or,, , ., , M=, , 5/fK* ffo dimensions on both sides must be the same,, , a-36+c, 0+c = 3, , So that,, Hence, Or,, Le. 9 the, , mass, , M, , 0,, , 1), , and, , =, , 1,, , and, , -a-2c = 0,, And, , 0+2c = 0., = JCKV*- 8, , M, M oc K, , varies with the sixth, , ,, , we have, or, , a-\-, , .*., , c, , 2c, , =, , =, , D,, , -3 and & ~ 6r, , ., , f, ., , power of the, , EXERCISE, , velocity, , offlow., , 1, , 1., If 10000 gms. be the unit of mass, 60 sees., the unit of time, and the, acceleration due to gravity (981 cms.lsec*.), the unit of acceleration, what, Ans. 3'465 x 1C 12 ergs., would be the unit of energy in ergs 1, , \j), dals (1 //., 3., , Convert by the method of dimensions, 4*2 x 10 7 ergs into foot -poun*, Ans. 96 6ft. poundals*, 30'48 cms., and 1 Ib. =453'6 gms.)., Deduce the dimensions of (/) specific gravity, (a) surface density and, , =, , *, , (Hi) angular velocity., Show that the kinetic energy of a, 2, by kmv , where k, , city v, is given, , Test, by the, , 4., , relations, , is, , body of mass m, moving with a velo-, , a constant., , method of dimensions, the accuracy of the following, , :, , (i) v, , (//), , 2, , u2, , =, , S, , 2aS,, , ut, , +, , a, , Jflf, , _ V, p=, 4^0, , /\, , (in), , *, , connecting initial velocity u, final velocity, tion a and distance S covered by a body., , v,, , accelera-, , connecting distance S with initial velocity u, time, and acceptation a of the body,, , where v is the w^aw density of the earth, g the acceleration due to gravity and (7,, y, , t, , t, , r, its radius, the gravita-, , tional constant., , that the excels pressure (p) inside a soap bubble depends, on (() the surface tension (T) of the soap film and (//) its radius (n, show, by, your knowledge of dimensions, that it is directly proportional to the former and, inversely proportional to the latter., 5., , Assuming, , [Hint, as, , Simply show that p^k.Tfr, whence, , T and inversely as r.j, 6., A drop of liquid, , it, , follows that, , p, , varies directly, , in another liquid of the same density, drop is distorted fiom the spherical, shape and released, deduce, by dimensional methods, a formula for its period, of oscillation (/), given that the latter depends on surface tension T, density?, , but with which, , and drop-radius, , it, , is, , is, , suspended, , immiscible., , If, , the, , _, , r., , Ans., , tk \I P, , -JL~,, , where, , fc, , is, , a constant., , /Ch Convert, by the method of dimensions, a pressure of Impounds wt., Ans. 7*912 x 10 4 dynes Jem, per square inch into dynes per sq. cm., 8., Show that when bodies of geometrically similar form and of the same, material, differing only in dimensions, vibrate in the same manner, the vibrations being due to,the elasticity of the material, their periods are proportional, to their dimensions., tfE, where I /*, Proceeding in the usual manner, show that t, klj, the linear dimension of the body, p, 1/5 density and E, the elasticity of the material., Since p and E are the same for all bodies, t varies directly as /.], Calculate, by the method of dimensions, the number of foot-pounds, 2, 7, in me calorie., (Given that 1 ca/0rie=4'2x 10 ergs ; #=32/r./,sec ; 1 /^.=453'6', Ans. 3'1 15., gms., and 1 iwcA=2'54 cms.)., If in a system of units, the unit of length be 1 mile and that of time,., 10., 1 hour, what will be the value of, Ans. 14*88 miles Isec.*, ?, 21, , ., , _, , ., , ^, , ., , y, , 11., , The time of oscillation, , tension depends, , dimensionally that, , /, , upon the density, t, , oc, , of a small drop of a liquid under surface, d,, , radius r and surface tension S., , *., \J, V S, , Prove-, , (Punjab, 1947),
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UNITS AND DIMENSIONS, 12., , Explain what you, , mean by, , 19, , the dimensions of a physical quantity;, , calculate the dimensions of Young's modulus., , Assuming that the period of vibration of a tuning fork depends upon the, length of the prongs, and on the density and Young's modulus of the material,, find, by the method of dimensions, a formula for the period of vibration., (Calcutta, 1950), Ans. [ML" 1 ! 2 ] ; t oc iVdIY,, 1, , the period of vibration ; /, the length ; dt the density and Y, the value, (where, of Young's Modulus for the material of the fork.), *13., Using the method of dimensions, obtain an expression for, (/) the acceleration of a particle moving with a uniform speed v, in a, circle of radius r ;, / is, , m, , circular wire of radius r and mass, (//') the tension Tin a uniform, per, unit length, rotating in its own plane with an angular velocity o>, about am axis, passing through its centre and perpendicular to its plane ;, of a planet round which a satellite completes its orbit of, (*ii) the mass, radius r, in a time-interval T., , M, , Ans., , (i), , K.v*lr, , ;, , (ii), , K.mrW, where Ki$, , a constant, , ;, , (///), , M oc r*!GT, , 2, ., , Obtain an expression for the height h to which a liquid, of density p, and surface tension Twill rise in a capillary tube, of radius r, given that /zocl/r., *14., , T, , Ans. h~k. ------ , (k being a constant)., r, -, , Assuming that the viscosity, free path X of us molecules, show that,, *15., , independent of the density, , p, , 73, , if, , P, , g, , of a gas is proportional to the mean, the temperature be kept constant, it is, , of the gas., , First obtain an expression for ?), in terms of p, X, c, (the root, velocity of the molecules) and />, the diameter of a molecule., Then, since r, oc x, we shall have 73 fc.p.c.X, (where A: is a constant). Again,, since p is inversely proportional to X, / e., p=A;'/X, (where &' is another constant), we shall have f\=k.k'c^ showing that ?j is independent of p.], , [Hint., , mean square, , Show that if the linear dimensions of the whole of Cavendish's or, *16., Boys' method fjr the determination of G be changed, the sensitiveness of the, apparatus remains the same., Show that the volume of a liquid, of coefficient of viscosity *j, flowing, *17, 4, per second through a tube of circular cross-section is given by K=--wpr /8r</. where, p is the excess pressure between the ends of the tube, r, its radius and /, its, length., *18., , If the resistance of a liquid to the motion of a body through it with a, 2, velocity v, be proportional to v , show that it is quite independent of the viscosity of the liquid., *19. A Nicholson's hydrometer of mass w, floating in a liquid of density p,, given a slight downward displacement and then released. Obtain an expression for the time-period Tof its oscillation. (Assume the area of cross-section, Ans 7=2", of its neck to be a.), *20. A (/-tube of uniform cross-section contains mercury up to a height, h in either limb. The mercury in one limb is depressed a little and then released., is, , Vm/tW, , Obtain an expression, [Hint, , for its time-period of oscillation., , Just put T=*k.d*tfg and, , show, , Ans, , r, , that, , T is the time-period of oscillation of mercury and d, its initial displacement, K being the usual const airtof proportionality. For small values of rf, a=0, Substitute their values and obtain the, and experiment gives K=* w\/2., where, , result.], , Note., character and, others., , The questions marked^ith an, may be attempted whin some, , asterisk are of rather an advanced, confidence has been gained with
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CHAPTER, iviOTION, , II, , ALONG A CURVE THE PROJECTILE, , Suppose we have a rigid body, with a fixed, it., Then, if a force be applied to it, it cannot, move bodily, as a whole, relatively to the axis, i.e., no motion of, but it simply moves round or rotates about, translation is possible, the axis, such that every particle of it undergoes the same angular, displacement. A body, so rotating about a fixed axis, is said to perform rotatory or circular motion., The force, producing rotatory motion about the fixed axis,, called the axis of rotation, is said to have a moment about that axis,, which is measured by the product of the force and the perpendicular, distance between its line of actio-i and the axis of rotation., Obviously,, therefore, if either of these be zero, the moment, or the turning tendency, 7., , Rotation, , axis, within or without, , ;, , will be zero, for the prod'ict of the force and perpendicular distance between the axis and the line of action of the force is,, , of the force,, , then, zero., It* fie rotation, produced bs anti-clockwise, the moment of the, force is said to ba positive, *ad if it b3 in the clockwise d ration, the, >m3nt of a force is, moment is said to be negative. And, since th^, a vector quantity, it follows that if a number of forces act simultaneously on a body, the algebraic sum of their individual msmints about the, given axis of rotation will be equal to the moment of their resultant, , m, , about, , it., , Angular Velocity. Let a body rotate about a fixed axis, through 0, (Fig. 1). Then, the particles composing it, at any distance, from 0, such as at A, B C, etc., complete, one rotation in the same time i.e., they, describe the same angle in the same time,, 8., , 9, , and, therefore, the angle described by, them per unit time is the same. This, angle described by a rotating body per, unit time is called its angular velocity and, usually denoted by the Greek letter a>., if the rate of rotation of a, body be, uniform, i.e., if its angular velocity be, constant, and it describes an angle 9, , is, , Thus,, *, , (radians) in tima, , /, , (seconds),, , we have, , =, , it is, , angular velocity of the body, a>, 6/t., If the body makes n rotations in time /, the angle described, 27T/I., Or,, equal to 2irn., , And,, , =, , =, , by, , its angular velocity 01, 2irn/t., the, If, however,, velocity be not constant, it may, at a given, instant, be expressed in the fornotai, d0/dt, where d0 is the small, angle described by it in the small iAryal of time dt<, , therefore,, , =
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MOTION ALONG A CUBVE, , THE PBOJECTILE, , 21, , Now, although the angle described by all the particles of the, body in a given time t is the same, the linear distances ^travelled by, them are different. Thus, the particles at A, B, and C, (Fig. 1), cover, the linear distances AA', BB' and CC' respectively, (which are arc$, of radii OA, OB, OC), depending upon their respective distances from, the axis of rotation through O., If OA =rl5 OB, , = r and OC = r, = radius* angle, =r, |~v arc, subtended by, _ '2, r U, Q, u, CC = r 8., A = rrf't, that of B = r 0/f, and that of, , arc, , clearly,, , ?, , A A', uu, , t,, , 9, , l, , it., , ', , ,,, , and, .-., , 3, , linear, , C=, , v, , velocity of, , t, , //., , Or, in general, linear velocity v, the axis of rotation is r6/t., v, , Or,, i.e.,, , linear, , velocity, , =, , distance, , =, , of a particle, , at a distance r, , [v, , roj,, , from, , the axis, , of, , rotation, , x, , 0\t, , from, , =, , o>., , angular, , velocity., , If the angular velocity of a rotatnot, is, said, to have an angular acceleration,, be, it, constant,, ing body, which is defined as the rate of change of angular velocity. It is, usually denoted by the symbol dwjdt. Thus, if the angular velocity, to a/ in time /, its, of a particle about a given axis changes from, rate of change of angular velocity, or its angular acceleration is,, in angular velocity, clearly, (a/, co)/f, or dw/dt, ifdfo>be the change, in time dt., 9., , ^Angular Acceleration., , >, , Now, if the distance of the particle from the axis of rotation be, linear velocity changes from ru> to ro/ in time t, and, therefore,, rate of change of its linear velocity, or its linear acceleration, is given, r, its, , by, r, , rw, , da>, w\, ", ~~ T, = roj(, ~dt, linear acceleration = distance from axis of rotation, , ~, a _/o/, , ', , ~, , ', , t, , t, , Thus,, , X angular, 10., , Couple., , collinear forces act, , When two, on a body,, , equal,, , (Fig. 2),, , acceleration., , opposite,, , parallel, , and non-, , bringing about rotation, (with, , no motion of translation), they are said to, constitute a couple, the turning moment of, the couple be.ng measured by the product of, one of the forces and the perpendicular distance, between them, or the arm of the couple, as it is, called., , Thus, moment of a couple, C, = one of the forces x arm of the couple., Fig. 2., The moment of the couple (also sometimes referred to as the, torque), acting upon a body is quite independent of the position of the, axis of rotation., For, if the t w^P*ces F and F, (Fig. 2), constituting a couple, act at points P andK, and if the axis of rotation passes, through P, there is no moment IS the force acting at P about it and, the moment oi the force acting mt Q is FxPQ, and therefore, the, ,
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PROPERTIES OF MATTER, , 22, , moment, , of the couple is FxPQ. And, if the axis of rotation passes, through any ojher point 0, the moment of the couple about it is, equal to the algebraic sum of the moments of the forces P and Q, about it, i.e., equal to (FxOQ)-(FxOP)=FxPQ, as before., The same will be true for any other position of the axis., Work done by a Couple. Work is done by a couple in, 11,, the, body on which it acts, the amount of work done being, rotating, equal to the product of the couple and the angle of rotation "of the, body, as will be clear from the following, :, , a body, acted upon by a couple,, of, of one of the forces, constithe, P,, application, point, pass through, , Let the axis of rotation, , of, , tuting the couple, (Fig. 3)., Now, if the body rotates through an, angle d&, the point Q moves through a, distance PQ.dQ, where PQ is the perpendicular from P on to the line of action of the, force T7 acting at Q., Therefore, the work, done by this force is equal to FxPQ.dB., And, since the point P does not move, no, work is done by the force at P. Thus,, Fig. 3., the work done by the two forces, i.e., by, the couple, in rotating the body through an angle dO, is equal to, ,, , Hence, work done by the couple in rotating the body through, the whole angle, is obtained by, integrating this expression, for the, , =, , and = 0., work, done by the couple in rotating the body through, Or,, , limits 6, , the whole angle Q, , is, , given by, , W -P, , W=, , Or,, , Now,, ", , F.PQ, , is, , the, , moment, , Now,, , C.0, , ., , F.PQ f, , ^ix work done by the couple, , W as, , F.PQ dd =F.P0 1, e, , T, , =, , d0., , F.PQ.8., , of the couple C, acting on the body., in rotating the, , body through angle, , $, i.e.,, , = couple x angle of rotation., , in one complete rotation, the, , body describes an angle, , 2tr;, , co that,, , And, , work done by the couple in one full rotation of the body =2?rC., work done by the couple in nfull rotations of the body*=*ZvnC., , .-., , 12., Relation between Couple and Angular Acceleration. When, the resultant couple acting on a body is not zero, it produces an, Let us deduce the relation betangular acceleration in the body., ween the two., , In Fig. 3, the couple C, acting on the body, causes it to rotate, about the axis of rotation through P., Breaking up the couple and tHjk body into small elements, let, ah element SC of the couple cause tilrotation of an element of mass, 8m of the body situated at Q. Thei^ince couple = force x distance,
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MOTION ALONG A CURVE, , ^, , THE PROJECTILE, , w, , at Q is == 8C/r, where r is the arm 01, the farce acting on the mass, the couple. And, since a couple consists of two equal, opposite and, an equal, opposite and parallel force if, parallel forces, it follows that, also acting at P., acceleration of, Again, since force mass X acceleration, the linear, the particle 8m at Q~8C/r.Sm. But, if angular acceleration of th, be, its linear acceleration is also equal to, , dw/dt,, , particle, [see, 9]., , SC, -sr-, , .c., , =r, , ~, n, Ur, SC, , da), ,-j~., , C=, , Or,, , Now,, , J?r a .Sm ==, , axis of rotation, (see, , /,, , the, , ~, , dot z, ,, ~j-.r .dw., , (da>ldt)Z.r*.8m., , moment of inertia of the body about the, , 27)., , =, , moment of'inertia X angular acceleration., The Hodograph. When a body describes a curvilinear, 13., so, its motion is accelerated and also changes in direction,, that, path,, its acceleration and its path may easily be determined by means of, what is called the hodograph of its motion., Couple, , Or,, , The hodograph may be defined as an auxiliary curve, obtained by, joining the free ends of a moving vector representing the velocity of a, moving particle along any path., For instance, if a point P moves along a curve ABC, [Fig. 4 (a)], such that its velocities are v,, v 2 and v s ..respectively at A, B and, etc., then, if we take any point O and draw straight lines, i.e.,, vectors, Oa. Ob and Oc, [Fig. 4 (&)], representing the velocities of Pat, A, B and C, in magnitude as well as in direction, the curve passing, through a, b and c is the hodograph of the motion of P,, ., , ,, , C, , (a), , Fig. 4., , Now,, , different cases arise, , :, , If the point P be moving with a uniform velocity along tfa, same direction, the points a, b, c, etc. will all lie in the same plac<, and the hodograph will, therefore, be a single point., (/), , (ii), , If the point, , P be moving with, , a variable, , velocity,, , but, , in the, , sapie direction, the hodograph will be a straight line, passing through, 0, For example, in the case of a body falling freely under the action, of gravity, the hodograph will b a vertical line, passing through, , O
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24, , PROPERTIES OF MATTER, , (tit) IfP be projected with a horizontal velocity, the patfo, described will be a parabola, (see, 20), and both the direction and, magnitude of the velocity will change. The horizontal velocity will, throughout remain constant and equal to the initial horizontal velocity, because the acceleration due to gravity acts vertically downwards. The points, a, &,, etc. will, thfrefore, always be at the, same horizontal distance from O, and the, hodograph, in this case,, will thus be a vertical line, not, passing through O., (iv) If the path of P be a closed curve, the, hodograph will alsobe a closed curve. For, example, if P moves in a circle with a uniform, speed v, the hodograph will also be a circle of radius v, because all the, lines, Oa, Ob, Oc< etc. will be of the same, length v. If on, the other, hand, it moves in a circle with a variable speed, the hodograph might, be an oval curve about the point O., ,, , An important property of the, Velocity in the Hodograph., is that the acceleration of P at, any point on the curve, is represented, in, magnitude as well as in direction, by the velocity of the corresponding point on the hodograph, as can be seen fronu, the following, Let A and B be two points, close together,, [Fig. 4 (a)], and let, move from A to B in time t lt such that its velocity v v at A is changed, l, h, to V 2 at B., 14., , hodograph, , ABC, , :, , P, , (b)},, , Further, let another point/? describe the hodograph abc, [Fig. 4, while P describes the curve ABC., , clearly, the point p moves from a to b in time t, and its, therefore, equal to ab/t., But, since oa represents the velocity of P at A and ob, that at, B, ab represents, in accordance with the law of triangle of velocities,, the change in velocity of P in time t,, and, therefore, the, of change, of velocity, or the acceleration of P, is represented by abjt i.e., by the, velocity ofp in the hodograph., , Then,, , velocity, , is,, , we, , We, , thus see that, at any instant, the acceleration, of, the velocitv,ofp in the hodograph of its motion., , P, , is, , given by, , IS/ Uniform Circular Motion. The above affords us a very, simple method of determining the acceleration of a body, moving in a, circle., , P move in a, with cei.treOand?, radius r, with a uniform, speed v, [Fig. 5 (a)]., Then, the hodograph is, also a circle, of radius v,, [Fig. 5 (b)}., Let, , circle,, , of, at, , P, , Now, the velocity, any instant is, , at, , right, , radium, Fig. 5., , OA, , its, , circular, , path, passing through P,, aaJ ob is perpendicular to O, , Therefore, oa is perpendicular to, AOB as / aob as B fin circular measure)., , *nH /, , angles to the, , of
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MOTION ALONG A CTTBVE, , =, , If, r0/t,, , THE PROJECTILE, , P takes time t to describe the arc AB,, whence, 6, vt/r., , its, , =, , velocity v, , And, the velocity of the corresponding point p,, graph,, , of, , is, , abjt, , =, , in the hodo-, , v&jt., , Since the velocity of/; in the hodograph gives the, actual path, we have acceleration ofP, , acceleration, , -, , P in its, , =, , v0 __, , v, , And, since a6, , is, , it is, in, , small,, , v2, , __, =, , j, , r, , t, , t, , parallel to, , vt, , x, , the limit, perpendicular to oa, or, , AO., , Thus, the acceleration of Pis v 2 /r and is directed along the radius, or towards the centre of the circular path in which it is moving., -, , Further, since v, , =, , r.aj,, , we have, , (where, , acceleration of P, also, , Alternative Method., a uniform circular motion, , o> is, , =, , the angular velocity of P),, , r 2 .o> 2 /r, , =, , roA, , The, , may, , acceleration of a body, executing, also be found out directly as follows, :, , Let a particle move with a uniform linear velocity v, in a circle, to B, r, (Fig. 6), and let it cover the small distance from A, in a small interval of time bt, describing an angle, 80. Then, clearly, its angular velocity, o = 86 //., $", of radius, , The direction of the linear velocity is at, every point, tangential to the circle at that point, and is, therefore, represented by the tangent AC, at the point^, and by the tangent BD at the, BD., point B, whtre AC, Now at A, the entire linear velocity is, along AC, there being no component of it along, AO, which is at right angles to AC. And,, revolving the velocity at B into two rectangular, components, one along AO and the other, at, right angles to it, we have the component along, , =, , v sin 8n,, or parallel to AO, represented by BE, and the component at right angles to AO,, , represented by, , BF=, , pj g, , 5., , v cosSti., , If 80 be very small,, , sin, , 80, , =, , 86 (in radians), and cos 8$, , AO =, , So that,, , component BE,, , and, , component BF, perpendicular to, , parallel to, , =, , 1., , v.S0,, , AO =, , v., , Thus, if B be very close to A, there is no change in the velocity, of the particle along the perpendicular to AO, for it remains the same, And,, v, but an additional velocity v 80 is acquired by it along AO., since this velocity is acquired in time 8t, the acceleration imparted, Ml** =,.. the angular velocity, to the, is, Vo>, where, particle, , of the particle., , v.80/8t }, , =
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PROPERTIES OF MATTER, , :26, , Or,, , acceleration of the particle s= tto, , =, , V, v,, , =, , V^, ., , Now, because the magnitude of the velocity remains the same, at every point on the circular path of the particle, it follows that the, acceleration must be acting in a direction perpendicular to the direction of the velocity at that point, i.e , along the radius of the circle, or, else it will also have a component along the tangent at the poiit, or, along the direction of the velocity at that point, which will, therefore,, , no longer remain constant., , Since this acceleration acts along the radius of the .circle, or, circle, it is called radial or centripetal, I seek)., acceleration, (from 'peto*, meaning centre-seeking, , towards the centre of the, , Thus, centripetal acceleration, , And,, , if, , =, , n be the number of revolutions, , per unit time,, , we have, , w, , =, , centripetal acceleration, also, , =, , or,, , ,, , --, , = rco, , made by the, , 1, ., , particle, , 2irn., , =, , a, , r.(27r/i), , =, , 47rVr., , Even if the path be not exactly a circle, but any other curve,, the value of the acceleration is v 2 /r, where v is the linear velocity, and, r, the radius of curvature of the path at the point considered., 16., According to Newton's first law of, CentrigetgLEttcce., motion, a body must continue to move with a uniform velocity in a, straight line, unless acted upon by a force. It follows, therefore, that, when a body moves along a circle, some force is acting upon it, which, continually deflects it from its straight or linear path and, since the, body has an acceleration towards the centre, it is obvrous that the, force must also be acting in the direction of this acceleration, i.e.,, along the radius, or towards the centre of its circular path. It is called, the centripetal force, and its value is given by the product of the mass, of the body and its centripetal acceleration. Thus, if, be the mass, of the body, we have, ;, , m, , centripetal force, , =, , mv<o, , =, , wv 2 /r,, , or,, , =, , mrof, , *=*, , AnWrnr., , centripetal force are met with in daily, in the case of a stone, whirled round at the end of a, string whose other end is held in the hand, the centripetal force is, supplied by the tension of the string ; (') in the case of a motor car or, , Numerous examples of, , life., , Thus,, , (/), , a railway train, negotiating a curve,, , it is supplied by the push due to, the rails on the wheels of the train and (Hi) in the case of (a) the, planets revolving round the sun, or (b) the moon revolving round the, earth, by the gravitational attraction between them., , If this force somehow vanishes at any point in its circular path,, the body will fly off tangentially to it at that point, for it will no, longer be compelled to move in the circular path., 17., Centrifugal Force. The equal and opposite reaction to, the centripetal force is called ihe centrifugal force, because it tends to, I flee). Centripetal, bake the body away from the centre, (from fugo*, force and centrifugal force being just action and reaction in the sense
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MOTION ALONG A OTTBVH, , THJB, , 2T, , PBOJBOTILB, , of Newton's third law of motion, tfio>4mmerical values of the two are, , =, , =, , 47rWftr., mrof, viz., mv*/r, Thus, in the case of a stone, whirled round at the end of a string,, not only is the stone acted upon by a force, (the centripetal force),, exerts an equal, along the string towards the centre, but the stone also, and opposite forc3, (the ce^tjfcfugal force), on the hand, away from the, centre, also along the string., - 18. Practical Applications of Centripetal and CentrjfugalJorcS>, , the same,, , 1., Road Curves. The centripetal force being directly proportional to the square of the linear velocity of the body and inversely, the radii of curvature, proportional to the radius of its circular path,, of road curves must be large and the speed of the vehicles negotiating, the value of the centripetal force, them slowed, in order to, , keep, , down,, , required within reasonable limits., 2. Rotating Machinery. The centrifugal force being proportional, to w 2 where n is the number of rotations made by the body per, second, the spokes of a wheel, joining its outer revolving parts to, the axis of rotation, experience an outward force, away from the, if, centre, and are, therefore, in a state of tension, and may give way, other, of, with, the, case, the, So is, the value of n is very large., parts, outer revolving parts to its axis, rotating machinery, connecting its, of rotation. In other words, there is a limit set to the value of n by, the tension these connecting parts can withstand. This fact is, ,, , like, always kept in view while designing highly rotating machinery,, armatures of motors and dynamos etc., Let us, as a specific example, discuss the case of a belt or a, etc., string rotating at a high speed over a pulley, , Let the string rotate in a circle of radius'r , (Fig. 7), andJet its angular, be o>. Consider a small portion AB of the string, of length / and, subtending an angle 20 at the centre O, of the circle. This portion is obviously, subjected to a tension T, at either end, by the, rest of the string as shown., Resolving these, tensions T and T at A and B into two rectangular components along and at right, angles to PO, (where PO passes through the, mid-point of AB), we find that the compoat right angles to PO are equal, nents T cos, and opposite and thus neutralise e?ch other,, but the components T sin $ along PO act in, the same direction. So that, we have, velocity, , resultant tension on portion, , And,, , AB of the, , Fig. 7., , string, , = 2Tsin in the direction PO., the centrifugal force acting on portion AB of the string, mass of AB x r, , If, , m be the, , mass per, , unit length, , .*., , centrifugal force acting on portion, , For equilibrium,, If B be small,, , therefore,, , we have, , 2T sin 9, sin B, , 2T &, whence,, , ,, , in the direction, , of the string, clearly,, , mass of AB, , And, , 8, , T, , = mx, , /., , AB, , of the string, mx/xr<w*, in the direction, , m/ro, , = rn.2r0.ri*, , So that,, 0., m.2rQ.r<** 9, , mrV., , 1, ., , p.*, , OP, , 9, , clearly,, , OP.
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PBOPBBTIES OF MATTER, , 28, , It will thus be seen that due to the centrifugal force, the tension in the, string is very Hgh. Indeed, if the rapidJy rotating chain or belt be pushed off, the pulley, it will run along like a rigid hoop., *, , The same, , is true about other rotating bodies which are always under, a state of elastic stress. It is this stress which sets a limit to the speed up to, which the flywheels can be rotated safely. Again, it is as a consequence of, this stress that the tyres of racing cars get stretched and there is a danger of, their being cast off the rims and flung out, at very high speeds., , 3., Revolution of Planets and the Length of the Year. In the case, of a planet revolving round the sun, it is the gravitational force of, attraction between the two which supplies the centripetal force,, necessary to keep it moving in its neatly circular orbit. Now, the, gravitational force between two bodies is directly proportional to the, product of their masses and inversely proportional to the square of, the, bet ween them so that, if m and, be the masses of the, distance], planet and the sun respactively and r, thj distance between them (or, the radius'of the planet's orbit round the sun), we have, , M, , ;, , ', , M,, , grarfationa! pull, , = '"f .G =, r^, , 4n 2n 2 rm, , =, , k, , Or,, , ^, Or,, , n, , =, , -, , ', , pttin, La constant., , r, , n 2 ==, , -,, whence,, 2, 1, , '-, , *-., 2, , k, 23.--, , T, , A /, -A/, m, V mr*, o, , 2?r, , =, , -,- =, \V /~mr^, , "mr*, , A, , 2ir, , ,^, MG, , K:, , where, , t is the time taken hy one revolution of the planet round the sun,, or the length of the year for that planet., , 3, Thus, / varies as \/ r i.e., the smaller the value of r, or the, smaller the distance of the planet from the sun, the smaller th3 valuo, of /, or the length of the year, for it. A planet will, therefore, have a, shorter year if nearer to the sun than when at a distance from it., ,, , 4., Banking of Railway Lines and Roads. When a railway train, a level curve on a railway track, the necefesary ceiitiipetal, round, goes, force is provided only by the force between the flanges or the 'rims of, the wheels and the raits, the normal reaction Of the ground or the, track acting vertically upwards and supporting its weight. This, results in a grinding action between the wheels and the rails, resulting in their wear and tear. Not only that, it may also prove dangerous, in the sense that it may bring about a displacement of the rails and, hence a derailment of the train., , To avoid, , these eventualities, the level of the outside rail, , is, , raised, , above that of the inside one. This is known as the banking of, railway lines, and the angle that the track makes with the horizontal, is called the angle of banking., , a, , little, , With the track thus banked,, above the, , with the outer, , rail thus raised, acts perpendicularly, inclined to tlie vertical at an angle, , i.e.,, , level of the inner one, the reaction, , to the track, as before, but, , is, , now, , R
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MOTION ALONG A, , CURVETHE, , PBOJECTILE, , 29, , equal to the angle of banking and its horizontal component (and not, the lateral thrust of the wheel flanges on the outside rail) now supplies the necessary contripetal force to keep the train moving along the, curve, thereby eliminating all unnecessary wear and tear., Thus, if 6 b3 tha angle of banking (Fig., reaction acting psrpendiculaily to it, we have, vertical, , and, , componet of, , horizontal component, , R, , ofR, , 8),, , and R, the normal, , = R cos $,, = R sin 6., , mg of the train and, 2, supplies th'j required ceritripstal force mv /r where v is, tha, curve, it, of, of, the, train, r, radius, atid, the speed, the, negotiates., So, The former component balances the weight, , the latt, , ?r, , t, , = mv jr, R cos $ = mg., , that,, , and, , Rsin, , 8, , R sin, , Q, , tan 9, , Or,, , 2, , wv 2 /r, , =, , v8, ~, , rg, , =, , Or,, , rg, , The angle of banking thus, dopends upon the speed (v) of the, train and the radius (r) of the curve, of the track. Obviously, therefore, a, track can be banked correctly only, in practice, naturally for its, for a particular speed of the train,, average sp3od. At higher or lower spe3ds than this, thore is again a, lateral thrust due to the wheel flanges on the outer or the inner rail, of the track respectively., Cleanly, the angle that the track makes with the horizontal is, equal to 0, i.e., equal to th3 a*igb of inclination of the train with the, vertical, (Fig. 8)., , Tails, , we, Or,, , Further, it will be readily seen that if the distance between the, be d and the height of the outer rail above the inner one be A,, , also, , have, , sine, , sin 9, , = --y, , ., , a, , of the angle of banking, __ height, ~~, , of the outer rail over the inner one, distance between the rails, , ., , Similarly, in the case of a car moving round a level corner, the, centrifugal force is largely provided by the friction between the road, and the tyres of the wheel. That is why, when the road is slippery, and the frictional force not enough the car begins to slide or skid., Here, too, therefore, ,the roads are *banked\ the slope being generally, more or less like a saucer the outer parts being, steeper outwards,, meant to be used at higher speeds and the inner ones, at lower speeds.
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PROPERTIES OF MATTER, , 30, , Again, an aeroplane, in order to turn, must also bank, the, centripetal force here being supplied by the horizontal component of, the lift L, (Fi. 9)., , The same, , applies to a cyclist,, negotiating a curve or a, corner, and he has to lean inwards,, towards the centre of the, (/.e.,, tan- 1 v 2 /rg ;, curve), by an angle, so that, the faster his speed and, the sharper the curve, the more, must he lean over. This will be, clear from the following, , when, , =, , :, , Let Fig. 10 represent a, , cyclist, , turning to the left in a circle of, radius r, at a speed v., Then, the, normal reaction R of the ground, acts vertically upwards, with the force of friction F between the, 2, ground and the tyres and the centrifugal force wv /r in the directions shown,, , where, , R = mg, , F=, , and, , Then, for equilibrium,, , 2, , /??v /r., , clearly,, , moment of mg about P equal and, to moment of /nv 2 yr about P., , we have, opposite, , mgxPQ =, , Or,, , mg x PG.sin, , Or,, , sin 9, -, , whence,, , ~, , =, , =, , mv, ., , tan, , A, , PG.cos, , =, , cos 6, , 6,, , v1, , rg, , In other words, in order to keep himself in equilibrium,,, the cyclist must lean inwards from the vertical at an angle, tan- 1 (v 2 /rg)., , =, , If he were to remain vertical, his weight would act through P,, 2, having no moment about it, so that the moment of mv /r about P, would remain unbalanced. In fact it will be readily seen that thesystem of forces acting on the cyclist form two pairs of couples, one, due to F and mv 2 /r and the other due to R and mg. So that, in theevent of the latter couple vanishing (i.e , if the cyclist were vertical),, the former alone will remain operative, resulting in the cyclist toppling, over., , =, , Further, since the maximum value of F, itfng, (where M is theof friction between the ground and the tyres), the cyclist, will skid when mv 2 /r, nmg, or when v*>urg., coefficient, , >, , Thus, skidding will occur (i) ifv is large, i.e., if the speed of the, (ii) if n is smallf i.e., if the road is slippery and (Hi], , cyclist is large., , tfris small,, , i.e.,, , if the curve is sharp.
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MOTION ALONG A CT7BVE, , THH PROJECTILE, , 31, , Similar conditions apply in the case of a motor car or any other*, For, here too, if we imagine it to be turning to the left r, (Fig. 11), the various forces acting, on it are the normal reactions Rl, and R t the frictional forces Fl and, F2t its weight mg and the centrifugal, force mv*/r, as shown, the whole, system being in equilibrium., vehicle., , ,, , Obviously, in the event of, the car being about to be upset, it, will be moving on the wheels on, one side only, so that the normal, reaction on the wheels on the other, side will be zero., So, 0., say R l, that, it will overturn as soon as the, moment of mv 2 /r about P is greater, than the opposing moment of mg, , =, , ;, , about P., as soon as, , i.e.,, , mv z, , mv 2, , when, , Or,, , where h, , is, , .GQ, Jt, , > mg, , >, , ., , PQ,, , mg.d,, , the height of the e.g., <7, of the car above the ground and?, between the two wheels., , 2d, the distance, , For the car to be upset, therefore, we have, , The, , car, , is,, , v2, , >, , '--., , therefore, not likely to bo upset if 2d, the distance, is large and if /i, tho height of the, e.g., , between the two wheels, from the ground is small., Again, the, , maximum, , value of the total frictional, force, , So that, as before, skidding, , when, , Or,, , v, , 2, , >, , will occur, , when wv 2 /r, , >, , iimg., , urg., , To avoid, , skidding, therefore, while taking a turn at a fast, speed, the corner must be cut so as to move along a comparatively, flatter curve than that of the actual turning., , J&., , Other Effects and Applications of Centrifugal Force., Rotation of the Earth- Its Effect. As we know already,, 1., the earth rotates or spins about its axis once during a day. It is, this rotation of it which is responsible for its getting flattened at the, a direct consequence of the, poles and its bulging out at the equator,, centrifugal force m<u*R acting on each particle of mass m of it, where, to is its angular velocity about the axis of rotation and R, the distance, of the particle from this axis. The value of a> is obviously the same, for each: particle, but the distance R increases from zero for particles, at the poles to a maximum for those at the equator. The centrifugal, force pulling the earth outwards, as it were, is thus zero at the
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PBOPBBTIES OF MATTER, , 32, , and the maximum at the equator and it is this force which has made, the earth (behaving like a plastic body) to bulge out at the equator, and to flatten at the poles, thus bringing about an incr3ase of about, 13 miles in its equatorial, as compared with its polar radius., This, , of the rotation of the earth had been first predicted, verified by a French expedition to Lapland, under the leadership of Maupertius. whose undue pomposity provoked Voltaire into making the caustic remark that he behaved as, though he had flattened the poles himself/, effect, , by Newton and was duly, , *, , The Centrifuges. These are simple devices used to separate, 2., ou f substances of different densities suspended in a liquid, by rapidly, rotating the liquid, when particles, whose density is greater than that, of the liquid, are driven away from the axis of rotation, whereas, those, with a density lower than that of the liquid, are drawn inwards, towards it. Thus, for example, in the familiar cream -separator, when, the vessel containing milk is rotated fast, the cream, being lighter,, collects in a cylindrical layer round about the axis, whence it can be, /, , easily, , drawn, , off., , Since the centrifugal force (wo>V), , increases with, , r,, , the pressure, , on the rotating liquid progressively increases as we move away from, the axis, with the result that, on a heavier particle, the centrifugal, force outwards is greater than the inward thrust of the liquid,, The problem is, whereas, on a lighter one, the reverse is the case., more or less akin to the sinking or floating of a body in a liquid at rest,, depending upon the difference in the magnitude of the forces acting, on the two sid^s of the body, the inward thrust on it corresponding to, .the upthrust in the case of a stationary liquid., Since centrifuges have as h><*h speeds of rotation as 40,000, revolutions (or more) per minute, the difference between the outward, and inward forces acting on the heavier and lighter particles exceeds, more than a thousand times the difference between their weights, so, , that quick and effective separation results. Sediments, precipitates, bacteria etc., may all be thus separated speedily., , and, , is the centrifugal drying machine,, in its walls. When,, just a cylindrical vessel, with perforations, with damp clothes placed inside it. it is rotated fast, the centrifugal, force acting on them forces the water out through the perforations, .and the clothes thus get dried up quickly., , Another familiar example, , which, , is, , Also known as the Turbine Pump,, of three essential p^rts, viz., (i) an outer drum-shaped, 'casing\ having an inlet near its axis and an outlet near its periphery,, wh*el (i.e., a hollow wheel, fitted with vanos) called the, (//) a paddle, can be rotated inside the casing, and (Hi) the, which, 'impeller',, is transmitted from the driving motor, 'spindle'* through which energy, 3., , The Centrifugal Pump., , it consists, , to the impeller., If the casing be filled with water and the impeller rapidly rotatsets the water into similar rapid rotation, which, due to its, outer wall of the, centrifugal force, exerts high pressure a<?a nst the, the outlet at the periphery into, out, water, the, through, forcing, casing,, ed,, , it, , ;
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MOTION ALONO A CtJBVl, , THU PROJBOTILB, , 33, , the rising discharge-tube connected to it. At the same time, there, decreased pressure near the axis, so that the atmospheric pressure, This is, forces fresh water in, from the reservoir, through the inlet., then again flung out through the "outlet, in the manner explained, and, the process goes on repeating itself over and over again., is, , The pump starts working only when the casing is full of water,, but, once it has started working, it gi ves a continuous supply of water,, unlike the ordinary piston pump, where we get only an intermittent, supply., , Further, as there are no valves to operate, the pump can be used, safely even if the water contains sledge or any other suspended matter, including sand or small-sized stones etc,,, , The Projectile Motion of a Projectile in a non-resisting, Before Galileo's time, it was supposed that a body thrown, horizontcally, travelled in a straight line until it had exhausted its, force and then fell vertically down., It was he who first showed that, it must take a parabolic path*, realizing, as he did, die physical independence of its horizontal and vertical motions, so that each could, be considered separately., 20., , medium., , Such a body, subjected simultaneously to a uniform horizontal, motion and a vertical uniform acceleration, is called a projectile, &nd, the path it describes is called its trajectory. Let us study its motion, in some detail., Let a body be projected upwards with a velocity w, at an angle, 6 with the horizontal. Then, resolving u into two rectangular components, along the verticalf and along the horizontal, we have (/) t he, vertical component (along (7*7) = us in 9 and (//) the horizontal com-, , =, , u cos 6. The latter component, being perpendiponent (along OX), cular to the direction of gravity, is not accelerated, and hence, , dx, , =, , u cos, , 0., , at, , Since at, , And,, , /, , = 0,, , x, , =, , 0,, , we have, x =s. ut cos, , because the vertical component, , acceleration due to gravity,, , **, , where, , Cl, , Now,, , at, , is, , t, , we have, , ... (/), , is, , subjected, , to a, , downward, , we have, ~~~, , dt*, , integrating which,, , 9, , ~, , -, , gt, , gf+Cj,, , a constant of integration., , =, , dy, 0,, , ,, , dv, , J, , s= u sin 6, , =, , u sin, , ;, , so that, , C =, 1, , u, , sin 9., , 6gt., , *He dropped objects from masts of moving ships, which fell vertically iq, relation to the ship but along parabolic paths in relation to the sea., f /.<?,, along the dirftfjpg in which th<? force due to gravity acts,
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34, , PROPERTIES OF MATTBB, , ut sin 8, Integrating this again, we have y, where C8 is another constant of integration., , Since, , y, , =*, , at, , t, , =, , we have Ct, , 0,, , y^utsine, , Or,, , Now, from, , relation, , Substituting this value of, , j, , =, , t, , =, , ..., , *-, , w cos, , .sin, , U CO^, , (i7), , t, , in relation (),, , w,, , 0., , Igt*., , we have, , (/),, y, v, , ], , 6, , we have, , Q\g., J, , ~, , (, \, , A}, , C05 0/, 1, , tl, , V2, , This is clearly an equation of the second degree in x and the, degree in y and thus represents a parabola, with its axis vertical,, The trajectory of the particle is thus & parabola., , first, , 21., , Horizontal Range of a Projectile., to reach the maximum height, , by the body, , velocity being u sin 0., And, since time of ascent, , Clearly, the time taken, u sin 0/g*, its vertical, , =, , is equal to time of descent, the total, time taken by the body for the whole flight, 2u sin dig., During this time, the horizontal distance covered by the body,, with its uniform horizontal velocity u cos Q is given by, , U COS, , A, , 2u, , sin 6, , 2u 2 .sin Q.cos 6, , =, , 6., , g, , ,, , =, , u*.sin 20, ~, , =, , g, , r, , ., , ['' 2 sin, , n, , .., cos Q=*sin 29., ., , g, , This horizontal distance covered by a projectile is called its horizontal range, or, more usually, simply, its range., Denoting it by R,, w 2 sin 20/g., therefore, we have, , R=, , 22., , va, , Height attained by a Projectile. We have the, w8, 2aS where the symbols have their usual, , g, , (the, , Maximum, , kinematic relation,, meanings., Here, a, , =, , t, , body being projected upwards), and, at the, , =, , highest point, obviously, v, attained by the projectile be h,, sin 6)*, , 0--(, , =, , h **, , if, , the, , S~h). we have, , f, L, , 2.(-~g)./j,, 7v2, , whence,, , So that,, , 0., (i.e.,, , c/2 *, /), , -^, 2g, , ., , maximum, , height, , v, , the initial, upward, velocity here is u sin $, and not u., , 23., Angle of Projection for Maximum Range. It is obvious, that for a given initial velocity (u) of the body, its horizontal range, (R) will depend upon its angle of projection (d)., , Now, the, , horizontal range, , D, , ^R,, , u*.sin, , as, , we know,, , is, , given by, , 20, , ~^g, ., , Putting x for R,, , we have x, , g, , *We have the kinematic, (here equal to u sin, pointj. So that,, , 9),, , -, , a==, , u sin, , relation, v, , w-f at, where u, , is, , the initial velocity, highest, , g and v, the final velocity, equal to 0, (at the, Q~gt ; or, gt *u sin fl. Or, t * u sin Qlg.
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MOTION ALONG A CTJBVE, , THB PBOJJROTILB, , 35, , It is thus clear that the value of, for maximum (horizontal) range, of the projectile would be that for which sin 20, 1, i.e., when, 28 =90, and, therefore,, =45., , =, , Thus, for, , maximum, , range, the angle, , of projection should be 45, , N.B. The following interesting result follows, howevw. from the relation, , R=, , We, an angle, , is, , u* sin 2$/g,, , know, , that the sine of, the same as- that of, , its, , supplement. And, therefore,, sin 2$ ^ sin (180-25),, from which it is clear that the, projectile will have the same, range (not the maximum), for, the angles of projection $ and, the two paths taken, (90~0),, being, however, different, and, called the high (H) and the low, (L) trajectories, , shown, , in, different, , Fig, , respectively,, , 12, in, , maximum, , as, , view of the, heights at*, , tained by the body., Fig. 12., , 24. Range on an Inclined Plane. We have seen above,, that the equation to the trajectory of a projectile is, , ~.~ _, , y=x tan 0$g., , a, , (20),, , ., , f, , Let us consider a plane inclined to the horizontal at an angle, (Fig. 13), , ;, , so that,, , y, , =., , x tan, , a,, , a., , Now, to obtain the range on the inclined plane, we must determine the point where the trajectory of the projectile will meet the, and to do this, we must, plane, solve the above two equations., So, ;, , that, substituting, , =x, , y, , tan a in, , the equation for the trajectory,, , we, , have, , x tan a, Or,, , =, g*, , M^J =, _, , Fig., , v, , x tan, , 2(tqn 6, , have, , y, , i, , ,, , o, , the range, pp tbat f, , tane - tan, , R is clearly given by the relation,, , -, , tan a).u* cos* 6, , Ms value of x irry = x tan a,, cos 2 B, 0tan a),w, 2(tan, ---------.tan a., 2, , ,, , ^^-, , M- COS 8, , ^, , 13., , And, therefore, substituting, wre, , J.
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PROPERTIES OF MATTER, , 36, , ac, , a, , P!^J?ITJ^, , 2, (14 fan, , Now,, , a, <Ji, , c, , <w2, l, , r2(tan, , *_6~]*, , =, , r, , a), , =, , sec 2 a, , jv, , _, , T2(/^ 0-tan, L, , p, , XV, , ~, , -, , _, , n -tan*, ., , ^, , ,, , a], , [1, , ., , n, , --a), , w 2 ros*, , --, , I 2/, I, , ,, , CL/o, , (A*, , Jy, , tan a ) " 2 coja *, , 2 (' flW ^, ~, , "I, , a.u*co$* 6 1*, , cos z a, , --, , ', , y.)tan, , 2, , 2, tan a).u 2 cos, , 2(tan Q, , =, , Qtan, , ---, , ~ -----, , ', , g, , ~, , *, , ~, , -, , cos a., , Resultant Velocity of a Projectile at a given instant., The, velocity of a projectile, t, sees, after its projection, are clearly dyjdt and dxjdt., So that, its, resultant velocity v, at the instant, is given by the relation,, 25., , and horizontal components of the, , vertical, , \dt, , dt, , Or, putting the values of dy/dt, , Or,, , and dxjdt from above,, , v, , =, , i/usmti-gty+fa casoy., , v, , =, , ^u*-2ugt.sinT+g*t, , 2, , dx, , dy, , u, , - 6~gt -, , sin, , /, , * *, ,, , P=^-, , Or,, N.B., /5, , Let, , gt, , 4-?, , 3, , =, , /,, , 0, v/e, , this value, , it, , have, , of, , f, , be denoted by, , gt' -- aw t, 6, tan, , cos, , by, , -, , makes with, , p is positive ; but, for, t, tan (3, and, therefore,, acquires a negative value. >4/H/, obviously, when (3=0, //re, moving horizontally, i.e., at the very peak of the parabola. In this case,, , since tan p, , ^', , Or,, , ., , ,, , tan e, , j/, , 0,, , whence,, , 0=, , /', , /'., , Then, we have, , sin, , ---, , ~, , ., , Clearly, the vertical distance covered by the, la/*., y** ut, , body at an instant, , =, , A,, , the, , maximum, , Substituting these valuss in the above relation,, , h, , when, , height attained by the body,, , =, , w, , j/, , $.r'~t^', , ,, , F, I, , *, , ', , f, , ~, , given, , ~, , I/, , C/H, , /Q, , K, , we have, ', , 3, , f, , is, , t, , +, , .y, , 1., , For small values of, , large values of, /><?*/>>, , =, , 1, [/ jw tf+co^ 2, , ., , And, the angle, p, say, that this resultant velocity, the horizontal, is obviously given by, tan p, , we have, , 20),, , (, , P ward velodt y, , initia, |, , SB, , M5i, , fl.and a, , =, , j?,
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MOTION ALO&Q A CtTEVE, , =, , P&OJEdTlLU, , T?H1S, , 37, , u sin 9, , ., , u sin, 2, , 2, , 2, , ^, , w stn Q u, _ ---sin*, , a-,., , -, , 2#, as obtained above in, , the same result,, Again, the horizontal range, , may, , sin, , i, , QJ-, , ., , 2, , ssr, , fi, , $, , 2#, 22 (page 34)., be easily obtained by equating the \alue, , of>toO., Let, , f, , -, , when y, , /*,, , Or,, , u sin, , Or,, , */, , ", 0.f, , -, , *, , Then, we have, from above,, u sin Q.t"-lgt"*., = \gt"., w j/, Or,, \gt"\, , 0., , =, -=, , 2u sin, , *, , whence,, , 0,, , /, , 2w, , *, , J//l, , =,-, , ., , #, , Now, as we know, the initial horizontal velocity u cos 0., And, therefore, the horizontal range R is given by the relation,, , RD =, , A, u cos O./*, 2, , = u-, , 2 sin, -, , u cos, , cos 6, , 2, , u', , o, , the, , same, , finally, if, , we, , f), , =, , (on 6, , tan p, , Or,, , -, , ., , --, , =, , cos, , 2, , [v, , ,, , /,, , (i.e, , ,, , t", , =, , 2u sin, , we have, , -, , ran, , -^r0-2, , /an, , =, , Mfl 20., , 21, (page 34)., , substitute this value of, , expression for tan p above,, tan, , sin 20, , o, , result as obtained before in, , And,, , ., , to, , -, , in, , the, , 2 " '"" 9, -, , 14, , ^-tan, , ff, , 0/0),, , COS, , #, , o,, , showing that the projectile comes back to the horizontal surface at the same, angle at which it was projected upwards. And, it is a further simple deduction, that its tangential velocity at this moment is the same as at the instant of projection., It ijmst be emphasized again, however, that the above treatment applies to the motion of a projectile, only in a non-resisting, medium, i.e., in vacuo. The presence of a medium, like air, offers a, frictional resistance to its motion, which depends, to a great extent,, upon the velocity of the body and is, for moderate velocitiesf directly proportional to the square of the velocity, in accordance with the, law of resistance given by Newton, in the year 1687. This alters the, very character of the trajectory of a projectile, which no longer, remains a parabola but becomes what is called a ballistic curve**, with, its descending part much steeper than its ascending part and the height, ,, , and range of the, , projectile considerably reduced,, , particularly at high, , initial velocities., , At higher, , altitudes, well above the average level of the earth's, surface, however, the air pressure, and hence also the air resistance,, becomes much smaller and, therefore, if a projectile be shot up to, , maximum, , quite possible to obtain a high, most probable explanation of the, the, perhaps, German canon, usad with such conspicuous success in the, historic Great War, and which could fire shots to a maximum height of, about 54 kilometres and hence had a range of about 130 kilometres., , such great, range for, long range, , it., , This, , heights,, , it is, , is, , *, Another possible value of t" is 0. We reject it, however, as it refers to, the time when the body is just starting on its trajectory., fFor example, from a velocity of a few metres per second to about the, velocity of sound, in the case of air,, ** Ballistics is the special name given to the science of the motion of
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38, , PROPERTIES OF MATTfiR, , SOLVED EXAMPLES, A, , particle, moving in a circle of radius 105 cms., has its velpcit, increased in one minute by 120 rotations per minute. Calculate (i) its linea, acceleration (//) its angular acceleration., 1., , in velocity of the particle in, , Change, , Now,, , in, ,,, , 1 rotation, distance covered, 2 rotations, ,,, , =, =, , min., , 1, , =, =, , 2nr., , = 2x2x*x IQScms.jsec., , 2x2*r, , in velocity of the particle per, , minute=420 x n cms, , change, /sec., change in velocity per second 420x^/60 = 7rr cms. /sec., 22 cms. jsec 1, rate of change of velocity = 7 x 22/7 cms. /sec*., 22 cms. I sec*., linear acceleration, a, , Or,, Or,, i.e.,, , Now,, , linear acceleration,, , where r, So that,, 22, log, log 105, , Antilog, , is, , a, , i.e.,, , r.du/dt,, , =, , =, =, , ', , 1*32 1 2, , Hence, the linear and angular accelerations of the particl<, 0*2095 are 22'0 cms., and '2095 radianjsec 2 ., respectively., /sec*,, , =, , from, , Let the present distance of the Earth, R., orbit round the Sun) be, , =, , If, , how man:, , the Sun,, , be in one year ?, , will there, , So, , from the Sun, , the radius of, , (i.e.,, , =, , we have, , If v t and v 2 be the linear velocities of the Earth, corresponding, two values of the acceleration, we have, , = -*, that ', , 2^7*, , -, , T, , a, , XT, , JNOW,, , i, , And, therefore,, , Now,, , Vt, , ', , =, =, , =, , a, , Or, , %" x 5?, , ', , _, , 2"R, 365, , P*, , ^2v t, , =, , oZc~, , -, , -, , -1, , and, , 1, , Vl, , f., , ,, , aay., , i.e.,, , \, , =, , -^- per, , I", , a, , Earth's orbit, circumference, of, ~., ., time taken, , *~*-, , L, -V2., , =, , is, , to" these, , ?&!, , -, , ~, , -, , -, , day., , circumference of the Earth's orbit, in the second case, , And, since one year, Sun,, , its, , that, half its present distance from the sun would be, R/2., a t and a* be the accelerations (linear) of the earth in the two cases res, , pectively,, , 80, , ., , the radius, and dajdt, the angular acceleration., angular acceleration, d&jdt, a/r., _ __, ., 1-3424, ,., ,. = 22 =, A, ,, ^r, *2095, radian/ sec., 2 0212, d^jdt, , If the Earth be one-half its present distance, , 2., , days, , 120 rotations per min., = 2 rotations/5^*, 120/60, , =, , 2n.R/2, , the time taken by the Earth to go once round the, , we have, , log 365, 2, log, | log 2, , - 2-5623, - 0-3010, - OJ_505, 6'45f5, , Subtracting, , (//), , we have, Antilog, , from, , -, , 2*1108, 2*1108, 129*0, , ..(/), , one year,, , in the second cave, , ..(//), (/), , in one year, when, half its present distance from the Sun,, , Hence, the number of days, the earth, to 129,, , is
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MOTION AL6tfG A OURVli, 3., , Assuming, , -TfiE PEOJECTlLli, , that the Moon revolves uniformly in a circle round the, the acceleration due to gravity at the Earth's surface, , Earth s centre, calculate, from the following data :, , = 6'4 x 10 cms., = 3*84 x 10 10 cms= 27*3 days., , Radius of the Earth, Radius of the Moon's orbit, Period of rotation of the Moon, of, , its, , (The force of gravity on a particle, distance from the Earth's centre)., , is, , (Oxford, Here, velocity of the Moon, v, , =, , 8, , inversely proportional to the square, , &, , Cambridge Higher School, ^ircui? feence of he orbit, time taken, , Certificate), , time taken, , 2*x3-84xl0 10, , &, .., , where, , r is, , cms., , 273x24x60x60, , r2*rx3;84xl0, ;, , 2, , 10, , X, 3x24x366oJ, 4*2 x 3-84 xlO 10, ~, , &, , 3-84 xlO 10, , due, , we have, , 6-4, 10, 27-3, , 24, , 3600, , ", , 6021, 0-9944, , ^, , i, , 33-3494, 0-8062, , Moon's, , Moon's, , cms.Jsei, , orbit), , 8, , 2, , orbit), , 2, (radius of Earth), , F, 8, L (6-4 xlO ) 2, , g, , Or,, , ('), , (radius of, , (radius of, , *, , =1-7529, =30-0000, , be g, , earth's surface, , and *m **, , (radius of" Earth)*, log 4, 2 log rr, 3 log 3-84, 3 log 10 10, , on the, , to gravity, , iay., , x 24 x 3600)', , (27-3, If the acceleration, , 2, , v /r, , 1, , ], , L27, , log, 8 log, log, log, log, , Isec., , acceleration of the Moon towards the centre of the Earth, the radius of the Moon's orbit, , J', , =8-0000, , = 1-4362, = 1-3802, = ^5563, , \, , 6-4, , KA, , 10* J, , 2, , L(27'3x24x3600) J, , ;, , 1(, , :, , 15 1789, , x, , 4ir^x_(3-84xl0, , __, , **, , I, , V, , x "iOx 27 3x 24 x 3600) 2, , ', , :, , (6*4, , 2x15-1789, , =30'3578..(/0|, 2, 980-9 cms./sec, Subtracting (//) from (/),, 2-9916, we have, Hence, acceleration due to gravity, 2-9916, Antilog, surface = 980 9 cms.lsec., = 980-9, ., , at the Earth's, , 4. The radius of curvature of a railway line at a place when the train is, moving with a speed of 20 miles per hour is 800yds., distance between the rails, being 5 ft. Find the elevation of the outer rail above the inner rail so that there, (Bombay), may be no side-pressure on the rails., , Here,, , we have, , v, , 20 m.p.h., , =, , 20x1760x3, 60x60, 800x3, , j^ ^, i, , 5ft., , rg, -0112, , and, , =, , wtw rail ihpul4 b, , 88 f, , T, , ,, , ', , 2400 ft., , _, , 88, , x 88, , _, , 3x3x2400x3T, , _ -0112., , 10800, , 38-33'., , sin 38*33', , And, Tbtrefore, tbt, , ~, , .*., , h, , -0112., ==, , [0 being imall., , dsinQ., , ralitd '6120 fecftt* above tbt, , innw
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*U, , PROPERTIES OF MAtff fift, , A, , stone of mass 10 Ibs. is revolving in a vertical circle at the end of 2, bag, the other end of which is fixed. When the stone is at the top o], the circle, the velocity is 16ft. per sec. Assuming g to be 32 ft./sec 2 ., find the, stretching force in the string when the stone is (/) at the top, (//) at the bottom, 5., , string,, , 8, , ft., , at a level with the centre., , (iiV), , m=, , Here, mass of the stone,, , and, (/), , OA,, , 10 Ibs., , velocity at the top, , ;, , radius of the circle,, 16 ft. (sec., , r, , =, , 8 ft.., , of the circle, , Therefore, force acting outwards,, , string, or, , upwards along the, , i.e.,, , alone, , (Fig. 14),, 2, _A, = mv = 10x16x16 =320, r, o, , A, , -*, , ., , ,_, , ,, , ;, , poundah., , And, downward force due to weight of the stone, , = mg, , 10x32 = 320 poundah., , resultant stretching force along the string, , .*., , 320-320 = 0., By the time the stone reaches the bottom, ., , (/"/), , it has acquired, additional kinetic, energy due to its having fallen a vertical distance, 16/f (the diameter of the circle)., , of the ciicle,, ,, , Therefore, kinetic eneigy ot the stone at the, circle--- |mi 2 4- tngh., , bottom of the, , = [xlOxl6xl6i+[10x32xl6], =-1280-1-5120 = WWft.poundals., This should be equal to \mvf, where, , So, , the circle., , | mVl, , 2, , =, , 6400., rz, , Or,, , Now,, , And, .-., , =, , =, , 12800/10, , centrifugal force acting, , 10x32, , mv t2, , Or,, 2, , the, , mv, , f,, , 6400 x 2, , =, , [v, , -, , =, =, , total stretching force in the string, , weight of the stone, , 1600^320, 1920/32, , =, , Here, the additional K.E. acquired by the stone, through a distance 8/f., the radius of the circle, and is, total K.E. of the stone, , 10 Ibs, , OB, , is, , equal to, , \92Qpoundals., , = 6Qpounds, , (///), , .-., , m=, , string, />., along, , 1600 poundah., , g, , to the, , force due, , (as before)., , 12800., , 1280., , downwards along the, 10x 1280, , and, therefore,, , downward, , 320 poundah, , =, =, , 16/r,, , the bottom of, , V L is the velocity at, , that,, , =, , [v h, , weight., , is, , due, , to the, , fall, , = 10x32x8 = 2560 ft. poundah., = i/wv 2 -h2560 = Jx lOx I6x 164-2560., = 1230+2560 = WWft.poundals*, , 2, Obviously, this must be equal to imr a, , ,, , where, , v,, , is, , the velocity of the, , stone, in this position., , Jmva, , 2, , =, , 3840,, , whence,, , vt 2 **, , A5, , ^?/, , 768., , centrifugal force acting outwards along the string,, , = ?>". "xl, r, , o, , ., , i.e.,, , along, , OC, , 10x96 = 960poUndah., 960/32 = 30 Ibs. wt., , And, since the weight of the stone is acting vertically downwards, it is, acting at right angles to this centrifugal force and has no component along it., Therefore, the stretching force in the string, in this case, is equal to 960 poundals, or 30 Ibs. wt., 6. Four masses, each of 10 Ibs., are fastened together with four strings,, each 3 ft. long, so as to form a square. This square rotates in a horizontal plane, on a smooth table at a speed of one revolution per second ; find in pounds-weight, tbt tension in tte string., , (Cambridge Higher School, , Certificate)
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41, , THE PROJECTILE, , MOTION ALONG A CTTRVE, , of 10 Ibs. each, forming a, /fh, mi, wa and m, be the four masses, shown in Fig, 15., The radius of the circle in which they rotate will obviously be equal to, is, the, 0w 2 -=- Om$ Ow 4 r, where, , Let, square, as, , Om =, l, , point of intersection of the diagonals of the square., If P be the mid-point of m l w 4 we have, , /^3/2, , ,, , ', , p, , %, , ,, , OP =, So, , -, , that,, , Pnii, , =, , 3/2 //., , +, , V( v, , rbe the tension in each string,, rotating on the smooth table., , Let, is, , square, , Then, representing T, in, direction, by the straight line Pm^, force acting along Om^, , when, , the, , magnitude and, , Fig. 15., , we have, =\/T*~+'f, , 2, , ~, , \/2T, , 2, , = V2T., , This, clearly, represents the centrifugal force acting along Otn^, NO.V, the centrifugal force acting along m^O is also equal to mra\ i.e.,, , So, , ITT, ^21, , that,, , lo ? 15, , 2 log n, , = M76i, = 0^9944, , ==, , T, , whence,, , 120X* 2, , =, , --, , per, , i.e.,, , sec.,, , cribes an angle, radians per sec., , desof 2n, , Ha. wt., , =, , 18*51, , -, , =, , 8, , The tension, , A, , 7., , tion, ,, , 60?r 2 poundals., , 2674, , 1, , =, , where m is the mass of, - 10 Ibs., r - 3/V2 //.,, wij, and to = 2n, because the, square makes one revolu-, , 120 n 2, , ^, , 2T, , Or,, , :, , Antilog, , =, , 2 1 705, 0-9031, , =, , log 8, , **, , 18'51 Ws. wt., , in each string, , wt., is, therefore, equal to 18 51 Ibs., under a load of 50 k.gms. A mass of 1, k.gm. is attached to the end of a piece of the string, 10 metres, long, and is rotated in a horizontal circle. Find the greatest, number of revolutions per minute which the weight can make, without breaking the string., Here two cases arise, viz.,, , certain string will break, , (/) when the fixed end of the string, is itself the, centre of the circle in which the load is rotating, i.e., the, radius of the circle is the length of the string, [Fig. 16 (/)], and, , when the string hangs vertically and a circular, given to the load at its end, the circle described, being in the horizontal plane, [Fig. 16 (//)]., First am?.The centrifugal force, acting on the, (/O, , motion, , string, , is, , is, , given by, , (W, r, , Fig. 16., , Now,, So, , the, , maximum, , that,, , value of, , 50 x 1000X980, , F is, , 50 k. gm. wt. (given)., , =, , 50 x 1000 x 980 dynes., , -, , 2, lOOOxv, ____, , -y^, , where m is the, mass of the load, =, , V*, , 1, , Therefore,, , k.gm., , 10 metres*, , 7000 cms. I sec., 7000x60cws./, 420000 cm$./wto/e., , 1000 cm*.,, , and, , v
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PROPERTIES OP MATTER, Mow,, log 42000, log 6283, Antilog, , =, , the distance covered by the load in, , 2nxlOOO =, , 5*6232, , rotation, , 1, , 2x3-1416x1000., , 6283 cms., say., 6283-2 cms., number of revolutions made by the load per minute, without breaking the string = v/2*r = 420000/6283, 66 83., , =1-7982, 1-8250", 66-83, , =, , Or, the number of complete rotations made by the load is 66 per minute., Second case, Here, two forces are acting on the load, viz.,, ., , (/), , wv 2 /r,, , horizontally=mr^\ since v = rco, where v, and co are the linear and angular velocities, the, of the load respectively, and, r,, radius of the horizontal circle in which it, , (centrifugal force), , rotates., , mg, (weight of the load), , (//), , cally, , verti-, , downwards., , Obviously, for the load to be in, steady motion, the resultant of these two, must act along the string, as shown in, Fig. 17., , Fig. 17., tant, , F, , force, , (the, , tension in, , tan, , rco', , Q=~, , -, , r=AB, , m, , sin, , k., , 1, , ..(1), , 8, , tng, , Now,, , Let $ be the angle, that the resulwith the vertical. Then,, , makes, , the string), , gm., , =, =, , 1000 sin e,, 1000 gms., , The maximum value of the tension of the, , T=50x 1000x980, If n, , velocity, , Or,, , =, , string,, , i.e.,, , dynes., , be the number of rotations made by the load per minute, the angular, 2nnper minute., 2nn, nn, , -, , '60, , Substituting the values of r and, , ,^, , fl, , w, , -*, , 2, , per, , sec., , we have, , in relation (1) above,, 2, , -10005/*0, , :V sin, 9g, , 10, , Or,, , ', , Or,, But, from the figure,, , we ha, 1000 g, , cos, , F, , |, , ', , g_, , 10, , __, , 50x1000x980, g ___, 50x980, , =, , 9, , 50x980*, , ', , 1, , "50x980, , ', , 9x50x980, , V9x5x980, 66*83 per minute, Hence, Dumber of rotations, made by the loed per minute is 66., , rotttioni, , ;, , or,, , number of complete
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MOTION ALONG A CtfRtE, , A, , S., , bf length, , /, , particle, , and hangs, , city greater than, , of mass, , V*#/, , it, , m is attached to a fixed point by, , Show, , freely., , THI PROJECTILB, , that if, , it, , means of a, , is projected horizontally, , will completely describe, , a vertical, , string, , with a velo-, , circle., , Let OB, (Fig. 18), be the string, fixed at Oand suspended freely, with, m at B. Let it be given a horizontal velocity u, when at rest at B. It, will naturally move, along an arc of radius /,, the length of the string. Let B' be its position, on the arc at a given instant, when its velocity, a mass, , is v., , v, , Then, clearly,, , =, , 2, , u*-2gh,, , where h is the vertical distance through which, the mass has been raised up., The weight mg of the mass is acting vertically downwards at B', and the centripetal force, , mv 2 //, along the string, in the direction B'O., The component of mg, acting along, , the, , string in the direction OB', i.e , opposite to that, mg.cos 0,, of th? centripetal force, is thus clearly, , =, , .'., , If, , T be, , B, , tension of the string, , the, , we have, , -, , = T mg cos 0., , --, , Fig. 18., , mv*, Or,, , mg cos, , -f, I, , cos 8, , Now,, , oc, OB, , OB-CB, , m(u*~2gh), , 1, , l-h, , OB', , substituting the values of, , v*, , and cos, , (I), , Q., , I, , in relation, , (I), , above,, , we have, , ,, , -f **, , I, , Or,, , (II), , At the highest point on the circle,, 21., half the vertical circle, we have h, , =, , i.e.,, , This will clearly become equal to zero, if u*, , at A,, , when the mass completes, , = 5#/., , 2, , h, v, 2/, here., ['., A,, u*-4gl., Obviously, therefore, if the mass is to continue in motion along the circle,, the tension T should not vanish, i.e., should not become zero, which means that, u 2 >5gl. For this value of u 2 , its velocity at A will also not vanish, and hence the, mass will describe a complete circle of radius /., , Now,, , at, , Thus, the condition necessary for the mass to complete a vertical circle, , is, , that if>5gl, or that u, , 9. Assuming the law of Gravitation, and taking the orbit of the Earth, round the Sun, and of the Moon round the Earth as circular, compare the masses of, the Sun and the Earth, given that the Moon makes 13 revolutions per year and that, the Sun is 390 times as distant as the Moon., , Let mass of the Sun be, , and, And,, *ft, , let distance, , that, , _., , of the, __, , Moon, Sun, , M, , mass of the Earth, t>, mass of the Moon =, m,, from the Earth be ~ JR,, _, , M, , ., , wm, , 300 IL, , Af*.
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**, , PROPERTIES OF MATTEtt, Then, force of attraction between the Sun and the Earth, , G is, , where, , the gravitational constant., , centripetal acceleration, , =, , of the Earth, , /?-c?-, , -, , t, , mais, , (6, , __ Ms_ G, ~, , ', , (390 R)*', Similarly, force of attraction between the Earth and the Moon, , And, , centripetal acceleration, , .'., , Let, pectively., , co^, , and, , co, , ~, , of the Moon, , w be the angular, , J\, , 2., , u, , jrJ, , ,,, , ,,, , and the, , Moon, , ^, , (390)2, , ~Me, , c), , OT, , ,,, , 390.tf.ov, , ", , Or,, , = 390 R.& e, = Rw 2, , also, , is, , Moon, , 3t, , [Mj/(390 R)*]G, , res-, , Moon, , *., , ., , 2, , ', , *z, , the Earth goes round the Sun only once in one year, 2x per year., angular velocity of the Earth, , the, , G., , ., , </v, , velocities of the Earth, , centripetal acceleration of the Earth, , Now,, , ^, , -G, , Then, clearly,, , and, , And, , **/TJ, , ;, , and, therefore,, , goes round the Earth 13 times in one year so that,, angular velocity of the Moon = 13 x 2rr /?er year., ;, , Mr, Or >, , if*-, , Mass of the sun, show that in, 10., , r.e.,, , :, , -2r--i3v-, , Mass of the Earth, , :, , :, , (390), , 3, :, , (13), , ., , the case of a liquid, rotating with a uniform angular, velocity, (/) the pressure varies directly as the distance from the axis of rotation, and (//') the free surface of the liquid is a paraboloid of revolution., , density, , (0 Imagine a closed,, p, to be rotating about, , cylindrical vessel, just full of a liquid of, with a uniform angular velocity co., Now, consider a ring of the liquid,, of radius x, width $x and vertical height, O on the axis of, 8/r, with its centre at, Then, if the pressures, f>+6f> rotation, (Fig. 19)., in the liquid at distances x and x+8x, from the axis of rotation be p and (p+8p), outwards and inwards respectively, we, , vertical^, its axis, , have, resultant, , and, where, , inward thrust on the ring, , centrifugal force outwards on the, , m is the mass of the liquid, , Clearly,, , m-= volume of liquid, , in the ring., , in the ring x, , p=2, , So that, centrifugal force outward on the ring = 2KX$x.$h.p.(**.x [v, = 2nx$x S/j.p.w 2 .*., And, therefore,, />.2rrx.8/z, whence,, , =, , pcAx.&e., &p, Integrating this expression, we have, , r=;c, here,
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MOTION ALONG A CURVE, , =, , p, , Or,, , C is, , where, , THE PBOJBOriLl, , ip, , a constant of integration., , This, , is, , then the expression for the pressure at a point distant x from the, , axis of rotation in a rotating liquid., , The value of C is obviously equal to that of p at x = 0, i.e., equal to the, pressure at 0, which, as we know, is zero at the surface of the liquid, but, increases with depth as in the case of a liquid at rest., (//) Consider a particle P in the liquid surface, of mass m, whose coordinates are (x, y) with respect to the axes, and, t (Fig., 20;, the liquid, being supposed to be rotating with a, uniform angular velocity co in the direction, , OX, , OY, , shown, about OY., , The, , forces acting on this particle, (/) its weight mg,, vertically, downwards, and (//) 'the centrifugal force, wto 2 *,, outwards., The resultant R of, these two forces must act at right angles, to the liquid surface, since there is no, flow of the liquid taking place, and,, obviously, it is counter-balanced by the, thrust due to the rest of the liquid on P., , P are,, , clearly,, , be the angle that the, If, therefore,, centrifugal force makes with the tangent at P,, , =, , =, , tan, , Fig. 20., , we have, , mg, Now,, , =, , tan, , obviously,, , slope at, , ., , ., , dx, Integrating this,, , P, , dyldx., , ,-^.dx., , Or,, , we have, -~~, , /*=/--, , \x.dx., , *~-2T +Cf, , Or,, , where, , C is, , again a constant of integration., , =, , Since, , y, , And, therefore,, , y**-*, , 0,, , when x, , =, , 0,, , whicli, , we have C, , is, , EXERCISE, , 0., , the equation to a parabola., , The free surface of a uniformly rotating liquid, boloid of revolution., , A particle of mass, , =, , is, , thus clearly a para-, , II, , whirled uniformly at the end of a string,, Find the tension in the string., 2ft. long, and makes 3 revolutions in 1 2 sees., Ahs. 15-43 Ib. wt., !., , 1, , Ib. is, , 2., A half-pound weight is being whirled in a horizontal circle at the end, of a string, 2 feet long, the o'her end of the string being fixed. If the breaking, tension of the string is 112 Ib. w/., find the greatest speed which can be given to, Ans. l\91ft./sec., the weight., , 132, , //., , 3., At what angle should a cyclist lean over,, radius at 15 miles p*r hour., , 4., , when negotiating a curve of, , A person skating on ice at the rate of 20 ft. per, , cle of 20 ft. radius., , What, , is,, , Ana., se<, , his inclination to the vertical ?, , 6* 32'., , ond describes a cirAns. 32*0,
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PROPERTIES OF MATTER, , 46, , A, , round a curve of 1000 ft, radius and the distance, the outer rail be raised above, the inner one, so that a train running at 45 miles per hour may exert no lateral, Ans. 8'094*., thrust on the outer rail ?, 5., , between the, , train is going, , rails is 5 feet., , By how much should, , A stone is suspended from the roof of a railway carriage by means of, 6., a string 5 //. long. The angle through which the mass moves from the vertical, Calculate the speed, is 10, when fhe train moves along a curve of radius 600 //., Ans. SSl/f./iec., of the train., 7., Assuming that the Moon describes a circular orbit of radius 3*84x 10, metres in 27'3 days and the outer satellite of Mars describes a circular orbit of, 2*35 x 10 7 metres in 1*26 days, find the ratio of the mass of Mars to the, radius, mass of the Earth., (Cambridge University), Ans. '1076: 1., , =, , A curve on a railway line has a radius 1 600 //., and tne distance, 8., between the inner and outer rails is 5 //. If the outer rail be 6" above the inner, one, calculate the maximum speed of a train going along the curve, so that no, Ans. 48 89 mileslhr., side thrust is exerted on the outer rail., 9., Calculate the increase in leagth of an elastic string of original length, 10 /f., at the end of which a stone of mass -5 Ib. is whirled at the rate of 4 revolutions per second, if a load of 25 Ibs. increases the length of the string by 2%., Ans. '8576 //., , A merry-go-round is revolving in a horizontal circle of radius 3ft. at, 10., the rate of 7 revolutions in 11 seconds. A child of weight 20 Ibs. rides a wooden, horse suspended by a vertical string. Find the tension in the string and its, Ans. (i) 36 Ib. vr/., nearly, (//) tan' 1 3/2., inclination to the vertical., [Hint, , :, , See solved example 1, , (second case, page 42.)], , A sea-plane of total mass 1000 Ib. (including the pilot) rounds a, 11., Draw a, pylon in a circular arc of radius half a mile at a speed of 300 mp.h., diagram showing the forces acting on the sea-plane, and calculate the resultant, force at right angles to its direction of motion exerted upon it by the air., Assuming that the pilot weighs 12 stones, calculate the force with which he is, pressed against his seat during the "turn/ (Cambridge Higher School Certificate), Ans. 8x 10* poundals 30 stone-wt., 1, , :, , Calculate the angle at which a curve of radius 352 //. should be banked, so as to avoid side-slip when a motor car is travelling round it at a speed of, Ans. 9 45'., m.p.h., 12., , W, , A, , road over a bridge has the form of a vertical arc of radius 60 //., the greatest speed in m.p.h. at which a car can cross the bridge without, Ans. 30 m.p.h., leaving the ground at the crest of the road ?, 13., , What, , is, , 14. A skater is moving on one foot in a circle of radius 20//. at 10 m.p.h., At what angle with the vertical will the line passing through his centre of gravity, and the edge of his skate be inclined ?, Ans. 18 35'., , In a 'loop-the-loop* railway, the cars, after descending a steep, run round the inside of a vertical circular track, 20 ft. in diameter,, making a complete turn over. Assuming there is no friction, find the minimum, height above the top of the circular track from which the cars must start., 15., , incline,, , Ans., , 5ft., , A symmetrically loaded lorry, , weighs 5 tons, and the height of its, centre of gravity is 5ft. above the ground in a vertical plane midway between, the wheels. The breadth of the wheel base may be taken to be 6ft. 3 in. If, there is no side-slip, what is the maximum speed at which the lorry can take a, curve of msan radius 6 yards without beginning to overturn ?, Ans. 1 3 m.p.h., [Hint : It will overturn only when the moment of the centrifugal force about, the wheels on one side is greater than the moment of the weight about them, (see, 13, case 4, page 31).], 16., , 17. ,, , An, , India rubber band has a mass of 4 gm* per metre when* stretched, radius, the stretching force being, , on the circumference of a wheel of 10cm.
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MOTION ALONG A, , OTTEVJB, , THB PROJHCTILB, , 47, , 20,000 dynes. Find how many revolutions per second the wheel must make to, that the band may not press upon the wheel., Ans. 1 1*3., , [Hint, , :, , See, , 18, case, , 2 t page 27.], , Discuss the possibility of a motor cyclist riding round the inside, surface of a vertical cylinder., (Cambridge Scholarship), 18., , 19., Explain why a motor-cycle combination (side car on left) is liable to, overturn when taking a left hand corner at speed. Assuming that the centre of, mass of the combination is 2 ft. from the ground and 1 ft. to the left of the, motor cycle, calculate the maximum speed 6f the combination in a circle of, Assume that the road surface is horizontal and that there is no, radius 50 ft, (Oxford Scholarship), skidding., Ans. 19" 3 m.p.h., ., , 20., , A closed cylindrical can, , fluid, , of density, , axis,, , held vertical., , p., , of radius a and height h is first filled with a, then rotated with angular velocity < about its own, Prove that the total thrust on the top of the. can will he, , It is
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CHAPTER, , III, , MOMENT OF INERTIA ENERGY OF ROTATION, 26., Moment of Inertia and its Physical Significance Radius of, Gyration. We know that, according to Newton's first law of motion,, a body must continue in its state of rest or of uniform motion along a, This inertness, straight line, unless acted upon by an external force., or inability of a body to change by itself, its position of rest, or of, , uniform motion, is called inertia*, and is a fundamental property of, matter. Thus, it is by virtue of its inertia that a body, at rest,, resists or opposes being put into motion, and a body, in linear or, translatory motion, opposes not only being brought to rest but also, any change in the magnitude and direction of its motion, And, we, know, by experience, that thegreater the Tpass_of a bqdy^ the greater, its, _inertiaj)r opposition to the desired _chaiige for, the greater is the, force requireJToTTa "appFed for the purposeT^ The mass of a body is, thus taken to b2 a measure of its 'inertia for translatory motion'-, as it, is this that opposes the acceleration,, (positive or negative), desired to, be produced in it by the applied force., ;, , Exactly in the same manner, in the case of rotational motion, find that a body, free to rotate about an axis, opposes any, change desired to be produced in its state of rest or rotation, showing, that it possesses 'inertia* for this type of motion And, obviously, the, greater the couple or torque, (see, 28), required to be applied to a, body to change its state of rotation, i.e., to produce in it a desired, angular acceleration, the greater its opposition to the desired change,, also,, , we, , or the greater its 'inertia for rotational motion', It is this 'rotational, inertia' of the body which is called its moment of inertia** about the, axis of rotation,, Him name being given to it on the analogy of the, moment of the couple, which it opposes., ., , It will thus be seen that the moment of inertia of a, body, in the, case of rotational motion, plays the same part as, or is the, analogue, of the mass of a body in he case of translatory motion ; and we, may,, , therefore, for purposes of analogy, describe the moment of inertia of, a body, in rotational motion as the 'effectiveness of its mass.', Or,, , pushing the analogy a little further, we may define mass as the, 'coefficient of inertia^ for translatory motion', and the moment of, inertia, as the 'coefficient of rotational inertia'.**, Yet, with, , all this, , between the two, , cases., , seaming similarity, there, , is, , all, , the difference, , For, in the case of translatory motion, the, , *That is why the comparative slackness or sluggishness of the, people SF, Eastern countries, a consequence of climatic conditions is dubbed by the, 9, Westerners as the 'Inertia of the East., **It is also sometimes referred to as the 'Spin inertia' of the, body, its axis of rotation., a, of, m^ss, as, fThe, body being usually referred to, its inertia coefficient,, , 48
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MOMENT, , INBHTIA, , Off, , BNEEGY Of ROTATION, , 49, , inertia of the body depends wholly upon its mass and is, therefore,, measured in terms of it alone. In the case of rotational motion, on, , the other hand, the rotational inertia, or the moment of inertia, of, the body, depends not only upon the mass (M) of the body but also, upon the ^Jfr^jv? ditifw' (K-) of its particles from the axis of rotation, and is measured by the expression MK*, (see next Article)., , This 'effective distance' (K) of the particles of a body from, , its, , axis of rotation is called its radius of gyration about that axis, and, is equal to the root mean square distance of the, particles from the axis,, i.e., equal to the square root of their mean square distance (not the, square of their mean distance) from it. Or, to give it a clear cut, definition, the radius of gyration of a body, about a given axis of notation, may be defined as the distance from the axis, at which, if the whole, mass of the body were to be concentrated, the moment of inertia of the, body about the given axis of rotation would be the same as with its, actual distribution of mass., , Now, it is obvious that a change in the position or inclination of, the axis of rotation of a body will bring about a corresponding, change in tho relative disfcancas of its particles, and hence in their, 'effective distance , from the axis, i.e., in the value of the radius of, gyration of the body about the axis And, so will the transference of a, portion of the matter (or mass) of the body from one part of it to, another, or a change in the distribution of the mass about the axis,, the total mass of the body remaining the same, in either case., 9, , Thus, whereas the mass of a body remains the same, irrespective, of the location or inclination of the axis of rotation, the value of its, radius of gyration about the axis depends upon, (/), , its, , the position, , and, , the distribution, , (ii, , of the axis of rotation, and, mass of the body about this axis, , direction, , of the, , value for the same body, , is diifererit, , so that,, ;, for different axes of rotation., , Further, it follows, as a converse of the above, that the, of gyration of a body about a given axis of rotation gives an, tion of the distribution of the mass of the body about it and, also, the effect of this distribution of mass on the moment of, of the body about that axis., , Expression for the, , """27., , Moment, , of Inertia., , radius, indica-, , hence,, inertia, , Suppose we have a, , body of mass M, (Fig. 21), and any axis YY'., Imagine the body to be composed of a large, number of particles of masses m v m 2 m 3, ,, , at distances, YY'. Then, the, , etc.,, , m, m, particle, particle, fore,, , the, , r,,, , moment, , YY, , l, , about, , a, , is JW 2 r aa ,, , from the axis, , r2 , r s ...etc, is, , of inertia of the, that of the, , mj^,, , and so on, , moment of inertia,, , /,, , and, thereof the whole, equal to the, , ;, , body, about the axis YY', is, 2, /w 2 ra 8, r 2, sum of, etc,, ,, 3 8, sss, /, Thus,, m^f + #ya *, , w^, , ,, , W, , +W, , r\, , Ur, , >, , K beinjj tlie, , 2, 3 r 3 -f, , . ., , ., , M, , is the mass and Ml, Twhere, JEVwr*., the summation 2'Mr 1 for tL, r ** MY*, *, &, Lwhoie body,, radius of gyration of the body about the axis YY',, ss=, , M, , .
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PBOPBKTIBS Ot MATTER, , 5C, , If we wish to accelerate the rotation of a body,, 28. Torque., The, free to rotate about an axis, we have to apply to it a couple., moment of the couple, so applied, is called torque, and we say that a, , torque, , is, , applied to the body., , Obviously, the angular acceleration of, , the particles, irrespec-, , all, , of their distances from the axis of rotation, is the same, but because, their distances are different from the axis, their linear accelerations, are different, (the linear acceleration of a particle being the product, of the angular acceleration and the distance of the particle from the, tive, , axis of rotation)., If, therefore,, , its particles,, , dwfdt be the angular acceleration of the body, or, , we have, , linear acceleration of the particle distant rt, ,,, , ,,, , ,,, , from the axis, , ra, , ,,, , r^dwldt,, r^dcoldt., , and so on., Hence,, , if, , m, , on the different, moments o these, , Therefore,, , be the mass of each particle of the body, the forces, etc., and the, particles are mr^d&jdt, mr2 .d^jdt, forces about the axis of rotation will, therefore, be, t, , x rl9 (mr2 .da>ldt) x, total moment for the whole body, , =, , (mrv da)ldt), , =, , (d&ldt).mr*., , But Zmr, , 2, , X ^, , +, , rz, , (mr2 .da}jdt), , and, , x, , r, , f, , so on., , +, , ......, , [d&fdt being constant., , =, , /, the moment of inertia of the body about the, And, therefore,, moment for the whole body = I.d^ldt., This must be equal to the torque applied to the body., , axis of rotation., , So that,, , torque, , =, , Ldaj/dt., , It will at once be clear that this relation corresponds to the, familiar relation, force, x a, in the case of linear motion, where, , =m, , m is the mass and, , a,, , the acceleration of the body., , Thus, in the case of rotatory motion, torque, moment of inertia, and angular acceleration are the analogues of force, mass and linear, acceleration respectively in the case of linear or translatory motion., , Now,, , if, , Or, the, , dwjdt =s, , 1, clearly,, , torque, , =, , /., , moment of inertia of a body about an, , torque, producing unit angular acceleration in, , it, , axis is equal to the, about that axis., , Incidentally, the expression for torque, obtained above, furnishes us with a method of deducing an expression for the moment of, inertia of a particle of mass m, about an axis, distant r from it., , For, if, , F be the force applied, we have, torque = F x r., torque, , is, , also
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MOMENT OF INEBTIA, , Fx, , torque, , And, therefore,, , r, , \, , ..., , F -a m x, , Now,, , 81, , ENERGY 0V BOTATlOH, , (0, , a,, , where a is the linear acceleration of the particle., of the, And, since a, dvjdt, (where v is the linear velocity, F = m.dvjdt., particle), we have, , =, , Again, since v ==, particle,, , ro>,, , where, , is, , co, , the angular velocity of the, , we have, , Now, the component, (drldt).a), plays no part in the rotation of, the body and may, therefore, ba, so that, F, mr.dco/dt., ignored, , =, , ;, , Substituting this value of, T, /==, , F in relation, , above,, , we have, , = mr*., , mr.(dcoldt).r, L, \, , (/), , t, , ,', , dwjdt, , Thus, the, axis distant r, , moment, , from, , it, is, , of inertia of a particle of mass m, about an, 2, equal to wr, ., , General Theorems on Moment of Inertia. There are two, general theorems of great importance on moment of inertia, which,, in some cases, enable us to determine the moment of inertia of a, body about an axis, if its moment of inertia about some other axis, be known. We shall now proceed to discuss these., 29., , The Principle or Theorem of Perpendicular Axes., According to this theorem, the, () For a Plane Laminar Body., moment of inertia of a plane lamina about an axis, perpendicular to the, plane of the lamina, is equal to the sum of the moments of inertia of, the lamina about two axes at right angles to each other, in its own, plane, and intersecting each other at the point where the perpendicular, axis passes through it., (a), , Thus, if /,, and / be the moments of inertia of a plane lamina, about the perpendicular axes, OX and OF, which lie in the plane of, the lamina and intersect each other at '0,, (Fig. 22), the moment of inertia about an, axis passing through O and perpendicular to, the plane of the lamina, is given by, , /, , - 4+V, , m, OX, , For, considering a particle of mass, at P, at distances x and y from, and, respectively, and at distance r from 0,, , OY, , we, , -have, , / *=, , lm, , =, , /.+/,, , *=, , Zmr 8, , So that,, , ,, , Zmy* and 7y, my*+Zmx*., Zmr*., , Or,, , /.+/,:=/., , [v y*+x*, Rg., , 22.
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PROPERTIES OF MATTER, , 52, , Suppose we have a cubical, , For a Three-Dimensional Body*., , (ii), , or a three-dimensional body,, , OY, , in Fig. 23, with OX,, as its three mutually, perpendicular axes, represent-, , shown dotted, and, , OZ, , ing its length, breadth and, height respectively., Consider a mass, of the, , m, , body, at a point P, somewhere, inside it., Drop a perpendifrom P on the xy, cular, Join, plane to meet it in, , PM, , OM, , draw MQ, and MN,, , Fig. 23., , also,, , P draw, , from, , of the point, , x, , ;=, , P are, , PjR, parallel to, , ON = QM, , ;, , Since the plane xy, , Then,, , = OQ = NM, , y, is, , OM., , M, , and OP,, , PR, , and, , PM, , parallel to the j-axis ;, clearly, the co-ordinates, , = MP = OR., , and z, , (v, , PR, , is, , OMP is a right angle,, , /., , parallel to, , is, , M, , perpendicular to the z-axis, any straight, it, and, therefore,, , Obviously, therefore,, parallel to, , ., , from, , parallel to the x-axis, , drawn in this pLane is also perpendicular to, OM and PR are bath perpendicular to the z-axis,, parallel to OM)., line, , and, , because, Hence, we have, , OR., , OM +MP* = OP*., =r, , drawn, , OM, , is, , 2, , Or,, , where,, , OP =, , = QM>+OQ*., OM = * +>>, 1, , Or,, , 2, , 2, , ., , Therefore, substituting the value of, , we have, Join, , x*+y*+z, , PN and, , to the axes of, , y., , angle between the axes, , /_PMN, , For,, , y and, , =A, , PN and PQ, , PQ. Then,, , x and, , 2, , z,, , is, , OJf2, , in relation, , and, therefore,, 2, , j>, , 2, , 2, ., , 2, , 2, , 2, , ., , 2, , 2, , ,, , from which, , it is, , clear that, , /.PM?, , PN is perpendicular to the x-axis., , Similarly, in the right-angled, , *Not, , strictly included in the, , above,, ... (ii), , 2, , Or,, , (/), , are the respective normals, a right angle, being the, , PN = MN*+PM* = +z, x*+PN* = x +j> +z = r, ON +PN* = r, , So that,, , (0, , a right, angle, being the, between, angle, the axes x and y., is, , OAP, , But, , r., , is, , [From, l\-, , x, , (//), , above, , - OM, , a right angle, and, therefore,, , &PMQ, we have, , Q.c. (Pass gr Geacral) course.
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MOMEtf 1? 0# INERTIAlfilfBlItGUr, , Now, moment, , of inertia of mass, , =*mxPR, because PR, the axis., , moment, , .*., , = OM is the, , 2, , =*, , Off, , m at P,, , ROfATIOl*, , about the z-axis, , m.OM\, , perpendicular distance of the mass from, , of inertia of the whole body about the z-axis,, , I, Or,, , /,, , Similarly, the, , moment, , 2, , ., , of inertia of the body about the y-axis,, , L = Zm.PQ, - Zm, Sm (xX +, 4-z, Z, '*, 2, , p-', , ., , Or, ur, , 2, , '\, , \, >, , T, , '*, , ', , ^, , * m(y, , moments of, , adding up the, , three axes,, , is, , i.e.,, , the J_ dis-, , L, , Zm.PN\, 4=, _, ~~ ymlv*-\-7%\, /., , PQ, , tance between ^e, mass and the axis., of inertia of the body about the x-axis, i.e. 9, 2, , /, , ', , And, the moment, , Or, Ur, , i.e.,, , = Zm.OM, = Zm(x*+y*)., , Z, , p.-, , L, , >, , inertia of the, , 1, , P-V is the, distance between the, , mass and the, , axis., , body about the, , we have, , Ix +lv+Ia, , Or,, , Hence, , sum of, , --, , moments of, , inertia of a three-dimensional, body about its three mutually perpendicular axes, is equal to twice the, summation Z"mr 2 about the origin., , the, , the, , This theorem, ^f) The Principle or Theorem of Parallel Axes., is, a, laminar, to, true, both, for, Steiner), (due, plane, body as well as a, three-dimensional body and states that the moment of inlertia of J, body about any axis is equal to its moment of inertia about a parallel, axis, through its centre of mass, plus the product of the mass of the body, and the square of the distance between the two axes., of a Plane Laminar Body. Let, (/) Case, of a body of mass M, (Fig. 24). and Ic, about an axis through (7, perpendicular to, the plane of the paper., ,, , Now,, , let, , C, , be the center of mass, moment of inertia, , its, , be required to deterinertia / of the body, , it, , mine the moment of, , about a parallel axis through 0, distant, from C., Consider any particle P of the body,, of mass m at a distance x from 0., r, , 9, , Then, the moment of inertia of the, , body about, , is, , given by, I, , From P drop a, PC., , And, , Theri,, , .%, , Hence, , OP, , 1 ss=, , m.OP, Emx*, 2, , /, , =, , Zwjc2, , [Since, , ., , perpendicular, , Fi S- 2 *-, , PQ, , OP =, 1, , on to OC produced, and, , CP*+OC*+20C.CQ., , ., , join, , [By simple geometry., , = w.CP*+w.0C +2m.0C.C#,, = 2m.CP*+2m**+%rZm.CQ. [v OP *, 2, , Ic +Mr*+2r2m.CQ., , [v, , a?, , & OC - r.
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PROPERTIES OF MATTBft, , 64, , since a body always balances about an dxis passiiig, centre of mass, it is obvious that the algebraic sum of the, of the weights of its individual particles about the centre of, , Now,, through, , its, , moments, , zero., Hence, here, Emg.CQ, (the algebraic sum of, such moments about C) and, therefore, the expression Sm.CQ is, equal to 0, g being constant at a given place. Consequently,, , mass must be, , 2r.Zm.CQ, So that,, (ii), , about, , /, , =, =, , 0., , Ic +Mr*., , Case of a Three-Dimensional Body. Let AB be the axis, which the moment of inertia of a body (shown dotted), is to be determined, (Fig. 25)., Draw a parallel axis CD through, the centre of mass G of the body,, at a distance r from it., Imagine a particle of mass, , m, , any point P, outside the, plane of the axes A B and CD and, let PK and PL be perpendiculars, drawn from P on to AB and CD, respectively and PT, the perpendicular dropped from P on to, , KL produced., , Fig. 25., , Put, , PL =, , Then,, , AB and, , Ic, , if, , d,, , LK =, , / be the, , moment, , its, , clearly have, , =, , LT =, , r,, , at, , x and Z.PLK, , =, , 6., , moment, , of inertia of the body about the axi, of inertia about the axis CD (through G), wa, , =, , 27w.PI 2 = Zm.d*., Now, from the geometry of the Figure, we have, /, , Zm.PK* and, , =, =, , PK*, , PL*+LK*-2PL.LK cos PLK., , in the right-angled, , And,, , d*+r*2d.r cos 0., &PTL, we have, , cos, , where, , PLT =, , == (180, , /_PLT, , cos, , Or,, , ^, , If, , LT/PL,, , PLK), , (180- 0), , ,, , d, , whence,, , cos &, cos Q, , Substituting this value of, above, we, , d, , =, , =*=, , =, =, , (180, , =, , /, , cos, , =, , Em.PK*, , =, , Jc +Mr*+2rZmx,, , =, , So that,, , x)d,, , ", , And, therefore,, , )., , x/d., x., , in the expression, , Zm(d*-}, , ftor, , PXT, , r 2 +2rx)., , M, , Mr 2 where, is the mass of tHe whole, because mr*, body and r,,, the distance between the two parallel axea and hence a constant.., 0, being the total moment about an axis through, Clearly, Zmx, the centre of mass of the body., ,, , =, , We4 ^therefore,, the, , same, , body*, , have, , result as obtained, , / a /^-f Mr*,, above in case (/) for a plane laminar
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MOMENT OF INERTIA, , ENERGY OF ROTATION, , 00, , Calculation of the Moment of Inertia of a Body. Its Units, In the case of a continuous, homogeneous body of a definite, geometrical shape, its moment of inertia is calculated by first obtaining an expression for the moment of inertia of an infinitesimal mass, of it about the given axis by multiplying this mass (m) by the square, and then integrating, of its distance (r)from the axis, (see page 51), this expression over the appropriate range, depending upon the shape, of the body concerned making full use of the theorems of perpendicular, r- 30., , etc., , ,, , and parallel axes, wherever necessary., In case, however, the body is not homogeneous or of a definite, geometrical shape, the safest thing to do is to determine its moment, of inertia by actual experiment, as explained later, in 34 and in, Chapter VIII., Now, it will be seen that since the moment of inertia of a body, about a given axis is equal to MK* 9 where, its, is its mass and K,, radius of gyration about that axis, its demensions are 1 in mass and 2, in length, its dimensional formula being [ML 2 ]., If the mass of the, body and its radius of gyration be measured in the C.G.S. units, i.e.,, its mass in grams and radius of gyration in centimetres, the moment, of inertia of the body is expressed in gram-centimetre2 (i.e., in gm.cm 2 .). And, if the two quantities be measured in the F.P.S. units,, i.e., the mass of the body in pounds and its radius of gyration in, , M, , ,, , feet, the, , moment, , of inertia, , expressed in Pound-feet*, (i.e., in Ib.-fP), carefully noted that since the moment, of inertia of a body, about a given axis, remains unaffected by reversing, its direction of rotation about that axis, it is just a scalar quantity.*, Thus, the total moment of inertia of a number of bodies, about a, given axis, will be equal to the sum of their individual moments oi, inertia about that axis, in exactly the same manner as the tota', mass of a number of bodies will be equal to the sum of their individual masses., Note. The argument is sometimes advanced that since the moment o, inertia of a body changes with the direction of the axis of rotation, it is not, , And,, , finally, it, , is, , must be, , i, , scalar quantity; and, since it is independent of the sense or direction of rota, tion about that axis, it is not a vector quantity either, and that it is what i, called a 'tensor'., , The author begs to differ. For, the term, 'moment of inertia of a bod), has hardly any meaning unless clear mention is also made of the axis of rotati, of that body. And, once the axis of rotation is fixed, the moment of inertia, the body, about that particular axis* becomes a scalar quantity, being independe, of the sense of rotation about that axis. Indeed, it would be misleading to cz, it a tensor ; for, the fact is that the moment of inertia and the products of inert, (see below), at a point, together constitute the components of a symmetric tens, of the second order, which simply means that, knowing the system of momer, and products of inertia at a point about any three mutually perpendicular axe, we can, by means of certain simple, transformations, obtain their values for ai, other set of three mutually perpendicular axes at tbat very point., A general tensor, of the second order, in three-dimensional space, has,, Ctl , CM CM CS1 , Ct? Ctl But, f, general, nine components, say, Cn , C,,,, C)8 so that it has only s, a symmetric tensor, C, a, C21 , C18 CM and CS, distinct components, viz., three moments of inertia and three products of inert, ^, about the three perpendicular axes., \\ \\, (, J, , =, , =, , C,, , i, , =, , ,, , ,, , ,, , ., , ,, , O, , '*, , *, , Scalar quantities are those which possess only magnitude \ but no direc, tlon,e.g. 9 mass, time etc. On the other hand, vector quantities are those whicl, possess both magnitude as well as direction,, , f .,, , acceleration, velocity, force, etc.
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66, , pjtopUBTiEis otf, if x,, , us,, , We have, (/), , y 2 be the co-ordinates of a, , particle of, , t, , moments of, , inertia, , mass w,, , at /*, in Fig. 23., , about these three perpendicular axes respectively, , given by, , Ix, (//), , =", , 2m(y, , 2, , + z), , 2, , =, , Iv, , ,, , 2m(z*+x, , 2, , =, , /0, , ),, , 2, , 2, , 2Vtt[# -h}>, , ),, , and, , the products of inertia about these axes defined by, , 2tnyz, P*x, 2mzx, Pxy = 2mxy,, Pys, Pv g, Pey- and Pxv are the six components of the, Then, /, /, 70,, symmetric tensor at point P., It will thus be seen that it is, at best, only a half-truth to say that the, moment of inertia of a body about a given axis is a tensor., , Cv^/1., , Moments, , Particular Cases of, , 11., , Moment, , of Inertia of, , of Inertia., , a Thin Uniform Rod, , :, , its, (i) about an axis through its centre and perpendicular to, Let AB, (Fig. 26), be a thin uniform rod of length /and mass, length., M, free to rotate about an axis CD through its centre O and per-, , pendicular to its length. Then, its mass per unit length is MIL, Consider a small element of length dx of it, at a distance x from O., Its mass is clearly equal to (M/l).dx, and its moment of inertia about, z, the axis through O, (M/l).dx.x, , =, , ., , The moment of inertia / of the whole rod about the axis is,, therefore, obtained by integrating the above expression between the, limits x =, or between .v=0 and jc=//2 and, 7/2 and x = +//2, ;, , multiplying the result by, , to include both halves of the rod., , 2,, , C, , Thus,, , ,, , 7=2 f//2, JO, , !, , M- x*.dx., *., , T, , t, , /, , _ Mr, , *n//;, , "~, , L, , /, , Jo, , -i, , ~TL, , i, , 01=, , -, , f, Flg, , 3, , 26., , ~", , F), , 2M, , /3, , /, , 24, , = M|8, , ', , 12, , (//) about an ixis passing through one end of the rod and perThe treatment is the same as above, except, pendicular to its length., since the axis, here, that,, *, passes through one end B of the, rod, (Fig. 27), the expression for, the moment of inertia of the ele('", ment dx of the rod is now to be %, limits,, integrated between the, \x, 0, at B and x, /, at A., , CD, , y, , ', , T, , =, , =, , Thus,, , if, , 7 be the, , inertia of the rod about, ==, , F te-, , moment of, CD, we have, , F, , M, , Jo, , /, , -, , r, , Af, , r, , /, , *, , .*, , *, f, , A, , .lfit, , = Af^ pcv~ 7, /, , 3, , y", , Ml, , 2, , 3^", , LS Jo, , 27, , -
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MOMENT, , JEHB&Glf OF ROTATION, , UN1BT1A, , Otf, , ft<9, , Or, we could have arrived at the same result by an application of the, principle of parallel axes, according to which the moment of inertia of the rod, about the axis through B is equal to the sum of its moment of inertia about a, parallel axis through its centre of mass and the product of its mass and the, square of the distance between the two axes., , TK, , T, I, , Thus,, , -, , M, , /2, , Moment, , 2., , ,./, Y - MI, M~, , 2, , /, , +L, , +,M/, , 2, , of Inertia of a Rectangle., , its, (/) about an axis through its centre and parallel to one of, sides., Lot A BCD be a rectangle, and let / and b be its length and, breadth respectively, (Fig. 28). Let the axis of rotation YY' pass, or BC., through its centre and be parallel to the side, , AD, , If, , M be the mass of the rectangle, (supposed uniform),, , per unit length will be, , its, , mass, , MIL, , Consider a small strip, of width dx of the rectangle, parallel to, the axis. The mass of the strip will obviously be (mjl).dx t an,, therefore, the moment of inertia of, }, c, ..., the strip, about the axis YY' will be /\., j, ;r, *jo.. a#, ,.., , ,, , ,;., , The whole rectangle may be, supposed to be composed of such like, the axis, and thereinertia / of the, whole rectangle about the axis YY' is, obtained by integrating the expression, 2, for the limits x=0 and x=//2, (Af//).dx.JC, , strips, parallel to, , fore,, , the, , moment of, , Fig. 28., , and multiplying the, , ,, , result, , by, , 2., , 7=2A, r, , i.e.,, , f, , ;/, , 2, , M, ., , Jo, , 2M- r, , f//2, = 2Af, -H x*.dx, , ,, , ,, , ,, , 2, , .x*.dx, , ~i/, , /Jo, , /, , 1, , Ur,, , _2M, ;, , /, , _ Ml, -, , 3, , -, , L, , x3, ^r~, 3, , ~|//2, , Jo, , 2, -, , 12, , 2|, , It will be seen at once that if b be small, the rectangle becomes, a rod, of length /, whose moment of inertia, about the axis YY', passing through its centre and perpsnclicular to its length, would be, M/ 2 /12, fas obtained above, 31, case 1, (/')]., We majr proceed as above (in case we want, about one side., (//"), an independent proof) except that the expression (Mjl).dx.x z may, and x, /., here be integrated for x, Thus, the moment of, or BC is given by, inertia of the rectangle about the side, , =, , =, , AD, , (V, , SB, , 2, , -, , ,, , [, , ,--, , Jo, , i^ M, , Or,, , M,~,x*.dx = M, /, ,, , r x 3 -iC, , I, , ., , x 2 .dx., , 1, , /Jo, , i/, , /, , Mi 2, , 3, , Alternatively, proceeding on the basis of the previous article, we may, apply the principle ofparallel axes, according to which the moment of inertia of, the rectangle abouc side AD or BC is given by, , +M / -^ V, ), /, , / =*, , MJ., , about. a, , I!, , axis through its cent re, , Ml, M, '-+-"', 8, , ,, , or,, , r, , Ml*, , 2, ', , (^
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MATTES, , Off, , to, (Hi) about an axis passing through its t$nit& and perpendicular, plane. This may be obtained by an application of the principle of, perpendicular axes to case (i) above, whence the moment of inertia, of the rectangle about an axis through its centre 0, perpendicular torts plane, is equal to its moment of inertia about an axis through, r, parallel to its breadth b, plus its moment of inertia about a perpendicular axis through O, parallel to its length /,, its, , O, , _, 1-, , i.e.,, , The above, , Ml*, , +-~ _, , 12, , equally valid in the case of thin (/..,, laminar) or thick rectangular plates or bars, no stipulation with, regard to its thickness having been made in deducing it. And, after, all, a thick rectangular plate or bar may be regarded as just a pile of, thin (or laminar) plates or bars, placed one above the other., relation, , is, , The same argument will hold good in all other cases of a similar, type, [see cases (iv) and (v) below],, (/v) about an axis passing through the mid-point of one side and, perpendicular to its plane., *, , Suppose the axis of rotation passes through the mid-point, EC, (Pig. 28). Then clearly, in accordance with the prinof parallel axes, we have, (a), , of, , Ap, , ciple, , or, , moment, , of inertia about, I, , =, , this axis, i.e.,, , moment, through, , where, , //2 is, , of inertia about a parallel axis, centre, x (//2) 2 ,, , the distance between the, , *M, , (, , ", , """12", , _, , 4, , 12, , '= M, , or., (b), , point of, , +, , its, , M, , two, , parallel axes., , ~, 12, , -"=-12, , (T+-H>, , Similarly, if the axis of rotation, , passes through the mid-, , AB or DC we have, >, , b, , a, , ^, , "2 }, , ', , f v the distance between, L ^e two axes is now 6/2., fl, , Or,, 3., , Moment, , of inertia of a solid uniform bar of rectangular cross-section,, its length and passing through its, middle point,*, , about an axis, perpendicular to, , ABCDEFGH (Fig. 29) be the rectangular uniform bat of, breadth b and thickness d, whose moment of inertia about the, XX', passing through its centre and perpendicular to its length, is desired., Let, , length, , am, , /,, , "This is really covered by case 2 above, but, understanding of the student., , is, , given here for a clearer
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EttEftOY 0*, , OF INERTIA, , Imagine the whole bar to be made up of a large number of thin rectanthe face, ing through the centre of mass, of each sheet, Consider one, such sheet, (shown dotted),, of mass m, of length and, MfffUJlJ'K / and d respectively,, and centre of mass O, through, which the axis XX' is passing, perpendicular to its plane., feular sheets, parallel to, , sheet, , x, , 2, , (/, , CDEHand, , perpendicular to the axis XX', pass-, , JT, 4*, , ^, , ^_, , Then, the M.I. of this, about XX' = its mass, 2, -f</ )/12, as can be seen, , H, , from the following, Let PQ be an axis through, O\ in the plane of this sheet, and, :, , parallel to its breadth, , CH or DE., , ,, , Fig. 29., , Take a thin strip of width dx of this strip, parallel to, and at distance x, Jrom, the axis PQ Then, mass of the strip, (w//).</Jcand, therefore, its moment, of inertia about the axis, is, , =, , PQ, , (mil) dx.x\, , moment, , of inertia / of the whole sheet about PQ, 7/ 2, , if//2, 2f, Jo, 2m, , m, ., , __, , 2m [U2, ftf, ', , #, , T VJo, , I, , 7/2, , is, , given by, , x\dx, , Jo, , " 2m, , r, , 8x3', , Ml 2, , Or,, , 12, , Similarly, the moment of inertia of the sheet about an axis through O,, in its own plane and perpendicular to PQ, i.e., parallel to its length, or, wiL, 2, be, /12., , DC, , Md, , EH, , Therefore, by the principle of perpendicular axes, the moment of inertia, of the sheet about the axis XX' through O and perpendicular to its plane, , ml 2, ."", , 12, , +, , md 2 "~, _ m /Pd, , V, , 12", , 12, , Hence, moment of inertia of the whole bar about the axis, , mass of the bar, , x f, , V, , 12, , M, , being the mas., of the bar., , Or., 1., , one, , XX', , side., , Moment, , of Inertia of a Thin Triangular Plate or Lamina about, (Pig. 30), be a th?n, triangular plate or lamina,, of surface density or mass pet, unit area, p, whose moment, of inertia is to be determined, about the side BC., , iittABC,, , Then, if the altitude, the plate be AP, H,, area =* J base x altitude, , =, , [v, , it*, , BC, , and, therefore,, Fig. 30., , its, , mass, , M, , *.., , (i
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Off, , MAfTUft, , Now, let us imagine the triangular plate to be made up of a, number of thin strips, parallel to BC, and placed side by side and,, let us consider one such strip DE of width dx, at a distance x from, the base BC. Then, clearly, the area of this strip, (which may be, ;, , t, , considered to be almost rectangular,, small) =5 DE,dx. And, therefore,, , mass of the, , New,, , =, , strip, , in the similar triangles, , DQ =, , whence,, Similarly,, , BP., , width being infinitesimally, , its, , DE.dx.p, , AQD, , ......., , (#), , and APB, we have, , -fr ., ri, , from the similar triangles, , AQE and APC we, , have, , t, , = 4' whence Q* -*>.--., therefore, DQ+QE = BP.-^+PC. 4-, , M-=^, , And,, , ri, , ri, , =, , Or,, , (BP+PC)., , mass of the, , Now,, , clearly,, , ., , ., , = a~A ~.dx.p., H, , strip, , moment, , =, , [From, , DE about, , of inertia of strip, , (//), , above], , the side, , 5C, , h, -=, , mass of the, , strip, , xx*=a., , -, , .dx.f.x*., , Hx, g, And, therefore, moment of inertia of the whole triangular, BO is equal to the, of this expression, between the, = and x = //. So integral, that,, , plate about, limits x, , M.L of the plate, , _, ~, , about BC,, , i.e.,, , I, , = Pa. (*~\*.&.dx., , H _, ~, , H \, T~T"y, #l~3, ~4"Jo, ^ 7T"w = ""12"", ~ff(, i2~~), , a.p rff.x 3, , x4, , ~|, , a.p / /?*, , "jtfA, , _H, , But, .-., , ^ a./f, , M., , I., , p., , =, , Af, ffte, , maw, , of the plate., , of the triangular plate about side BC,, , [Sec, i.e, , ,, , I, , =, , (/), , '-r, , above.
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MOMENT OF INBBTIA, $., , Moment, , BK1BGY OF BOTATIOH, , 61, , of Inertia of an Elliptical Disc or Lamina., (?) about, elliptical plate or lamina, of, , oneof its axes. Let XYX Y' be a thin, mass M, and surface density (ic., mass per unit area), P, and let its, major axis XX and minor axis YY', be equal to 2a and 2b respectively,, f, , 9, , f, , (Fig. 31)., , PQSR, , Consider a strip, , of the, , to the, plate of width 1 dx, parallel, minor axis, and at a distance x, from it. Then, if 2y be the length of, the strip, its area is clearly equal to, 2y.dx and, therefore, its mass equal to 2y>dx.?., , YY, , ~y, Fig. 31., , -, , then, M, I. of the strip about the minor axis, 2y.dx.p.x* and, therefore, M. L of the whole elliptical plate, about the axis YY' is equal to twice the integral of the above, a., 0, and x, Or, denoting it by, expression, between the limits x, , Obviously,, , YY', , =, , ;, , =, , Iv, , ,, , =, , we have, 2y.*x*.dx, , =, , a, , 4P, , [, , y.x*.dx, , (I), , Now, with the centre of the ellipse as the origin, and with the, co-ordinate axes coinciding with its major and minor axes respectively,, , we have, , 2+, , =, /,>, , fa, bz, , as the equation to the ellipse, , 1>, , = !-, , -ii., a*, , So that,, , y, , =, , or, , -, , whence,, , l, , b ^/i^x^ja^T, , Substituting this value of, , =, , y*, , ;, , y, , in relation (I) above,, , we have, , 4P, , Jo, ...(II), , Now, putting x, dx, , Or,, , =, =, , a sin, a cos, , 8,, 0., , we have, , T-, , aQ, , = a cos, , 6., , d9., , Substituting these values of, , x and dx, , in expression, , (II), , above,, , we have, /2, , .a, , cos, , Or,, , cos 6.dd.
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or MATTIE, PTC/2, , lo, f, 2 1, , 1, , cos 40, , .dg., , 1-2, , 2, , I,, , sm, pfi.t;, , 'Jo, , *2|, , Now,, , Tr.a.b.p, , =, , 1, , TT, , 2, , 2", , Af, the, , Similarly, the, , XX', , 7T, ~, , mass of the, I,, , the major axis, , ^, =, , elliptical plate., , M.a 2 /4., , of inertia (Ix ) of the elliptical plate about, given by the expression,, , moment, is, , =, , I,, Mb*/4., the centre of the plate or lamina, (//) about an axis passing through, The axis in this case will pass, and perpendicular to its own plane, of the, through O, (Fig. 31), and will be perpendicular to the plane, Hence, if /be, paper, (or the plane of its two axes, XX' and 77'), the moment of inertia of the elliptical plate about this axis, we have,, by the principle ofperpendicular axes,, ., , !-, , 0,, , ^fff Moment of Inertia of a Hoop or a Circular Ring., about an axis through its centre and perpendicular to its, (i), Let the radius of the hoop or circular ring be 7?, and its mass,, plane., , M,, , (Fig. 32)., , Consider a particle of it, of mass m. Then, the moment of, of the hoop,, inertia of this particle about an axis through the centre, , O, , and perpendicular to its plane, will obviously be mR 2, And, therefore, the moment of inertia /of the entire hoop about, the axis will be ZmR*., ^, m* 2, P-' 2'^=Mand R is the same, ., , -r, , I, , Or,, , = MR, , ., I, , for all particles., , about its diameter. Let it be required to determine the, of inertia of the hoop about the diameter AB, (Fig., Obviously, the moment of inertia of the hoop, will be the same about all, the diameters., Thus, if / be the moment of inertia of the, hoop about the diameter AB, it will also be, its moment of inertia, about the diameter, (ii), , moment, , perpendicular to AB., Then, by the principle of perpendicular, axes, its moment of inertia about the axis, through the centre O, and perpendicular to, its plane, is equal to the sum of its moments, of inertia about the perpendicular axes AB, , CD,, , and -CD,, , in, , its, , own, , plane,, , and intersecting
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63, , MOMENT of INBBTU--BNBEQY of BOTATKW, And, therefore,, , 7+7, , Or, , I, , MR, = MR, , 2, ., , MR*., , Or, 2 7, , [Seecaie (0., , 2, , /2., , ', , 1., , Moment, , of inertia of a Circular Lamina or Disc., , about an axis through its centre and perpendicular to its, (i), Let, be the mass of the disc and R, its radius., Then, since, plane., the area of the disc is 7T.R 2 its mass per unit, , M, , ,, , area will be, , Consider a ring of the disc, distant x, i.e., of radius x and of width dx,, (Fig. 33). Its area is clearly equal to its, circumference, multiplied by its width, or, equal to 2wx x dx, and its mass is thus, , from O,, , Hence, moment of inertia of this ring, about an axis through O and perpendicular, to its plane, , Fig. 33., , 2Mx*dx, , M.2<xx.dx, , Since the whole disc may be supposed to be made up of such, to jR, we can get the, rings of radii ranging from, moment of inertia / of the disc by integrating the above expression, for the moment of inertia of the ring, for the limits x=0, and x=/Z., like concentric, , ..., , *,T, MJ., , Aofr*u, the dzsc, , ., I, , Or,, , = MR, , 2, , /2., , about its diamet-er. Let AB and CD be two perpendicular, (//), diameters of a circular disc of radius R and mass Af, (Fig. 34)., Since the moment of inertia of the disc, about one diameter is the same as about, any other diameter, the moment of inertia, about the diameter AB is equal to the, moment of inertia about the diameter CD,, perpendicular to AB. Let it be /., , Now, we have, by the principle of, perpendicular axes, M.L of the disc about, , AB+its M.L about CD, = its L about an, , M, , O, n, Or,, , axis through, its plane., , and perpendicular to, r, , ., , ., , ,/+/, , - MR*, , -%-OT,, , = MR, -g-., 2, , 27T, , MR*, Or,, , AB, , be, about a tangent to the disc in itsi&vn plane. Let, which, and, mass, about, it*, radius, disc, of, circular, to, the, M,, tangent, (ii), , R
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PROPERTIES, , 64, , Off, , MATTER, , moment of inertia is to be determined, (Fig. 35). Let CD be a, diameter of the disc, parallel to the tangent AB. The moment of, A inertia of the disc about this diameter is, clearly,, equal to MR-J4., , So, , that,, , by the principle of parallel, , we, , axes,, , have, , MJ., , of the disc about, , AB = MJ., , of the, , CD+MR*., , about, , disc, , = MR +MR* =, 2, , I, , Or,, , MR, , 2, , D, , about a tangent to the disc and perpendiThis tangent will obviously be, plane, parallel to the axis through the centre of the disc and perpendicular, to its plane, the distance between the two being equal to the radius, of the disc. Hence, by the principle of parallel axes, we have, (iv), , Fig. 35., , M.L, , cular to, , about the tangent, , its, , =, , M.L, , about the perpendicular, , =, , Or,, , ;:, , MR, , axis+MR*., , 2, ., , Moment, , of Inertia of an Annular Ring or Disc., an, axis passing through its centre and perpendicular to, about, (/), its plane., An annular disc or ring is just an ordinary disc from, which a smaller co -axial disc is removed, so, that there is a concentric circular hole in, it., Let 7? and r be the outer and inner radii, of the disc, (Fig. 36), and M, its mass. Then,, clearly,, , =, , face-area, face-area of the annular disc, of disc of radius R face -area of disc of radius, r,, , And, , .., , mass per, , unit area, , =, , of the disc, , M/7r(#, , 2, , r, , 2, )., , p., , 2$, , Imagine the disc to be made up of a, number of thin circular rings, and consider one such ring of radius x, and of width dx., Then,, , face-area of this ring =27ix.dx,, , and, , its, , And, therefore,, ,, , ,., , ,, , x, , and perpendicular to, , mass, , moment, , its, ., , x, , ., , its, , M, __, , =, , plane, , =, , of inertia about an axis through, -, , 2Mx -, , ,, , 9, , ax.x*, , =, , 2Mx3, , O, , -, , rnz^TW, , The moment of inertia of the whole annular disc may, therefore,, be obtained by integrating the above expression for the limits x, r, and x, R, Or, moment of inertia of the disc about the axis through, , =, , =, , O and perpendicular to, f, , *, , jrr, , its, , plane, , 2^*> dX, *-r*)-, , ~, , 2M, , (W=, , f*
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MOMENT OF OTBBTU, , ENERGY 0! ROTATION, -, , -, , (IP-r), , r, , mp+ r*).(/l_rn, , 2M, , r, , ^IrSjL, , -M, , I, , Or,, It follows at once, , from the above that, , moment of inertia, , = R,, its M.L =, , .Again, if r, , R, and, , = 0,, , if r, , disc, or that it is just a plane,, , no hole in the, disc, its, , ~J, , MR /2., 2, , is, , [Case 7, , we have a hoop or a, , i.e, t, , i.e., if, , ^, , is, , (/),, , above., , circular ring, of radius, , ^ MR, , A, , there, , (and not an annular), , *, , [Case 6, , t, , (/),, , above., *, , Obviously, due to its symmetrical shape,, (it) about its diameter., the moment of inertia of the annular disc about one diameter will be, the same as that about any other diameter. Let it be 7. Then, the, sum of its moments of inertia about two perpendicular diameters will,, by the principle ofperpendicular axes, be equal to its moment of inertia, about an axis through O and perpendicular to its plane, i.e., equal to, , MGR2 +r, , 2, , )/2., , *, Aff/P+l, M( * 2 + r *> /* 2/), 74-7i.e., LL, /-j-7 =, , Or, ur,, , ,, , -=, , whence,, if r, , Now,, , M.L, Or,, its, , -, , -, , =, , 0, i.e., if, , -, , -, , --, , the disc be a plane one,, , of the disc about a diameter, , if, , moment, , =, , r, , R, we have a hoop, , =, , --, , -, , = MR, , 2, , /4:., , its, , we have, [Case 7, , (//),, , .--, , = MR*, r--., , ., , ....., , [Case 6, , (//), , R, , I, , Or,, , the tangent, , =, , and, , diameter, , in its own plane., The tangent, (Hi) about a 'tangent,, parallel to the diameter of the ring or disc, and at a distance, it, we have, applying the principle of parallel axes,, , M.L about, , above., , or circular ring of radius R,, , 442, , of inertia about, , ,, , =, , M.L, , u, above., , being, , from, , about the diameter -{-MR*., , +MR* = M, , M, , about a tangent, perpendicular to its own plane. The tan(iv), in, this, case, is parallel to the axis through the centre of the ring, gent,, or disc and perpendicular to its plane, the distance between the two, being equal to jR. Hence, by the principle of parallel axes, we have, , M.L about, , the tangent, , Or, , I, 9., (i), , cylinder, , about, , the perpendicular, , axis+M/? 2, , ., , -**, A, , own axis, or its axis of cylindrical symmetry., thick circular disc, or a number of thin circular, , its, , just a, , M.L about, , of inertia of a Solid Cylinder., , Moment, is, , =
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86, , PROPERTIES OF MATTER, , discs, piled, , about, , its, , one upon the other, and, therefore, its moment of inertia, is the same as that of a circular disc or lamina &bout an, , axis, , and perpendicular to its plane, i.e., equal to, mass and R, its radius., [Case 7 (i) above., , axis through its centre, , MR /2, where M, 2, , (ii), , is its, , about an axis passing through, , centre, , its, , M, , and perpendicular to, , own axis of cylindrical symmetry. Let, be the mass of the cylinder, R its radius and /, its length, (Fig. 37)., Then, obviously, if it be, homogeneous, its mass per unit length will be M/l. Let YY' be the, and perpendicular to its own axis, axis, passing through its- centre, its, , XX' about which, ,, , the, , moment, , of inertia, , is, , to be determined., , Imagine the cylinder to be made up of a number of thin idiscs, and consider one such disc at a distance x from O, and of thickness dx., Obviously, the mass of the disc, , is, , (Mjl).dx and, to, , its radius,, , R, , ;, , so, , moment of, about, is, , /.Jj, , U~., , ,.l-~-l., , SI, , 4---J, , its, , equal, , that, its, inertia, , diameter, , equal to mass of, , the disc, , x(radius)^., , -?*' f, And,, , its, , ', , mo-, , inertia, of, about the axis YY', iftent, , Fig. 37., , by the, , 9, , principle of parallel axes,, , = M, ,, , Therefore, the moment of inertia of the whole cylinder about, the axis YY' may be obtained by integrating this expression for the, result by 2. Thus,, limits x, and x, //2 and multiplying the, , =, , MJ., , =, , of the cylinder about the axis, , 2M, , YY, , f, , J!_l, Jx3J, , =, , -i, , +-*-J, , Or,, (HI) afeowf, , from the above, , a diameter of one of its faces. It is an easy deduction, for, by the principle of parallel axes, we have, , ;, , M.I. of the cylinder about the diameter of one face, , MR* .Ml* .Ml*, 4,, , I., , M, , \*, , MR*, , Ml*
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MOMENT, , M, 4T Moment, , Off, , ENERGY Of ROTATION, , INERTIA, , 67, , of Inertia of a Solid Cone., , M, , its vertical axis. Let mass of the cone be, (i) about, height, h and radius of its base, R, (Fig. 38)., , ,, , its, , vertical, , =, , z, Then, its volume, ^nR h., And, if p be the density of its material,, , M = knR*hp,, , mass, , p =-, , whence,, , its, , ---..-, , TT/v, , II, , Imagine the cone to be made up of a, discs, parallel to the base, and, placed one above the other. Consider one, such disc at a distance x from the vertex,, and of thickness dx., , number of, , If r be the radius of this disc,, [where a, , is, , the semi-vertical angle of the cone., , =, its mass =, , z, , volume of the disc, , And,, , we have, , = x tan a,, , r, , Fig. 38., , 7ir .dx., 2, , 7rr .c/x.p., , Now, moment of, ing through, , its, , AO, , inertia of the disc about the axis, y, passcentre and perpendicular to its plane, i.e., about the, , vertical axis of the cone,, , its, is, , clearly equal to, , And, therefore, the moment of, vertical axis, , x, , =, , /., , .,, , and x, J/.7., , AO, , =, , mass, , ^, , ts radius, , ?, , ., , inertia of the whole cone about, , will be the integral of this expression, for, , its, , the limits, , /?., , of bhe cone about, , -, , its vertical, , axis, , is, , given by, , h, [, , Jo, , '*-]*, .5 Jo, , 2, 7TP./?, , 4, , ~, , A5, 5, , Or, substituting the value of p from above,, 1, , =, , ', , 7t~R*.h.2h*, , T, , we have, , == ", , 10, , (ii) about an axis through its vertex and parallel to its base., Again, considering the disc at a distance x from the vertex of the cone,, , we have, M.I. of the disc about, , its, , diameter, , =, , 2, , 7rr .Jx.p., , r2, ., , ., , 4, , And, therefore, by the principle of parallel axes, its moment of, inertia about a parallel axis, passing through the vertex of th, , XX, , cone, r*, , 1, , ',
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08, , *ftoEfeTiB$, , oir, , Therefore, the moment of inertia of the whole cone about the, axis passing through the vertex and, parallel to the base, i.e., about, XX', is obtained by integrating this expression for the limitSj x, , = h., , and x, , =, , Thus,, , M.I. of the cone about, , >, , XX*, , tan* a fh, , 4, , -f*, , ^, , ,, , **.</* +irPtoi, , fh, of*, , 9, , 4, , ^, , A*, f, , /?, , TTP, , 4, , A5, , M.L, , XX', , of the cone about, , ., Or,, , =, , _, , 1=, Moment, , r, , hb, , R*, , ,, , substituting the value of, , Or,, , ^^, , J, , '0, , P,, , ^, , we have, , ., , ,- ,, , /i*, , -, , -, , 5, , 3MR*, , --+ 3Mh-, , of Inertia of a Hollow Cylinder., , own axis. A hollow cylinder may be considered to, number of annular discs or rings of the given inter-, , aftowf its, , ((-/), , consist of a large, , and external radii, placed one above the other, the axis of the, cylinder passing through their centre and being perpendicular to, their planes, (Fig. 36)., , nal, , The moment of inertia of the hollow cylinder about its own, axis is, therefore, the same as that of an annular disc of the given, external and, radii, internal, about an axis through its centre and perpendicular to its, 2, 2, plane, i.e., equal to, -fr )/2,, where, , M, , M(R, , the mass of the, and r, its external, , is, , cylinder, R, and internal radii respectively., Let, Alternative Proof., , M, , be the mass of the cylinder, , and, , /,, , its, , R and, , Fig. 39., r, its, , external and internal radii, , length, (Fig. 39)., , = Tr^ r, =, r, volume of the cylinder, 7r(7?, M, its mass per unit volume =, ^, 3, , face-area of the cylinder, , Then,, , f, , )., , 2, , And,, , 2, , )/, , BO that,, , co- axial cylinders,, , thickness dx., , ;, , -^, , cylinder to be made up of a large number of, and consider one such cyluvfer of radius x and, , Now, imagine the, thin, , ;
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MOMENT 0* INERTIA, area, , its face, , Then,, , =, =, , and, , its, , Since, , all its particles, , -., , .., , ,, , mass, , .,, , ., , ., , of inertia about the axis, , ENERGY OF ROTATION, , 27tx.dx, , M, , volume, , its, , 9, , ,, , =, , =, ,, , /m~ -jr-, , /j? 2_^/x27rx.a.x./, , from the axis,, 2 Jf **.</*, -, , are equidistant, , 2Mx.dx, --, , f, .x 3 ==, , ~2, , ,, , (A, , D2, , its, , moment, , r2), , (K, , r-), , 69, , And, therefore, the moment of inertia of the whole cylinder may, be obtained by integrating the above expression for the limits, x, r, , and x, , s=, , jR., , J/.7., , Or,, , of the cylinder about, t, CR, , Jrr, |, , 2M.X, , 8, , 5Jf, , V-OL, , x, , a, , (^^H7^J r, , ), , r^-iJ, , (R*, , *M_[ R, , _, ~, , ,, , a, , (j?2-Z7, , its axis, i.e.,, , n^-, , 2jj/, , ^, , 4 Jr, , 14 )", , -'], , 2, , (^ -r2), , Or,, , an axis passing through, its, , As, , own axis., , fore,, , let, , M, , centre, , and perpendicular, , to, , be the, , mass of the, der,, , its, , be-, , cylin-, , its, , length,, and/? andr, its external and internal, radii, respectively, /,, , x, , ;, , and, , let, , YOY'bz, , the, , axis through its centre 0, and perpendicular to its own axis, , V, Fig 40., , XX',, , (Fig. 40)., , and, , Then, face-area of the cylinder, its volume, its, , mass per, , unit, , = 7r(jR, = 7t(R, , volume ==, , a, , r f ),, , z, , r 2 )/., 8, , Jf/7r(J?, , a, /, , )/., , Imagine the cylinder to be made up of a large number of, annular discs of external and internal radii R and r, placed one by, the side of the other, and consider one such disc at a distance x from, f, the axis YY and of thickness dx. Then, clearly,, ,, , surface area of the disc, its, , volume, , =, , =, , 2, , 7r(/?, , 2, , 7r(jR, , r*).t/x,, , and, , ~~r a ),, .-.its, , mass, , =, , Jf.rfx//., , Now, moment of inertia of an annular disc of external and, internal radii, jR and r, about its diameter, is equal to its massx, 2, [Case 8 (),, +R* r )/4., , +, , .-., , M.I. of the disc about, , its, , (}i$iter
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PROPERTIES OT MATTJBH, , 70, , YT', , And, therefore, its moment of inertia about the, by the principle of parallel axes, given by, , axis, , parallel, , is,, , M, , ., , ,w^c, , l, x (R*+r ), , M, , ~, , *, , ,, , ,dx.x., , ~l, , And, clearly, therefore, moment of inertia /, of the whole cylinder, about the axis IT', is twice the integral of this expression, for, the limits, x, , =, , =, , and x, , 7/2., , tip., i.e.,, , 2P, 2Jff//2P, i, /, JO L, , +, , 7?2, , ,-a, , --dx+x*.dx \~, , \, , I, , Or,, , =, , 2, , 2Afr(/? -fr, /, , M.L, , ^2r, (/), , about, , its, , )/, , that, , of the cylinder,, , Moment, , L, , ", , =, , if r, , 0,, , i.e.>, , the cylinder bo a, , if, , about an axis through, , (solid),, , own axisM(R 2 /4, , its, , Jo, , "3, , 2, , 4x2, , L, , It follows, therefore,, solid one, we have, , and perpendicular to, , -, , -., , J, , + F/12)., , centre, , its, , [Case, , 9, (//), , above., , of Inertia of a Spherical Shell., , diameter., , First Method., through its centre, , O, , Let, , ABCD, , and, , let, , be the section of a spherical shell, and its, the mass of the shell be, , M, , radius R, (Fig. 41)., Then, area of the shell, , ,, , equal to, , is, , 47T/?, , 2, ,, , and, mass per unit area of the shell = MI&nR 2, Let it be required to determine its, moment of inertia about the diameter AB., .*., , *, , Consider a thin, , EF, diameter AB, , cular to the, , and, , of the shell, lying, , slice, , between two planes, , (x-\~dx) respectively, , 9, , and OH, perpendiand at distances x, from, , its, , centre O., , This, obviously a ring of radius, width, and, EG, (not PQ, which is equal, PE,, , Fig. 41., , slice is, , to dx)., , area, , of this ring => its, =*=, , and, hence, , its, , mass, , Or., , OE, , and OG, and, , =, , XM tin R*., , 2n.PExEGxM/4>7rR*., COE . and l_EOG, let, , Then,, , PE, , Similarly,, , OP = R sin, x, , X width., , 27T.PEXEG,, , == its area, =c, , Join, , circumference, , OE.cos, , -* J?, , $m, , OEP, , ..., , *, , (/), , d0., , R cos, [v LOEP, , - iCOE -, , e., , O/ -, , *., , Q, e,, , ['', , OE -, , J?, , ni/
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ENERGY OF ROTATION, , MOMENT OF INERTIA, Now,, , differentiating, , x with, , respect to 0,, , we have, , = R cos g, dx = R cos Q.dB *= PE.df)., G == 0".rf0 = #.^0., , dxjde, Or,, , And,, , t, , =, , the ring, , flrc, , subtended by the arc., , L, , 27r.P.^.rf0.M/47r#, , X cos o JP&, = radius* angl*, , [v, '', , I, , mass of, , 71, , 2, , [from, , ., , (/)., , Hence, moment of inertia of the ring about AB, (an axis passing, through its centre and perpendicular to its plane), is equal to its, , mass x (its radius) 2, /., , moment, , ,, , =-M *.PE\, , i.e.,, , of inertia of the ring about, , And, therefore, the moment of, shell, , AB =, , about, , x, , limits,, , =, , where, , ZK, , inertia, , /,, , x\, , of the whole spherical, , x 2 ), between the, , 2, , D, , AjK, , (R*-x*)., 2, , .(R, , >, , twice the integral of, , =, , 2, , AB =-AJ\, -V, ", , and x, , PE =, , .(R, , R., , '-, , ie, i.e.,, , I, l, , M, , ~, , -^, Second Methad.~Let, , 2, , ~~, R3 y<, , ., , ^, , 2, , y, , M, , be the mass of the shell and R, its radius., Consider a particle, of mass m, anywhere on the shell. Then, since the, thickness of the shell is negligible, the distance of the particle from the centre, of the shell is the same as the radius of the shell, i e., R., Obviously, therefore, the summation / for all the particles of the shell,, centre 0, is given by the relation,, TAll particles being at distance R, a, = M., 2mR*., /o, Or,, /., rom Q an d, ,, , about, , its, , -, , MK, , [j, , Jm, , Now, /be the moment of inertia of the shell about one diameter, it, be the same about any other diameter also, from the sheer symmetry of tha, if, , will, , Hence, in accordance with the principle of perpendicular axes for a three, dimensional body, [29 (a), (//), page 52] the sum of the moments of inertia of, the shell about its three mutually perpendicular diameters must be equal to, twice the summation / for all its particles, about their point of intersection,, i.e., the centre of the shell O ; so that,, ", 9, 3/ - 2MR*,, /+/+ / 2/o,, [V /, Or,, , shell., , ,, , - MR, , -, , whence,, , I, , --, , ., , MR*., , about a tangent. Obviously, a tangent, drawn to the shell, any point, must be parallel to one of its diameters, and at a, distance from it equal to J?, the radius of the shell. Hence, applying, the principle of parallel axes, we have, (ii), , at, , A/./,, , of the shell about a tangent, 2, its M.L about a diameter -{-MR, , =, , Or,, , /, , |, , MR*+MR*, , =*, , JMR*., , .
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OF MATTBH, , 71, 13., , Moment, , of Inertia of a Solid Sphere., , Let Fig. 42 represent a section of the, (i) about its diameter., sphere through its centre O., and its radius, R., Let mass of the sphere be, 47T# 3 /3., Then, clearly, its volume, , M, , t, , =, , And, , .*., , its, , mass per, , unit volume, , Consider a thin circular slice of the, sphere at a distance x from the centre O, and of thickness dx., t, , This slice, radius \/ R*, .-., , surface area of the slice, , volume, , area, , And,, , .-., , its, , mass, , = its, , obviously a disc of, , ,, , = TT^/^-X^ =, , == 7t(R*, , x thickness, , is, , x 2 and of thickness, Tr(R*, , x, , 2, , dx., , and, , ),, , its, , x 2 ).dx., , volume x mass per unit volume of the, , sphere, , Now, the moment of, passing through, , moment, , its, , inertia of /Af5 disc about ^4., centre and perpendicular to its plane), , = its mass x (radius), , of inertia of the disc, , 2, 1, , (an axis, , 2., , AB, , ;#"', , *, , ."(I), , /. moment of inertia /, of the sphere about the diameter AB, equal to twice the integral of expression (1) between the limits x, and x, R., , is, , =, , =, , ", , 3, , "*", , 5 Jo, , 3M, Or,, , Now, the moment of inertia of the sphere about one diameter, the same as about another diameter, so that we have the moment, of inertia of a solid sphere about any diameter given by I =c MR*., is, , Alternative Meth6d., , Let, , M be the, , mass of the sphere and, , p,, , the density, , of its material., , Imagining the whole sphere to be made up of a number of thin, concenone inside the other, and considering one such shell of, radius x and thickness dx, we have, tric spherical shells,, , surface, , arm, , of the shell
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MOMBNt OP INERTIA, and, , /. volume of the shell, , moment, , .'., , = 4*rx*.djc, , 73, , INjBRQY OF ROTATION, , and, , its, , mass = 4*x*.dx.p., , of inertia of the shell about a diameter, , npx* dx, [case 11 (/),, |x(its tfttm)x(its radius)*, $.4nx*.dx.pxx*, And, therefore, the moment of inertia /, of the whole sphere, about its, diameter, is obtained by integrating the above expression between the limits., x, and x = R., (, , -J, , R, , 8, , rw, , 8, , --T" P, , t, , 437^3/3, , =, , M.L, , L, , U, f x, , R*, , ~]R, , 8, , Jo, , --3, , the v0///w? of the sphere, , of the sphere about, , its, , ;, , 8, , _., , -"'-T-Tr-P-*, , and, therefore, 4rtR*?l3, , =, , /.*., I, , diameter,, , = M,, , its, , 2,MR/5., , A, , (n) fl&0w/ a tangent., tangent, drawn to the sphere at any, will, be, obviously, point,, parallel to one of its diameters and at a, , distance from, , it, , equal to R, the radius of the sphere., , with the principle of parallel axes, we, of the sphere about a tangent, , Therefore, in accordance, , have, , =, x^14., (/), , Moment, about, , its, , + MR*., , about a diameter, , 1-2 MR*/!>, , Or,, , vx, , M.L, its M.L, , -f-, , MR* = 7MR, , 2, , /5., , of Inertia of a Hollow Sphere or a Thick Shell., , diameter, , A hollow, , just a solid sphere, solid sphere has been, , is, , sphere, , from the inside of which a smaller concentric, , And so, the moment of inertia of the hollow sphere is, moment of inertia of the bigger solid sphere minus the, to, the, equal, moment of inertia of the smaller solid sphere removed from it, (both, about the same diameter). If R bs the radius of the bigger sphere, and r be the external and, and r, that of the smaller sphere, i.e., if, internal radii of the hollow sphere, and p, the density of its material,, removed., , R, , we have, volume of the bigger sphere, smaller, , and,, .-., , volume of hollow sphere, , And, and, , /., , M.L, , M.L, , =, =a, , M.L, , 3, , 3, -J-Trr, , = ^(J?, , and, , ,, , .. its, , mass, , and, , 3-, , r8, , of the bigger sphere about, , ), , and, , its, , its, , =, =, , mass, , diameter, , of the smaller sphere about the same diameter, , .-., , ^TtR, , H7rr*.p).r*., , of the hollow sphere about that diameter, rr.p).^, ..., , Now, mass of the hollow sphere,, s, And .-. p, 3Af, 47r(JR -r ),p,, !l, , Or,, , M=, , 8, , i.7r(jR, , r s ).p., , (1)
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rftOFEBTlBS Of MATTER, Substituting this value of p in relation (1) above,, of inertia / of the hollow sphere about its diameter, , wo have, , moment, , _, Or, Ur, , -1, , *, ', , 6, , 3, , w, , ', , M J^'-., i_A, 5, (R-r), l, , ', , iV1 *, , Alternative Method. As in the case of the solid sphere, so also here, we, can imagine the sphere to be made up of a number of thin, concentric spherical, shells, and considering one such spherical shell of radius x and thickness dx, we, the, mass of the shell = 4nx*.dx.p., fp, being, have, as before,, density of the, of the shell about a diameter = &Anx*.dxj.&., .-., (, 6, material, of, .**.</*., |.7r. P, [ the sphere., , ML, , Hence, the moment of inertia of the whole sphere about its diameter, r and x = R., the integral of the above expression, between the limits, x, , =, , M.L, , Or,, , of the sphere about a diameter, 8, , But, , ^(R*, , r 8 ).p, , 8, , I"*, , Af, the, I, , i.e.,, , I, , fR, I, , o, --, , TC, , p.x*.dx., , r x *-R, , mass of the sphere., , =, , =, , is, , [See case, , (/), , above.], , --., , (ii) about a tangent. Again, as in the case of a solid sphere, the, tangent to the sphere, at any point, will be parallel to one of its diameters, and at a distance equal to its external radius R from it., Hence, by the principle of parallel axes, we have, , M.I. of the sphere about a tangent, , = its M.L, I =, , Or,, , about a diameter -{-MR*., , ["-|-M(R, , 5, , ~r 5 )/(R 8 -r 3 )"l+MR 2, , ., , Moment of Inertia of a Flywheel and Axle. A flywheel, a targe heavy wheel, with a long, cylindrical axle, passing, through its centre. Its centre of gravity lies on its axis of rotation^, so that, when properly mounted over ball-bearings (to minimise, friction), it may continue to be at rest in any desired position., 15., , is, , just, , Let, , M be the mass of the flywheel, and, , m, that of the axle, be their respective radii., Then, for our present purpose, we may regard the flywheel to, be a disc, or a small cylinder, from which a smaller, concentric disc or, In, cylinder, equal in radius to that of the axle, has been cut off., sther words, we may take it to be an annular ring, (or hollow cylinier) with an outer radius equal to R> and an inner radius equal to r,, iose moment of inertia is to be determined about an axis passing, ough its centre and perpendicular to its plan*., , and, , let, , R and, , ;, , r, , '
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ENERGY OF EOTATION, , MOMENT OF INERTIA, , 75, , The face area of this wheel or annular disc is clearly equal to, the area of the whole disc of radius R minus the area of the disc of, radius r., a, a, 2, 7rr =7r(jR, face area of the wheel ==7r#, And, if its mass be M, clearly,, mass per unit area of the w/zee/=Jf/7r(JR 2, , r 2 )., , i.e.,, , r 2 )., , x from tho, , Now, consider a thin circular ring at a distance, centre, and of width dx., Then, face area of the ring=its circumference x, , And, therefore,, , Now,, mass x, , s.ince, 2, , (radius), , M., , I., , the, , mass, , ,, , ==, , its, , width=27rx.dx., , 2, , 27rx.dx.M/7r(R*r )., of inertia of a ring about an axis, , moment, , centre and, , its, , through, , its, , perpendicular to, , its, , plane, , we have, , of the wheel about, , its, , axis, , f, , =, , equal to, , is, , M, , R, , ,- ^, , j _., , its, *, , .2nx.dx.x*., , TT(R*-1, , 2M, 2, , Or,, , ALL of the, , -r'), , wheel about, , 4, its, , axis, , M, v, , .,, , ., , ,, , ,, , The, , moment, , M, , ., , JL, , z>, , axle, again,, just a disc, (or solid cylinder), and its, of inertia about its axis is, therefore, just the same as that, is, , of a disc or a cylinder about, , So that,, Hence,, , M.L, , its axis, i.e.,, , M.L of the axle =, wheel and axle =, the., of, , =, , its, , massx(radius), , 2, , /2., , 2, , w.r / 2, , -, , M.L of the, , wheel -\-M.L of, , the axle., I, , Or,, , - [M(R+r)/2]+iM, , 2, , /2., , Table of Moments of Inertia. The values of moments of, inertia for the cases discussed above, together with some other important ones are given in the Table below for ready reference of the, student, the mass of the body being taken to be M, in all cases., 32.
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FBOPKETIBS OF MATTE*, , 3., , 4., , MOMENT, , AXIS, , BODY, , (Position, , and Direction), , lar bar,, , Thick uniform rectanguof length / and, , Through it* mid-point, and perpendicular to its, , thickness d., , length., , Thin triangular plate or, , About one, , side., , About, , one, , OF, , INERTIA, , lamina, of altitude H., 5., , Elliptical disc or lamina,, of major and minor axes, , (/), , of the, , or, , axes, (major or minor)., , 2a and 2b., , Through, , (ii), , and, , centre, to its, , its, , perpendicular, , plane., 6., , or circular, of radius R., , Hoop, , Through, , (/), , ring,, , and, , centre, to its, , its, , perpendicular, , plane., , About a diameter., About a tangent, , (ii), (i//j, , own, , its, , About a tangent, per-, , (iv), , pendicular to, 1., , Circular lamina, of radius R., , or disc,, , its, , plane., , Through, , (/), , and, , in, , plane., , its, , 2MR*, , centre, , perpendicular to, , its, , plane., , About a diameter., About a tangent,, , (//), , (///), , in, , own plane., (iv) About a, , 5MR*I4, , its, , tangent perpendicular to its plane., 8., , Annular ring or disc of, outer and inner radii R, and r., , perpendicular, , plane., 07), , centre, to its, , About a diameter., , About a, , (/i7), , own, , its, , Through, , (/), , and, , tangent, in its, , plane., , (iv) About a tangent perpendicular to its plane., , 9., , Solid cylinder of length, and radius R., , /, , (/), , About, , lindrical, (ii), , its axis, , of cy-, , MR*/2, , symmetry., , Through, , its, , centre, , and perpendicular to its axis, of cylindrical symmetry., About a diameter of, (///), one face., 10., , Solid cone, of altitude h, , and base radius R., , (/), , About, , its, , v*rtical, , axis., (//), Through its vertex, and parallel to its base., , 11., , Hollow, of, cylinder,, length / and external and, internal radii, and r., , R, , (/), , About, , its, , own, , axis,, , about its axis of cylindrical symmetry)., (i.e.,, , Through its centre, (i7), and perpendicular to its, , own, , 3MJK*, , 3MfP
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Of INE&TiAENBRGY 6f, , 77, , 33., Routh s Rule. This rule states that the moment of inertia, of a body about any one of the three perpendicular axes of symmetry, passing through its centre of mass is given by, its mass and one-third of the sum of the squares, (i) the product of, in the case of a rectangular lamina or paratwo, the, other, semi-axes,, of, llelopiped, , ;, , the products of its mass and one-fourth of the sum of the, of a circular or an elliptisquares of the other two semi-axes, in the case, cal lamina ;, its mass and one-fifth of the sum of the squares, the, (//), , product of, (til), a, of the other two semi-axes, in the case of a sphere or spheroid., easily, , Quite a few of the cases, dealt with in the proceeding pages, may be, deduced by an application of this rule. Thus, for example,, (/), , moment of inertia of a uniform, , rec-, , (angular lamina (of mass M, length /and breadth, about an axis passing through its centre, ), and perpendicular to its plane, , O, , 12, for, here, the two semi-axes of the lamina are, clearly, //2 and 6/2 respectively, (Fig. 43).
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78, , PKOPERTIJES OF, , MAtTBR, , M, , Moment of inertia of a uniform, , (it), , circular lamina or disc, (of mass, and, radius R), about an axi, passing through its, enire and perpendicular to its plane is equal to, , -"*"-'--, , #-;'", because here the two semi-apes of the lamina or, disc are obviously R and R> (Fig. 44)., And,~again, moment of inertia of a uniform, , M,, , and with 2a and 2b t, elliptical lamina, (of mass, as its major and minor axes respectively), about, a perpendicular axis passing through, is equal to, , Fig. 44., , because, , (a), , and, , (6) are, , its, , centre,, , the two semi-axes of the, , lamina, (Fig. 45)., , moment of inertia of a, , (///), , mass A/and radius R) about, , its, , solid sphere,, , diameter, , is, , (of, , equal, , to, , because here the two semi-axes of the sphere are, R and R., , Fig. 45., , Practical methods for the Determination, , 34., , Moments, , of, , of, , The, , principle underlying the experimental determination of, the moment of inertia / of a body, about a given axis, is to apply a, known couple C to it and to measure the angular acceleration doj/dt, Inertia., , produced in, , it., , Then, from the relation,, , C=, **, , whence,, (/), , /may, , Moment, First, , we have, , Ldwldt,, , /, , (2, , =, , ,, , ., , ,, , ,, , u to I at, , be easily calculated., , of Inertia of a Flywheel., , The flywheel, whose moment of inertia is to be, mounted on ball-bearings (to minimise friction), and, , Method., , determined,, , is, , its axle is arranged to be in the horizontal position at a convenient height, from the ground, (Fig. 46)., j~:...J:r, , _., , r, , trn, , T_-, , mg, ;, , Fig. 46., , A small loop at the end of a, small piece of fine cord is then slipped, on to a tiny pag on the axle and the, entire length of the cord wound evenly, round the latter, with a suitable mass, , m, , suspended from, , its, , free end,, , and, , properly held in position,, , As the mass, , is released and, allowed to fall, under the, weight, the cord starts unwinding itself round the, axle, thereby setting the wheel in rotation. The length of the cord, is so adjusted that the moment the mass reaches the, ground, the, of it gets just unwound from the axle arid, off the, , action of its, , own, , slips, , Hbviously, the, , rotation of the wheel, (with the descent of
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MOMENT OF INERTIA- ENERGY OF BOTATION, the mass), is due to a couple T.r. where, and r, the radius of the axle*., , is, , the tension in the cord, , If, therefore, / be the moment of inertia of the flywheel about, axis of rotation and dwjdt, the angular acceleration produced in, , its, it,, , T, , 79, , Ldw/dt = T.r., The downward force due to the weight of the mass, when, , we have, , it has, no acceleration, is mg but when it has a vertical acceleration a, the, force due to it is equal to m.a., and this must clearly be equal to, ;, , mg-T., , =, , m.a, , Or,, , /.dot/at, , But, , =, , dw/dt, , / =,, , Or,, , mgT, , whence,, , 9, , m.(gd)r., , And, , tf/r,, , mr\, , (g, , .-., , ~~, , I.a\r, , =, , wr, , T=, , =, , m.(g, , a),, , m(gd)r. [v a, -, , =, , r.da>jdt], , -1, , ...(1), , The time-interval between the, , release of the mass and the slipping of, the cord from the axle is r rpfully noted. Let it be, and let the, distance through which the, falls down during this interval be S., Then, since the mass starts from rest, we have, ,, , m, , =, , S, , i, , =, , 2, a, 2S/t, So that, substituting this value of a in relation, -I, , at 2 ,, , whence,, , ., , (1), , we, , above,, , have, , whence, , /,, , the, , moment, , of inertia of the flywheel about, , its, , axis of, , rotation, can be easily calculated., , Second Method. Proceeding as above, the loss of potential, energy af the falling mass is equated against the gain in kinetic energy, of the wheel, the K. E. of the mass itself and the work done against, Thus, \vheri the mass falls through distance S, the potential, friction., energy lost by it is equal to tng.S. And, if a> bo the angular velocity, of the wheel at the time, the K.E. gained by it is | 7oj 2 the K.E., 2, acquired by the mass being \ wv where v is its velocity on descending through distance S., ,, , ,, , =, , 2, , -f lmv*+the work done against friction., (2), To determine the work done against friction, we note the number of tunn made by the whoel before coming to rest, after the mass, has been detached from the axle. Then, obviously, the kinetic energy, 2, | 7o> of the wheel, is used up ia overcoming; the fricuional forces at, the bearings. If the couple due to friction t>3 G and the number of, turns made by the wheel before coming to rest be n, work done by, Ms couple is equal to STTH xC, (v work done = couple xangle and, the angle, described by the wheel in one rotation is equal to 2v)., So, .., , mg.S, , J, , 7o>, , ., , ., , ,, , t, , that,, , i, , 7o>, , a, , =, , 2nnC., , Or,, , C=, , 2, , 7co /47rfl., , The couple due to friction being thus determined, we can easily, work done against friction during the descent of the, , calculate the, , *If the cord be appreciably thick, half of its thickness, added to, radius of the axle, gives the effective value of r., , t!
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90, 5., For, clearly, the number of turns madt, of the mass through this distance is, the, fall, the, wheel, by, during, S/27rr and, therefore, the total angle turned through by it is equal, to 27r.5/27rr, S/r., , mass through distance, ;, , =, , Hence, work done against friction, .*., , our energy equation, , is, , equal to C.S/r, , now becomes, , (2), , Or, , Now,, tance S,, , if/ be the, , its, , time taken by the mass to, , =, , average velocity, , (initial velocity -{-final velocity) l'2, , [, , final velocity, v, the initial velocity, , Sjt, , through the, , fall, , since average, , dis-, , velocity, , =, , we have, , t, , = 2#/f., is zero,, , and, , ;, , v*, , Or,, , =, , 45 2 // a, , ., , the mass starting from rest.), , 2, Substituting this value of v in expression (3) above,, , we have, , (, , _, "~, , Or, 1, , 2S( 1, Let the number of rotations made by the"wheel,, before the cord and the mass slip off from the axle, (i.e., after the mass has, fallen through a distance S), be N.* Then, taking the fractional force to be uniform, and the work done against it p? r rotation of the wheel to be w, we have, Alternative Calculation., , werk done against, , friction during AT rotations of the wheel, , Thus, our energy equation, , (2), , =, , N.w., , becomes, , mg.S = J It**+ Jwv +JV.w., .(5), after, the, of the mass from the axle, the wheel cornea, detachment, Now,, to rest after n rotations, and, therefore, work done against friction during these, n rotations of the wheel, n.w and this must obviously be equal to i /a>, the, K.E of the wheel at the instant that the mass gets detachedfrom it. Thus,, 2, , _, ., , ,, , n, , w =i, , __, , /eo, , 2, , Substituting this value of, , mg.S, , Or., , -, , J, , whence,, , ,, , w, , w, , 2, J 7w /., , in equation (5) above,, , we have, , *This is obviously equal to the number of turns of the cord on tb* axle, it the very start.
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MOMENT OF INERTIA, , ENERG* OF ROTATION, , Of,, , .., [Smce v, , whence,, , by dividing both the numerator and the denominator of, , Or,, , ", , this expression, , by, , w 8 we have, ,, , _, , (2mg 5/6>*)~, , Now, the angular velocity of the wheel at the instant that the mass gets, and becomes zero when the wheel conies to rest, after, detached from it is, time t'> say. Hence, if the fractional force uniformly retards the rotation of the, wheel, its average angular velocity, during this interval of time f, may be taken, to be equal to (to4-0)/2, i.e., equal to co/2. And, since the wheel makes n rotations before coming to rest, it describes an angle equal to 2w in time t',, ,, , ,, , ,, , ,, , co/2, , So, , ,, , whence,, , 2rc/i//',, , that, substituting this value of, , co, , co, , =, , 4-nnjt'., , in relation (6) above,, , we have, , (n+N)ln, , wX'S?", , 1, , ), , Or,, , ..(7), , whence /, the moment of inertia of the, be easily calculated., , flywheel, about its axis of rotation,, , can, , Accurate value ofu>. In the above treatment, the angular velocity w of, the wheel has been obtained on the supposition that the factional force remains, constant during the time t' that the value of o> falls to zero, after the detachment of the mass from the axle. Obviously, this is by no means a valid assumption, because, as we know, the frictional force decreases with increase of veloso that, the value of/, the moment of inertia of the wheel, deduced on, city, the basis of the above calculations, cannot possibly be quite accurate., ;, , If we aim at accuracy, therefore, we must adopt a sensitive method for, determining the value of w, and the one method, which at once suggests itself,, is to make use of a tuning fork, as explained below, :, , A, , is arranged horizontally, (Fig., tuning fork, of a k no wit frequency, 47), with a slightly bent metallic style, attached to one of its prongs, such that,, when desired, it can be made to, lightly press against, or taken, off, a strip of smoked paper,, wrapped round 'the rim of the, wheel., ,, , Now, with the style, kept off the paper-strip, the, mass m is allowed to fall down,, thus setting the wheel in rotation, and just a second or so, before the mass is due to get, detached from the axle, the, Fig. 47., tuning fork is set into vibration, {by smartly drawing a bow across it), and the style pressed lightly on to the, strip, taking care to take it off soon after the detachment of the mass. A long, wavy curve is thus traced out by the style on the smoked strip. The mean wavelength A of this wave is then determined by dividing the tota distance occupied, by the wavy curve by the total number of waves constituting it., Since one wave is traced out by the style_j!uring one vibration of the, prong or the fork, we have linear distance covered by the wheel during on*, So that, distance covered by the wheel during *, vibration of the fork, x., - if*., vibrations of the fork
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Again, since n vibrations are made by the fork in one second, it foliowi, that distance covered by the wheel in 1 second, i.e., the linear velocity v, n\., , =, , But, city, , ;, , v, , ~, , where, , -Rco,, , we have, , so that,, , Thus, knowing, , R is the radius of the wheel and, ~ n\; whence, to = n\jR., , ,, , its, , angular velo-, , ,/fo, /t,, , X and R,, , we can, , easily calculate the value of, , w, , for the, , wheel., This value of, , co,, , substituted in, , then gives a, , relation (6) above,, , more accurate value of /, the moment of inertia of the flywheel about, , much, , its axis, , of, , rotation,, , Note. The student may, as an interesting exercise, show that expression, above can also be reduced to the same form as expression (7). This may be, easily done by remembering (/) that when the wheel makes one full turn, the, mass descends through a distance 2^r, tne circumference of the axle, and,, therefore, when the mass descends through a distance 5, the number of rotations, and further (11) that, made by the wheel is equal to S/2rcr so that, S/2-nr =, t = 25/v, 25/ro, where o> = 4-nn/t', (see page 81)., (4), , N, , ;, , Moment, , (//), , its, , (a), , metal, , on, , of inertia of a disc about an, , centre and perpendicular to, , its, , plane, , ~, , ;, , axis, , passing through, , Disc suspended by two parallel threads. The disc, with a, is supported on two cords, wound uniformly on the axle, , axle,, , either, , side,, , (Fig., , On, , 48),, , releasing, , the disc, it begins to fall down until the, whole cord is unwound from the axle, say, through a distance S., , P E. lost by the disc, the mass of the disc and, This energy will obviously be, the axle., gained by the disc in the form of kinetic, energy of rotation and translation., Then, , clearly,, , mg.S, where, , m, , is, , Fig- 48., , be the angular velocity acquired by it after falling through, 2, where 7 is, *S, its K.E. of rotation will clearly be |/o>, its moment of inertia about an axis passing through its centre and, and its kinetic, parallel to the axle, (i.e., perpendicular to its plane), 2, energy of translation will be \mv, If, , a}, , this distance, , ,, , ;, , ., , mg.S, , =, , i/o>, , 2, , + Jwi>, [v, , Or,, , whence,, , |/v, , 2, , =, / =, 2, , /r, , 2, ,, , o>, , 2, , (v, , being, , =, , v /r 2 ,, 2, , its final linear velocity),, , where, , r, , =, , radius of the disc., , mg.S-lmv*,, , (mgS, , Jmv, , 2, , 2, , ).2r /v, , 2, ., , =, , Now, average velocity, S/t, where t is the time taken, disc in falling through distance S ; and, therefore, velocity v, disc =z 2S]t, and .-. v 2, 452 // 2 So that,, , =, , ., , __, ., , ~~, , Or,, , / SB, , m, , VS, , _ mr, , by the, of the
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MOMENT, Di, , (b), , disc,, , c, , inclined rails, as, , acquire a, velocity, , the, , shown, , when, , o>,, , 83, , BOrATtON, , Here, the, t, rolling on inclined rails., inertia I, is allowed to roll down along, , v, , descends a vertical, a distance S along, , it, , it rolls, , Let it, 49., and an angular, , in Fig., , linear velocity, , distance h, as, , Off, , mounted on axle, , M and moment of, , of mass, , EtfEBOY, , ItfERTlA, , Off, , down, , /'<&*', , rails., , h, , clearly, loss, , Then,, ofP.E. of the disc, K.E. of translation gained by disc, K E. of rotation gained by disc., , =, , +, , Or,, , Mgh = \M v 2 f /oA, , So that,, , Mgh, , =, , ~ Mv *+, , -r, 2, , Fig. 49., , L, , ^, , r*, , r where r= radius of the, LAnd .-. o,=v 2 /r 2, , axle, , ., , /., , Or,, , 2, , /, , Or,, , =, , M(gh-\v*) whence, /, , -, , ^, , 9, , =, , ~, , ., , (*A-Jv), , (*gh-v*)., , Or, substituting the value of v =~ 2S/t, (see page 82), where, the time taken by the disc to cover the distance S, we have, , whence the value of, , /,, , th3, , moment, , /, , is, , of inertia of the disc can be easily, , calculated., , Note : For other methods for the determination of moment of, underjbrsional Pendulum, (Chapter VIII), , inertia, see, , 35., Angular Moment and Angular Impulse. In the case of, linear motion, the momentum of a body, as we know, is the product, of its mass and velocity. On the same analogy, we have, in the, case of rotational motion, the product of the moment of inertia and the, , momentum bfa rotating, angular momentum = /.<o,, , angular velocity as the angular, , Thus,, where I is the moment of inertia and, about the axis of Dotation., , o>,, , body., , the angular velocity, , of the body, , For, suppose we have a body, rotating about an axis with a, velocity w. Then, all its particles will have the same angular velocity, but their linear velocities will depend upon their respective diso>,, tances from the axis of rotation, being equal to the product of the, angular velocity and the distance from the axis. Thus, the linear, of that, velocity of a particle, distant r x from the axis, will be r^, distant r g from the axis will be r 2 cu and so on., ;, , m, , be the mass of each particle, we have,, if, of the particle, distant rt from the axis, equal to, m.^w and, therefore, the2 moment of its mttmentim about the axi, would be m.rl .cuxr=m.rl .oi. Similarly, the moment of momentum, a, of the particle, distant rg from the axis, would b3 w,r4 .cu and so on, , And, therefore,, , linear, , momentum
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PBQPERTltS.Oir MATMft, , 84, , moment of momentum of the whole rotating body, = wr1 2 a>+wr22ai+' ........., 2, i:mr*a>=/.a>,, [v ZVwr = /., the moment of inertia of the body about the axis of, , Therefore, the, , about the axis, , where /, , is, , rotation., , Thus, the angular momentum of a rotating body about its axis, of rotation is the sum of the moments of the linear momenta of its, it is also referred to as it*, particles about that axis. For this reason,, moment of momentum about the axis., , Now, we have, where, , C, , Ldt*>\dt, , =, , C., , Or,, , I.dw, , =, , C.dt,, , the torque or the couple acting on the body., Integrating this with respect to t, between the limits, is, , and, , t,, , we have, , =, , angular momentum, I, , co, , an expression which, , true,, , If, , C be, , is, , C.dt,, , 1, , JO, , constant,, , however, , we have, , C may, , vary with time., , =, , C.dt =~ C.t t, , 7.o>, , I, , which gives the angular momentum acquired by the body, If, , t, , be very small and, , C, , in, , time, , quite large, the expression, , I, , t,, , C.dt, , stands for the angular impulse given to the body, which again be*, comes equal to C.t, if C be constant., 36., Law of Conservation of Angular Momentum. Just as we, have the law of conservation of momentum for linear motion, we, have, for rotational motion also, the law of conservation of angular, momentum, which states that the angular momentum of a rotating, body about an axis remains constant, if no external torque be applied, to, , it., , For, suppose the angular velocity of a body is changed by d<# 9, to it for a very small interval of time dt., , by a torque C, applied, C.dt, Then, we have, where /, Hence,, , ia, , the, , I.d<*>,, , moment of inertia, , C, , =, , of the body about the given axis., , l.dw/dt, assuming, , /to remain constant., , =, , i.e.,, , If, however, /also change*, we have C, d(Ia>)ldt,, the torque is equal to the rate of change of angular momentum., , 0, i.e., if there be no external, Obviously, therefore, if C, torque applied to the body, dw/dt or d(Ia>)jdt is also equal to zero, or, the rate of change of angular momentum remains constant., It is obrious from tha above that in the case when / is not con*, stant, and no external torque ia applied to the body, the angular, velocity must change in the inverse ratio to /, in order to keep its, , angular momentum constant., This may be clearly seen by whirling round a stone tied to one, end of a string, whose other end is held in the hand. On stopping the, Application of any force to it, /.*., on removing the external torque,
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MOMENT Of IHBBTIA, , ENERGY OF EOTATION, , 85, , the string besrins to wind its If on the hand, with continuously increasing velocity, because as the distance of the stone from the hand, decreases, its moment of inertia about its axis of rotation also, decreases, resulting in a proportionate increase in its angular velocity., , Another good, air,, , illustration is, , provided by an acrobat executing a, , we know, he instinctively curh himself up in, thereby decreasing his moment of inertia and consequently, , somersault., , For, as, , incr .Basing his speed of rotation. But, before his feet touch the ground,, he slows it down by straightening himself up and increasing his mo-, , ment of ineitia., 37., Laws, in, , of Rotation. Corresponding to Newton's three laws, the case of linear motion, we have also three laws of rotational, , motion,, , viz.,, , Unless an external torque be applied to it, the rate of rotation, a, rigid, body, about a fixed axis in it, remains unaltered., of, An obvious example of this is the constant rotation of the Earth, about its axis. The force of attraction due to the Sun is certainly, there, but it acts at the centre of the earth and hence produces no, effect on its rotation., 2., The rate of change of rotation of a body, about a fixed axis, 1., , in it, is directly proportional to the external torque applied and takes, place in the direction of the torque., 3., If a torque be applied by one body upon another, an equal, and opposite torque is applied by the latter upon the former, about the, same axis of rotation. In other words, a change in the angular, momentum of one body brings about an equal and opposite change in, the angular momentum of the other body., , It is useful to remember that the moment of inertia (I), in, rotational motion, corresponds to mass, (m), and the angular velocity (w), to linear velocity (v), in the case of linear motion., The following Table gives the linear and rotational analogues at, , a glance, , :
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PROPERTIES OF MATTEB, , 86, , Kinetic Energy of Rotation __, , 38., , (a) Kinetic energy of a body about an axis through its centre of, mass. Suppose u e h ive a body of mass, rotating about an axis AB,, parsing through its centre of mass O, (Fig. 50)., It, obviously, possesses kinetic energy due to, its motion, this energy of the body is called its, energy of rotation, because it is due to its motion, of rotation., , M, , r, , ;, , Imagine the body to be divided up into a, of small particles, of masses m lt, , number, , large, , w2 w, ,, , distances, , at, , etc.,, , 3,, , r l9, , r2 ,, , linear velocity, , w =, , of, , a, , =, , r 2 co, , =, , ofm, of, , v, , rs, , ...., , etc.,, , Then, we have, , respectively from the axis AB., , rlW, , =, , v1, , =, , w3 =, , ;, , and, , v, , so on,, kinetic energy of, , .-., , mass MI, , =, , Or,, , W, , J, ., , 2, , v2, , w 3 = Jw 3 v3 2, , of mass, , 2, ;, , of the, , body, , mass m 1 =, *nd so on., , of, , =, = W[w/ +"V a4"V'3, i, , ......, , 1, , iw*mr*, , K.E. of the body, , Or,, , Now,, , if aj, , =, , 1,, , /, , velocity, is, , /A^, , 2, , co, , 2, , -, , 27mr 2, , 2, , ., , ,, , then, obviously, K.E. of the, , Or,, , rAw,, , =, , \<JMK\ [v, , MK*, [, pa>, AB., axis, about, of the body, , = |MK, , where /is the moment of inertia, , --=, , body, , =, , \, , MK*., , -, , /., , /., , #.., , 2, , rotating with unit, , moment of inertia of a body,, , equal to twice its kinetic, , angular, , energy of rotation., , K.E. of body which is not only rotating but whose centre of, A body which is rotating as well, also a linear velocity v., has, mass, of kinetic, as moving forwards with a velocity v, has both types, of its motion of rotation, because, rotation,, energy, viz., {/) energy of, about a perpendicular axis through its centre of mass, and (//) energy, therefore,, of translation ^bez&use of its linear motion. 2And, clearly,, we have K.E. of rotation of the body == $ /w, (b), , ,, , and, , its, .-., , because, , total, , w2, , =, , K.E. of, , -, , K.E. of the body, , v^/r, , 1, , where, , Or, total kinetic energy, 39., , translation, , =, , } Mv*., , K.E. ofrotation+K.E. of translation,, , the radius of the body., ?, s, body =| Jf v [(X /r*)+l]., , r is, , of the, , Acceleration of a body rolling, , Let a body of mass, , M, , roll freely, , down an, , nation a to the horizontal, (Fig. 51),, be rough enough, so that thero may be, , no vork done by, , frict on., ;, , inclined, , plane., , inclined plane, of incliThe plane is supposed to, , down an, , no, , slipping,, , and hence
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ENERGY OF ROTATION, , MOMENT OF INERTIA, , 87, , Then, if v be the velocity acquired by the body after traversing, a distance S along the plane, we have, vertical distance through which, it has descended = S.sin a., , And, therefore,, P.E., , lost, , by the body, , =, , sin a., , Mg.S., , This must, obviously, be equal to, the K.E. gained by the body., Now, K.E. of rotation of the body, i/w2, , =, , w, , Fig. 51., , angular velocity about a perpendicular axis through, centre of mass., , where, , is its, , And,, , its, , because, , its, , K E., , of translation, , =, , {Mv*^, , centre of mass has a linear velocity, , total, , K.E. gained by the body, , v., , = f 7o> +| Mv*., = I Mv\(K jr*) + l]., 2, , 2, , Since gain in, , K.E, , o r the, , body, , its, , is, , equal to the loss, , ISee, , 38., , in its P.E.,, , we, , have, , iMV<[(K, , 2, , Mv*[(K, , Or,, , 2, , lr, 2, , )+l], , 2, , lr ), , + }], , 2, , Or,, , v*(K +r")lr*, 2, , whence,, , v', , =, , 2, , 2(r, , 2, , =, , Mg.S sin, , =r, , 2Mg.sin a S., , =, , 2g.sina.S,, , /K +r, , 2, , a., , ).g sin a.S., , Comparing this with the kinematic relation,, body starting from rest, we have, acceleration of the body down the plane, i e., a = (r 2 /K 2 +r 2 ).g sina., , v, , 2, , =, , 2aS, for a, , t, , Or, the acceleration, of inclination a., , is, , proportional to, , r l j(K 2, , +r 2, , ), , for a, , given angle, , This show that, (/) the greater the value of K, as compared with r, the smaller the, acceleration of the body coming down the plane and, therefore, the, greater the time it takes in rolling down along it and vice versa., (//} the acceleration and, therefore, the time of descent, dent of the mass of the body., , is, , indepen-, , K =, , 2, 2r 2 /5, will roll down, Thus, a solid sphere, for which, 2, r 2 /2, and, similarly, a disc will, faster than a disc, for which, 2, 2, roll down faster than a hoop, for which, is equal to r ., , K =, , K, , K, , Since 2 for a hollow sphere about the diameter is greater than, that for a solid sphere of tho same mass and radius, they can be distinguished from each other by allowing them to roll down the plane,, Obviously, the solid sphere will roll down faster than the hollow one., The same test may be applied in the case of a hollow and a solid, cylinder etc., , Some, , particular cases, , :, , (0 Case of a Spherical, has, , Shell., , Let a be the angle of inclination of the, rolling and let the velocity be v when U, , down which* the spherical shell is, moved a distance S along the plane,, , plane,, , (Fig. 51) ,
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PROPERTIES OF MATTER, , 88, , the vertical distance covered by the shell, loss in P.E. of the shell == Mg.S sin, , Clearly,, , 5 s in, , ., , a., %, , K, , work, , is, , This loss in\?E must be equal to the gain in, E. of the shell, for no, done by friction, as there is no slipping., K.E of the shell - j- h^ + \Mv*, Now,, JMff'w'-f J Mv., 8, for a shell., [v A:*, r'^+iMv 2, |r, , =, , W.f, , =, , ., , -}MrV + iMv = JAfv 2 + JAfv a, t, 5Mv*, 2Mv*--3Afv, __, 2, , ., , ., , ^, Since, , m, , #fl/Vi, , we have, , 5Mv~/6, v, , Or,, , 2, , K.E. of shell, , Mg.S.sln, , = ..., , =, , a =*, , 2(|, , = 6# 5.J//I, , 5r*, , Or,, , a,, , fltf, , loss in its P.E.,, , j/w, , a., , a)^, , 2, Comparing it with the relation, v = 2a S, (when // =, of acceleration a of the shell, down the pla*ie = ^g sin, , 0),, a., , we find, , that the valua, , We, , Case of a SaMd.) Shere., know that the acceleration (a) of a body, 2, 2, inclined plane =(/ ~JK + r ) g sin a, where A' is the radius of gyration of, the body, and a, the angle oi inclination of the plane., (ii), , down an, , a, , ,*., , = (r/4'', = (r r r, z, , 7, , !, , 8, , 2, , 4-r, , z, , )g sin a, , ), , g, , =, , -, , sin, , -, , 9, , |r, , ,, , in this case*, , *..g sin a., , Thus, the acceleration of a solid sphere, to, , [v K*, , sin a., , down, , the inclined plane, , is, , equal, , ., , 40., Graphical Representation of Plane Vectors. We are, already familiar with the two types ot physical quantities, viz.,, scalar and (//) vector,, the former poswssmij only magnitude, but, (i), no direction and the latter, possessing both magnitude and direction,, (see foot note on pai^e 55). Theso latter can, as we know, be, represented by a straight line, drawn to a chosen scale, whose length, and direction respectively represent the magnitude and direction of, the quantity., %, , ;, , other quantity, either derived from a vector, or obtained, vector with a scalar quantity, is also veetorial in, nature., Thus, for example, the acceleration of a body, depending, upon the velocity of the body, (a vector quantity), is also a vector, , Any, , by combining a, , quantity., , The vector quantities referred to above are, strictly speaking,, and must be clearly distinguished from what are called, plane vectors, a term applied, in rotational dynamics, to such, quantities as angular velocity, angular momentum and torque etc, linear vectors,, , ,, , which are, , all directional, , in, , the sanso that they are confined to, , one, , plane., , Such a plane or two dimensional vector, , is, , also represented, , by a, , straight line, drawn normal to its plane of rotation, or parallel to its, axis of rotation, its clockwise or anticlockwise rotation being indicated, according to an agreed and established convention, by the, straight line being directed towards, or away from, the observer, respectively., , Further, corresponding to the parallelogram law for the compowe have, here, a modified form of it to determine the resultant of two plane vectors, viz., that, sition of linear vectors,, , two, , "if there be two plane vectors acting simultaneonsly on a body in, different planes, such that they can be represented in magnitude and
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MOMENT OF INERTIA, , ENERGY OF ROTATION, , direction* by the two adjacent sides of a parallelogram, drawn perpendicular to those planes, their resultant is represented completely by the, , diagonal of the parallelogram, passing through their point of intersecthis diagonal representing a plane vector in a third plane, pertion,, pendicular to itself.", , OA, , and OB,, , (Fig. 52), represent two couples, in two, simultaneously on a body, whero OA and OB, are drawn perpendicular to those, planes, their resultant is given, completely by the diagonal OC of, the, parallelogram, which represents a couple in a third plane, per-, , Thus,, , if, , different planes, acting, , pendicular to, , itself., , And, obviously, what, , is, , true, , about the composition of couples, is equally true for the composition, of any other plane or two-dimen-, , Fig. 52., , sional vector quantities., 41., , Precession., , Just as in the case of linear motion,, , we may, , have a constant acceleration acting on a body, without changing, , its, , constant speed, (e g., the centripetal acceleon a, ration acting, body, moving with a, uniform speed in its, t, , circular orbit), so also,, in rotational motion,, , r----r ----- ---;, , -~~Jt, , we may have a constant, acceleration, anS^ar, acting on a body, having a constant angular, , speed. This is rendered, possible bv the plane of, rotation changing direction at a given rcte ,, without, in any way,, Fig. 53., affecting the rate of, rotation of the body about its axis of rotation, or axis of spin, as it is, also sometimes referred to., This change in the plane of rotation is, called 'precession', and is caused by a couple or torque, called the, , precessional torque, acting: in a plane, perpendicular to the immediate, In other, or instantaneous plane of rotation (or spin) of the body., to, the, at, axis, the, instant,, perpendicular, words,, torque is,, of, any given, the rotation-axis of the body, as will be clear from the following, :, , DD, (Fig. 53), be the edge of a disc, with its plane revolvits geometric axis, with an angular velocity w., Then, if its, moment of inertia about this axis be /, its angular momentum will, Let this be represented by the straight line OA,, clearly be lw., drawn perpendicular to the plane of rotation of the disc., , Let, ing about, , *{n accordance with the convention, stated abpv$.
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PROPERTIES OP MATTER, , fO, , Now, let the axle of the disc also rotate, i.e., let there be a precessional motion, about an axis, perpendicular to the plane of the, so that, after a small interval of, paper at a (prece:?sional) rate, ;, , the disc takes up the position D'D', making an angle <f>.dt, Its angular momentum, again equal to, its original position, is now represented by the straight line OA\, , time, with, /co,, , dt,, , The change in the angular momentum of the disc is thus reprearc = radius x angle., sented vectorially by, A A', /to <j>.dt., [, This change has, clearly, been brought about in time dt, and, , =, , therefore,, rate of change of, , =, , the disc, , I w.<f>.dtjdt, , =, , of change of momentum of a rotating, equal to the torque applied to it, we have, , And, since the, , is, , momentum of, , r ite, , TI, , -, , ICO, , 7o>.0., , body, , (f),, , where 7^ is the torque applied to the disc., --So that, the rate of precession,, TJfw., <f>, , since the change in the angular momentum of the disc is, it, is clearly parallel to its plane of rotation, or perpendiAA',, along, cular t its axis of rotation, and A A' is thus the axis of the torque, , Now,, (, , >, , applied., , In other words, the axis of the torque, , Thus, we see that, , and, , if the axis, , lies, , along, , OX., , OY, , of rotation of a body be along, along OX, the body 'precesses', , the axis of the applied torque, , about the, third, mutually, perpendicular axis OZ. This, will be readily understood, , from Fig. 5*, which shows, the disc, , 7/2, , perspective., , Here,, , of rotation, , OY, , is, , the axis, , arid,, , therefore,, is the plane of rotation, is the axis of the, torque or, couple applied,, is the, and, therefore,, plane of the torque and, since, , XOZ, ;, , OX, , YOZ, , the axle of the disc turns toOX, i.e., about the, axis OZ, the plane of pre-, , wards, , In other, cession is XOY., words, the axis of rotation, (OY) turns in tins plane,, cular, , OX,, , which is, clearly, perpenditwo planes, its direction of rotation (towards, depending upon the direction of rotation of the disc and, , to the, here),, , first, , that of the torque or the couple applied., 42, The Gyrostat. A gyrostat is just a disc or a flywheel, having, a large moment of inertia, rotating at a high speed about an axle,, passing through its centre of mass, and mounted, as shown in Fig. 54,, so that the wheel and the axle are both free to turn, as a whole, o-bout, , any, , axis, perpendicular to the axle itself,
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MOMENT OF INERTIA, , ENERGY OF ROTATION, , 91, , As explained above in 41, if a torque or couple be applied to, the wheel, with its axis perpendicular to the axis of rotation of the, wheel, the wheel 'precesses' about the third mutually perpendicular, T /Ia), where Tl is the, axis, at a processional rata, given by, moment, of, inertia, of the wheel about, the, or, /,, couple applied,, torque, </>, , it, , rotation-axis and, , o>,, , its, , }, , angular velocity about this axis., , Clearly, therefore, for a given torque (Tj) applied to the wheel,, the precessional rate is inversely proportional (?) to the moment of inerits axle, and (//) to the angular velocity of the, tia of the wheel about, wheel so that, the larger the moment of inertia of the wheel about, its axle, and the higher its angular velocity, the smaller the rate of, precession of the axle, and vise versa., ;, , The following simple experiment, above results, , will, , beautifully illustrate the, , :, , fairly larse awl heavy disc, (Fig. 55), free to rotate, axis YY' passing through its centre, and fitted inside two, sockets at the ends of the horizontal diameter of a bigger ring, suspended by nvans of a string vertically above its centre of gravity., , Take a, , about, , its, , (/), , with the disc quite stationary, a weight Mg be, the torque due to it will tilt the ring, the end Y', down and the, , Now,, , suspend fd at, , if, , Y',, , moving, end Y moving up,, , /, , e, , ,, , the ring will turn about, , OX., But, the, , instead of, the weight,, bo, simply, , if, , ,, , suspending, ring, , pushed horizontally, y, from in front, behind,, , it, , at, , or, , turn, , will, , about OZ., Let the, , disc, rotation about its axle, in, the, direction, shown,, (//'), , be, , now, , set, , into, , y, , with the, weight Mg, kept properly supported,, so as to exert no down-, , ward, , pull at, , y., , Fig, , It will, , 55., , be found that the ring, remains quite steady and a twist, given to the string either \\ ay,, hardly produces any tendency in it to rotate about OZ, as it certainly, would, if the disc were stationary., (///) \With the disc in motion let the weight Mg be released, so, as to exert a downward pull at Y, thus producing a torque about, OX. It will be found that the ring at once rotates about OZ, with, the end Y' slightly tilted downwards. On pushing the ring horizontally at y, as before, the axle, instead of turning more rapidly about, OZ, as might be expected, simply gets tilted a little, raising the, weight Mg slightly upwards, clearly showing thereby that the horizontal, r, , 9
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PROPERTIES OF MATTER, , 92, , rotation of the axle YY' opposes the torque due to Mg, which, therefore,, So long, however, as the downdescends comparatively slowly now., ward descent of the weight continues, just so long does the rate of, rotation of the axle about OZ also continue to increase, thereby increasingly opposing the torque duo to the weight, until a stage is, reached where the two exactly balance each other. After this, the, downward descent of the weight naturally ceases, and the ring conat a constant rate, with the axle YY slighttinues to turn about, , OZ, , f, , ly tilted., , found that the greater the moment of inertia of, rotation-axis or the axle, and the higher its angular, velocity about it, the smaller the rate of rotation of the axle about, OZ, i.e.. the smaller the processional motion about it., (iv), , It will be, , the disc about, , its, , (v) Since the torque or a couple is needed to produce this processional motion of the rotation-axis of (he disc, it is clear that a rotatThis, ing body offers resistance to a processional motion of its axis., , resistance to a processional motion is called gyrostat ic resistance, and, is equal and opposite to the prccessional torque., 43., Gyroscope. In a majority of cases, a body, subject to, preccssional motion, is supported at a point, away from the vertical, line through its centre of gravity., , A, , gravitational torque or couple thus acts upon the body, which,, simply tends to rotate it into a position of, a lower potential energy, ie., simply tends to lower its centre of, But, if the body be rotating obout some axis, this gravitagravity., tional torque supplies the necessary processional torque equal in value, in its stationary condition,, , to its own, provided there is no other couple acting on the body. The, rate of precession <, maintained by this gravitational torque jP 2 is, ,, , given by the relation,, , where /and o> stand, as usual, for the moment of inertia of the body, and its angular velocity about its axis of rotation., Such a body is called a gyroscope, its motion being appropriately, ter, , m ed, , ', , 'gyroscop ic, Thus, consider a heavy disc D, revolving with a high angular, velocity o> about its physical axis POQ, itself resting on a vertical, pivot at P, (Fig. 56)., ., , Then clearly,, ft, , JL, //Q\\, , weight Mg, acting vertically downwards at its, c g'i O, exerts a gravitational torque T z on it,, - /]., given by T = Mg.OP, Mg.l. [Putting OP, 9 So that, if, be the rate of precession of the, disc maintained by it, we have, , its, , =, , y, |, ;, , (/>, , ~, , -"", , "LaT, , MK, , K, , 2, is the radius of, where, putting 7, gyration of the disc about the axis POQ., Hence, if t be the time-period of its preif it takes time t to, cessional motion, i e., ,, , 9, , one, we have, , complete, Fjg. 56., , tion,, , its, , full cycle, , of processional mo-
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MOMENT OF INERTIA, ', , -, , ", , ENERGY OF ROTATION, , * 2ir, , 93, , ', , gl', gltK'.w, This precession, once started, can be maintained, at this very, A higher rate of precession, rate, by the gravitational torque alone., than this will make axis POQ rise and a lower rate will make it fall., This rise and fall of the axis of rotation, or its oscillation up and, down about its position of dynamic equilibrium, accompanied by a, , "f, , correspondingly changing preccssional rate, is termed nutation., Further, there is a centrifugal force acting on the disc along, POQ and an equal centripetal force in the opposite direction QOP, their net effect, if they act along the same line, being to increase the, fricticnal resistance at the pivot P. If, however, their lines of action, be different, we have yet another couple T 3 formed by them, aptly, known as the centrifugal torque., In order to prevent the disc, or a precessing body, in general,, from moving outwards from the centre of precession, it is necessary, that the centrifugal torque on it must be balanced by an equal and, t, , ,, , opposite centripetal torque, this balancing effect being supplied by, part of the gravitational torque, the remaining pan of it r producing, Thus, if T 3 be the centripetal torque and l\ and T a, precession., the gyrostatic and gravitational torques, we have, T3, T1, " (11/, *2, ", where the different torques are given their proper sings, (i.e., anticlockwise, positive and clockwise, negative), all acting in the same, direction in the case shown., ,, , T, , \, , A general rule to determine the sense of the torque, producing, precession in a given direction, is given by Lanchester's rule, which, may be stated as follows, If the gyrostat be viewed from a point in its own plane, with the, line of sight perpendicular to the axis of the given precession* it is seen, to describe an ellipse, the sense of whose path gives the direction of the, precessional torque, with the line of sight as its axis., :, , The Gyrostatic Pendulum. A gyrostatic pendulum is a, 44., small and heavy disc or gyrostat (Z)), revolving with uniform angular, velocity (co) about a light rigid rod,, (SD) as axis and precessing about the, vertical (SO) at a uniform rate (<^) as, , shown, , in Pig. 57., , Obviously, there are the three, following torques acting on the pen-, , dulum., , A, , (/), gyrostatic, to the gyrostat or disc, , torque, Tj,, , duo, , D possessing two, , simultaneous rotatory motions., Since the plane of rotation of D, rod, is always perpendicular to the, , SD, , t, , its, , rate of precession, , <, , is, , the same, , as the angular velocity of SD 9 i.e^, SD, For, in time dt,, equal to vji, traoas out an are v.dt, where v is its, velocity in the horizontal circle, , /', (, *, , pt|, 57.
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OF, of radius r, which is described by it in its processional motion about, SO or, the angle described by it in time dt is equal to v.rfr//, and, hence the angle described by it in unit time is clearly equal to, ;, , (v.dtfi-dt, , == v//., , Thus,, , we know that, , But,, , whence,, , <, , v//., , =, , T, , <, , >, , .., , =, , 1/1, , =, , So that T"1, -^Iw, la), , =, , -, , ,, , ----*, , ~7, I, , ', , ---, , the ra^to of gyration of the gyrostat about the axis SD., let, us apply, of this torque,, Lanchester's rule, i.e., let us look at a point B on the edge of the disc, and #Z), when B, to both, along LB, where LB is perpendicular, that, clearly appears to move in the anticlockwise direction, indicating, anticlockwise., T, is, the, of, the direction, torque l, If t be the periodic time of precession of the disc or the gyrostat,, So that, substituting this value, we have v.t, 2irr//., 27rr, or v, , # is, , where, , To determine the direction, , OD, , =, , =, , of, , T, , the relation for, , v in, , T =, , MK*a>.2vrjt.L, , l, , =, , since r/l, , And,, , MI, , 0,, , we have, , T!, the, , -f-ve, , :-, , A, , its, , torque, , gravitational, , 2, , +Jf, , sign indicating that, , (ii), , we have, , above,, , 1, , .o>., , is, , anticlockwise., , T, , to, , the weight, , ?,, , clue, , gyrostat acting vertically, supposed to be concentrated)., , between, , =, , Mg, , of the rod, , SD, , of the, , mass, , is, , = MgxBO =, , moment, , of this gravitational torque, where BO is the perpendicular distance, Mg.l, and an equal and opposite reaction at S /, the length, and 0. the angle that it makes with the vertical., , Clearly, the, , MgJSD, , Mg, , at D, (where its whole, , downwards, , sin, , sin 0.(27r//),, , direction, , sin 0,, , ;, , T 2 = Mg. I sin 0,, So that,, ve sign indicating that its direction is clockwise., the, due to the centrifugal force Jfv 2 /r,, (Hi) A centripetal torque T 3, outwards, the, on, along OD., gyrostat,, acting, ,, , And, the moment of, , this, , torque, , T3, , is, , obviously equal to, , a, the perpendicular distance between Af v /r and an equal, we, cos, have, since, SO=l, S., at, reaction, Q,, and opposite, Or,, , where, , SO, , is, , Mv, = --, , 3, , *, ~, , ., , Icosfi, cos v, i, , ,, , ve sign again indicating that the direction of the torque, the, clockwise., Or,, , Tt, , substituting the value of v=27rr// [see, , ~*-g)\, , I, , cos ,, , *f, , ^=sin, *, , i, , I, , (?1 )'., or, , r=/, t, , (/), , above],, , co, 9, , sin 0., , =-, , we have, , is
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ENERGY OF ROTATION, , MOMENT OF INERTIA, Hence, , 3, , Now, from, , = -M, , 2, , .jm, , 0. cos, , we have, , relation (1), (page 93),, , T2 +T 3 =T, , Or,, , 1, , ., , So that, substituting thoir values, we have, , - Mgl sin e +M1*, , sin e, , ., , ., , -/+/, , Or,, , 2, , ^, , cos 6 (, , cos e, , =MK w.sin, 2, , ), , (-y-, , ^Wo, (-^L)., , (, , [Dividing by Af s//i, , throughout.*, , Now, putting (2irjt)p, we have, , -glip 2, Or,, , /?, , which, , is, , 2 2, /, , coy, , cos0^pK 2 w., ~pK2 w - /= 0,, , a quadratic equation, , T, ,, Therefore,, , !, , 2, , in p., , Pn, , ,, , which, obviously, gives two values of J9., To decide between the two values, we put w=0, so that there, and the whole arrangement, is no rotation of the disc about SD, reduces to a conical pendulum, with, t, , _^ l_^, , P ~~, , 4, , ~, , 2/ 2, , c, , ., , ^, , ^l-~-a.A /* gi4, , V, , coils, , ^/, , 7g, , COs, , I--A.\ I, , cos*, , ^", , ^/, , cw, , /, , 1, fl*, , "^, 2?r, , But,, , since, , 2?r//, , naust, , necessarily be positive for, , a conical, , pendulum, the negative value^becomes inadmissible and, we, therefore,, have, , It follows, therefore, that, in the expression for, So that,, , p, , above, only, , the positive value must be taken., , p ~~~~, whence, , ', , Case of a Rolling Disc or Hoop. A simple and a familiar, 45., example of gyroscopic motion is that of a thin circular disc or a hoop,, If its, set- rolling over a plane horizontal surface., it continues to roll along a straight path in, , enough,, , velocity be large, a stable vertical, , But, as its velocity decreases, due to friction between it, position., and the horizontal surface, its plane inclines progressively to one side, and its path becomes curved towards the 'side of few', the curvafcure, , of the path constantly increasing with the decrease in, , its, , velocity
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PROPERTIES 0? MATTER, it follows a spiral path, until, finally,, Let us study this motion of the disc in some, , so that, , on the surface., , it f&llajtat, , detail., , Let, , D, , be the, , circular disc,, 58), of, , mass, , (Fig., , M and, , radius, r,, rolling, along a horizontal, surface with v, as, the linear velocity, of its centre 0, and, with its plane AB, , angle, , an, , at, to, , inclined, Fig. 58., , 6, , the, , Then, the three torques acting on it are, rotation about, (i) Gyrostatic torque T ls due to its simultaneous, its point of contact and about E such that 7^=7.60. $, where 7 is its, and permoment of inertia about the axis OE through its centre, vertical., , :, , t, , plane co, its angular velocity about the same axis, of precession., Now, I=MK*, where K is its radius of gyration about OE,, and, [See 44 (/)., o>=v/r, <=v//., , pendicular to, , and, , its, , ;, , its rate, , <f>,, , So that, , where r//=tan, , 0,, , V, , V, , T=--if*: 2, , v>, , =z-MK*, , T, , ~-MK*, , V, -?-, , tan, , and the ~ve sign of Tx shows that the torque, , is, , clockwise., (ii), , cally, , Gravitational torque, , T, , due to, , 2,, , weight Mg, acting verti-, , its, , downwards at O, such that, "v, , in the right-angled, , &OCB, , _,_, , CB, , CB, , also acting in the clockwise direction., (Hi), , Ta, , Centrifugal torque, , ,, , due to, , its, , rotation about, , ,, , such, , that, , v, .., , OC=,, , ., , COS B, , I, , where, , 93), , r cos $,, , the rt.-angled, , cos Qa/l., and in the rt, -angled, , AOCB, , EC=a., , cos, , Mv*, , Or,, , the, , in, , AOCE, , Mv*, , ., , ~=, , Jl/v, , 0=OC/r., , 2, ., , ve sign again indicating the clockwise direction of the torque., Substituting these values of Tlf T, and T s in relation (1), (page, , we have,, , for equilibrium,, 8, , Jtf.r sin, , Or,, , Mg.r, , v, $(Mv*.tan g)*=MK*. -r, , sin, , o+Mv*.tan, *F', , tan 0., , JfA^.-r-. tan 9, , v*, , Or,, , ., , Jlfjf.r
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ENERGY OF ROTATION, , MOMENT OF INERTIA, +K*., , tan g, , Or,, , v*, , Or,, , v*tang(l+=gr.sing., , 9, , whence the angle of, clearly given by, , 'lean, , p/", , its, , velocity, , by, , v2, , 0= gr., , tan, , sin g, , v, , Or,, , vl+*r.o0., , Or,, , and, , ., , 97, , c, , Or,, , of the disc for a given velocity v of, , it is, , c, , =i rv/-, , for, , a, , the, , in, , equilibrium, , leaning, , position., , Now,, , for, , the, , critical, , velocity v c, , ,, , i.e.,, , the, , minimum, , which the disc can move along a straight path, with, clearly,, , And,, , .-., , 0=0,, , so that cos, , in this case,, , =, , its, , velocity at, , plane vertical,, , 1., , v^=_, , Or,, , v,, , =, , Y~, , ja/TT-, , For a value of v less than v c the upright position would obfor, on the slightest displacement, it will be, viously be unstable, tilted over by the force of gravity until, attains the value given, above, (by expression A), corresponding to the leaning position., ,, , ;, , Now,, , for a (uniform) disc,, , K =r, z, , for a disc,, , And,, , for a hoop,, , 2, , /2., , Hence,, , ?/2r-, , V, , =, , vc, , R, , Let us now calculate the radius of curvature, the disc on the horizontal surface. It is clearly equal to, And, in the right-angled triangle EOB, we have, sin g, , = OBjEB =, , R=, Or,, , sin g, , == ^7-, , substituting the value of cos, , n, , y, , r/R, , ;, , so that, r, , ^-~~. LTV, , lcos 2 o, , of the path of, , EBR., , = R sin, , sin g, , g., , = \/icos, g., v, 2, , deduced above, we have, V, , ^_, , which, with the substitution of the appropriate value of AT, gives the, radius of curvature of the path of the disc or the hoop along the, horizontal surface., 46. Gyrostatic and Gyroscopic Applications. The tendency of a rapidly roto preserve its axis of rotation, disc or wheel, (and, in fact, any rigid body),, endows a gyrostat with a stability of direction, which is made use of in a, tating
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PROPERTIES OF MATTER, number of ways, , for the steadying of motions., Among the more important, and familiar applications of this may be mentioned the following, :, , (0 The Gyrostatic or the Gyro-Compass. It is a special type of compass, used in aeroplanes and ships, and, more particularly, in submarines., In, essentials, it consists of a disc or a flywheel, of a large moment of inertia, (/ <?., a, gyrostat), suspended in fnctionlcss gimbals inside a supporting frame, which is, kept rotating at a high speed by means of an electric motor about a horizontal, ax's, lying in t^e geographic meridian (i.e., in the vertical plane passing through, the geographic north and south of the earth), Its directional stability and the, conservation of its angular momentum make its axis always lie in the direction, of the metndian, i.e., along the geographic north and south. And. since the, arrangement is such that the disc or the flywheel has three degrees of freedom,, irrespective of any porsition of the supporting frame, a movement of the latter, produces no deflecting toque or couple on it, and this particular direction of its, axis continues to be maiatain^d in space all the time, despite any changes in the, direction of the ship or the submarine, or any tossings or pitchings of it. It is,, therefore, preferred to the ordinary magnetic compass and is more dependable, than the latter, in view of the additional advantage of its remaining altogether, unaffected by any type of magnetic diturbanccs., , The Pendulum Gyro-Compass., , The above, , arrangement, with a small, , rnass, suitably suspended below the rotating disc or flywheel, constitutes what, , the pendulum Gyro-compass,, small mass supplying the necessary, restoring torque to bring its axis back:, to its original direction, should it get, displaced due to some disturbance. In, the absence of this simple but ingenious, device, the instrument would lack its, restorative action, due to the inherent, G stability of a gyrostat in any position., The essential features of the, construction, of the Pendulum Gyro59,, compass will be clear from Fig, where the rotating disc or gyrostat, has its axle PQ mounted in a horizontal, ring R, free to rotate about the axis EF, inside a vertical ring C which, in its, turn, rotates freely about the axis AB, within a frame work M, carried on, forms one pair), to ensure the fullest freedom, is, , called, , the, , D, , **, Fig. 59., horizontal gimbals, (of which, , GG, , of movement., The horizontal ring R has a stirrup S, fixed rigidly to it, which is loaded, with a weight W, immediately below O, the centre of the disc or the gyrostat., It can be shown that this arrangement would be stable, at any given, place, only along true north and south, i.e., when the end P points truly north,, any accidental displacement of it calling into play a directive force, restoring it, back to its original direction., (//'), Rifling of barrels of Guns and Rifles. This is another well-known, application of the directional stability of a rapidly revolving body. For, it is, found that if a shot or a bullet be given a rapid *spin\ about an axis along its, direction of motion, its uniformity of flight is greatly improved by making it, less responsive to small deflective forces during its passage through air., This, is achieved by 'rifling* the barrel, i.e., by cutting spiral grooves inside it so that, the shot or bullet is first forced to move along these, before it emerges out into, the air, thus acquiring the necessary 'spin* to ensure an almost uniform linear, motion., (///), Riding of Bicycle and Rolling of Hoops or Discs. These are both, cases of what is called 'statical instability for, neither of the two, at rest, can, possibly remain in equilibrium in the position in which it does, when it is in, motion. Here, again, it is the gyroscopic action that does the trick, by appro*, priately deflecting their axes of rotation and thereby changing their planes of, rotation, to counterbalance the disturbing effect due to gravity., ;
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MOMENT OF INERTIA, , ENERGY OF ROTATION, , 99, , Thus, when a person rides a bicycle, without holding its handle, he has, simply to tilt to one side in order to turn to that side ; for, by so doing, ho, produces a couple about the horizontal direction of motion of the front wheel, of his bicycle, which, here, acts as a rotating gyrostat. This couple, then, turns, the axle of the wheel about the vertical, and hence its plaie of rotation, into the, desired direction., , The same is true about a hoop or a disc, projected, with its plane vertical,, to roll over a horizontal surface, which we have discussed fully in $45, above. As, explained there, so long as its linear or translational velocity remains above a, certain critical value, it continues to advance along a straight path, but as soon, as its velocity falls below this critical value, its plane gets inclined to the vertical,, or it begins to 'lean' from the veitical and its path gets curved towards, its 'side of lean'., And, then, as its velocity goes on progressively decreasing, due to friction, the curvature of its path goes on increasing correspondingly, so that it follows a more or less spiral path until, finally, it falls flat on the, surface., Precession of the Equinoxes. The earth, as we know, is not an, but bulges out slightly at the equator, (or has the shape of a, 'flattened ellipsoid of revolution"), Further, the Sun and the Moon do not usually, He in its equatorial plane but rather in the plane of the ecliptic, which is inclined, at an angle of 23 5 to the former, with the result that the gravitational attraction due to the Sun and the Moon, on this equatorial bulge gives rise to a, torque, bringing about the precession of the axis of the earth, which, acting, as a gigantic top*, describes a comrr, relative to the fixed stars, e.g., the pole star,, similar in manner to the cone described by the axis of a precessing top, due to its, the phenomenon being spoken of as the 'precession of the equinoxes'., M>e/>/tf,, Tins couple on the earth due to the attractive force of the Sun and the Moon, earth's axis to desis, however, very small, so that it takes 25,800 years for the, cribe the complete cone, at which rate of rotation, the star Vega will be the pole, star in about 12,000 years hence., (iv), , exact sphere,, , It is interesting to observe that atoms too have the mechanical properof tops, and, at least in one special case, their gyrostatij moment has been, demonstrated experimentally by Einstein and De Haas., , ties, , Other Recent Applications. The modern aircraft appliances, like, (v), the automatic pilot, the artificial horizon and the turn and bank indicators etc., all, depend for their ction on gytoslatic principles., The function of all these instruments is to record the effects of a change, of orientation between a relatively fixed plane, provided by a fast rotating gyrostat, serving as the reference or the datum plane, and some other movable plane, in the machine, and this they do with a degree of precision which makes their, indication far more safe to rely upon than mere human judgement, howsoever, trained or mature., , SOLVED EXAMPLES, A flywheel, , of mass 500 k. gins, and 2 metres diameter, makes 500 revolutions per minute. Assuming the mass to be concentrated at the rim, calculate, the angular velocity, the energy and the moment of inertia of the flywheel., 1., , (/), , And, , No. of revolutions made by the flywheel, Angle described in one rotation, .'. angle described by the wheel per minute, ,,, , the angular velocity, , ,, , =, , second, , 2 Tr.500, , ,,, , 2 ir.500/60., , = 50ir/3, =, , radians., , of the flywheel, 50:r/3 radiansfsec*, 2, Moment of inertia / =*, , Or,, , (o>), , MK, , (ii), , Here,, , And, , ,,, , ,,, , = 500 per minute., =2n, radians., , mass, , ., , M = 500 x 1000 gms., , K, the radius of gyration =, , 1, , metre or 100 cms., , *A *top\ in Physics, is the name given to a rotating body, either completely free to move, or fixed at the most at just one point with absolute freedom, of rotation, and it must not, therefore, be confused with the toy that goes by, that name., fin the clockwise direction, as seen from the north*
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PROPERTIES OF MATTER, , 100, , because the mass is concentrated at a distance, equal to its radius from the axil, of rotation, which passes through its centre., moment of inertia of the flywheel == 500 x 1000 x(100) 2, = 500 x 1000 x 10000 5 x 10 gm-cm*., The energy possessed by the flywheel is due to its rotation, i.e., it, (iil), 2, possesses only rotational energy, which is equal to i/w, ., , ., , 9, log 10, log 1250, 2 log n, , =, =, =, , Or, , 9-0000, , >, , ener*y of the flywheel, ', , -., , 9944-, , y, , 13-7903, , =0 9542, , log 9, , -, , X, , X 1 250^, , 1, , 9, , ', , 6857x10*, 68 57, , 5, , _, , T, , 12 8361, , Antilog, , =, , 3-0969, , 6857xlO u ^r^., , I, , xlO 11, , ', , A, , 2., flywheel weighs 10 tons, and the whole of the weight may be considered as concentrated at a distance 3 ft. from the axis. What is the amount of, energy stored in the flywheel when rotating at a speed of 100 revolutions per, , minute, , ?, , log 5, log 2240, log 100, 2 log *, , (Punjab, 1934), , ==0 6990, , =, , Here,, , 3*3502, , log 32, Af;ir>, , ~, , and, , 7 0436, , 2, , 2, , 1-5051, , ^"^8^", 5, 5385, , JRT.E., , x, , 10-, , co, , (Given), , 2, , ,, , of the flywheel, , = ^x, "5 x, , ,, , 34 55, , 10 x 2240 Ibs., 100 x '2r> "", radian f/min., "', , 100x2rr/60 = 107T/3 radians/sec., K = 3ft., Since K.E. of rotation of a body, = * x ^ = 4 M/C, we have, , =09944, , =, , to, , i, , =2*0000, :, , Antilog, , M == 10 tons =, , ', , I, , ]0x 2240x 3 x (lOTr/3) 2, 224 x 9 x 100 x 7i 2 /9 fi.poijgals., 2, , ., , 2, 5x2240x9xlOOxnr, ", 9x32, , -, , Or, kinetic energy stored up in the flywheel, , 34*55, , x, , 10* ft. -Ibs., , A, , flywheel of mass 100 k. gins, and radius of gyration 20 cms. is, mounted on a light horizontal axle of radius 2 cms., and is free to rotate on bearings, whose friction may be neglected. A light string wound, on the axle carries at its, The system is released from rest with the 5 k. gms., free end a mass of 5 k. gins., mass hanging freely. Prove that the acceleration of this mass is g/2001 cm. /sec 2, 3., , ., , If the string slips off the axle after the weisht has descended 2 metres, prove, that a couple of moment 31*8 k. gms. wt.-cm. (approximately) must be applied in, order to bring the flywheel to rest in 5 revolutions, (Cambridge H. S. Certificate), , (0, , and, , The mass of the, its, , So, , that, its, , flywheel, , (M), , radius of gyration (K), , moment of, , inertia (I), , = 100 k. gms. =100 x 1000 gms., = 20 cms., = MK = 100xlOOOx20 2, = 100x1000x400 4 X 10 gm.-cm*., 2, , ., , 7, , Let angular acceleration of the flywheel be, d&l dt., Then, linear acceleration of the mass of 5 k. gm.^r.d^Idt t uhere, radius of the axle., acceleration of the mass, i.e., a, Or,, r.da>ldt,, , whence,, , d^jdt, , a\r, , =, , r is, , the, , 0/2., , on the flywheel =* Ld^fdt., = 4xl0 7 xa/2w.H>/.-cw. 2xl0 7 xa^m. wt.-em., This must, obviously, be equal to the couple applied to the wheel by the, rotational couple acting, , tension in the string., If Tbe the tension in the string, the couple due to it =7>., If the mass of 5 /r. gms. had no acceleration, the tension in the string, would be equal to its weight == 5 x 1000 x^ dynes., But, since it has an acceleration a, we have, 1, , ma, , ', , mg, , T., , Or, , T, , m(g, , ).
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ENEBGY, , MOMENT ot IHBBTU, , couple applied to the wheel, , ROTATION, , T.r, , 101, , m(ga)r., , = 5 x 1000(#-- a) x2 dynes-cm., = Tr., 5xlOOOte-a)x2 - 10000^-100000., = 10000^. Or, a(2x!0 + 10000) = #x!0, , Now,, , /.rfco/cfr, , 2xl0 7 a, , a, , whence,, , -, , 4, , 7, , 2xlO T a-HOOOOa, , Or f, , off, , ,, , -, , j?x 10*, , =, , i?x 10*, , (Tx fQ7, =*, , Thus, the acceleration of the mass, , " */ 2001, , T' Or, , #/2001 cw./sec*., , the weight has descended 2 metres, it has lost some P.E. Thii, must be equal to the gain in K.E. of the wheel and the weight ; so that,, , When, , (//), , =, , K.E. of the wheel +-K.E. of the weight, , P.E. lost by the weight., , =, , Now P.E. lost, , work done by it in falling through 2 metres, by the weight, distance = mgh, (where h = 2 metres = 200 cms ) = 5 x 1000 x 981 x 200 ergs., = 1000000x981=981 x 10 6 ergs., .*. K.E. of the wheel and the weight, This must, therefore, be the work that the couple applied to the wheel, in order to stop it. If C be the couple required for the purpose, we, , must do, have, , work done by the couple = CO, (where o s the angle of rotation)., Since the wheel comes to rest after 5 revolutions, it describes an angle, , = 2^x5 radians., , work done by the couple, , -, , Or C, , *, , 2 log, log, , 10, , ,., , ir, , -'", , 981x10, , =, -, , * ", , 981, , 2-0000, 0-4972,, , ADtilog^ ^1-5028, , ,, , j, , 10 *, , -, , 1000x7r, , =, 31, , =, , ,_,, , ., , 83, , 2ir, , x 5 x C., , And,, 981, , ., , io, , =, , .'., , 2rc, , X, , xlO 5, , 5, , x, , C =981 x, , 10*., , 10 5, , a, , -, , k.gm.wt.-cm., , ^m .^ cm, , ., , Hence, a couple of moment 31 '83 k.gm.wt-cm., , will bring the flywheel to, , rest in 5 revolutions., , A flywheel of weight 200 Ibs. which may be regarded as a uniform disc, 4., of radius 1 ft. is set rotating about its axis with an angular velocity of 5 revolutions, At the end of 40 sees., this velocity, owing to the action of a, per second., constant frictional couple, has dropped to 4 revolutions per second. What constant, couple must now be applied so that in further 20 sees., the angular velocity will be, 8 revolutions per second., Find the total angle turned through during the minute., (Cambridge Higher Secondary School Certificate), , M =2000, , Here mass of the flywheel,, Since it is a uniform, /.*.,, , /, , =, , j, , Mr -, , Us angular velocity,, and, .-., , ,,, , i,, , Ibs., , and, , circular disc, its, a, , 2, , 100, , i.200./, , to start with, , =, , after 40 sec., , change in angular velocity in 40 ,,, rate of change of angular velocity, , its, , radius r, , moment, , lft., , of inertia about, , its, , axis,, , Ib.ft*., , 5 revolutionslsec., , 4 revolutions [sec., , =, =, , 2rrx5 radians/sec*, 2::x4 radiansjsec., , = 2nx52n x4 = 2n radians/ sec*., = 2rc/40 = 7t/20 radiansjsec, 2, , ., , = re/20 radiansjsec*., Or,, angular retardation, i.e., dujdt, =, couple, I.d^ldt., Now,, .*., frictional couple acting on the wheel = 100x7r/20 = STT poundal-ft., *, , Again, the velocity of the wheel, revolutions per sec., in 20 seconds., f, , %, , initial, , is, , now, , desired to be raised, , angular velocity * 2rcx 4 radians/sec*, , from 4, , to I
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fBOKBRTlES, , 102, , And,, , = 2rc x 8 radians! sec., = 2n x 8 2* x 4 = STT radians fsec., rate of change of velocity = 8?r/20, 2r;/5 radiansl sec, =, radians, sec., angular acceleration,, /<#, 2^/5, , angular velocity in 20, change in velocity in 20, , final, , And,, , Oft, , sees., , sees., , 1, , .., , ., , Or,, , </co, , J, , And, the couple required to produce this acceleration, = 100x2^/5 = 40-npoundal'ft. [ v couple, , =, , Ldujdt., , total couple required to be applied, , log 45, log TT, , =, , 1-6532, 0-4972, , Antilog^2-T504, , =, , Now,, , =, \, , i, , 141-4, , this couple, , of 40rc 4 a couple of SK, , by the wheel in the, , d-, , Then, from the relation, , *, ==, , (27ix5)x40~}., , And,, , if 0,, , =, , 40:r-}-5rr, , overcome fric-, , 45?r, , ^14\'4 poundal-ft., , tional couple)*, , ', , let the angle described, , G!, , =, , Or, total couple required, , (to, , r -f 1-, , 2, ., , 2, , first, 2, , (40), , ^./, , ,, , 40 sec. be, , Oj, , S, , we have, , ", , - 400nr-407r =, , C, , n, , sees.,, , we, , J g^, , [~, , 360rr radians., , be the angle described by the flywheel, , in the next, , 20, , have, as above,, 0,, .*., , -, , 2, , *, , (27tx4)x20-f, , -, , 2, , (20), , ^, , 160^f 80rr, , =, , 2407T radians., , the total angle turned through by the flyweel in one full minute, =,, , X, , 4, , 3, , =, , =, , 3607T+2407T, , 600 radians., , since a rotation through 2* radians means one revolution, a rotation through 600:r radians means 600^/2^, 300 revolutions., , Now,, , =, , Thus, the flyweel makes 300 revolutions during the minute., 5., Ibs., , and 52, , A, , pulley of radius 2 ft. has hanging from it, a rope with masses of 60, attached to its two ends, the masses being kept at rest initially by, holding one of them. If the moment of inertia of the pulley, be 320 Ib -ft 2 ., what will be the velocity of the masses, when, they have moved a distance of 6 ft. from their position of, rest ?, It may be assumed that there is no slip between the, rope and the pulley and that friction at the axle of the, , Ibs., , pulley, , is negligible., , Here, obviously, the motive force, i.e., the force, which makes the masses and the pulley move, is the, weight of the excess mass of (60 52) or 8 Ibs. wt. at one, end of the rope = 8x32 = 256 poundals. (Fig. 60)., When the masses have moved through a distance, of 6//., the loss oj potential energy suffered by this excess, mass is clearly == 256x6 = 1536 ft. poundals., , This loss of P.. of the excess mass is equal to the, gain in the K.E. of the system consisting of the two, masses and the pulley., Fig. 60., , Let v be the velocity of the masses, Then, K.E. of the two masses, , at, , = 4.112.V = 56v ft. poundals., i/w = Jx320xv /r - ix320xv /4, And, K.E. of the pulley, - v/r and, = 40v ft. poundals., [v, = 56v 40v = 96v ft. poundals., total'gain in K.E. of the system, , =, , 2, , 2, , instant., , this, , 2, , i (60-{-52)v, , =, , 2, , 2, , 2, , 2, , 2, , co, , 2 --, , 2, , r, , 1ft., , z, , .-., , Since, , we have, whence,, , loss in P E. of the excess mass., gain in K.E. of the system, 2, 2, v, 1536., 96v, Or,, 1536/96 =16., v, , =, = ^16*", , =, , 4 /'-/ 5ec, , -, , Thus, the velocity of the masses when they have, tance of 6ft. will be 4ft.jsec., , moved through a, , dis-
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ENERGY, , MOMENT OF IN&RTII, , 103, , fcofATlON, , off, , 6. A flywheel of mass 65*4 K. gms. is made in the form of a circular disc, of radius of 18 cms. ; it is driven by a belt whose tensions at the points where it runs, on and off the rim of the wheel are 2 K. gms. and 5 K. gms. weignt respectively. If, the wheel is rotating at a certain instant at 60 revolutions per minute, tincl how long, While the flywill it be before the speed has reached 210 revolutions per minute., wheel is rotating at this latter speed, the belt is slipped off and a brake applied., Find the constant braking couple required to stop the wheel in 7 revolutions., (Cambndge Higher School Certificate), (/), , Here, obviously,, tension 7\, where the belt runs on the rim, , and, , r,,, , off, , ,,, , the resultant tension in the belt, , ,,, , = 2K., =5, , gms. w/., , ,,, , ,,, , = Tt -7\ =, , (5-2) K. gms. wt., 1000x981 dynes., ~ 3 x 1000 x 981 x 18 dynes-cm., .*. moment, And, if dujdt be the angular acceleration of the wheel, the couple acting on the, /., , =, , 3x, =-3 K. gms. wt., of the couple due to this tension, , wheel, .'., , =, , ., , l.d<*ldt., , X 1000 x!8 2 ;/2 = 65400x18x9 gm cnr., moment of the couple = 65400 x 1 8 x 9 x dujdt dynes-cm., This must be equal to the moment of the couple due to tension, , = M.r, , /, , Now,, , belt., , =, , 2, , /2, , (65'4, , in, , the, , Hence,, , 3x1000x981x18., 3x1000x981x18, a, <, S, radians Isec*., dldt~ 65400xJ8x9, Now, we ave the relation, a = c^-f (r/co/J/)/, .(/'), 65400xl8x9x</o>/</f, , ~, , ,, , Or,, , ,., , ,, , <.>, [See page 85., the final angular velocity t^, the initial angular velocity, c/u>/Jf ,, the angular acceleration and t the time., ., , where w 2, , is, , ;, , ;, , t, , Here,, , a> a, , =, , 210, , rev. I, , =, , mm., , =, , ^ = 60, and, .'. from relation, , (/), , whence,, , 210x27t/60, 60x2rc/60, , =, =, , 2, 5 radians/sec, d&ldt, =, above, we have In, 2x-\~5t., t == K ~ 3*142 sees., , Jr., , radians/ sec.,, , 2rr, , ., , Or, 5*, , =, , 5/., , So, after, , that, the flywheel will obtain a speed of 210 revolutions, 142 seconds., , 3', , per minute, , (//) Let the angular retardation produced in the wheel by the braking, couple be dte/Ut, the angle turned through by it before coming to rest being, 14^ radians., equal to 7x2rc, , ~, , Then, applymg-the relation oj^-wj = 2(d>ldt)$,, we have O a -(77r) 2 = 2(d<*fdt) x 14nr. Or, 2^.d^dt = -~49^ 2, 2, , Or,, , d<*Idt, , -, , -49r;, , 2, , /287T, , Or, the angular retardation required, , Now,, , since couple, , =, , Ld<^ldt,, , ==, , =, , -7^/4, , 327-25145, 63-17993, , ^, Or, , !, , !, , I, , 0=, , log ,o, ^;, , ", , 4, oi 17736, , =, , 5937, , ', , ['/, , =, , 14*., , radians I sec\, , 77T/4 radiansjsec*., , we have, , C = J x 65400 x 18 x 18 x 7rr/4, 65400x18x18x77:, n, C==, , braking couple required,, log, log, , ISee page 85., ., , 2x4x981, 65400x18x18x7, 2X4X981X1000, , dynes-cm., , ~ "'", , "'"^, , i, , 327 x, , 63rt, , 1090, , _, , 59 37 j^, , g tnSt, , 59 37 K. gms. wt.-cm., Hence, the required braking couple, 7., A flywheel, which can revolve on a horizontal axis weighs 900 ibs. and, its radius is r ft,, A rope is coiled round its rim and a weight of 90 Ibs. hung from, Find the speed at which the weight is, its free end, turns the wheel by its descent., moving after descending 20 ft. from rest.
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104, , PROPERTIES OF MATTER, Let the acceleration of the weight be a ft. per, , mass of the weight suspended,, , and, , sec*.,, , let, , M, , .be, , the, , (Fig. 61)., , Then, if T be the tension in the string, we have, M.a=(Mg-T). Or, T^M(g-a) = 90(32 -a) pom dais, Now, moment of the couple acting on the wheel due to, tension, , =, , Tin the rope, , T.r., , =, , ', , T.r, , 90(32-ra). r poundal-ft., Also, rotational couple on the flywheel =* I.d^/dt., 8, 2, 7, Here., JAfr, J.900.r, f Considering the flywheel to, 450r 2, ^ be a uniform circular disc of, And, dujdt =a\r., \jnass Mand radius r., 2, .'. rotational, couple on the wheel-=450r (a/r)=45Q a.r poundal-ft, have couple due to tension T, rotational couple., , =, , =, =, , ., , ., , We, , = 450 a.r. Or, 90(32-) = 450 a., = 450 a. Or, 540 a = 2880., a = 2880/540 - 16/3 // /sec*., r, , 90(32-0), , 2880-90 a, , Or,, Fig. 61., , be the velocity of the weight after, , If v, , log, log, , whence,, , v, , 640-2-K062, , 3=0, , 4771, , x 2-3291, , 1, , 1-1645, , Antilog, , 2, , -, , 2, , =, , it, , has descended 20//.,, , v'-O =2x(16/3)x20., 2, , Or,, Or,, whence,, , v, , v, , =, , 640/3,_, , =, , ^640/3, , [v S, , =, , 20 ft., , 14-61 ft. /sec., , Therefore, the speed of the weight would be 14'61, , -14-61, , we have, , 2.aS., , ft. /sec., , A, , sphere of mass 50 gms., diameter 2 cms., rolls without slipping, with a velocity of 5 cms. per sec. Calculate its total kinetic energy in ergs., 50 gms. and radius of the sphere, /, mass of the sphere,, 1 cm., Here,, Now, moment of inertia of the sphere (solid) is given by, 8., , M=, , /, , =, , 2, , J.Mr, , =-, , =, , 9, , |x50xl, , =, , 2Qgm.-cm*., , As, , the sphere rolls, it rotates about its own diameter as axis as well as, its centre of mass moves with a velocity of 5, cms /sec. It has, therefore, both, kinetic energy of rotation as well as kinetic energy of translation ; and, therefore, its total energy is the sum of both., , Now, K.E. of rotation = }/ <o 2 = J/.v 2 - |x 20x 5 /l 2 = 250 ergs, [v o>=v/r., And KE of translation = iMv = |x50x5 2 -25 x25 = 625 ergs., .*., total kinetic energy of the sphere =250 + 625 = 875 ergs., A flywheel of mass 10 K. gms. and radius 20 cms. is mounted on an, 9., 2, , 2, , //', , 2, , A, , axle of mass 8 K. gms. and radius 5 cms., rope is wound round the axle and, carries a weight of 10 K. gms., The flywheel and the axle are set into rotation by, Calculate f ) the angular velocity and the kinetic energy of, releasing the weight., the wheel and axle and (//) the velocity and kinetic energy of the weight, when the, , weight has descended 20 cms. from, , its, , original position., , The, , flywheel, here, (Fig. 62), is just a hollow circular disc or cylinder, (as, it has been cut in, the centre for the axle to pass, its moment of inertia about its axis, therethrough), 2, 2, fore, is equal to MtR + )/2, where, is its mass and R, and r its outer and inner radii, (r being the radius of, ;, , M, , /', , the axle)., , (See page 78)., for the flywheel, is equal to 80 K.gms., or 80 x 10 gms., and R and r, equal to 20 cms. and 5 cms., respectively ; so that, the moment of inertia of the wheel, , Now,, , M, , 8, , 2, =80xl03x(20M-5 )/2 = 80xl0 3 x425/2., = 1 7000x10"= llxlW gm.-cm*., , And, the axle, , moment of, , is, , just a disc or cylinder, is equal to, , about its axis, where, is its mass and r, its radius., So that, moment of inertia of the axle, , M, , inertia, , whose, Mr*,, , [See pp. 63, , &, , 66., , = ix8x!0 x5 = 10'xl0 = 10 w.-cm, total moment of inertia of the wheel and, e, axle,, * 171 x lO'^m.-cw, I - 17xlO+ 10, Or,, 3, , 2, , 2, , 8, , 5, , ., , /., , /, , s, , ., , ,, , 1, .
Page 117 : MOMENT 01 INERTIA, , ENERGY OF ROTATION, , 105, , When the weight descends through a distance h, it loses potential energy, * mgh, and this loss in P E. of the weight is, obviously, equal to the, gain in, K.E. of the wheel and axle and the weight itself., Now, since m = 10 x 10 3 gm. g, we have loss in P.E. of the weight = lOx, t, , =, , 2, , 20 cms.,, 981 cm.jsec and h, 10 8 x981 xlO, 1962xl0 5 ergs., ., , =, , If to be the angular velocity of the wheel and axle, when the weight has, descended through 20 cms., the velocity v (linear) of the weight will be rco,, , where, , r is, , the radius of the axle, , v, , i.e.,, , ;, , K.E. of the wheel and axle = I, K.E. of the weight = Jmv 2, and,, , 5w., , =, , [, , r, , ., , =, , 5 cms., , L o> 2 = i x 1 71 x 10 5 x w 2 =* 855 x 10* x w 2 ergs., = Jx 10 x 10 3 x(5o>) 2 =125x 10 2 Xo> 2 ergs., , .'., , K of the wheel and the axle and the weight., -=855xlO*Xco 2 -J-125xl0 3 xco 2 = 8550x I0 3 xo> 2 -fl25x 10 2 x<A, = (8550+ 12:>) x 10 2 x w 2 = 8675 x 10 8 .o> 2 ergs., , total gain in, , .*., , ., , Since total gain in K.E. of the wheel and axle and the weight, the loss in P.E. of the weight, we have, 3'2927, 8675 xlO 3 co 2 = 1962x10'., log 1962, 6, log 8675, 2 ~~, _ 1962 xlO __ 1962*10*"", 3_9383, Or,, 3 """"8675, XlO, 8675, JxT'3544, 10 x, :, whence,, v/1, 962/8675"., 6772, T, Antilog, , =, =, , is, , equal to, , ., , =, , = 10x-4755., , -4755, , 4 755 radianslscc., , = 4'755 radians/sec., = 5 X 4 755 = 23', , and axle, i.e., co, Or,, angular velocity of the wheel, rw =-- 5o>, and linear velocity of the weight, i.e., v, is, axle, and, given by, of the wheel, Now,, , =, , i/.w, , a, , = }xl71xl0 xo)*., = 855xl0 x(4-755) 5, , 2, , 4, , 2, , 2, , i, , 19'34x 10 7 ergs., , K E. of the weight is given by, = i m(5w) - Jx 10*.(5x4 755)*, = 5xl0 x25x(4755), = 28 '27 xlO, , And,, , /iiv, , 2, , 3, , ., , 5, , <?/'#*., , Thus, (i) the angular velocity and kinetic, energy of the wheel and axle are 4 755 radians/sec., and 19 34 xlO 7 ergs and (//) the velocity and, kinetic energy of the weight are 23-775 cms. I sec., and 28'27x 10 5 ergs respectively., If the pulley in an Atwood's machine be of moment of inertia 1500, and radius 5 cms., what should be the acceleration of the system in, which the weights at the two ends of the string passing over the pulley be 200 and, 250 gms. respectively ? (Given that g = 981 cm. /sec 2 .), Let a be the acceleration of the system and v, the velocity of the, weights, when they have moved a distance S cms. from the starting position,, , e.g.s., , 10., units, , (Fig. 63)., clearly,, , Then,, , And, , v, , 8, , /., , Now,, , loss, , through distance, , in, , S, , -w a =, , 2aS., , a, , v /25., , =, , P.E of, , .*., , mg.S, , =, , a, , =, , 2aS., , the heavier weight, , = Mg.S =, , [v w=0., , M,, , falling, , 250 x 981 x S ergs,, , and, gain in P.E. of the lighter weight, , =, , v, , Or,, , 2, , 200 x 981 x S, , m, , ^^, , ergs., , o?n, , net loss in P.E. of the system, =, , (250x981x5) -(200x981x5), , 50x981x5er^., , 200 gm., , This must be equal to the gain in K.E. of the pulley, as well as the weights themselves., , =, , =, , 2, 2, Clearly, gain in K.E. of the pulley, } 7w*, | /.v //*, v, rw, where w is the angular velocity and r, the, radius of the pulley., , v, , =, , ., , |M|, ZSOgm., Fig. 63.
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PROPERTIES, , 106, , =, , Or, gain in K.E. of the pulley, , *, , = iMv, , of the two weights, , in K.E., , And, gain, , 8, } 7.v /25, , Otf, , 2, , -fimv, , =, , 2, , = ix450xv =, 2, , .*., , #.., , total gain in, , Since, , we have, , 255v, , =, , 327, , log, log, , =, , 5, , 2 5145, , 2, , =, , Now,, , Antilog, , =, , Or,, 11., , v /25., , What, (//'), , is, , the K.E.,, , if, , one extreme end, , M =3 K.gms. =, , 3, , M.L, , 96-18 cms, , is, , 1, , ?, , =, , x 1000, , 3000 #mj., and, , =, , /, , 8, 1, 3000x(100) /12^m. cm, , - M/ /3 2, , Now,, , ., , ., , o>, , of a rotating body, , of the bar in case, , (/), , =, , Jf, , case, , (//), , =, , ., , 2, /3#. cm, , ['.', , ., , it is, , 1, , rotation/sec., , g, , X, , o>*., , ix[3000x, , 2, Jx[30DOx 10000/12] X4n, , its ^T.E. in, , f, , 2n radians/ sec., , =, , 100 cms., , f, , 3000x(100), , =, , =, , metre, , 1, , its e.g., , about an axis through one end, , Angular velocity of the bar, , I sec*., , metre long, weighing 3 K. gms., rotates, the axis of rotation passes through (i) its, , narrow uniform metal bar,, , of the bar about an axis through, , Aad,, , 2, , a, , i.e.,, , 17, , the acceleration of the system of weights, , A, , 2, , And,, , 255, , 96-18 cms. I sec*., , - M/ /12, its, , 50x981x5, Or, v, , 96-18, , Here,, , M.L, , of the system and the weights,, , "25"5x25, , 983f, , once per second., centre of gravity,, , /., , 225, , 50x981x5^327x5, , *, , 1-2304, :, , 2, i(Af-f m) v ,, , the acceleration of the system,, , OJ5990, , l, , in P.E., , loss, , 50x981x5., , 32135, , 17=, , log, , 2, , 30v ergs., , of the pulley and the weights,, , gain in K.E., , this, , ~, , 2, Jx 1500x v /25, , (100)*/12]x (2n)\, , 500X 10000 Xn z, , J(3000x 1 0000/3) x4*, , 2000 x 10000 xn =, 2, , = 5xl, , 8, ., , 20 x 10 6 xn, , ergs., , Find the moment of inertia of a homogeneous circular cylinder of, length 2/, radius of cross-section r, about (/) the axis of the cylindrical symmetry ;, (//) a generating line ; (*//) a diameter of cross-section at a distance x I, and 21 (or, 0) from one base., 12., , (/), , The moment of inertia of the cylinder about the, symmetry is the same as that of a, , axis of cylindrical, disc about an axis, passing through its centre and perpendicular to its plane,, (for a cylinder is nothing but a thick disc), and is equal, to MR*I2, where, is the mass of the disc or cylinder and, , M, , R,, , its radius., , .'., ifMbc the mass of the cylinder, and r, its, radius of cross-section, (Fig. 64), we have moment of inertia, of the cylinder about its axis of cylindrical symmetry equal to, 2, A/r /2., , ^, , line is parallel to the axis of, e.g. of the cylinder, and, Therefore, by the principle of, is at a distance r from it., parallel axes, moment of inertia of the cylinder about the, generating line is equal to its moment of inertia about the, (11), , The generating, , symmetry, passing through the, , axis of, , x (distance from, symmetry plus massa of thea cylinder, 2, , the axis) 1,, , i.e.,, , - iMr + Mr - 3Mr, , /2.
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ENERGY OF ROTATION, , MOMENT OF INERTIA, , 107, , an axis, passirg through, (Hi) (a) The moment of inertia of the cylinder about, centre and perpendicular to its length, g\, , its, , =, , 2, , a, , M[(4/ /12) f(r, , [v, , /4)], , =--, , length, , A, , 2/., , A, , (6) Since an axis at a distance x from, one base is at a distance (lx) from the axis, through the centre, we have, by the principle, , k, , (/-.r, , of parallel axes,, I. about a, M.I. about this axis =*, parallel axis through the centre-f M(l x)*., , M, , -, , 2, , Mf(/ /3)i, 2, 2, M[(4/ /3)f (r /4H (x*-2lx}\., , Fig, 65., , the principle, , of parallel axes,, Similaily, by, M.I. about the diameter of cross-section, (c), , =, , M[(/, , 2, , M/ 2 -, , 2, , /3) + (r /4)]-f, , f, , 2, , A4[(4/ /3) + (r /4)]., , Find the moment of inertia of a sphere about a diameter., , 13., , You, , are given two spheres of the same mass and size and appearance, but, one of them is hollow at the centre and the other is solid throughout. How will, (Delhi), you find which is hollow and which is solid ?, , For answer to, , first, , part, see, , 31, (case 13),, , pages 72., , inertia of a solid sphere about its diameter is, as we, is its mass and R, its radius, and that of a hollow sphere, know, 2M/T/5, where, 5, 5, 8, 3, /, r )], where R and r are Us outer and inner radii respectively,, )/(/, 2/5[A/(7?, and, being the same in the two cases., , The moment of, , M, , M, , R, , ~ \/ 2R, , the radius of gyration for the solid sphere, gyration for the hollow sphere is, ."., , e= \/2/5[/2 6, , -r >/\rt, ), , J, , a, it is, , =, , n, , na, , is given by, f a being the inclination, [ of lhc plane> (pagCf .), , v-\, , (R*IR*+K~) g sm, >, , ., , and the radius of, , /5 t, , masses being the same., down an inclined plane, , /)], their, , Since acceleration of a body rolling, /, , t, , a,, , ., , clear that the greater the value, , of, , A', , 2, , compared with, , as, , R*,, , the less the, , acceleration of the body., , K\, , Now,, , j*, , and, , t, , 5, , 5, , 5, , j, , K*, , .'., , _r, , .-_, , ., , 7?, , 2, , 5, , //^, , U1, 5, , 6, ), , -'', , >, , 3, , 8, , /* J, (l-r, , 5, , ', , 3, , ri-r*/*'!, r //?, , (1, , *1, , i> 3 ~__~"Y, , 5, , -, , that, the fraction,, , ., , 2, , 3, , >[l--r /#<J, 3, a, 3, 3, And /., Obviously, r /K < r //^, , rT^j- 5, , 2, , =, , * [l-r /K ]_, ", , 2, 5, , And, , 27? /5,, , for the hollow sphere, , ~, , So, , a, , for the solid sphere, , ,-, , >, , 1., , /fl quantity greater than, , 1., , ^-, , K* for a hollow sphere, , Or,, , 2, , is, , 2, , greater than 2/? /5., , hollow sphere than for a solid sphere, and, therefore, the acceleration of the hollow^sphere is less than that of a solid, sphere. In other words, the solid sphere will come down the inclined plane, faster than the hollow sphere, and the two can thus be easily distinguished from, each other., , Thus, the fraction (R-jR'+K, , ), , is, , less for a, , EXERCISE, Moment, , III, , of Inertia and Radius of Gyration. Explain their, physical significance. State the laws of (i) parallel and (//') perpendicular axes, and prove any one of them., (Bombay, 1945), 1., , Define
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108, , JteottBfcTite oir, , M, , Calculate the moment of inertia of a thin circular disc of mass, add, about its diameter (ii) about a parallel axis to the diameter and, 2, Ans. Mr /4 ; 5Afr"/4 f, tangential to the disc., 2., , radius r, , (/), , Define Moment of Inertia of a body about an axis. Show that the, of inertia of a body about an axis through the centre of gravity is less, than that about any other parallel axis., 3., , moment, , A uniform circular disc of radius r is free to oscillate in a vertical plane, about an axis perpendicular to it and distant x from its centre. Calculate the, periodic time., (Madras, 1950), Ans., , - 2^(r* + x*)l2gx., , /, , A, , flywheel of mass 2 I tons and diameter 8//. makes 250 revolutions, Find (/) its angular velocity (ii) its energy (Hi) its moment of inertia., Assume its mass to be concentrated at the rim., Ans. (/) 25n/3 radians/sec, (ii) 3 07x 10 7 ft. poundals. (Hi) 89600 Ib.-ft*., <4), , per minute., , Show that, , 5., , of angle, (, , is, , oP, , 2g, , A, , the acceleration of a disc rolling down, sin 0/3, while that of a ball is 5g sin Q/7., , uniform rod 4 //. long and weighing 9, , minute^about one end., , Calculate, , revolves 60 times a, Ans. 29 61 //. Ibs., , Ibs.,, , kinetic energy., , its, , an inclined plane, , A, , I., hoop of mass 5 k. gins, and radius 50 cms. rolls along the, at the rate of 10 metres per second., Calculate its kinetic energy in ergs., , ground, , Ans. 5x10* ergs., , Explain clearly what, , 8., , you understand by 'Moment of, , Inertia', , 'Angular momentum*. State the principle of conservation of angular, tum, illustrating your answer by an example., , and, , momen-, , Find the moment of inertia of a circular lamina about a tangent in, , its, , (Patna, 1949), , plane., , Ans. 5Afr/4., , A, , 9., , solid spherical ball rolls, , kinetic energy, , is, , on a, , table., , What, , fraction, , rotational ?, , of its total, Ans. 2/7th., , 10., Show that the K.E. of a uniform cylinder or disc of mass Af, rolling, so that its centre has a velocity v ii f Mv 2, In the case of a sphere, show that, 2, the K.E. would be 7Mv /10., ., , M, , A thin hollow cylinder, open at both ends and of mass, II., (a) slide t, with a velocity v without rotati tg, (b) rolls without slipping, with the same speed, 2., Ans. 1, Compare the kinetic energies it possesses in the two cases., :, , radius of gyration, (//) moment of inertia. Find the, of inertia of a circular dire about the axis perpendicular to its plane., , 12., , moment, , Define, , A circular, its, , (/), , disc of, , angular velocity,, , mass, , show, , that, , m and, its, , radius, , r is, , E=i, , mr 2 .co a, , on a, , set rolling, , E is, , total energy, , table., , If, , <o is, , given byj, (Punjab, 1950), , ., , Derive an expression for the kinetic energy of a body rotating about, , an, , axis., , A, , is in the form of a uniform circular, its radius is 2 ft.,, disc, Find the work which must be done on the flywheel to increase, speed of rotation from 10 to 20 revolutions per second. (Madras B.A., 1947)., , flywheel, , and mass 2, its, , ;, , Ibs., , Ans., , Five masses, each of 2 k. gms., are placed on a horizontal circular, disc, (of negligible mass) which can be rotated about a vertical axis passing, through its centre. If all the masses be equidistant from the axis and at a distance 10 cms. from it, what is the moment of inertia of the whole system ?, 14., , (, , Hint.:, , i, , and, , C, 15., , M.L, , of each mass about the axis, , .. total M.I. of the, , =, , = Mr, , system, , sum of the MJ. of the masses., Define 'Moment of Inertia' and 'Radius of, , of parallel axes* and prove, , it., , 9, ,, , }, r, ), , Ans. 10 8 gm. cm*., , Gyration.' State the law
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ENERGY OF ROTATION, , IIOMIHT 01 IHERTIJL, , A, , is mounted so as to rotate about art horizontal, string of negligible mass, wrapped round its circumgms. attached to its free end. When let fall, the, descends through 100 cms. in the firsts seconds. Calculate the angilar, , wheel of radius 6 cms., , axis through its centre., ference carries a mass o!, , mass, , 109, , A, , 2W, , acceleration of the wheel and its moment of inertia., 2, *. radians I sec, Ans. Angular acceleration, ;, ., , 16., , sary theory, , What is meant by moment, how the moment of inertia, , (Bombay, 1947), , M-L, , 8*748, , x, , 10 4 gm.-cm*., , of inertia of a body ? Show with necesof a flywheel may be determined., , (Allahabad, 1948), free end of a string wrapped round the axle of a flywheel, of, 2, 5, moment of inertia 27*61 xl0 #w.-cw ., carries a weight of 5 k.gms., which is, allowed to fall. What is the number of revolutions per second made by the, wheel, when the weight has fallen through 1 metre ? The kinetic energy of the, 17., , weight, , The, , may be, , Ans,, , neglected., , 3., , the wheel be mounted on an axle of half its, moment, cm 2 .) and radius 5 cms., and the K.E. of, the weight be taken into account, what will be the number of revolutions per, second made by the wheel ?, Ans. 2-413., If in question 17,, of inertia (i.e., 13-80, , 18., , xWgm., , Masses of 95 gm. and 105 gm., hanging freely are connected by a, which passes over a pulley of mass 20 gm. when icleased, the system, moves with an acceleration of 46 7 cm. per sec 2 Calculate a value of g if the, mass of the pulley is (a) neglected, (b) taken into account. Regard the pulley as, a simple disc of moment of inertia i Mr 2 and assume that no kinetic energy is, lost in friction., (Northern Universities Higher School Certificate), Ans. (a) 934 cm.se<r 2 ., (b) 980-7 cm.sec~*., 19., , light string, , ., , ,, , 20. (a) Four spheres, each of diameter 2a and mass m, are placed with, their centres on the four corners of a square of side b. Calculate the moment of, inertia of the system about one side of the square., 1951), , (Punjab,, , A flat, , thin uniform disc of radius a has a hole of radius b in it at a, distance c from the centre of the disc, [c <(-/>)]. If the disc were free to rotate about a smooth circular rod of radius b passing through the hole, calculate, its moment oi inertia about the axis of rotation., (Punjab,, (b), , Ans., , where, , M, , is, , (a), , m(4a' + 5i>, , 2, , ), , ;, , (b), , M, , the mass of the disc., , Describe the experiment to determine the moment of inertia of a, Derive the formula used in the experiment, without, flywheel., neglecting the, friction at the bearings of the flywheel., (Allahabd, 1948 ; Gujrat, 1951), 21., , 22., A flywheel, which can turn about a horizontal axis, is set in motion, by a 500 gm. weight hanging from a thin string that passes round the angle., After the wheel has made 5 revolutions, the string is detached from the axle, and the weight drops off. The wheel then makes 7 revolutions before, being, brought to rest by friction. The radius of the axle is 2-0 cm., and at the instant, when the weight drops off the angular velocity of the wheel is 10 radians, per, , sec., , Assuming, , that the, , the same, calculate the, rotation., , work done against, , friction in each revolution is always, of inertia of the flywheel about its axis of, (Oxford and Cambridge Higher School Certificate), Ans. 3 59xlO*#m.cm 8 ., , moment, , 23., A pair of rails is supported in a horizontal position and the axle of, a wheel rests on the rails. A thread is wrapped round the axle and a, weight, hung on the end of the thread. As the weight falls the wheel moves along the, rails., How would you determine the moment of inertia of the wheel with thii, , arrangement ?, , A, , 24., circular disc, starting from rest, rolls (without, slipping) down an, inclined plane of 1 in 8, and covers a distance of 5*32, //. in 2 sees. Calculate the, value of V., Ans. 31*92 ft [sec*., , 25. Two gear Wheels, of equal thickness, of the same material and, having, radii in the ratio 2 : 1, are mounted on, parallel frictionless spindles, but are, so, not, as, to metii with ono another. The larger vbeel is sot, separated, spinning
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PROPERTIES Of MATTER, , 110, , at a speed of 10 rev. per sec., and the wheels are then brought into mesh. What, is the resulting speed of each wheel ?, (Cambridge Schorlaship Examination), Ans. 8-9 and 17-8 rev. sec* 1, ., , 1, , 26., What do you understand, the axis of the torque applied to a, , rotation,, two axes., , by the term "precession ? Show that if, body be perpcnJicular to its axis of, the body precesses about an axis perpendicular to either of the first, , 27. What is (/) a gyrostat, illustrate their action., , and, , (//), , a gyroscope, , ?, , Describe suitable experi-, , ments to, , What, , is, , meant by the term nutation, , ?, , Explain the theory underlying a, expression for its time-period., 28., , gyw static pendulum and, , obtain an, , Discuss in detail the case of a thin disc or hoop set rolling over a, 29., plane horizontal surface and obtain expressions for (i) its critical velocity, and, (it) the radius of curvature of its path on the surface., , Write short notes on the following, , 30., (i), , funs and, , (jy), , ;, , f'endulum Gyro-compass, (Hi) Rifling cf barrels of, 'Precession of the Equinoxes., , Gyro-compass,, , (ii)
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CHAPTER, , IV, , SIMPLE HARMONIC MOTION, Definitions. A simple harmonic motion is a particular case, 47., of periodic motion, i.e., a motion which repeats itself over and over, again after regularly recurring intervals, called its time-period, and is, so called becauss of its association with musical instruments. Common, in nature, it is in fact th^ most fundamental type of periodic motion,, as all other periodic motions, (harmonic as well as non -harmonic),, can be obtained by a suitable combination of two or more simple, , harmonic motions., If the acceleration of a body be proportional to its displacement, from its position of equilibrium, or any other fixed point in its path and, be always directed towards it, the body is said to execute a simple harmonic motion, (written, for short, as S.H.M.)., Now, a simple harmonic motion may be (/) linear, or (//) angular,, according as the body moves along a linear path, under the action of, a constraining force constantly acting upon it, or rotates about an, axis, under the action of a constant torque or couple., The time-period of a body, executing a 5. H. M., is quite independent of the extent of its motion to either side of its mean position, (i.e., of its amplitude), and the motion is, therefore, said to be, isochronous., , Mathematically, a linear S.H.M. may be regarded as the projecof a uniform circular motion, or of a rot at ing vector, on the diameter of the circle, or any other fixed line in the plane of the circle,, this circle being refer red to as the circle of reference, and may, in many, a case, be purely imaginary., , tion, , Thus, if a particle P (Fig. 66), moves with a uniform speed v, along a circle of radius a, and another particle M, along the diameter, is at O,, YOY', such that when P is at X,, and, as P starts along the circle in the anticlockwise direction,, starts along OK, so that, when P reaches Y,, As P, also reaches Y., continues to travel further along YX',, starts back towards O., , M, , M, , M, , M, , And, when Preaches X',, , As Pnow, , traverses the, , M, , M, , reaches O., lower half of the, , proceeds downwards, along X'Y', reach Y', along OY', so that both Pand, and, finally, when P travels further, together, on along Y'X,, starts back along Y'O, reaching, whe& P reaches X., circle, , 9, , ;, , _, , M, , M, , M, , Y, , Fig 66., its, , mean position, , moves along the diameter YOY' from O to, Thus, the particle, F, from Y to Y' and back to O (i.e., completes one vibration), in, the same time in which P moves once round the circle, such that, at, 111
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PROPERTIES OP MATTE!, , 112, , of? and M, , instant, the line joining the positions, , any given, , M, , is, , perpendi-, , on the diameter, or, the position of, YOY' corresponding to the position of P on the circle of reference, at, on to YOY'., any instant, is given by the foot of the perpendicular from, cular to, , the, , diameter, , TOY',, , P, , M, , is said to be performing a linear S.H.M. along, uniobviously, the projection of the particle P, moving, or, is the projection of, circle, the, XYX'Y',, reference, of, formly along, on the diameter YO Y of the circle., the rotating vector, , This particle, , YOY', and, , is, , OP, , 1, , M, , be due, not to P, but to any other force, If the motion of, along its path, the circle of reference will, as indicated above, be purely, need not necessarily be, an imaginary one. Further, the path of, curved., be, as, well, and, may, straight, , M, , Since a force acting on a body is proportional to the acceleration it produces in it, it is obvious that the force acting on a body, the changes in its acceleexecuting a 8. H. M. must correspond to, In other words, it must also be proportional to the displaceration., ment of the body from its mean position and must always be directed, , towards, , it., , familiar examples of simple harmonic motion., , Some, , Linear., , (a), , The up and down oscillations of the piston of a cylinder, con(/), solved, a, gas, when suddenly pressed down and released, (see, taining, example 1)., Tli 2 oscillations of mercury or water contained in a U-tube,, (//), when the column in one limb is depressed and released, (see solved, , example, , 7)., , The, , (Hi), , vertical oscillations, , spring) suspended, , from a, , rigid support, , of an elastic, and loaded at, , string (or a spiral, lower end, (see, , its, , solved example 10)., Angular., , (b), , The, , (/), , oscillations, , of a pendulum, provided the amplitude be, , small, (see Chapter VI)., , The, , (//), , (see solved, , oscillations, , example, , o^ a magnet suspended, , in, , a magnetic field,, , 8)., , Torsional oscillations, in general, (see Chapter VIII)., , (Hi), , Characteristics of a Linear, , 48., , S.H.M., , 1., Amplitude. The maximum distance covered by the body, on either side of its mean or equilibrium position is called its ampliIt is, obviously, equal to the radius of the circle of reference., tude., , Thus, the amplitude of the particle M, in the case above,, the radius of the circle XYX'T., , is, , OY, , *=OY'=a,, , The distance of a body from its mean posiinstant, measured along its path, gives its displaceat that instant., 2., , tion, at, , ment, , Displacement., , any given, , Thus, the displacement of M, in the position shown, (Fig. 66),, equal to OM, orj, such that, , OP sin, , $., , Or,, , y**a, , sin 0., , (where, , is
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SIMPLE HAEMONIO MOTION, , 113, , If a> be the angular velocity of P and t, the time taken by it in, traversing the distance OP along the circle, i.e., in describing the, 0=tof so that, y=a sin wt., angle 0, we have, This relation, giving the value of the displacement of a body,, executing a S.H.M., in terms of its amplitude and the angular velocity of the rotating vector, (or of the particle in the circle of reference), is referred to as its equation of motion., Thus, the equation of motion, ;, , of, , M along YOY', , is represented by y= a sin cot., we consider the motion of a particle W alon?, , the diameter XOX',, such that both P and AT are together at, and as P goes round the circle in the, anticlockwise direction,, starts along XOX', so that when P reaches Y,, reaches 0, and when P reaches X',, also reaches X', and when P goes along, the lower half of the circle,, starts back along X'O, reaching O when Preaches, F', and finally both arrive together at X, then the motion of ATalong XOX' is, also a S. H. M. And its displacement ON, x is clearly given by, If, , X, , N, , N, , ON = OP, , Or,, its, , equation of motion, , The, , N, , N, , is, , x, x, , cos, , o., , a cos, , 0., , a cos, , cot., , [, , v ON = x and OP =, , M, , [v, , =, , a., o>f., , executing a S.H.M. along the, position of the particle, diameter YO Y may at any time, be found with the help of its displacement curve, which is a graph, showing the relation between the, time that elapses since the particle was at its mean position O, and its, displacement from O during this time., f, , DISPLACEMENT CURVE, Fig. 67., , Let time be represented along the horizontal axis AB and displacement along the vertical axis DC, (Fig. 67)., Let the circle XYX'Y' be divided up into an equal number of, the, parts, say 8, representing equal intervals of time T/8, where T is, time taken by tho particle P to go once round the circle. Let these, intervals of time be also marked along the axis AB, taking A as the, Then, the perpendiculars drawn from, origin or the starting point., the points on the circle on to XOX give the displacements of, along YO Y', corresponding to the intervals of time represented by, them, as shown in tabular form below, 1, , :, , M
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114, , PROPEKTIES OF MATTER, , Ordinates equal to these perpendiculars are then erected at the, corresponding points on AB. Thus, the ordinate/? represents the, after a time T/8 of its starting from O; the ordidisplacement of, nate a, after time T/4 the ordinate q after time 5 T/8, and so on., The extremities of all these ordinates are then joined and a smooth, curve AJKLB is obtained, which is a harmonic or a sine curve, because it is of the same form as would be obtained for the relation, between angles from, to 360, and their sines, the maximum value,, 1 at 270 and the least,, viz., J, being at 90, and, i.e., zero at 0,, 180 and 360. The displacement curve shows at a glance how, the displacement of the particle, changes along the diameter YOY', and its value can be readily obtained from the curve at any given, , M, , ;, , M, , instant., , M, , 3., The velocity of the particle, is clearly given, Velocity., by the component of the velocity of P, along the diameter YOY, 1, , ,, , (Fig. 68)., , Now, the velocity of P is v in a direction tangential to the ^circle at P., , resolve, , PN, , it, , and, , along, , Fig. 68., , Now, , clearly,, , Since cos, , we have, Or,, , tf>, , component, , PM, , along and perpendicular to, YOY as shown. The com1, , PN, , ,, , represents the* velocity of, , M, , YO Y'., , PN =, , v, , cos, , <f>, , =, , v cos wt., , -, , = OQjOP, , component, , and, , magnitude, , by the straight line PK, we may, into two rectangular components, , the diameter, , ponent, , in, , it, , Representing, direction, , V QP-y/*., = OM, f, , [v, , PN =, , vV, , 2, , v, , >*/<*, , =, v-, , [v, , =, velocity of M, , M, , may be obtained by differentiating its, Alternatively the velocity of, is rate of change of displacet, displacement y with respect to (because velocity, = a sin <*t, we have, ment). Thus, since y, = av.cos of. Or, dy\dt =*a<*.Va*-y*la =, dyfdt, , oW-/-, , M, , 2, , 2, , -= wv/^ -^, velocity of, Or, be different at different points, would, of, the, velocity, Thus,, or distance from its, alone its path, depending upon its displacement,, is a minimum, and a, when, maximum, a, y, being, mean position O,, , M, , -, , minimum, when y is a maximum., So that (0 wheny =0, i.e., when, , M, , is, , at O, (or, its displacement, , = wV^ ~= w a ^ v ^ e same as ^at f**, is zero), its velocity, its dis=, when M is at Y or T,, a,, and, '() wheny, ^, =, >-0 =, is maximum), its velocity, w-v/fl, placement, 2, , -, , ;, , >, , (i.e.,, , i.e.,, , 2, , fl, , 2, , -, , varies inversely as its displacement., Or, the velocity of the particle, varies from a, of, It will thus be readily seen that the velocity, from zero, then, at, increases, 7,, a, minimum, (zero), maximum (v) at to, , M
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SIMPLE HABMONIO MOTIOtf, , 115, , to v at 0, decreasing to zew again at 7', and again becoming v when, it comes back to O., In other words, at time 0, it is maximum at, tima 274, a minimum, at 3T/2, a maximum at 32^/4, again a minimum, and finally at time T, again a maximum, as shown in the table, are shown at different times., below, where the velocities of, ;, , ;, , ;, , M, , If,, , therefore, a, , the particle, , M, , we, , ,, , graph be plotted botween time and velocity of, get the velocity curve of the particle, shown in, , where time, , Fig. 69,, , shown along the, , is, , hori-, , zontal axis and velocity, along the vertical axis., The curve obtained is, a cosine curve, for it is, of the same form as the, curve plotted between, to 360*, angles from, , and their, , t, r#-, , T/2, TIME, , cosines., , Fig. 69., , It should be noted that the, is, , side, or, , when, , its, , has attained its, , maximum, , and occurs when, , aa> or v,, , particle, , *-, , it, , value of the velocity of the, /Ys mean position to either, and its velocity is zero, when it, , passes, , displacement is zero ;, displacement on either side., , maximum, , As in the case of velocity, so also here, the, 4. Acceleration., is the resolved part of the acceleration of P along, acceleration of, 2, 2, 2, 2, YOY', (Fig. 67). Now, the acceleration of P is v /a or # o> /tf, or aa>, and is directed towards O. Resolving it into two rectangular com-, , M, , ,, , PM, , MO, , and MO, we have the component along, equal, ponents, along, z, aj y., to auP.sin $ or == aw 2 .yla, And, as is clear from the figure,, it is directed towards 0, the mean or equilibrium position of M., , =, , acceleration of, , Thus,, , M=, , &*y,, , the negative sign being put to indicate that it is directed toward 0,, direction opposite to that of y, its displacement., , in, , a, , M, , Alternatively, we may obtain the acceleration of, by differentiating its, velocity with respect to time, for acceleration is the rate of change of velocity., au.cos <of,, Thus, since, dyjdt, [see page 114., , =, =, a^-sin^t, d*yldt =*, acceleration of M =, , we have, , 2, , Or,, , Or,, , we may put, , where, , co, , 2, , *= M,, , tlon, , So, , that, here,, , cof, , =, , 2, >>,, , o>V, , M=, , ofM, , *Sinrilarly, in the case of angular, = /*,0, where, is its, , of the particle, , sin, , as acceleration of, a constant of proportionality., it, , acceleration, , Or,, , ^.a, , MJ>*,, , oc y., , S.H.M we, ,, , shall, , have angular accekra*, , angular displacement., angular acceleration oc Q.
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PROPERTIES OF MATTER, , 116, , M, , is proporThus, we see that the acceleration *of the particle, y its displacement from O, and is always directed, , tional to, , towards, , t, , it., , Obviously, this acceleration has, , same as, its, , that, , minimum, , ofP,, , its., , value,, , i.e.,, , maximum, , extreme positions Y and, 0., zero, at O, where y, , at the, , =, , ~ a, and, , It should be noted that the constant of proportionality, 2, equal to o> or the square of the angular velocity of the particle, the circle of reference., ,, , M=, , =, , Further, if y, 1, acceleration of, Thus,, p,., defined as the acceleration per unit displacement of M., , Tabulating acce]eration of, , 774, , T/2, , 3r/4, , (m/|f>), , (nmx), , is, , in, , may, , be, , o, (min), , .., , If,, , /z, , P, , M against time, we have, , Time, , A, ,, Acceleration, , the, , value, aof,, , where y, , Y',, , i, , (min), , (max), , therefore, a graph bo plotted, , between time and acceleration,, , we, , get the acceleracurve, of the, particle, as shown in, , tion, , Fig. 70, which is of a, similar to the, , type, , displacement, (Fig. 67),, procal in, , curve ,, , but is, form,, , reci-, , for, , acceleration is directed, in the opposite sense, to displacement., , Fig. 70., , Time-Period and Frequency. The time taken by the particle, 5., in completing ono vibration, (or one cycle), i.e., in going from, to Y,, to Y', and finally back to O, is called its lime-period, period, of vibration or periodic time, usually denoted by the letter T., Obviously, it is the same as the time-period of tho particle P, i.e.,, equal to the time taken by P in making one full round of the circle,, , M,, , (from, , O, , Y, , X, , back to X), or in describing an angle, 27T, , 2?r, , 2-7T, , rv, l~, , Therefore,, , 2ir., , =, , **, the acceleration, of, per unit displacemei, ment., , to, , M, , whence,, , Or,, , time-period of, , M=, acceleration per unit displacement', , The number of vibrations made by the body per second is called, the frequency of vibration of the body, and is denoted by the
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IIMPLE HARMONIC MOTION, letter n., , 117, , Clearly, therefore,, , frequency, n, , l/T, , =, , \/u, , /27T., , _ y acceleration per unit, """ displacement, 27T, , The term,, , Phase., 'phase' applied to a vibrating particle,, has a meaning similar to the ono associated with it when, we talk of, the 'phases of the moo:i. Just as tho phase of the moon i.e., whether, it is a crescent, (or new moon), half moon or full moon, tells us about, its position etc., so also th3 phase of a paroicle, executing a S.H.M.,, enables us to form an idea about its state of vibration., 6., , 1, , Thus, the phase of a vibrating particle, at any given instant, may, be defined as its state or condition as regards its position and direction, of motion at that instant. It tells us in what stage of vibration the, particle, , is., , is indicated either (i) in terms of the angle 0, described by, the rotating vector, measured as a fraction of the whole angle 2-/T, that it describes in one full rotation, or (//') in terms of time t that, has elapsed since the particle last passed its mean position, in the, positive direction, measured as a fraction of its time-period T., , It, , M, , Thus, taking O as the starting position of the particle, 66), if its phase be zero, it indicates that the particle is at O,, tending to move towards Y. And, if the phase be Tr/2, or T/4, it, indicates that it is at Y, the position of the maximum positive displacement for, the radius of the circle of reference, or the rotating, vector, has, up to this instant, described one-fourth of the total angle, or that one-fourth of the time-period,, 2-7T, i.e., an angle 2?r/4 or Tr/2, i.e., T/4, has elapsed sinco the particle last passed its mean position, O in the positive or upward direction, towards Y., (Fig., , ;, , ;, , Hence, when we talk of a 'phase difference' between two simple, harmonic motions, we mean to indicate how much the two are out of, step with each other, or by how much angle, (measured as a fraction, of 2?r), or by how much time (measured as a fraction of T), one is, ahead of the other., , Now, because the phase of a vibrating particle merely indicates, actual stage of vibration, it is clear that two vibrating particles, if, they happen to be in identical stages of their respective vibrations,, at any given instant, will be said to be in the same phase, at that, particular instant, irrespective of their amplitudes and velocities, being the same or different., its, , Thus, for example, they will be in the same phase, if they both, simultaneously attain their maximum displacements, positive or, negative or, when the two pass through their respective mean positions at the same time and in the same direction., Similarly, if one, of the particles attains its maximum positive displacement simulta*, neously with the other particle attaining its maximum negative displacement, or when the two cross each other simultaneously in, opposite directions at their mean positions, they are said to be |n, ;, , opposite phases,
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US, , PROPERTIES OF MATTEE, , We have deduced the reiatioii, 7. Epoch or Initial Phase., a sin ojt for the displacement of the particle, executing a, S.H.M., [ 48, (2)] on the assumption that the starting position of, the rotating vector is OX, or that the starting point of the particle, in the circle of reference is X, i.e., we start counting time when P, , y, , M, , =, , crosses the axis of, , ,, , x at X., , Sopaetimes, however, the starting position of the rotating, vector, or the position of the particle P in the circle of reference, is, fixed, not in some standard position, as on the axis of x or y, but, anywhere, in an arbitrary manner, such as at P', (Fig. 66), i.e., the, time is counted from the instant when P is at P' t such that the, , angle, , XOP', , =, , e., , =, , =, =, , =, , POP', cot, Then, clearly,, 6+e. Or,, (wf-e)., So that, y, a sin, a sin (a>te), where cot is the phase, 9, angle of P. This angle e is called the 'epoch' or the 'initial phase, of the particle. It may also be measured, in terms of the time, taken by the particle P in describing this angle, i.e., by the time, , =, , It should not be confused with 'phase', of the particle continuously changes with time,, remains the same all through., , whereas the 'phase*, epoch or initial phase, , for,, , ;, , its, , 49., Equation of Simple Harmonic Motion. Let y be the, displacement from its mean position of a particle, executing a S.H.M., Then, if v be its velocity at that instant, we have, v, , n, , ,,, , 1, , ,, , ., , si, the, , So that, acceleration of, , =, , dy/dt., , = dSy =, 2, , i, , dv, , -, , particle, , ,, , =, , dv, , x, , ., , dy, -, , =, , *,, , dv, V.T, , n.y, where /* is the consNow, acceleration is also given by, tant of proportionality and is equal to o> 2 (o> being the angular, velocity of the rotating vector, or the particle in the circle of, ,, , reference)., , = aA.y., = -a>\y,, dv = - a>*.y.dy., , Thus,, , d 2yjdt 2, , fthe negative sign, , Or,, , v.dvldy, ,, , {, , ', , i', , v., , whence,, , a, , Integrating this expression,, , Or,, , C, , v.dv, , =, , }v*, , =, , and displacement (y), { are oppo ^ tely directed!, , ,/, , ,, , indi-, , eating that acceleration, , i, , we have, , =, , aP.y dy, , -co 2, , ly.dy., , -jc, , the constant of integration, and has to be determined from, the condition of the particle at the instant considered., , where, , is, , Obviously, the velocity of the particle, a, i.e., v, 0,, displacement a, or, , =, , maximum, , So that, we have, from relation, , (/),, , - JwW+C., , Heaoe, Jv* =*, , And, , /., , v, , Or,, ico (a, , =, , zero,, , when y, , when, , =, , it, , a, or, , has, , its, , a., , above,, 2, , -Jo^+JcoV, , is, , 2, , C, , -^)., , w^/o^y^., , ** JciiV., , Or,, , v, ,., , 1, , =, , (), , cu, , 8, , (a*-^)., , [Sec, , 48, , (3).
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SIMPLE HARMONIC, , 119, , MOtflOtt, , llms, the velocity of the particle can be determined for any, value of its displacement y. Clearly, the value of v is the maximum,, i.e.,, coa, when y, 0, i.e., when the displacement is zero, or" the, particle is in its mean or equilibrium position ; and, it is a minimum,, a, or,, i.e., 0, when y, a, i.e., when the particle has its maximum, displacement, positive or negative., , =, , =, , =, , =, , since v, , Now,, , we have, from, , dyjdt,, , =, , dyjdt, , n, Ur, , dy, >, , oV**-, , =, , above,, , (//), , 1, }'*-, , = a sine, fPut y, then, dyj~ a, cos 0.^0,, a 2 y z =a cos Q,, and, , ,., , ;, , ai.at., , -y-~=.^, v a, y, , we have, , Integrating this,, , sin-, , Or,, , whore C', , =, , ~^^==_dy, , =, , y\a, , ^r 4, , w.dt., , {, , |, , J^a*y*, 1, , ^, , I, , =, , j, , Or,, , a, , that,, , sin (tt, , Or,, , =, , a, , C=, , 0,, , sin (Q, , + Cy., , whence, C', , substituting this value of C' in (Hi) above,, , .-., , (h) ]f,, , has, , its, , J, , ...(/), , =, , =, , a sin, , displacement,, , =, , t',, , i.e., t, , = 0,, , when y, , =, , (0+C), or, sinC'^a/a^l, y = a sin (cot -\-7ij'2). Or, y, , Again,, , when y, , if, , we, , start counting time from, , mean or, , =, , a sin, , Or,, , a,, , =a, , =, , cos, , an instant, , ?r/2, , = 0. Therefore, from, above,, = a sin (ut' + C), or,, +C, C = -o>f = -*,, , ;, , a>t., /', , before the, , we have, , (///), , =, , 0., , the epoch of the particle in the circle of reference., = a sin, (wte)., Substituting this value of C' in (///'), we have y, is, , (d), , And,, , //", , we, , start counting, , particle has passed through its, , y, , = 0,, , whence,, , And,, , ojt., , we have, , C', , equilibrium position,, a>t', , where e, , //*, , i.e.,, , 0., , or, , particle has passed through its, t, , direction,, , we have y, , so that, in this case,, (r), , w/i^n, , /.<?.,, , on the other hand, we start counting time when the particle, , maximum, , a, , y, , + C)., , = 0,, (a) Now, if we start counting time when y, particle is in its mean position, moving in the positive, y = 0, when t, 0, we have, from relation (in) above, a sin, , -/", , ., , |, , I, , 2, , __ stn^ ^, 9^9-, , r, , C",, , ~r$^**, , f, , Jva, , j, , another constant of integration., , is, , so, , J, , therefore,, , when, , t, , t', , ...(iv), , after the, , mean position, we have, , f,, , ut + C' =, y =, , time from an instant, , or,, , = a sin (ut' + C),, C = cot = e., 1, , or,, , 0,, , a sin (ut+e)., , A mere glance at the above relations for y indicates that these, simple harmonic vibrations of the particle are a case of periodic, motion., (e), , relation, , Now,, (I'v), , y, , if the, , time, , t, , be increased by, , 2ir/co,, , we, , above,, , = a sin, , [o>(f, , -f 27r/o>), , e], , = a sin (o>f-f 2ir-e),, , have,, , from
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PBOPERTJES OF MATTER, , 120, , whence,, , y, , =, , a sin (wte),, , i.e.,, , the, , same as be, , ore, in, , (z'v),, , showing that the pos'tion and direction of motion of the particle is, the same as 2ir/a> seconds earlier, i.e., the particle repeat? its move*, ments after every 2ir/o> seconds. In other words, the time-period of the, Further, since this value of the time- period is quite, particle is 2^/0)., independent of a and e, it is clear that the vibrations or oscillations, of the particle are also isochronous., , The above results will also be true for angular S.H.M., if we, consider angular displacement, acceleration, velocity, etc., in place of, the linear ones., Important Note. We have seen, a S<H.M. is given by, , how, , the, , acceleration, , of a, , particle executing, , The general solution of, , y, Thus,, , if, , =, , a, , sin, , a>, , this equation is of the, , +b, , cos, , form, , cut., , the displacement of a vibrating particle be given by a, form y, sin cot+b cos cut, it is executing a Simple, ja, , = **, , relation of the, , Harmonic Motion., Clearly, as, , t, , takes up the values 0,, , assumes the values, , 0, 2ir, 4?r, 2n-rr,, , etc.,, , etc., ajt, , 27r/o>, 47f/oj, 2n7r/co,, , with y assuming the same, , value over and over again., , =, , In other words, the time-period of the motion, , form, , 2irjaj t, , Further, this equation can easily be reduced to the simple sine, as follows, :, , Let, , (Fig. 71)., , Then, clearly, a, b, and, , Now,, , y, , = c cos Q = \/(a*+b*) cos, = c sin = i/(a*+K*).sinO., = a sin cot+b cos wt., , So that, substituting the values of a and, obtained above, we have, s ;n, , wt, , (sin, , cos, , Q+^(a^Wj.cos, , wt cos 6 +cos, , cut, , wt, , b,, , sin 0., , sin 0)., , sin (a>t+0),, , which, , is, , the usual form of the displacement of a, , body executing, , S.H.M., Obviously, the displacement will have, , = 90, , the, , and, therefore, sin, , maximum, , (a>t+0), (wt+0), And, since the maximum displacement of the particle, , value, , 1., is, , equal, , t
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SIMPLE HARMONIC MOTIOJf, its, , amplitude,, , we Have, , amplitude of the vibrating particle, here, Again, the velocity of the particle is given by, V, , so that, the, , And,, , =, , maximum, , -., value, , finally, acceleration, , Hence,, , =, , maximum, , o^v, , r=, , v/a, , 2, , of the particle, , +fe, , =, , 2, ., , v., , -vy., , value of acceleration of the particle, , 50. Composition of Two Simple Harmonic Motions. Just as a, particle may be subjected to two forces or two velocities simultaneously, so also we may have a particle under the action of two simple, harmonic motions at the same time. Its final motion will then be the, , resultant of the, , two simultaneous simple harmonic motions impressed, , mean that it will execute both the, motions simultaneously, any more than a particle,, having two, velocities impressed upon it, w ill move in -both the directions at, the same time. All it moans is that its resulting motion would be, one as though it were simultaneously executing the two motions, upon, , it., , It does not, of course,, , r, , together., It should be clearly understood, however, that the simultaneous, execution of two rectilinear simple harmonic motions by a, particle, is no guarantee that the resultant motion of the, particle will necessarily be rectilinear or harmonic., Indeed, if their time-periods be, incommensurable, it may not even answer to the definition of a, , vibration., , We shall now take up first the simpler case of the composition, of a S.H.M. along one direction with a linear motion in a, perpendicular direction and then pass on to the composition of, simple harmonic, motions along the same straight line and at right angles to each other,, both graphically and analytically., Graphical Method., , 1., , Composition of a S.H.M. with a Uniform Linear Motion perpendicular, resultant motion will, in this case, be a sine curve. This, may be, easily seen by attaching a small spike or style to the prong of a tuning fork (at, right angles to its length) and then drawing it* uniformly over a smoked plate of, glass, with the style just touching the plate, in a direction at right angles to that, of the vibrations of the fork. It will be found that a series of sine curves are, traced out on the plate, with the, direction of motion of the fork as the horizontal or the time-axis and the direction of vibration of the, prong as the vertical, or the displacement-axis., (/), , to, , it., , The, , Composition of two linear simple harmonic motions in the same direction., two simple harmonic motions take place in the same direction, their resultant, is also a simple harmonic motion, defined by the resultant of the vectors which, define the two motions, this resultant vector being obtained by the, ordinary law, of vector addition. This will be clear from the following, If, , :, , ", , *Qr, holding the fork in position and moving the platp.
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122, , 'KOFJEKTIES, , Olf, , MATTER, , Let two simple harmonic motions, having the same time-period tiit, and phases, be represented by the projections of the vectors, , different amplitudes, , ,--*-., , J, , Fig. 72., , OP and O, , respectively,, , on the, , axis of y,, , (Fig. 72),, , and, , let, , equations of, , their, , motion be, , y, , =*, , a sm, , cof, , and, , >>, , where a and 6 are their amplitudes, and, and AOQ respectively., , =, , <o/, , b sin, , anJ, , equal, , to angles, , AOP, , Then, if DEFG be the sine-curve for the first motion, and DffJK for the, for the resultant motion by adding up the, second, we obtain the sine-curve, ordinates of the two curves at all points, because the displacements of the two arc, in the same direction and can be added up algebraically., , DLMN, , Now, the curve DLMTVis the same as would be obtained for the rotation, of the resultant vector OR, whose projection on the axis of y, therefore, gives, the resultant of the two motions., , RB = CB+RC, , For,, , PA+RC., , = OP. sin AOP+PR.sin CPR., = a sin, sin (a>/-f0),, OP = a, PR = OQ = LAOP = w/ and LCPR = LAOQ, co/-|-/>, , because, , />,, , **, , w/4- $, , Thus, the resultant motion is also a S H.M. and takes place along the same, and, have the same velocities),, line and, (since the rotating vectors OP,, // has the same time*period as the two component motions., , OQ, , The amplitude, , a' of the resultant, , motion, , OR, is,, , clearly, equal to, , OR., , OR*~OP*+OQ*,20P.OQ.coS POQ., , Now,, , 2, , a, , Or,, , 0*-}-6 -f 2ab.cos f,, , a', , whence,, , Now,, motions, cos, , if, , =^, , <f>, , 1,, , 0, i.e., if there be no phase difference between the two, and, therefore,, , algebraic, , The phase, , ROB, such, , sum of the amplitudes of the two component motions., motion is, obviously, given by the angle, , angle of the resultant, , that, tan, , ROB-, , JRj?, , OB**, , CB+RC, OA+AB, , PA+RC
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SIMPLE SARMONIO MOTION, , 123, , 6r,, , Now,, , =, , where LPOR, first, , e , the, , phase angle by which the resultant motion, , is, , ahead of the, , motion,, , ROB =, , tan, , -, , tan, (at, sin of, , a cos, , Now,, , yiw/, , <rt, , the start,, 5/w, , Hence, , tan e, , =, , w/, , =, , +, , where e, , =, , tan-, , t, 1, , b sin, , =, a, , ;, , -f, , and, , Or, f, , is, , cof, , 6 cos, , and, , ~, , <f>, , bcos, , Thus, the resultant motion, ., , 0,, , == 0,, , b sin r, , --., , -, , a, , t, , _-j-_b_sin, , co/, , .-., , to/, , cogjj -f /> fas, cos ^, b sin, , wf, , c<?5 cuf, , =, , ahead of the, , =, , =, , ;, , cof, , MM ^, , <*t, , sin, , <f>, , so that,, , 1., , _, , b cos, first, , motion by a phase angle, , e,, , <f>, , -., -r*, b cos, , -f, , <f>, , Resolution of a S. H. M. into two components in the same direction., (//), The converse of the above is also tiue, viz. that a simple harmonic^ motion may, be resolved into two by resolving its rotating vector into two vectors, in accordance, with the law of resolution of vectors, each vector defining a component simple, harmonic motion., ,, , (Hi) Composition of two linear simple harmonic motions at right angles, The resultant of two, S.HM's, impressed simultaneously on a, to each other., particle, along directions at right angle* to each other, is a curve lying in the, , plane containing the two motions and its character depends upon the amplitudes, time-periods (or frequencies) and the phase difference of tee two component motions. Let us consider the different cases that arise., , When the time-periods (or frequencies) of the two motions and their, (a), phases are the same, but their amplitudes are different. Let the two motions be, defined by the rotation of the, vectors in circles (/) and (//), respectively, (Fig. 73), /<?., let, (i) and (//) be the circles of reference of the two motions, with, radii equal to their amplitudes, respectively,, , say, , Divide the two, , a, , and, , circles into, , b., , a, , number of equal, , parts in the, ratio of the frequencies of the, , two motions,, , in this case,, , 1:1,, , as shown, (each circle being, divided into eight equal parts,, for the sakg of convenience),, the starting point of the rotating, vector being marked zero. Then,, draw straight lines passing, , through points bearing the, same numerals in the two circles, and parallel to the axes OX, and OY respectively, along, which the motions take place, in the, , two cases*, , Fig., , 73,
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PROPERTIES OF MATTER, , 124, , Mark, intersection., , the points where these lines intersect and join all these points of, It will be found that, in this case, the straight line AB is obtained, the path along, , as, *, , 2, , which the, , resultant motion takes place,, , the arrow heads indicating the, direction of motion about 0., , And, as, , be readily seen,, , will, , this straight line is the, , diagonal of the rectangle with sides, , 2a and 1b, , ;, , and the amplitude, vibration of, , of the resultant, , the, is,, , or, , particle,, , OA, , i.e.,, , or, , OB, , therefore, clearly equal to, , (b) When the time-periods, frequencies are the same,, are different and, plitudes, , phase difference, , is, , TT., , Here, the, , starting position ,(Fig. 74), of, the vector in circle 00 is at, , 74., , the top of the circle, as shown, instead of at the bottom, (as in the first, the second motion being ahead of the first by a distance equal to half it*, , case),, , path, -the other numerals being shifted accordingly. Again, drawing straight, lines through the same numerals and parallel to the corresponding axes OA'and, Or, along which the two motions take place, and joining their points of intersection, we get the straight line CD, inclined in the opposite direction, showing, a straight line motion, about 0, but inclined the, that the resultant motion is, , again, , other way, (i.e., the other diagonal of the rectangle of sides 2a and 2b), the direction of motion of the particle being as indicated by the arrow-heads., , When, , (c), , difference is, , 7t/4., , the time-periods are the same, amplitudes different, and the phase, , We again, , exactly as above,, , proceed, with the only, , difference that, here,, , we, , ---;, , ^^-^~, , ........, , shift the, , zero, or the starting position of, the radius vector, by one-eighth of, its path in the case of the second, , (lower) circle of reference, (Fig., second motion being, 75), the, ahead of the first by r/4,, , of, , Joining smoothly the points, of the straight, , intersection, , lines, , through the same numerals,, to the two axes respec-, , parallel, tively,, , we, , get an oblique ellipse as, , the resultant path of motion of, the particle, the direction of motion along, , it, , being indicated by, , the arrow-bead., , 75., , .-
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S1MPLB HARMONIC MOTION, , When, , (d), , 125, , the time-periods are the same, amplitudes different, andtha, , phase difference is rr/2. In this, case, the starting point of the, radius vector in the second, , of reference, , circle, , original, , its, , difference being, , Then,, before,, , taken a, , is, , path ahead of its, the, phas, position,, , quarter of, , we, , get, , rc/2., , as, proceeding, an ellipse as the, , path of the resultant motion,, H ith its axes coincident with the, directions, , of the component mo-, , tions, the starting, , point being, , O, , and the direction anticlockwise,, as shown in Fig. 76, , 76., , When, , (e), , the, , I, , Ln > periods, , or frequencies are the same, amplitudes different, and the phase diff-, , erence, 0,8, , the, , is, , Here, (Fig., , 3*/2,, , 77),, , starting point of the radius, , vcuo., , in the, , second, , circle, , is, , taken three-fourths, of its path, ahead of the original position.,, , and we, an, the, , get,, , as, , in, , the last case,, , as the, , resultant path, , direction of, , motion being, and the, , ellipse, , clockwise,, , as, , shown,, , starting point being 0., , Fig, , 77., , (/) When the time-periods, or, , the frequencies are the same,, , amplitudes equal, and the phase, difference is 3^/2. In this case,, (Fig., , 78),, , circles, radii, , we, , and proceed, , when a, , take, , both, , the, , of reference of the same, circle, , is, , as, , in, , case, , (c),, , obtained as the, , path of the resultant motion, the, of, travel, direction, along it, , being anticlockwise and Jhe starting point being 0., , Fig., , 78,
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PBOPEETIBS OF MATTBB, , 126, , (g) When the time-periods, or frequencies are the same, am-, , and the phase diffHere, again, we, take the radii of the two circles, to be the same, and proceed as in, plitudes equal, , erence, , case, , is 3rr/2, , (d),, , when we obtain a, , circle,, , the path of resultant motion, the direction of travel along it, as, , being clockwise, in this case, as, , shown,, , (Fig. 79),, , and the starting, , point being O., , 79., Fig., the time-periods, or frequencies are in the ratio of, (h), , When, , 2 1, amplitudes are different,, and the phase difference is zero., :, , In this case, (Fig. 80), we divide, the two circles into equal parts,, 1/1 the ratij 2:1, (e.g.,, the first, one into 8 parts, and the second, one into four parts). Then, proceeding as before, we get the, path of the resultant motion of, the form, , of, , the figure 8, as, , shown,, , the direction of motion along, it being indicated by the airowheads, and the starting point, being O., , Fig., , 80., , When, , the time-periods, or frequencies are f n the ratio 2:J,, amplitudes are different and there, , 0), , an initial phase difference, equal to a quarter of the smaller time-preiod. As in the case, above, we divide the two circles, here also into equal parts, in the, ratio 2: 1, but shift the zero of, the second circle, one-fourth part, as, ahead., Then,, proceeding, before, we obtain, in this case, a, parabol-a as the resultant path, of motion of the particle, (Fig., 81), the direction of motion being as indicated by the arrowheads., is, , Ctf>, , Fig, , N.B, , The, , 81., , epithet 'initial* has been deliberately used here with 'phase, difference* to emphasize that the time-periods being different, the phase does not, remain constant, even though we start with the same phase originally. Inevidifference comes in between the two motions,, tably, therefore, a, , phase
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ilotiOtf, , 127, , Precisely in the same manner, we can obtain the path of the resultant, particle, subjected simultaneously to two simple harmonic motions,, perpendicular to each other, whatever their frequency ratio or the phase differ-, , motion of a, , ence between them., , The student may, as an exercise, try to determine the resultant path of, a particle, subjected to two simple harmonic motions, at right angles to each, PHASE DIFF, , 277, , Fig. 82., , other, with the time-poriods and amplitudes equal, but with phase difference, changing from to 2n, when he will find that, as the phase difference changes, from to n, the resultant path changes from a straight line, inclined one way,, through an oblique ellipse inclined the same way, a circle, and, again, an ellipse,, inclined the other way, to finally, a straight line, inclined at right angles to the first, one, as shown in Fig. 82. And, as the phase difference changes from n to 2n,, *the same figures are repeated in the reverse order, as shown., The superposition of such rectangular vibrations is of particular importance in the subject of sound, since it serves as a test for the equality of the, periods of two vibrating bo Jies like tuning forks etc. The method was first, adopted by Lissajjus and aeace the various curves thus obtameJ, v/z., those in, Figs, 73 to 81 and others, are usually referred to as Lissajous' figures., , Analytical method., , II., , Composition of two linear simple harmonic motions along the, Let two simple harmonic motions, having the same timeperiod^ be represented by the equations., b sin (o>f+<),, a sin ait and y 2, yl, (1), , same, , line., , =, , =, , where, the, , <f>, , is, , the phase angle by which the second motion, , is, , ahead of, , first., , The phase difference will throughout remain constant, because, the time-periods of the two motions are the same., Now, since the two displacements are along the same line, thd, resultant displacement y will, at any given instant, be equal to the, algebraic sum of the displacements of the two component vibrations., Thus,, Or,, , y = Ji+JV, =, a sin wt+b sin (wf-f <)., y, = a sin ajt-^b sin a>t cos, cos a>t sin, =, sin, (a + b cos (f>)+cos wt.b sin $., y, = a' cos e and b sin = a' sin e, we have, <f>-\-b, , Or,, , <f>., , cot, , Putting, , (a+b, , cos, , </>), , <f>, , y =a', , y, , Or,, , =, , sin ojt cos, , e+a', , cos, , tot sin e., , a' sin (a*t-\~e)>, , the resultant motion is also a S.H.M., along the same line > and, has the same time-period, its amplitude being a' , and its phase angte&e,, by which it is ahead of the first motion., i.e.,, , The values of, , We, , tove, , a', #', , and e may be deduced as follows, = (a^b cos, b sin, sin e, ^ and <?' cos e, :, , ^),
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PBOPBBTIES OF MATTER, , 128, , So that, squaring and adding the two, we have, b 2 sin 2 + a 2 +b* cos 2 <f>+2 ab cos <f>., 0' 2 sin* e+a' 2 cos 2 e, 2, 2, 2, =, a' (sin, a 2 +b 2 (sin 2 (/>+cos 2 (f>)+2 ab cos <., Or,, e+cos e, <f>, , a', , =, , tan e, , =, , Or,, ., , =, , a' 2, , Or,, , ,, , And, , a2 +, , 2, , cos ^, <f>., , +2ab, , Now,, , (/), , have e, , if, , =, , <, , =, , ;, , 1, , ;, , = L, , 8, , Vfl +>T2flftciw"^, e, , =, , e, , ', , -, , [_also,j/wV+c<wV, , <i', , sw--e, ----, , =, , e, , Or,, , sin*e+cos*e, , ['.', , tan-*, , ~*~ ^, , --.-, , =, , y, , -, , <f>, -, , 1, , a-{-b cos, [See page 123], , ., , fwo motions be, , 0, i.e., i/ fAe, , so that,, , b sin, ,,-, , same phase, we, , the, , in, , a' sin cut., , Or, the resultant motion will also be in phase with the two component, motions, with its amplitude given by, , i.e.,, , sum of the amplitudes of the two component motions., , e^W(?/ to the, , [See page 121., , And, , (ii) if, , we have again, , <, , ==, , TT,, , i, , e.,, , i/, , motions be, , //ze ^v<?, , ~, , a' sin, So that, again, y, Or, the resultant motion will be in phase with the, its amplitude now given by, , e, , 0,, , in, , opposite phases,, , cat., , motion, with, , first, , -* = ***, , / e.,, , /o, , ##/, , //ie, , t, , v, , cos *, , -, , -1-, , difference between the amplitudes of the two motions., , =, , &, '*.<?., //^Ae amplitudes of the two, Further, in this case, if a, 1, 0., component motions be also equal, we shall have a, Or, the amplitude of the resultant motion will be zero. In other words,, , ab =, , there will be no resultant motion at, , all., , Note. In the above treatment, we have, for the sake of simplicity,, taken one motion a phase angle, ahead of the other., The same result may be obtained, however, if we take the phase angles, of the two motions to be l and 2 respectively. For, in this case, we have, b sin, yi = a sin (u>t-\-$^ and >> a, So that,, y = y^y^ = a sin (wf-f ^) -|-6 sin, a sin w/ cos <f>ia cos at sin <^ t +b sin f cos fi+b cos wf sin ^ f, = sin ut (a cos $ L + b cos <f>t)+cos cor (a sin $ L +b sin ^ a )., <f>, , <f>, , <f>, , ., , Now, putting, we have, , (a cos <f>i+b cos, , fi t ), , y, y, , = a' cos e, = a' sin co/, , and, , (a sin, , fa+b, , cos e f a' cos, , co/, , sin, , == a' .y/, w /-f <?), as before ;, the resultant motion, is also simple harmonic, with the same, of the two component motions., , Or,, /, , r.,, , =, , a' cos e, , a cos, , <f>i, , -f, , time-period as, , b cos ^ g, , 1, Again, the values of tne amplitude a of the resulting motion, obtained in exactly the same manner, as before., , Thus,, Or,, , a', , z, , z, 2, sin*e+a' cos e, , w*(sin*e+cos*e), , =, , sin*, , Or,, , <?', , *, , 2, , (w, , 2, , ^^ f co^Vt), , a), , -f, , (a cos, , 2, h+b* sin fa+lab, 2, , cos, , -i-a*, , 2, , 2, , (a sin <f>i+b sin, , a2, , a' sin e $, , (, , _., , And,, , 8), , sin e., , f6, , 2, , (w, , 2, , <f>, , l, , +6, , fa+b, , maybe, , cos &)*., , sin fa sin fa., , cos, , 2, , fa+2ab cos fa cos, , fa., , 8, , 2, , ^i-f-co5, -f, , fa), , 2a6(5/# ^| 5/w fa i cos fa cos fa),
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129, , = a*+b*+2ab cos (&-0,), whence, a' - *Jf+b*+1ab cos, when (<j>i~<f>i) = 0, gives the same result, as above, (page 80)., , a', , Or,, , which,, , 1, , Proceeding again, as before, if,, two motions be in the same phase, we have, , - (+&), , a', if 01--, , (//), , &=, , w, j>., ;///ie, , we nave, , a', , =, , if, , a, ;, , =, , bt, , i.e., if, , or (0i, , a, , &), , 0,, , (<f>i-f*\, if the, , i.e.,, , ;, , we have, , /wo motions be in opposite phases,, , =, , a', , and, further,, , ^ =* ^, , (/), , =, , (fl-, , ;, , the amplitudes of the two motions be also equal,, the same results as above., , we obtain, , i.e.,, , Composition of two linear simple harmonic motions at right, (2), Let the two simple harmonic motions be along, angles to each other., the axes of co-ordinates 'XOX' and YOY' and let a and b be their, t, , amplitudes respectively and <f>, the phase difference between them.*, Then, if their displacements at any instant t be x and y,, , we have, , =, , x, , a sin, , cot y, , and y, , ...(/), , =, , sin ojt, , and, since sin*, , cut, , =, , cos* wt, , Or,, , Now,, , =, =, , y, , Or,, , + cos*, , y\b, , va, , fc(.sm cu/, , we have, , cos <f>+cos, , $, , -f, , cos, , [from, cos*, cot, , =, , a)t, , ==, , f, , 1, , 1, , (/),, , above., , sin* wt., , --x*\\, , ', , ^J, sin, , cot, , cot, , ("), , (cut +</>)>, , ;, , so that, cos, , sin ojt cos, , sin, , [from, , <f>),, , (//),, , above, , </>., , c, , jc, , sides,, , a, , i-+^, , ^., , 1,, , lx*/a*,, , So that, squaring both, , ', , =, , b sin, , /, , A, , Or, , Or, , a>t, , x/a, , =, , COS * +, , we have, , ~, , 2jcv, , ab, , COS, , |r, , Or,, ', , --, , .<Mh|, , the equation to an ellipse, inclined to th axes of, ordinates, and may be inscribed in a rectatnglp of sides 2a and, , This, , is, , Now, a number of, (a), , When, , the two motions., , <f>, , =, , special cases arise, , o, i.e. 9, , when, , In this case, sin, , So that, substituting these values, y*, , x*, , there, <f>, , =, , is, , no phase difference between, , and-c05^===, , in relation, , 2xy, , ^", , :, , ', , 0,, , ~, tlj, , (///), , 1., , above;, , ., , ', , 3., , ^
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FBOPtRTIE3 Of, , 130, , 0., , This, , is, , 0., , X, , y_, b, , Or,, , -t, , Or,, , JL, , Or,, , a, , the equation to a straight, , line,, , that, , a, , passing through the origin,, , XX' at, , meets the axis, , it, , tan- 1 b/a, [see case, , The, , such, , an angle, , I, (ill), (a)]., , resultant motion,, , is,, , therefore,, , along the straight line AB, (Fig. 83),, describes a S.H.M., the particle, i.e.,, along this line, with the same time-period, as that of the two component motions and, an amplitude equal to <\/a*+b 2 If the, -, , amplitudes of the two motions be equal,, the straight line AB is inclined at an, angle of 45, , to the axes of, , When, , (b), , =, , <j>, , x and, , i.e.,, , TT,, , two motions is JT. Here, sin, from (iii) above, we have, , y, , ^, +, , b, , y., , when, , =, , <, , *., , a, , the phase difference between the, 0,, , =0., , and cos, , <f>, , =, , So that,, , A, , -3-, , Or,, , 1., , x, , a, , This too is an equation to a straight line, passing through the, so that, the, bja, origin but inclined to the x-axis at an angle tan^, resultant motion is again a S.H.M. , with the same t'me-period but along, the straight line CD, (Fig 83), inclined the other way, [see case I,, ;, , (iff), (b)],, , the amplitude being again ^/ a *+b*., , When, , (c), , two motions, , <j>, , =, , ?r/2, i.e.,, , when the phase, , difference between the, , is Tr/2., , sin, , Here,, , <f>, , Substituting these values of sin, , =, , and, 1, and cos in, <, , </>, , we have, , cos, , <j>, , relation, , =, , 0., , (ill),, , above,, , 1., , This is the equation to an ellipse*, whose major and minor axes coincide, with tht directions of the two given, motions, and whose semi-axes are, equal to b and a respectively. The, resultant path is, therefore an ellipse,, (Fig, 84), which it describes once in, the time-period of each component, ,, , S.HM., , t, , [see case I,, , The, , (///),, , (d) 9, , direction of motion of the, , particle along the ellipse, determined aa follows :, , Since, , ^, , we have y, , above]., , ir/2,, , b, , sin, , may, , be, , and x = a sin <vt,, Fig. *4., bsin M+ir/2), whence, y * b cos, (ut+j), , *t,
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SIMPLE HARMOKIO MOTION, , A, , differentiating, , y with, , respect to, , =, , velocity of the particle, , 131, , we have, , f,, , =, , dyjdt, , u.bsin wt., , Now, x and, therefore, sin wt is positive in the right half of the, figure., And, therefore, the negative sign of dy/dt means that the, velocity of y is negative, i.e., it is directed downwards in the right half, 9, , In other words, the direction of the particle along the, , of the figure, ellipse, , If,, , clockwise., , is, , on the other hand,, , i.e.,, , <f>, , =, , the resultant motion, , Tr/2,, , we, , again an, , is, , x*, , v*, , have, again,, , ,, , ^- -f, , =, , a, , *, , ellipse., , =, , b cos a>t 9 we have dy/dt a).b sin cot,, sin (cot, But, since, IT} 2)=, i.e., the velocity of y is now positive in the right half of the figure, and, is, therefore, directed upwards. In other words, the direction of motion, of the particle along the ellipse is now anticlockwise., , y=b, , =, , sin, /., , =, , When, (d), 7r/2, and b, and the amplitudes are equal., <j>, , 7T/2, , <f>, , = sin ir/2 =, , 1 ,, , and cos, , substituting these, , above,, , =, , <f>, , a, i.e, , cos Tr/2, , when the phase, , ,, , In this, , difference is, , case, obviously,, , = 0., , \f, , values in relation (Hi), , we have, -- -4-, , Or, , ~ A, , =, , BC 1, , _, , \Jl', , ., , OP, , ', v, , fl*, , t, , ), , \, , whence,, This, , is the equation to a circle, whose, Fig. 85., equal to the amplitude of either, motion, so, harmonic, in, this, the, describes a, that,, case,, particle, simple, circle, (Fig. 85), once in the same time as that taken by any one of the, two component motions, [see easel, (///), (g), above] the direction of, motion along the circle being determined as explained above, in the, case of the ellfpse., , radius, , is, , ;, , A, , uniform circular motion, , may, , thus be regarded as a combination, , of two equal or similar simple harmonic motions,, other, and differing in phase by ?r/2., (e), , When, , <$>, , two motions, , is ?r/4., , Here,, , cos, , = Tr/4,, =, , <f>, , Hence, from relation, , j, , x1, , 2xy, , i.e.,, , cos-~r, , 4, , (Hi), 1, , =, , when the phase, , ^, , ;, , and so, , at right angles to, , difference between the, , also, sin, , $, , **, , v, , \/-, , above,, 1, , each, , 5., , *, , we have, _, , y1, , which Js the equation to an oblique ellipse., So that, the resultant motion, in this case,, , x1, , \/2xy, (See case, , is, , 1, , I (///), (c), , above.), , an oblique ellipse., , Thus, we see ihat the two perpendicular linear simple harmonic, motions compound into a straight line motion, when they differ in phase, or ir, and into an ellipse or a circle, when the phase difference, by
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PBOPBKTIES, , 132, , MATTJCB, , OJT, , For any other phase difference, the motion is still an, major and minor axes no longer coinciding with the directions of the two component motions, but being inclined to them., , with, , its, , When, , the amplitudes are different and the time-periods or, In the cases, dealt with above, where, are, frequencies, nearly equal:, the time-periods of the two component vibrations are identical, the, But if the, elliptical paths of the resultant vibrations remain fixed., two time-periods differ slightly from each other, there comes about a, gradual but progressive change in the relative phase (^) of the two, vibrations and the elliptical path consequently undergoes a corresponding cycle of changes, whose frequency is equal to the difference, between the frequencies of the two component vibrations., , (/), , Thus,, , when, , (i), , <, , = 0,, , the rectangle of sides la, , and, , the ellipse coincides with one diagonal of, %b, within which the ellipse lies ; for, here,, , a, , When, , (ii), , x*, , v, , <f>, , b, , --, , to w/2, the ellipse opens out to the, , increases from, , 8, , + 73-= passing through intermediate obliq ue positions.,, if a = 6, the ellipse is reduced to the form of a circle., -, , form, , 1,, , 2, , And,, , (Hi), , and finally, , When </> increases from ir/2 to TT, the ellipse closes up again, coincides with the other diagonal of the rectangle ; for now, , in the reverse order, when (f> inthe, to, 2?r,, ellipse, from, ultimately coinciding with the same, all these changes, first diagonal as in Case (/),, being shown in Fig, 82,, , And, the same changes take place,, , creases, , TT, , above., , When the frequencies are in the ratio 2 : I, or the periods, (g), arc in the ratio, 1 : 2, and the amplitudes are different., In this case,, the angular velocity of the particle in the circle of reference of, one will be double of that of the particle in the circle of reference of, the other., if, , /., , the two motions be represented by, , x, where, the, , <f>, , is, , first,, , Or,, , .'., , and, , the phase angle, , y, , = b sin, , (2wt-\- </>),, , by which the second motion, , is, , ahead of, , we have, , xja, , and, , = a sin wt, , sin wt,, , and, , = sin (2^-f, y\b, , <^), , .*., , cos, , cut, , =, , == sin 2a>t cos, , \/lsin* wt, $+cos 2wt sin, ;, , sin2, y\b s= 2 sin wt cos <*>t cos, 2 sin a>t.cos wt and cos 2wt, [v sin 2a>t, , ^+(12, , =, , substituting the values of sin wt, JC*, , I, , I i, , t, , \i-, , sin, , <f>., , <f>., , = (12 sin* wt)]., , and cos wt from above, we have, X^ \, , /, , jcos<f>+(, , .pj*, , a*t), , cos f-^.$in, , 1, , ^, , 2-~2-Jsw ^., , ~, , fin, , ^
Page 145 : 133, , SIMPLE HARMONIC MOTION, , t+^sin +, , -*/, , Or,, , 1--*, , --, , cos +., , Or,, -, , Or,, , ^ r>, , \, , h, , ~~ s, , $, , ), , n*, r~ s *, , +, , sin, , ^+2(-r, , =, \', , ., , Or,, , -y, b, (y, , sin, , r, , sin, , (, , \, , <b, , -, , sin^, , ", , d>-\~~, , fl*, , <b, , -j-, , ), , /, , t?, , 4jc 4, , 4jc^, , -L, , ), , J, , COS^, , - ,m, , ^+, , <A-|-, , [, , :, , <f>, , <f>)~\~, , sin, , ,5/w, , ^, , at), , -5/, , (-^-,/, , ^ ^ +-^ (f, +, , rt,, , +, , -, , -, , tp, , T*, , a2, , *_l), , 8, , --rih, , Or,, , .sin, , a, , =, , #+W5 #, , # _(,/, , <f>., , A y2, , 4, , Or,, , -, , >, , ^, , m, , }sin, /, , b, , /, , COS, , \, , -7, , \, , fl, , ^-(sin^ <b~\-cos^, , +, , j- sin, , 4x^ / v, cos^, , o, , ~, Or,, , Y, , <f>, , +rfi, *-l, , ., , 0., , 0., , = 0., , ., , .(A), , This is the general equation for a curve having two loops, for any, values of phase difference and amplitude., , Let us now take some particular cases, If the phase difference, i.e., </>, be eqnal to o or, (/), :, , Here, sin, which, , is, , r, , </>, , = 0,, , ~ +i;2, , and, therefore,, , b2, , 4.^2, , ', , If the phase angle, , a-, , </>, , == ir/2,, , {, , [See case, i.e.,, , sin, , <f>, , ir., , \, , x%, , a 2 \ a*, , the equation to the figure ofS., , (//), , /, , 1,, , 1, , ), , =, , 0,, , /, I (///), (h),, , above.
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134, , FBOFJCBT1BS OF, , MATTER, , This represents two coincident, parabolas, each having the equation, , or, , **._(_,), , ,, , Ot, , =, , x>, , ,, , -(, , y_b, , If, however, the frequencies differ, slightly from the ratio 2:1,, the time-periods differ from 1, 2), the variation in the resultant, path of the particle may be obtained by substituting the consequent, changes in the value of $ in the general relation above. The, (i.e.,, , :, , when, , changes, , changes from, shown in Fig. 82, above., , occurring, , to, , <, , and then from, , TT, , IT, , to 2ir are, , Note. Alternatively, the student may, without, deducing the general, equation (///), [50, (2), page 129], obtain the resultant motions in, simple casei, , as follows, , :, , Taking displacements of the two S.H.M's, at right angles to each other,, x = a sin cot,, and, y = b sin(ojt + fi),, , as, , where $, , is the phase, angle by which the second motion, have the following cases, , ahead of the, , is, , first, , we, , :, , When, , (/), , So, , We have x = a sin co/, = y/b, whence, yjx =, , o., , <f>, , =, , that,, , and, , b sin, , y, , /., , sin co/, , x\a, [See 5 J, II, (2), (a)., b\a,, the equation to a straight line,, passing through the origin ; and inclined, 1, to the *-axis at an, angle, tan- b/a (straight line AB, in Fig. 83>; /.*., the resultant, motion, here, is along the straight line AB., , which, , is, , When # = n. We have x = a, (//), S that, *1<* = sin co/ = -y/b, whence, y/x, which is, again, the equation to a, , and, , sin, , wf, , =, , -/>/,, , y, , *, , b sin w/., , [See ^ 50, II, (2), (fe)., straight line, inclined to the x-axis at an, angle, ta/r-i-A/0, (/. e f straight line CD, in Fig. 83), at, right angles to that in, case (i). The re suit ant mot ion is thus, along a straight line, at right angles to that, ., , :, , ., , in the first case., (Hi), , So, , When, , 5m, , that,, , And, Or, , =, , <f>, , ", , w/2., , We have, , co/ ==, , x/a, , 2, ^/o +, , a, , and, (sm, , y*/b*, , Wit*) f (^, , ', , x, , s, , 1, , s, , /fl ), , cc?5, co/ -f, , =, , and, , 5/11 co/, , o/, , ow*, , ^, , =, , 6, , c<?5, , w/., , yjb., a>/), , =, , 1,, , 1 ,, , [See, , 50, II, (2), (c), , which is the equation to an ellipse, with its major and minor axes, coinciding, with the directions of the two given, perpendicular motions, and whose semi-axes, are equal to b and a (Fig. 84). The resultant motion is thus an, ellipse here,' described once in the time-period of each, component motion., (iv), , When ^, , w/l and b, , x, So, , =, , and, , a, , x, , -^r+~, , fl, , ?, , *, , */n, , f, , <*t, , + cos*, , a, i.e., the, , and, , J/H co/, , <j, , sin co/, , that,, , And,, , a, , to/, , -, , 1., , >>, , amplitudes are also equal., , 6 co5, , o coi w/., , to/, , cos w/, , Or, y*+x*, , o 2 [See, ,, , 50, II, (2), (d)., , which is the equation to a circle, with a radius equal to the amplitude of either, of the two motions. The resultant motion, in this case,, therefore^ is $ drch, d$$in ffo, , timtywfad of each component motion,
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135, , S1MPLB HAKMOHIO, , in opposite direcComposition of two equal circular motions, Let twopwticles P, and Pa move with equal velocities along, , 51., tions., , XYX'Y' of radus 2a, in, opposite directions, as shown, (Fig. 86),, such that when, the point X,, l passes, Pa passes X'., and 9 be, Let the positions of, 3, as phowi, at any given instant t after, and X' respectively so, starting from, the same circle, , P, , P, , P, , ,, , that,, , y, , X, XOP^X'OP^t =, , ;, , <of,, , the angular velocity of P! and, , Now, we know, , where, , Pt, , o> is, , ., , a circular motion, , that, , equivalent to two equal linear simple harmonic motions with a phas? difference ir/2, and along perpendicular directions to each, other, (sae pige 131)., is, , The circular motions of both P, and P, may, therefore, be resolved along perpendicular directions XX' and YY'. Then, the displacements x l and JC 2 of P, and P.,, will, at the given instant, be, ,, , equal in magnitude but opposite in direction along XOX' and will,, therefore, cancel each other out, but their displacements y l and y t, along YOY' will be equal and in the same direction, so that the, resultant displacement, , Since, , y, , =, , is, , yl, , =, , yt, , yl, , -f, , y9, , y^y^y^ along YOY'., a sin $, a sin wt, we have, a sin <ot -f a sin wt =* 2a sin wt., , given by, , =, , =, , the resultant motion is a linear simple harmonic motion along the, diameter YOY', at the extremities of which the particles Pl and Pt, cross each other as they describe their circular motions., , Or,, , And clearly, the amplitude of the resultant motion is 2a, and its, time-period the name as that of the two constituent circular motions., 52., Energy of a Particle in Simple Harmonic Motion. The, acceleration of a particle, executing a S.H.M, is. as we know, directed towards its equilibrium position, or in a direction opposite to that, in which y, the displacement of the particle, increases., Hence, work, is done during its displacement, or the particle has potential energy., Also, the particle possesses velocity and, therefore, has kinetic energy., Thus, it has both potential as well as kinetic energy, or its energy, is partly potential and partly kinetic., And, if there be no dissipative, force at work, i e., if the energy is not dissipated away in any way,, the sum total of the two remains constant, although as the displacement increases and the velocity decreases, the potential energy increases and t!ie kinetic energy decreases., ,, , Now, when the particle has its maximum displacement, positive or, negative, its velocity is zero and, therefore, its kinetic energy is then, eero ; so that, in this position, the whole of its energy is present in the, of potential energy. And, when the particle is in the equilibrium, , form, , its displacement, who'e of its energy is, , position,, tfre, , is, , zero, , and, , its velocity,, , maximum, , MW present in the form of kinetic, , ;, , so that,
Page 148 : m, , If, be the ma?s of the particle a, its amplitude and 27r/w, its, if o> be the angular velocity of the rotating, time-period (T), i.e, vector, or that of the particle in the circle of reference, we have, ;, , ,, , of the particle, , velocity, , And, , its kinetic, , .%, , and, , its, , in its, , equilibrium posit ion =aa), a, , energy =*\m. (aw), , =wa w, , z, , potential energy=Q,, its total, , Or,, , = J/w7, , energy, , 2, , v, , its, , [, , 2, , 2, , co, , 2, , maximum., , ,, , displacement, , ii, , zero., , -f 0=|wa oA, 2, , In other words, the whole of its energy, here, is present in the, kinetic form., Similarly, when the particle has its maximum displacement, the whole of its energy is in the potential form, which,, tna 2 w* in this position of the particle,, therefore, is also equal to, 9, , -J., , For any other position of the, , particle, its, , displacement, , is, , given, , y=asinwt., , by, A.nd /., , its, , is, , velocity, , given by dy\dt =aaj.cos, , its kinetic energy =, Hence,, 4.nd, since its total energy =, its, , s, , potential energy, , we may proceed, , Alternatively,, , We have,, , 2, , \m.(aa) cos, 2, , fyna-to, , ,, , a>t., , =%m.a, , cot), , 2, , a)*.cos, , 2, , wt., , we have, , ^ma^^^ma, , as follows, , :, , executing a S. H. A/., given by, , acceleration of the particle,, <py[dt*, , =-, , coV-, , if m be the mass of the particle, ths force F required to maintain this dis*., >lacement y is equal to m wV, Knd, therefore, work done by the force for a small displacement dy is equal to, , Now,, , this work is also a measure of the potential energy of the particle, di8placement., '., P. E. of the particle for a displacement dy ** F.dy = mu>*y.dy., Hence, total work done for displacement y and, therefore, total P. E. of, ^he particle for a displacement y is given by, it, , this, , f>, I, , P. E., , 3r,, , moj 8, , mco'./.dfy, , fr, I, , y.dy., , of the particle for a displacement y, Potential Energy, , r.,, , oc, , 1, , Jwco .^, , 1, ,, , f, jv, , ., , Thus the, , P. E. of a particle, executing a S.H.M. is, at any given instant,, proportional to the square of its displacement from its mean or equiliposition, at that instant., , lirectly, , >rmm, , velocity, , v,, , of the particle at displacement y, -,, , at, , Vnd, /. K. E. of the particle, , ience, , tolal energy, , =, , - (a sin, , to/), , *, , oc.), , cos w/., , \m(a<* cos tot)*bma*c**.cos*<t., , of the particle=//j/?0te//0/ energy-f- its kinetic energy., , And, since to = 2?r/r = 2?r, where Tis the time-period of the, particle and w, the frequency of its vibration, we may also say that, total energy, , of the, , particle, , =, , \m (-, , }, , fl=
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137, , H1BMOHIO MOTIOW, , Now, since in any conservative system the sum total of the, kinetic and potential energies of the system must be a const ant it is, clear that the former can only increase at the expense of the latter,, f, , and, therefore, attains its maximum value, when the latter is reduced, to its minimum value or zero, and vice versa., Thus, the maximum, value of any one of the two forms of energy measures the total, energy of the system, (see page 136)., , Average Kinetic and Potential Energies of a Particle in, have seen above that, at any given instant, the P.E. of, a particle is equal to }mu*w* sitfojt and its K.E. equal to |m 2 o 2 cos z wt., Now, the mean or average value of both sin'wt and cos^ojt for a whole, to 2ir), i.e., for a whole time-period is equal to i*, and,, cycle, (from, therefore, the mean or average K.E. of the particle over the whole of, its period of vibration is equal to its mean or average P.E. over the, 2 2, whole period, each being equal to J x i wa 2 co*, Thus,, |wtf a>, 53., , We, , S.H.M., , =, , average, , And,, , K.E of the, , particle, , total energy of the particle, , .., , =, =, , its, , 2, , ., , average P.E. =*, , x Jwa2 o> 2, , =, , Jma, , We, , may express this by saying that the energy, executing a S.H M. 9 is. on the average, half kinetic and, inform, the whole being present in the kinetic form, equilibrium position, and in the potential form at its, tion,, , on either, , The, , 2, , co, , 2, ., , of a, , particle,, , half potential, at its mean or, , extreme posi-, , side., , above for linear S.H.M. are equally valid, Only, the linear displacement x or y of the, particle or the body, and its mass in, are replaced by their rotational, analogues, viz., the angular displacement 6 and its moment of inertia, / about its axis of rotation, respectively., results obtained, , S.H.M., , for angular, , SOLVED EXAMPLES, A quantity, , is enclosed in a cylinder, fitted with a smmooth heavy, The piston is thrust downwards to, the cylinder is vertical., compress the gas, and then let go. Is the ensuing motion of the piston as S.H.M. ?, If so, what is its time-period ?, , 1., , piston., , of gas, , The axis of, , Let original volume of the gas be, , = V and, , its pressure, , =, , P., , Let a be the area of cross-section of the piston, (and cylinder)., , *This, , is, , so, because the, , mean value, , of sin*, , for a, , /, , ", sin*<*t.dt, j, , to 2, , i., , given by, , ----- _, , J, , *=, P *=***, J, , dot., , JO, , i 2*, , r, <*tl1-sin 2o>//4, , ~the pa$c with, , wV, , *9f *, , ^, , ", , w^pjc time-pcri0d, , T, f, , Then,, , if, , whole cycle from
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PBOPBETIE3 OF MATTBB, , 138, the piston, , be displaced through a distance *, (Fig. 87), the change, produced is given by x.a, the correspjnding change, , By, , being p., , volume, , in, in, , pressure, , Boyle's law, therefore,, , PV - (P+p)x(V-x.a) - PV-Px.a+pV-p.x.a., =, , Or,, ties,, , P.x.a+pVp.x.a., , Neglecting p x.a as the product of very small quanticompared with the other terms in the expression, we, , have, , p ft, P.x.a \-pV., , pV, , Or,, , P.a, p.a., , y, , -.x.a, , -p-.x., force/mass, the acceleration of the piston * p.ajm., ^- fl2, __ ^ flf8 . -^, f substituting the value of, , -*, , t*>, , ' Jr., ', , Vm, , F, /w, t, L P-a-> from above., accel<ra'ion of the piston * A*-.x,, a constant of proportionality, which is equal to the acceleration, -, , P a*\Vm, , Pa*, , -, , =, , acceleration, , Or,, , where, , of the pistonper uwt displacement, (i.e., when x = 1)., acceleration of the piston is proportional to x,, Or,, Hence, the motion of the piston is a 5. H. M., , And, , -y-.x., , Now, the restoring force on the piston^ which is equal, to the disturbing force, is obviously equal to change in pressure into area of cross-section of the piston, p.a., , Fig. 87., , Since,, , p, , P.x.a, whence,, , .'., , its, , time-period,, , T, , 2n, , its displacement., , 1, , \f, , acceleration per unit displacement, , r., A, , 2., body describing a simple harmonic motion executes 100 complete, vibrations per minute, and its speed at its mean position is 15 ft. per second., What, is the length of its path ?, What is its velocity when is its half way between its mean, , position, , and an extremity of, , Here,, , time-period, , and, , velocity of, , its, , T of, , path ?, the body, , body, , at, , is, , 1/100 mt. -= 60/100, , mean, , position, , =, , =, , '6 sees., , IS ft. I sec., , Since velocity of a body executing a S.H.M. = aca, at, is its amplitude and to, its angular velocity, we have, , it*, , mean, , "*, , position,, , where a, , flo>, , Wh encc,, Now,, , =, , 15, , Or,, , a.lr.lT, , 15,, , .--_.-_., , [, , ., , or,, , -, , v, , <o, , 1-432, , 2rr/r., , A, , of path of the body = /we? 1/5 amplitude, because, same distance on either side of its mean position., Hince, length of path = 2* 1'432 = 2'864/r., Again, velocity of a body at a displacement y is given by, /e/t#//j, , v =, , Or,, , So, , 2, , 6>\/a*, , >>, , it, , goes the, , ., , its, Here, displacement of the bodv, - a/2 - 1-432 - -716//. amplitude/2., >>, , that,, , v, , -, , ~ V(f, , :, , 432), , 2, , ^716)^, , =, , \/(T432, , 1 716)(i'432-'71Q., , -V/2T481T71T-. 12-99 /r./^c., , Thus, the length of path of the body is 2 864 /Jr., and its velocity when it is, half way between its mean position and an extremity of its path, is 12-99 ft. I sec., If the earth were a homogeneous sphere, and a straight hole were, 3., bored in it through its centre, show that a body dropped into the hole will execute, a S.H.M. , and calculate the time-period of its vibration. [Radius of the eartl*, 4009 miles, aqd value of f op its surface - 32 ft. per sec. per $ec.J
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139, , SIMPLE HABMONIC MOTION, , We know that the force with which a body is attracted by the earth towards its centre is equal to the weight of the body, (m^), and also equal to, G.m.MIR*, where m is the mass of the body M, the mass of the earth ; JR, the, of the, radius of the earth, g, the acceleration due to gravity on the surface, earth and (7, the gravitational constant., G.m. MIR 9, Or,, g - G.MJR*., nig, 4* 8/3, and, thereSince the earth is supposed to be a sphere, its volume, fore, if A be its density, we have, ;, , ;, , ., , its, , mass,, , M * 4* R*.&/3., , So, , *, , that,, , -~, , ;~. G, , ^, , ., , ** 4.7t/?A.C7/3., , If the value of acceleration due to gravity at a distance, of the earth, (Fig. 88), be g' t we have, as above,, , Dividing, Or,, , r, , ., , .(0, , below the surface, , by (i), we have, 8'lg- l.*.(R-r)&.Gl.*.R.&.G., g'Ig=(R~r)lR., , (//), , = g(R-r)IR - (R-r).glR., Thus, the acceleration of a body at a distance, #, , Or,, , ', , (# r) from the centre of the earth is equal to, (R-r).glR and since g/R is constant, /A/5 acceleration, is proportional to (/, r), //ie displacement of the body, from the centre O of the earth. The body, therefore,, executes a S.H.M., and its time-period is given by_, ;, , _, , ", , IT//?, , R, , Now,, and, , ^, , 4000 m//e5, =*, , 32, , ft. I sec, , /?, , V, , 3 //., , and ^ in relation, , ., , A/, , 5105, , 4000 x 1 760 x, , ., , substituting the values of, , 2, , -, , 1, , 32, , 2ff, , (111), , above,, , we have, , A / 12^176x107, , V, , 32, , 85-07 m/roife*, , sec*., , Thus, the time-period of vibration of the body would be 85'07 minutes., If a body executes a simple harmonic vibration in time TI, under one, 4., constraining force, and in time T 2 , under another, what will be its time-period under, both forces together ?, , Let rrass of the body be, m,, let its acceleration under the first force FY, second, F,, ,*, , and, and, , M, , .,, , F,/m, , Then, clearly,, its, , ;, , fl t, , both the foices, , Ff /wi, , ;, , a, , a l$, at, , Fi+Ft bt, j, , '.', , a., , arc., , L, Also, the ac:eleration of a body executing a S.H.M. is proportional, displacement x from the equilibrium position, i.e., acceleration a oc jc., , where, , /* t , ^,, respectively., , and, , /*, , are the constants, , to, , of proportionality in the three caset, , Again^ because the time-period of a body executing a S.H.M., Vf/^ where A* is a constant of proportionality, we have, , T, , So, , f,, , be, , is, , given by, , 2n, , that,, , and, , where Til the time-period of the body under both the forces acting together
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PROPERTIES OF MATTEL, , 140, , Since acceleration a, under both the forces acting together, will obwe., viously be equal to the sum of the accelerations under the individual forces,, , have, , And, therefore,, , (, , *.)'., , a, , =, , ,, , =, , J___, - 1, , or, u, , i, ,, , -f-MT, , t, , r=, , ', , \I, , V, , ^, , ^, , (*- )'. * + (, , _, 8, , JVIV, -, , r,l r.', , is, , )'., , ~, , V, , r 2 -^l, , r, , ', , ', , ', , ', , v+r,', , under both the forces together will be, , of weight 6 gins and of external diameter 2 cms. is floated, water by placing 10 gins of mercury at the bottom of the tube. The, Find the time of oscillation., depressed by a small amount and then released, (Oxford and Cambridge Higher School Certificate)., 5., , vertically, , tube, , O'r, ur, , ., , A/, , TV+r,'", , Thus, the time-period of the body, , A, , x, , -, , we get, , Or, dividing by (2z)*.x throughout,, , Or, or,, , fli+fl|,, , test tube, , in, , =, , mass of the tube and mercury, , Here,, , and, , external radius of the tube, , f6, , ', , =2/2, , =-, , =, , 16 gms., cm., 2, n sq. cms., rr.l, 1, , area of cross-section of the tube =7rr 2 =, Let the tube be depressed through a distance x cms., f Taking density of, Then, volume of the water displaced = KXX = TCX c.cs., = :rx.l. = x.x gms.wt. {_ water = gm.jc.c., and, ,,, ,,, weight ,, ,,, 1, , Therefore, upward thrust experienced by the tube, , of the water displaced,, , Hence, , g dynes., , e., TT.X, , i, , acceleration of the tube, , Since ^r^/16, displacement, , ;, , Hence,, , //re, , a constant, say,, , Px, , and, therefore,, its, , ., , Or,, , is, , of the tube, , acceleration, , time-period, , 2,, , equal to the weight, , is, , = ^f, 16, /*,, , ^, , ==, , \'.'acc. =*, , x., , 16, , L, , '--., , mass, , we have, , i.e., acceleration, it executes a, , ;, , of the tube, , proportional to, , is, , its, , S.H-M., , is, , given by, , \/~T, V ng\6, , =, , 2rt, , /, , =, , 2?c, , \i, , =, A/I?, V *, , //me of oscillation of the tube, , =, , 2, , A/, V ^, , -', , '4527 sec., , A, , particle executing a S H.M. has a maximum displacement of 4 cms., and its acceleration at a distance of 1 cm from its mean position is 3 cm /sec 2, What will its velocity be when it is at a distance of 2 cms from its mean position ?, 6., , ., , amplitude of the particle, , Here,, , and, , its, , Now,, , acceleration,, acceleration -, , where x, , is, , when, x./, , 2, ., , its, , is, , equal to 3 cms. /sec*., , the displacement and co, the angular velocity of the particle., <o* * 3,, co, V3, whence,, , =, , Or,, , 4 cms., , displacement is 1 cm.,, 8, 3 = 1 to ,, Or,, , radians/sec., , .Now, the velocity of a particle executing a S.H.M., v, , where a, .*., , when, , *., , given by, , the amplitude and x, the displacement of the particle., the displacement of the particle, i.e., x ** 2 cms., we have, * ^3, 2, v, 1(3, ^ 3 A/12., -v/4^2, , is, , -, , Will, , is, , =, , ., , Vlo^ =, , the velocity of tbe particle at a distance of 2 pm^. frorn, , it|, , mean, , positipr
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MOflOU, water up to a height, vertical U-tube of uniform cross-section contains, its, that if the water on one side is depressed and then released,, calculate its, motion up and down the two sides of the tube is simple harmonic, and, \Uelnl. J"4/), tinuk nprinH, two limbs of, Let 'AA', (Fig. 89), represent the level of the water in the, to 0,, the U-tube, to start with, and let the column on the left be depressed up, through a distance x cms. Then, the column on the right, will naturally go up, say to the level C, such that the difference of levels in the two limbs is now, B'C, where B' is at, 7., , of 3D cm?., , A, , Show, , the same level as B., The weight of this column of length 2* will now act, on the mass of the water in the tube, as a result of which it, will oscillate, , six;, , up and d :>wn., , Now,, , obviously, the weight of this, , its, , column of water, , B, , = its volume x its density xg., = ax2xxl xg dynes., , massxg, , Or, fora acting on the mass of water in the tube, , =, , And, mass of water, .*., , acceleration, , both, , =2x30xaxl =, , produced, , in, , 2jca.g, , ~, , **, , 2.x.a.g. dynes., , in the tube (in, , its, , limbs), , 60a~, , 30, , '30, , *-**, , Where g P, , acceleration is proportional to x, the displacement, , Or,, , period, , Fig. 89., , 60a gms., the mass of water in the tube, , Hence, the motion of the water column, given by, , ""*' a constant-, , of the water column., , simple harmonic, and, , ft, , its, , time-, , is, , t, , 'V?, The water, , T, , in the, , ', , *, , -, , 09S,, , will thus oscillate with a, , time-period equal to, , 8., Show that the time-period for the swing of a magnet in the earth's, is the magnetic moment of the, \/i /MH, where, given by t, magnet, I,, moment of inertia about the axis of suspension and H, the earth's field., Lei a magnet NS, of pole strength m, be suspended so as to make an, angle with the earth's field H, v Fig. 90 >., , field Is, its, , U-tube, , Or, , ', , M, , =, , Then, clearly, the forces acting on its two poles are, and mH, as shown. These two forces being equal,, opposite and parallel, constitute a couple, whose moment is, equal to the product of one of the forces and the perpendicular distance between them., , mH, , So, , on the magnet, , that, couple C, acting, , - mHx NS sin, , m x NS, So, , mHxST., [v ST - NS sin, , a., , C*=MHsin*,, , Or,, If a, , ,, , M, , ,, , be small, we have, , that,, , the magnetic moment of the magnet., sin a, (in radian measure)., , =, , C =MH.*., , Since the magnet is in equilibrium, this must be, balanced by the restoring couple set up in the suspension, i.e , by I.d<*ldt, where dv>ldt is the angular acceleration of, the magnet and /, i is moment of inertia about the axis of, So that,, Ldu/dt *= A///.a,, suspension, , <M#//M =, , where /*, MH\1, a constant,, ccj*s\, *H should be noted that the time-period is the same as tliat of a simple, pendulum of the same length * s the height of the water column* i.e., of length, Or, dtafdt, Or,, , equal to 30 cms., , /*.,, , d<*\dt, , .'-.-
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MATTE*, , PBOFJBBTIXS Of, , 142, , the acceleration of the magnet Is proportional to its displacement. The motioh, of the magnet is, therefore, simple harmonic and its time-period is given by, , i.e.,, , A particle is moving in a straight line with simple harmonic motion., has the values 5 ft./sec. and 4 ft/sec. $ when its distances from the, Find the length of its, centre-point of its motion are 2 ft. and 3 ft. respectively., it is at a, path, the frequency of its oscillation, and the phase of its motion, when, distance of 2 ft. from the centre., (London Higher School Certificate), 9., , Its velocity, , We know that, , Length of path., , So, , that, in theory/ case,, , in the second case ,, , o, , v=5//./s<?c, , and x~2//., we have, , ,, , - o'.(a -4)., when v = 4 //. 'sec. and jc=*=3//., we, 16- V-9)., , 5=yV--, , 2*., , 2, , 25, , Or,, , And,, , when, , v, , ], , ,, , ...(/), , have, , 4<o y/ aZ3., fl, , 8, , Or,, /., , dividing, , (/), , by, , (//),, , 25a 3 -225, , Or,, , -, , we have, , **-, , -, , 16a'-64., , 2, , Or, 25(a -9)~16(a -4)., , 25a a ~16a*, , Or,, , 9a'=161,, , Or., , ..(//), , a, , *., , ', , Or,, , -, , ~, , 225-64., , ^, , -, , And/., , <yi|i-4-23/ir., Or, the amplitude of the particle, , 4'23 ft., , is, , Sioce the length of the path traversed by the particle, tude, (as it travels equal distances on either side), we have, length of path of the particle, , Frequency., period ; and since, , The frequency of the, /, , =, , 2*/w,, , =, , 2.a, , 2x4'23, , particle, n, , I//,, , is, , =, , twee the ampli8'46//., , where, , t, , is, , its, , -, , "-te/i-^r', (/), , or, , The value of w may be obtained by, Thus, we have from (/),, , ( 'v/), , substituting the value of a io either, , (//)., , 1 2)(4, , 23-2, , \/6 23 x 2'23, , And, therefore, substituting the value of w in, *ir-??~** -sy, 2", \/6 23x2'23, , n, , Thus,, angl.., , ......., , the/rtfgtteflcj', , of the particle, , Here,, , 2, , 2/r., , -, , and, , 4-23 sin $., , is, , above* we haro, , -2135., , a, fl, , Or, , -4729 *, , (///), , "* 0*2135., , We h.vc the relation, ,, x, , 5, , 5, , ^35, , t, , Muue, , time-, , we havo, , |., , 4'23/f., , gin 6, , -, , /w^ and, , -, , r,, , thi, , '4729., , 28' 13'., , Hence, the phase of the motion of the particle, when, is 28 13'., , its, , distance, , is, , 2ft., , from the centre,, , A light elastic string is suspended vertically from a, , point and carries A, lower free end, which stretches it through distance / cms. Show, that the vertical oscillations of the system are simple harmonic in nature, and of a, time-pwiod equal to that of a simple pendulum of length / cms., 10., , heavy mass at, , its
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143, , SMPtB HARMONIC MOTION, Let original length of the string ,45, (Fig. 91), be mass attached to its lower end be, mg dynes., the downward force acting on it =* mg dynes., Then,, And, if T be the tension of the siring (upwards), we have, , T, , mg, because the string, , Young's modulus for the string,, , Now,, Or,, , stress, , Obviously,, , stress, , where a, So, , L cms, , and, , ,, , lot, , the, , in equilibrium., , is, , Y, , i.e.,, , -. -, , Yx strain., , =, , T[a, , and, , t, , strain, , =*//,, , the area of cross-section of the string., , is, , ~^, , that,, , Y.4, J, , T-, , Or,, , ., , j, , 2, , H-, , K., , i*, ', , m^, , Since, , we have, , T,, , =, , =, , ./,, , JL, , 7a, , m^, , Ar, Or., , m^, , i.*, , *" (/), , "/"-, , If the string be pulled down a little through, in the string acting upwards will, clearly, ba, , =, , ., , a distance x, the tension, , ~.(l+x)., , (/ix), , [See, , ~ mg, , And, since the downward force acting on the string, force acting on the string will now be, , _, Or,, , ,, , ,, , retultant, , Now,, , f, , upward force, , -, , ', , above., , tht resultant upward, , t, , mg.x, ----, , ., , = ~~ =, , -, , acceleration, , mg(l+x)mgl, - * =, , =*, , (/), , where, , ^.x,, , ^// =-, , /*,, , a constant., , acceleration oc displacement., , Or,, , the oscillations are simple harmonic in nature, //. M. is given by /, body executing a, , .*,, , ;, , and since the time -period of a, have, , = 2^^^^ we, , ., , MJ/ ^/ a simple pendulum of, , A, , length, , 1, , cms., , a straight line with simple harmonic motion, of, 11., particle moving, period IT/CO, about a fUed point O, has a velocity ^3 6o>, when at a distance b from, 0. Show that its amplitude is 26 and that it will cover the rest of its distance in, time ?t/3c>>., in, , We know that, placement y>, , is, , the velocity of a particle executing a S.H.M., at a disgiven by <*\/a*^y* 9 where a is Us amplitude., , Here, displacement of the particle, , ment, , is, , i|3.ta>, , 36', , Or,, , *, , Or,, , y, Now, we havt, Hore,, y =6, and a, , Or/, , */n, , ":\x, , *, , liual, , t, , ., , l, , -, , CD, , /a *31, , and, , its, , velocity at this, , displace*, , Or,, , a 1 * 46*., the amplitude of the panicle, , -o -^,, 1, , whence,, , *, , 26, i.e.,, , =*, , a sin, , o>r., , 26., , And, therefore, 6, , =, -, , 6/26, ^, , Or,,, , is b,, , \3.o>., , Or,, , i,, , wf, , **, , 26 sin, i//!" 1, , is 26., , *f., , J, , *, , n/6., , ^/6ca., , Hence tho time taken by the, , particle in covering the distance 6, , from, , O, , it
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PROPEETIBS OF MATTBK, , 144, , Now,, 2*/co or,, , time, , since, 7t/2o>, , it takes time >/co to complete one vibration, it will take i of, to complete Jr/i of its vibration, i.e., its amplitude, 2b. And,, , time already taken by, .*., , the time, , it, , it is :t/6<o., , will take to cover the rest, 7t, , 2o>, , 7t, , TC, , STC, , 6o>, , of its distance, n, 2n, 6o>, , 6o>, , EXERCISE, Deduce the equation, , 1., , is, , clearly equal to, , 3o>, , IV, , for the simple, , harmonic motion of a, , particle., , Two simpl e harmonic, , motions, having the same period but differing in, Show that, phase and amplitude, are acting in the same direction on a particle, the resultant motion is simple harmonic, and deduce the expression for the, (Calcutta 1940), resulting amplitude and phase., 2, Find the resultant of two mutually perpendicular S. H. motions, which agree in period but differ in phase. Consider the important cases for, (Punjab, 1953), phase difference varying from to 2n, 3., A particle executes a S., Af- of period 10 seconds and amplitude 5, rt. Calculate its maximum acceleration and, velocity., Ans. 1-974 ft Isec*. ; 3'l42ft.jsec., , H, , 4., The path of a b^dy executing a S. H. M. along a straight line is 4, cms. long and irs velocity, when passing through the centre of its path, is 16, cms. /sec- Calculate its time-period., Ans. '7854 sec., :, , H, , 5., The maximum velocity of a particle undergoing a 5., acceleration at 4ft. from the mean position is Ibft.fsec*., Ans. (i) 4ft., amplitude and (ii) its period of vibration., , and, , how, , M. is, What, , its, , 6., Explain the characteristics of a simple harmonic, to find the velocity at any phase of the motion., , ;, , (//), , 8 //./sec., is (/) its, , 3'142 sees., , motion and show, , A, , particle executes simple harmonic motion of period 16 sees. Two, seconds after it passes the centre of oscillation, its velocity is found to be 4ft., per second. Find the amplitude., (Madras, 1949), Ans. 14-41 //., 7., Define simple harmonic motion and show that if the displacement of, t moving point at any time is given by an equation of the form, x, a cos o>f + sin /,, the motion is simple harmonic., If a, 2, determine the, 4, nnd, 3, b, period, amplitude, maximum velocity and maximum acceleration of the motion., (Madras, 1949), t, , =, , =, , =, , Ans. (/)3-142, (i7)5,, , (//), , 10,, , (iv)20., , Find the velocity aad acceleration of a point executing simple harM, monic motion., ^, A point describes simple harmonic motion in a line 4 cms. long. "Ita, velocity, when passing through the centre of the ljre, is 12 cm. per second., Find the period., (Calcutta, I94<r), 8., , Ans., 9., , 1*047 sees., , Define a S.H. motion, explaining the meanings of the terms, period,, , amplitude and phase., , A particle is subjected simultaneously to two S.H. vibrations of the, lame period but of different amplitudes and phases,, perpendicular directions., Find an expression for ttie resultant motion and show that the path traced by, the particle is an ellipse., For what conditions may the path be a circle and a straight line ?, , m, , (Calcutta,, , A mass of 15 Ibs., , suspended from a fixed point by a light spring., In the equilibrium position; the spring is extended by 15 inches. The mass isttien, pulled down by 4 inches and released from rest. Show that it executes a S.H.M., ana calculate its time-period Also calculate the energy of its mass., 4Q ft.-pound*l*., c., 10., , is, , ',
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SIMPLtt iUtttfOtflO, , Show, , 11., , the distance of, , compound pendulum would swing most, , that a, , its e.g.,, , MOflOH, , from the, , rapidly, , when, , axis of oscillation equals its radius of gyration., , A thin and square metal plate, of aside 2/, is suspended from one, 12., corner so as to swing in a vertical plane. Calculate the length of the equivalent, Ans. 4 A/2// 3., simple pendulum., 13., Calculate the time-period of a circular disc of radius r, oscillating, about an axis through a point, distant r/2 from its centre and perpendicular to, its plane., Ans. 2n\/3rl2f., 14., Find the velocity, acceleration and the periodic time of a point executing Simple Harmonic Motion., ", , A, , particle is moving with simple harmonic motion in a straight line., the distance of the particle from the equilibrium position has the values, x l and * g , the corresponding values of the velocity are u and a, Show that the, , When, , ., , period, , is, , Find also, , (i), , the, , maximum, , velocity and, , (11), , the amplitude., , (Madras, 1949), , A, , moves with uniform speed in a circle. Show that the, motion may be resolved into two simple harmonic motions at right angles to, each other. How do they differ in phase and amplitude ? Show how the, potential and kinetic energies ot a particle executing siniple harmonic motion, 15., , vary., 16., , motion, , is, , particle, , (Calcutta), that the total energy of a particle executing simple harmonic, proportional to (a) the square of us amplitude, (b) the square of its, , Show, , frequency., , Show how, on an, , average,, , its, , energy, , is, , half kinetic, , and half potential in, , form., 17. In ths HCl molecule the force required to alter the distance between, the atoms from its equilibrium value is 5 '4 x O 5 dynes per cm. What is the, fundamental frequency of the vibration of the molecule, assuming the vibration, to be simple harmonic, and the mass of the Cl atom to be infinite compared to, atom which is 1-66 x 10~ 24 gm. ?, that of the, I, , H, , (Cambridge Scholarship Certificate), Ans. 9'1 x 10"., 18., Find graphically the resultant of two simple harmonic motions at, right angles to one another (a) when the amplitudes and periods are equal and, one vibration differs in phase by */2 from the other, (b) when the amplitudes are, equal, the period of one is twice that of the other and the slower vibration is, w/2 ahead of the other., , The, , 19., , total energy of a particle executing a, , placement, , 2rc sec. is, , the swing its disand the mass of, (Oxford and Cambridge Higher School Certificate), Ans. 1 6 cms. ; 80 gms., , the particle., 20., , S.H.M. of period, , rc/4 sec, after the particle passes the midpoint of, is 8^ 2 cm., Calculate the amplitude of the motion, , 10,240 ergs, , Show, , that the motion of the piston of a steam engine is approxiif the connecting rod is long compared with the crank., , mately simple harmonic
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CHAPTER V, , MEASUREMENT OF MASS THE BALANCE, Mass and Weight. The mass of a body is the quantity o*, 54., matter contained in it and is an inherent, invariable and fundamental, property of it, quite independent of the presence or absence of any, other neighbouring bodies or of the place where the body happens to, be situated. Thus the mass of a given body will be the same at the, equator, at the poles of the earth, or, for that matter, anywhere else, in the whole of the universe., The weight of a body, on the other hand, is the force with, it is attracted by the earth towards its centre, and is equal to, the product of its mass and the acceleration due to gravity., Thus, if m be the mass of a body, and g, the acceleration due to, gravity, its weight is given by w = m.g., which, , Since the value of g changes from place to place, being inversely proportional to the square ol the distance from the centre of the earth*, the weight, of the same body differs from one place to another, being about half a per cent, greater at the poles than at the equator, twenty-eight times its weight on earth,, , on the sun and about one-sixth, , its weight on earth, on the moon., be seen that the weight of a body in a variable property of it,, depending not only upon its own mass but also on its distance from the centre, of the earth, i.e., on its position, relative to the earth., , It will thus, , Then, again, since the mass of a body endows it with the property of, nertia or of reluctance to chin^e of both rest and motion, we may also define it as, the digree of resistance of matter to changes of motion.. As against this, the weight, of a bady, being a force, directed towards the centre of the earth, tends to accelerate it ! own mjtion in that direction., Thus, whereas the one resists, the other, tends to produce, motion., 1, , Nevertheless, at a place, since g is constantf, at any rate, within a small, space, the weights of two bodies are directly proportional to their masses. For, if, w and w' be their weights and and m', their masses, we have, w, mg and w' m'g., , m, , =, , So that, , --=, , w\w', mg/tn'g^ m/m'., If follows, therefore, that the common physical balance may be, used to compare masses. For, although, strictly speaking, it really, compares weights indicating a measure of their equality or want of, but since the value of g, for the body as well as the, equality,, , standard weights, placed in its two pans respectively, is the same,, the forces exerted at the two ends of tho beam, in its equilibrium, ~, *SecChapt er VI, fThis was first shown by Galileo in 159), by his famous experiment of, dropping simultaneously two unequal masses from the top of tne Leaning Fower, of Pisa, wnen they reached the ground together. The same fact was confirmed, by Newton and Uter by Bes.se I, by using pendulums with hollow bobs, filled with, materials of different densities and, observing no variations in the value of g, beyond those within experimental error. And finally, it has been shown con., , clusively by Eotvos by his experiments, , Torsion Balance., , with an ingenious modification of the
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JHURA5UHBM&NT, , OJf, , MASS, , TUB, , **', , BALJLHUJi, , position, are evidently equal, thus indirectly establishing the equality, of the two masses, irrespective of the value of g., If it be desired, however, to determine the weight of a body,, we make use of a spring balance, the stretch of the spiral spring of, which, if riot unduly large, is proportional to the force applied to it, by the weight of the body suspended from it, and this, as can be, readily seen, will be different for the same body at different places,, depending on the value of #., 55, The Common Balance. It is, in essentials, an equi arm, lever of the first order and, of, depends, for its action, on the principle, , moments., The essential feature of its construction is a symmetrical rigid beam, usually in the form of a triangular lattice girder, as shown in Fig 93, (to ensure, lightness vviih strength), pivoteJ centrally, so a* to be free to rotate in the vertical plane about the-horizonUl axis, provided by a knife-edge of steel or agate,, resting on an agate plane carried by a stout vertical pillar., long and light, pointer, hxed at tight angles to it moves over a small ivory scale below, whose, central division marks/its normal position, when the beam is in equilibrium or, at rest., A screw, worked upwards and downwards, at the top of the pointer,, enables the e.g. of the beam (together with the pointer; to be rahed or lowered,, as desired*., , A, , Two other knife edg:s, similar to, and equidistant from, the central one,, are carried by tlie beam itself on either side, with two identical scale pans, of, equal mass, suspended Irorn the agate planes resting on them., The whole instrument is enclosed in a glass case, with side-windows and, a sliding front, to safeguard against disturb ince due to air draughts or temperature variations, all weighings being earned out with the glass case propeily, closed on all sides., The bodyf to be weighed is placed in the left-hand pan and standard, weights from the weight box, in the right hand pan*, starting with the seemingly, heavier ones, until the pointer swings evently on either side of the central mark, on the wofy scale If the ////$ (/ <*., the two halves of the beam, on either side of, the central knife edge) be of the same length and the scale pans be of the same, weight , the beam will come to ic^t in the horizontal position, but if the weights, of the scale pans clilfer even slightly, it will be tilted towards the side of the, heavier weight, with the pointer moving correspondingly over the scale below., The use of the Rider. Since the weight boxes are not- provided with, weights smallei than milligram, the final adjustment for the equilibrium of, the beam is made with the help of what is called a *rider\ which, is just a piece of wire, weighing 1 centigram, and bent into the, form shown, (Fig. 92), and can be moved over the right half of, the beam by a levei -device, manipulated from outside the case,, this arm of the balance being graduated into 100 equal divisio-s, from the central to the end knife-edge. With the rider at the, 100th division, the effect is equivalent to placing a centigram, weight in the right-hand pan so that, when it is, say, at the nth, division, ihc etfect is equivalent to adding a weight of w/100, _^, centigram or lOrt/100 or w/10 milligram to the pan., Essentials or Requisites of a Good Balance. There are three, 56., 1, , ;, , *, , essentials of a good physical balance, viz., (/) Truth,, (or Sensitivity), and (///) Stability or Quickness., , (//), , Sensitiveness, , *If the beam were to be pivoted exactly at its e.g., it would be in neutral, equilibrium, and will remain at rest at any angle with the horizontal. Its e.g., is, therefore, arranged to be below ihe central knife-edge, because as it tilts, one way or the other, the c g. rises upward, and the beam is thus, in stable, equilibrium., tToo heavy bodies, likely to break or bend the beam, should be avoided., Jit, , is, , purely a matter of convenience, with no principle involved io, , it.
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148, , BOFEBTI1S OF MATTltt, , A balance is said to be true, when, with, 1., Truth., unloaded, or equally loaded, the beam remains horizontal., , its, , scale, , pans, , Let a and b be, the lengths of the two, arms of the balance,, , S, , (Fig. 93), and S and, 8', the weights of its, two scale pans., , scale, , Then,, pans, , beam, , the, , horizontal,, , Fig. 93., , moments, , side of the central knife-edge, , C balance each, Sxa=S'xb., , obher,, , i.e.,, , with the, unloaded,, , will, , remain, , when the, on, , either, , when, ..(/), , pans be loaded with equal masses m and m., for, Then,, equilibrium, we have (S+m).a=(S +m).b., .('), Subtracting relation (/) from (//), we have m.a~m.b., , Now,, , let the scale, , t, , ., , whence,, Substituting this in relation, , (/), , Thus, a true balance must have, , a=b., we have, S=S'., (/), , arms of equal lengths and, , --("'0, (iv), (//), , pans, , of equal weights., 2., Sensitiveness., A balance is said to be sensitive when, for a, small difference ofhads in the two scale pans, the beam (and, therefore,, the pointer) swings through an appreciable angle, it beinj assumed that, the balance is true., , Thus, the ratio between the deflection of the beam or the, pointer and the difference of load, (usually 1 m.gm.) causing it,, measures the sensitiveness or the sensitivity of the balance., So that,, the greater the displacement of the pointer for a given difference of, load,, or, conversely, the smaller the difference of load required to, produce a given displacement of the pointer, the greater the sensitiveness of the balance., Usually, a balance is regarded to be quite, sensitive, when a difference of 1 m.gm. in load causes the pointer to, be displaced through 1 division on the scale., (/) Case of a Balance with the three knife-edges in one plane., Let Fig. 94 represent a vertical section of the balance through the, centre of the, beam,, , passing through the three, knife-edges at A, B and, C, all in one plane., , Let a be the length, of each arm, d, the depth, of the e.g. (0) of the, beam from the central, knife-edge C and Jf, its, mass. Further, let S and, (8-i-m) be the masses, of the two scale pans
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MEASUREMENT OF MASS, , THE BALANCE, , 149, , m, , between them, together with their loads, the difference of load, being small., Then, if 6 be the deflection of the beam from its initial horizontal position, so that it takes up the position A'B' with its e.g. shifted, to (?', we have, taking moments about C,, t, , + m)g.a cos = Sg.a cos + Mg.d sin, + M.daia 0., (S + m).a cos 6 = S.a cos, m.a cos = M.d sin 0., , (8, Or,, Or,, , ~, , m.a, , sin, , Or,, , And,, , if, , =, , be small, tan, 6, , =, , m.a, , ^, , Or, , =TT>, Md, , cos, , ,.., , tawfl^iTjMd, , >, , ('), v, ', , so that, in that case,, , ;, , m, m.a, , ,..., , <">, , where Q\m measures the, , 0., , ~, , -, , <*, -sf, , a, , ...., ', , <">, , ira, , sensitiveness of the balance,, , It is thus clear that to increase the sensitiveness of the balance., (/), , (ii), (///), , a must be, , large,, , i e.,, , M must be small,, d must be, , small,, , the, , i.e.,, , arms (or the beam) must be, beam must be light, and, , long,, , the, , i.e.,, , the e.g. of the, , beam must be, , close to the, , central knife-edge., , Now, a cannot be increased beyond reasonable limits. For, as, Blrge correctly pointed out, the bending of a beam being proportional directly to the cube of its length and inversely to the cube of, its thickness or depth, its thickness will also have to be increased in, the same proportion with its length if its original stiffness is to be, maintained, and this will inevitably increase its mass in a much, greater proportion, thereby seriously impairing sensitiveness., Nor can the beam be made light beyond a limit, or else it will, break or bend permanently so that, the only workable alternative is, to decrease d., This may be done with the help of the vertical movement of the screw, provided at the top of the pointer, (Fig. 93),, though, carried to an excess, this too has its own drawbacks, v/z.,, We, (/) loss of stability, and (//) a longer period of swing of the beam., have, therefore, to content ourselves with a judicious compromise, ;, , between, , all, , these factors., , Further, if / be the length of the pointer,, the scale will obviously be 10. And, therefore,, tan 00, we have from relation (//) above,, 5, , =, , ,, , l, , its, , if, , displacement s on, be small, so that, , m.a, , 'M~d', , To determine the displacement of the pointer, it is by no means, it can be easily, necessary to wait until it actually comes to rest, estimated from its swings to the right and the left., Thus, suppose we have a scale with its zero at one end and the, successive turning points of the pointer occur on it at the Wth, the, 2nd and the Sth divisions, (Fig. 95). Then, clearly, if S be its restingpoint, i.e., the division where it will eventually come to rest, it is, clear that the successive displacements of the pointer from this, sl, division are, .S)., (S~~2) and $t, (8, (10 S), s^, ;, , =, , =, , =
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PROPERTIES OF MATTBB, , 150, , Now, although, theoretically, as we shall soon see, (page 152),, the oscillations of the pointer muss be simple harmonic, it really, seldom happ3ns that the oscillations of any vibrating system remain, truly so. The oscillations always die down and their amplitude goes, on progressively decreasing due to air-resistance and other causes,, but the ratio between the successive swings to the left and the right,, very aptly called 'decrement', is found to remain constant., , Thus, with 5l9 s z and S3 as the successive swings of the pointer,, , we have, P, , ri, , -1, , 2, , ^=, , ~,, , and so on. So that,, , -S, , ^ 5~-2, = (10-5)(8-5), Or,, (8(^-2), O 5 45+4 = 5 -185'+80., 145 = 78,, Or,, S = 76/14., whence,, = 5-43., 10~s-, , 2, , 2, , 2, , r>, , J, 3, , Thus, the pointer, , will, , come, , at, the, , ultimately, the 5*43r</, , "AV,, , to, , division, , rest, , on, , scale., , Fig, 95., , with the end knife-edges in a different plane, (//) Case of a Balance, from the central one The balance, diseased above, with all the three, ideal one,, knife-edges lying in the same plane, is re illy only an, this condition being hardly ever attainable in ordinary balances., For, the beam does yield, however so little, to the forces acting, at its two ends, so that the end knife-edg^s do get depressed a, little below the central one, and no longer remain in the same plane, with it., , Let us see how the sensitiveness, knife-edges are not co- planar., Let h, be the, height of the end knife-, , is, , affected, , when, , the three, , edges A and B, above, the central one,, C,, (Fig., , 96),, , beam, , and, , be, , let, , through an angle, before,, , mass, hand, , m, , the, , deflected, , an, , for, , <i/", , 0, fas, , extra, , ., , the right-, , in, , j-^, ^i, /, , *_, , ^, , pan. Then, for, we have, equilibrium,, here,, , (S4 m).g.(a cos 0+h sin, , = S.g.(a cos, , 0h, , 6), sfn Q), sin 0., , Fig. 96., , +Mg.d, Or,, , (S+m)(a, , cos, , 0+h, , sin 0), , S(a cos, , 0h sin 0)+M.d sin, , 9,
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MBASTTBEMlMfT OF MASS, Or, S.a cos, , g+S.h, 2, , Or,, , Now,, , if, , S, , sin, , h sin, , 9+m.a, , cos, , ft, , +m.h, , mh0, , 2Sh.e+ma, , M.d, , and cos, , d, , =, , = -^, , three possible cases arise, , When h, , 1,, , we have, , =* M.d.O., , =, , =, , sensitiveness,, , 9., , sin g., , as the product of very small quantities,, M.d.6-2S h.$, Md.Q. Or, ma, , =, , whence,, , Now,, , +M.d sin, , =, , sin, , 9=0,, , be small, so that sin, , 151, , = S.a cos 0S,h sin, , 0+m a cos 0+m.h sin, , 2S.h.0+ma+mh0, Neglecting, , THE BALAffOJ, , n., ., , ~, , we have, , 0(Md-2Sh), , t, , ... (iv), , :, , =, , o, i.e., when the three knife-edges are co-planar., (/), Here, ti\m = a/Md, [See relation (Hi) above], and the sensitiveness "is, quite independent of the total load (2S)., is positive, i.e., when the end knife-edges are higher, (it) When h, , In this case, obviously, the sensitiveness, increases with the total load. But, as will be readily seen from Fig, 96,, the effective length of the arm, in the tilted position of the beam,, becomes greater on the side of the heavier, and smaller on the side of, the lighter, pan than its true length so that, for a given value of the, excess load m, and for a given deflection tf, the difference of moments, on either side of C is greater for a heavier than for a, due to the, than the central knife-edge., , ;, , pans, , the greater the load in the 'pans, the, lighter load, witL the result that, longer the beam takes to attain equilibrium., negative, i e., when the end knife-edges are lower, In this case, clearly, the sensitiveness decreases, with increase in the total load., N.B. We have seen above how, in the ideal balance, with its three knifeedges in the same straight line, the end knife-edges get depressed with the beam, a little below the central one when the pans are loaded. This results in a, decrease in the sensitiveness with increasing load. If, therefore, the end knifethe central knife-edge, the, edges could be arranged at the Correct height above, decrease in sensitiveness due to flexure could be just offset by its increase due to, The method, the latter, and the balance thus nvde equally sensitive for all loa'ds., has actually been used with the success in building balances whose sensitiveness, is quite independent of the total load placed in their pans., A balance is #aid to be stable (or, 3. Stability or Quickness, or equally loaded, the beam be disquick), if, with the pans unloaded, rest quickly,, it comes back to, and, small, is, its, time, of swing, turbed,, thus making for convenience in weighing., (Hi), , When h, , is, , than the central one., , Now,, , as, , we have, , seen above, with the three knife-edges in the, , same horizontal plane, the condition for equilibrium is that, [See page 149., (S+m)g.a cos $ = S.g a cos 6+Mg d sin 0., =, m, if, be, 0, or there, equally loaded, i.e.,, Therefore, if the two pans, be no extra load in the right hand pan, the only restoring moment, about C, tending to br<ng the beam baek to its original position, is, be small. This, obviously, tends to accele0, or Mgd 0, if, so that, if a be the angular, rate the motion or swing of the beam, and, /, the moment of inertia of, beam, in, the, it, acceleration, produces, central, the, about, C, we, knife-edge, the, system, , Mg.d sin, , ;, , moving, , Qt BBS, , Mg.d0, , ~,
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PROPERTIES OF MATTER, , 152, , / =* moment of inertia of the beam about, , Attd, clearly,, , C, , +, , moment of inertia of the two scale pans about C., /= Mk*+2S.a*,, , Or,, , where k, , is, , the radius of gyration of the, , -, , Thus,, ., , ,, , where ^, , is, , -0., ., , beam about, , = M., , Or,, , C., , Totting, a constant for a given, , ., , a constant., , L balance., , a, , Or,, , oc 6,, , beam is proportional to its angular, i.e.,, The swing of the beam is thus a simple harmonic, displacement., motion, and its time period t is, therefore, given by the relation,, of the, , the angular acceleration, , 0,, , /, , -_, , >, , A I, , r', , ,, , M, , S- d, , ', , ', , =, , -, , **, , V, , + M8 .d, , In order, therefore, that / be small, i.e., the balance be stable, k, S, and d should be large. We thus see that, and a should be small and, a balance would be stable when, its arms are short, (/), , M, ;, , (i7), , Us beam, edge, , (iii), , is, , heavy, with, , the radius of gyration, is, , its, , e.g., , far below the central knife-, , ',, , of the beam about the central knife-edge, , small, , stability diminishes with increasing load. It will be seen, at once that almost all these conditions are opposed to those for, sensitiveness. So that, sensitiveness and stability of a balance are, to, , and that the, , a great extent, mutually exclusive, and we have, therefore, to^trike a, working balance between the two., Determination of True Weights., 57. Faults in a Balance, , Arms unequal in lengths and pans unequal in weights. The, 1., co'mmonest fault in a balance is that it may appear to be true, i.e.,, the beam may swing evenly on either side of the central knife-edge,, with the moments on either side balancing each other, and yet the, arms may have unequal lengths and the pans, unequal weights., Thus, if $! and S% be the weights of the two scale pans and a, and b the lengths of the two arms respectively, we have, t, , The true weight w of a body may be determined with such a, balance by the method of double weighing, i.e., by weighing the body, first in one pan and then in the other., Let the counterpoising weight in the right hand pan be, is placed in the left hand pan., Then, clearly,, , wlt, , when the body, , ..., , ()
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And,, , .-., , 153, , THE BALANCE, , MBASUBBMENT OF MASS, , we have, , subtracting relation (f) from (//), w.a, wfi., , =, , ...(///), , the body be placed in the right hand pan and let the, So that,, counterpoising weight required in the left hand pan be vv a .-, , Now,, , let, , we have, , Again, subtracting relation (/) from (/v),, w.b, w^.a., , =, , (, , Multiplying relations (///) and (v), we, therefore, have, w2 = w lt w 2, w 2 .ab =. jvr w 2 .0&,, whence,, , w, , Or,, the true weight of the, Heights in the two pans., , i.e.,, , = vX^V, , body, , the geometric, , is, , mean of, , v), , ., , its, , apparent, , The same will be true if the pans be equal in weight and the, arms slightly different in lengths., Note. If we multiply relation (///) and (v) above, crosswise, we have, Or,, , ---b, , Or,, , 2, , Or,, , w,, , And, since from relation, , (/), , b, , M, , ,, , =A, , V/, , 2, , 5 2 /5i, we have, , above, a\b, ", , Thus, we can determine the ratio between the lengths of the two arms, or that between the weights of the two scale pans., , Scale pans unequal in weights. Another common fault in a, that whereas the arms may be equal in length, the scale, pans may not be truly equal in weights, so that the beam does not, remain perfectly horizontal., To determine the true weight of a body with such a balance, we, again resort to double weighing, i.e., to weighing the body first in one, pan and then, in the other., 2., , balance, , is, , apparent weights in the two scale pans respectively be, Then, if the length of each arm be a and the weights of, the two scale pans, Sl and $2 we have, , Let, , w and, l, , H', , its, , 2, , ., , ,, , in the first case, (S l, And, in the second case,, , Adding, , relations, , (i), , SL +Si+2w, ., , whence,, , and, , +w).a, (St, , =, , (# 2 -f-u\),0, or, , -\-w).a, , (//),, , = (S, , i, , S^+w = S^+w^.^i), , -\-w 2 ).a, or, , 82+^=^ + ^2., , ..("'), , we have, , = S^+^+H^+HV, W 4- Wo, w =, =-,, ~>, , Or,, , 2w, , =, , Wj+w,,, , t, , 2t, , the true weight of the body is now the arithmetic, apparent weights in the two scale pans., i.e.,, , A, , mean of, , its, , 3., Inaccuracy of the Brass Weights., possible source of error may, also be the inaccuracy of the brass or 'standard' weights, supplied in the weight, box, due to their getting worn out by use or getting slightly rusted by discuse., The probability of error due to such causes is presumably the least in the case, of the larger weights and the greatest in the case of the smaller ones. So that,, assuming the largest among them to be accurate, others, of smaller denominations, are counterpoised against it ; these smaller ones are then counterpoised, against others smaller than them, the process being continued up to the very, smallest ones, and, in this manner, the errors in the smaller weights are easily
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PROPEBTIES OF MATTER, , 154, , Thus, for example, a weight of 100 gms. is first counterpoised againsl, 5+2 i 2 -M, and, 10, then, the weight of 50 8 ms against 20 + 10 4- 10, agam the weight of 20 gms, against 10^5 + 242+1 and so on. "lo make, a weight, sure, the weights assumed to tx correct must also be tested against, known accurately in terms of the, Inter'national Standard., 4., Blunting of the Knife-Edges. Due to constant use, the knife edges, so that, with the tilt of the, get blunted or rounded off, in course of time;, beam, the point of contact with the plane of support may shift slightly. This, tantamounts to a slight change in the lengths of the arms and must also be, detected., , 50+20+20+, , -r, , -, , ;, , corrected for., , 4, , Correction for Buoyancy. Ordinarily, we make all our weighings, in air. But air, in common with all other gases and liquids, exerts an upward, in strict obedience to the Principle of, thrust on a body immersed, it,, Archimedes. So that, the body to be weighed, as well as the brass weights,, agamst which it is weighed, are subject to this upthrust or buoyancy due to the, air displaced by them, which is equal to the weight of the displaced air, in, 58., , m, , either case,, If the body weighed happens to hnve the same density as that of the, material of which the standard weights are made their volumes too would, obviously be the same, when they are counterpoised against each other, and the, volume of air displaced by both, and hence the buoyancy or upthrust due, tD it, would just be counter-balanced and the standard weights used would, straightaway give the true weight of the body in vacuum. This is, howBut, it makes it clear why, in realby true and accurate, ever, rarely the case., balances, we insist upon the arms and the pans being identical in length, volume, and mass., More often than not, the density of the body is quite different from that, of the material of the standard weights, and, therefore, even when they counterof the air displaced by, poise each other, their volumes, and hence also the weights, them are altogether different. Let us, therefore, deduce the necessary correction, , in this, , commonly occurring, , case., , M, , and its density p. And, let, Let the true mass of the body we'ghed be, the mass of the weights required to counterpoise it be M', an J the density of their, material p'. And, finally, let the density of dir be, Then, clearly, volume of the body and hence the volume of air displaced, ., , by, , it, , =, , =, , M/p'., , And, so the weight of, , this displaced air and, therefore,, , to, , it, , So, , that, their apparent (or observed) weight, , the upthrust due, , M.&.gl?., , of the body, , Similarly, the volume of the standard weights and, therefore, of the air, Af'/p' ;, displaced by them, M'.S.gfc'., and the upward thrust due to this displaced air, , =, , "^, , M'g, (Tkf/, , .8-g, , -,, , \, , Since the body and the standard weights counterpoise each other in air,, must be equal. And, therefore,, , their apparent weights, , Wh cnce,, ', , / S, , r, 1, , L, , Or,, , +, , VP, , (, , M, , -, , ,, , P, , ), , /J, , ., , fNeglectng the prouct o, { S/P and 6/p', compared with, $ othe terms.
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MEASUREMENT OF MASS, Prom, , the above, , it, , 155, , THE BALANCE, , follows at once that, , M>, , M, , = <, , ', , according as, , p, , < =, , >p',, , the true weight of a body (ie. its weight in vacuum) is greater than, equal, to or less lhan, its observed or apparent weigtit in air, according as its density, is Ijss titan, equal to or greater than that of the material of the standard, weights used., , i.e.,, , t, , SOLVED EXAMPLES, 1., The arms of a balance are unequal in length but, without the scale pans,, the beam and the scale-pan holders are correctly balanced. The scale pans A and, B are of weights 2w t and 2w 2 respectively. A body placed in pan A has an apparent, Show that the true, 2., weight Wj and placed in pan B has an apparent weight, weight of the body is, , W, , 1/[W X, , W, , a, , +2(Wi, , Wi+w, W,) +0"i +*>*] -(*!+*>, , f, ^, (London Higher School Certificate), and the lengths of the left-hand, , W, , Let the true weight of the body be, and right hand arms be a and b respectively., Then, since equilibrium is attained with the body in the left-hand pan, in tho right-hand pan, the moments on either side of the, and a weight, central knife-edge must be equal so that, neglecting moments due to the pan, holders, which already balance each other, we have, , W, , ;, , (2vv 1, , W, , %, , + PK).a, , =, , (2w 9, , -, , +W U., l, , ....., , (0, , attained with the body in the right-hand pan and a weight, in the left-hand pan ; so that,, , Again, equilibrium, , is, , (2w z + W).b, , ==, , (2w^ W}(2, , Or,, Or,, , 4w l H> a -r-2H', , 1, , W'-h2H' a W-r, , ^, , Or,, , Adding Ovj-Hv;), , The, , left, , 2, , 2, , -h2^(>Vt, , to both sides,, , 4-, , W =, l, , w2), , *, , ......, , (2^+Wj.a., , Multiply the corresponding sides of relation, , (/), , ("), , and (), we have, , 4w l w^2wJV n, , i-2w l lV l, , f, , we have, , hand expression, , is, , clearly the complete square of, , so that,, , Or,, , And,, Or,, , .-., , W, , the true weight of the body, , The arm? of a balance are similar and of equal length, a. The scale, 2., P. When the beam of the balance is, pans are similar and of equal weight,, horizontal the central knife-edge is a distance x vertically above the middle of the, is a disline joining the knife-edges of the scale pans, and the e.g. of the balance, Assuming that the weight of the moving, tance y vertically below the same point, for the angle of deflection of, system of the balance is W, derive an expression, the beam when weights w x and w, are placed on the scale pans. fw t > w 2 ]., (Joint Matriculation Board High School Certificate), Let AB, (Fig. 97) be the position of the beam, when the pans are yet unand p,, loaded, C, that of the central knife-edge, P and P, of the scale pans, that of the pointer, with G, as the e.g. of the beam., Let the heavier weight H'i be now placed in the right-hand pan and the, and let the beam, and, therefore, also, lighter weight w,, in thr left-hand pan, the pointer, deflect through an angle 9, into the positions A'B' and p' 9 with the, joining, eg of the beam at G', where OG'=~y (O being the mid-point of thebeline, now at C',, Let the central knife-edge, the knife-edges of the two scale pans)., , where, , OC'~x (given)., Then,, , beam are, , the different forces,, , all, , acting vertically, , downwards,-on the
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PBOPBBFIBS Of MATTEB, , 156, (/), , (P+Wj) at B', , t, , (//), , (P+w> 2 ) at A' and, , W,, , (///), , the weight of the, , moving, , system, at G'., Since the beam is in equilibrium in this position, the, knife-edge C', on either side, must be equal., , moments about, , the, , Fig. 97., , (P+\v*).DEW.G'L=(P-{ wJ.DF., cos Q + OC' sin Q=a cos Q+x, , Or,, , DE^OE+OD=OA', , But, , y, , 9+ x sin, , (x-f y) sin, , 9,, , DF^OF-OD^OB' cos 9 OC sin = (a cos 0-x sin, , and, , f, , So, , that, CPH-w> 2 ). a cos, , Or,, , P.a cos, , 0+x sin, , 0)-f W.(x+y).sin Q, , Q+w^x, , sin 9, , = H', , f w 2 .x, , 1 .a, , sin 9, , cos 9, , w2, , -f, .ci, , + W.(x+y).sin, w^x sin, , W.(x+y).sin, cos, , 0)., , iP+wJ.ia cos Q~-x, , 0+P.x sin Q + w 2 .a cos 0-f w 2 .x sin, =P.a cos Q~P.x sin Q-\-w v a cos, , 2P.x sin, , Or,, , sin 0)., , Q., , Q., , 9, , 6., , @==, , Or,, , cos 9, , whence,, , w, , sin, , sin $,, , 6*, , This, then, is the angle of deflection of the beam,, are placed on the two scale pans., , when weights, , and, , With a balance of which the arms were 10 cm. long, it was found that, 3., 0*010 gm. extra-load on one pan deflected the beam of mass 20 gms. through 1 and, What can you, that this deflection was independent of the loads placed on the pans., deduce from these measurements ?, (Oxford Local Higher School Certificate), Since the deflection of the beam for the given extra load of '01 gm. is, quite independent of the loads placed on the pans, it is clear that the three edges, are co-planar., , And, since, in the question, every other factor is given except the depth, of the e.g. of the beam below the central knife-edge or trte centre of the beam,, we are obviously expected to determine its value. Let it be h., Since the beam is in equilibrium at angle 9 -1 from the initial horizontal, it is evident that the moment about the central knife-edge due to its, , position,
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PBOPERT1BS Ot MATTEB, , 158, , We know, , that sensitivity of a balance is given by the relation,, , =, , m, Now,, , in the first case,, , / /&? second, , Sothat, , 6, , case,, , -, , dividing relation, , (/), , 3, *, , 270/1, , Or,, , ~, , Md2S.hr, $ = 3'0, m TT>Jr-*, , =, , 2'70,, , m=, =, , where 2S, , 1, , *001, , mg., , 1, , m.?., , => '091, , = --3M,, , gm. and 2S, , Ma, , 0.7, 7 *,/, 2, , Or, 3 Afa, , ., , ,, , whence, A, , =, , -'3, , gm. and 25, , 0., , -, , 100 gms., , .........., , )Y, -AfJ-lOOA, by (//), we have, MJ-100/i, -, , =, , [Page 151., , the total load., , is, , (//), , -> 7 AA, 270/r., , A/r// 270., , negative value of /z thus cbarly indicates that the end knife-edges are, below the central knife-edge., Now, let thi sensitivity of the balance be x divis ons per mg. for a load, , The, , of 200 gms., , Then, we have, x_, , __~, , 001, , x, , <*, , Md-2i)0h, , _a, , _, , ~~, , r, , ustt, Substituting, the val, value of, L// from above., a, , <*, , _, , ~'3 \fd\', MJ~200/~'3, \ 270 /, , ___, , 270a, , -001, , .*., , dividing relation, , x, 001, , (///, , -001, , X, , ^3, , by, , we have, 270, Md_, X '"*, 330V/J, , (/),, , ^, , 270x3, whence,, , "330, , ^, , o f) x, , 270, , _, ", , 330', , 3, , 27, , Ti, , =3, , 2455., , Thus, the sensitivity of tUs balance for a load of 200 gms., , is, , 2455, , di\i-, , sions per mg., , A, , 6., piece of metal weighs 300 gms in air If the densities of the metal and, the brass weights used by 19 gms. c.c , and 8 gms. /c.c., respectively, and that of air, 00123 gm./c.c., calculate the true mass of the piece in vacuum and the correction due to buoyancy., , We know, , that the true mass of a body in, , M-M, , 1, , *, , +8, , vacuum, , is, , -, , giv^n by the relation,, , [Sec page 154., , (, p, )J,, [l, the true mass of the body, p, its density, M', its apparent mass, p',, the density of the weights used and 3, the density of the air., 8 gms./c.c. and 8, '00123, Here, M' -= WQgms., p =- 19 gms lex., p', gms.lc.c. Substituting these values in the relation for, above, we have, where, , Mis, , =, , M, , -, , 300, , -, , 1--00123 x, , 300, , -, , 300--026706 - 299'91 3294 gms., 300(1- '00008902), Thus, the true mass of the body in vacuum will be 299973294 gms., the, buoyancy correction being obviously '026706 gm., , EXERCISE V, how are they secured, in actual practice ?, Show that the sensitiveness of a balance is independent of the load in tho, does the position of, two scale pans, if the three knife-edges be co-planar., the centre of gravity of the beam affect the working of the balance ?, 1., , What, , are the essentials of a good balance and, , How
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MEASUREMENT, , MASS, , Off, , THE BAUUSCJB, , ^, , 2", e arms of a Dalance ar eacn 7 ww. long, the length of the, pointer, ?, 12 cms. and the mass of the beam is 50 ms. If the, knife-edges are in a plane, and the centre of gravity ot the beam is 0*02 cm. below the centre knife-edge,, now much will the end of the pointer be deflected when the difference in load in, tne pans is 1, ?, School, *, is, , (Cambridge Local Higher, , milligram, , Certificate), , Ans 0*84 cm., 3., A certain balance has a beam weighing 200 gms., with knif-edges, carrying the pans 15 cms from the central knife-edge. What is the depth of, the centre of gravity of the beam bslow this, knife-edge, if a weight of 1 mg., placed on one of the pans displaces the end of the, pointer through a distance of, U'5 cm., the pointer, Br Inter ), being 15 cms. long ?, Ans. 0*0225 mm., f, , 4., Two balances, made of the same material, are alike in all respects, except that the linear dimersn ns of one are n times those of the other. Compare, the angular deflections of the beams for H, given difference in load, (Cambridge Local Higher School Certificate), Ans. 1/fl 3 : 1., , 5., A body is weighed first in the left and 'then in the right hand pan of, a balance, the respective weights being 9'842, gms. and 9'833 gms. Find the true, weight of the body and the ratio ot the lengths of arms of the balance., Ans. True weight 9 837 gms ; Ratio of arms 1 '0005, 1., 6., Discuss the points to be taken into consideration in the design of an, accurate, sensitive and convenient balance., If the arms are of unequal lengths,, show how the error on this account can be avoided. How would you, except the, sensitiveness to vary with the load ?, (Bombay, 1933), 1., What are the requisites of a balance ? Obtain the general expression, used for determining the conditions for these, requisites and show that the conditions for two of these are mutually, contradictory., (Punjab, 1933), 8., Sketch ihe essential parts of a balance in which the two end knifeedges arc h cms. below the centra) kiufe-edge and discuss the conditions of itt, :, , sensitiveness., 9., Obtain an expression for the true mass of a body in vacuum, when its, apparent mass in air is, gm its density p, the density of the standard weights, used P and the density of air 8, Would tne same treatment be, applicable, if the, body be weighed in a liquid ?, , M, , 10., , Known, you, , ', , ,, , Given the apparent weights of a body, , densities, similar standard weights being, proceed 10 calculate the density of the body ?, , A gla&s st, Jc gm. fc.c, ws, -and, 1, , PP er Density, , in, , two, , different liquids, of, how will, , used in both the cases,, , gms Ic c.) is first weighed in water of denapparent weights in the two liquids are, 8'6^mv. and 2') 4 #wv. respectively, the brass weights used being, similar in cither case. Calculate the, Ans. '8489 gm /c.c., density of oil., *, , sity, , ', , loand to be, , 1, , then in, , oil., , 2'5, , Its
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CHAPTER, , VI, , ACCELERATION DUE TO GRAVITY, 59., Acceleration due to Gravity. Galileo was the first person, to have performed in 1590, the then bold and spectacular experiment, of dropping a cannon and a musket ball from the Leaning Tower of, Pisa, which, contrary to the teachings of Aristotle 9 reached the ground, , simultaneously. He thus clearly showed that, at any given place,, all bodies, big or small, when dropped so as to fall freely, do so at, the same unijorm rate, neglecting, of course, the resistance to their, motion due to air. That is to say/ all bodies, irrespective of their, mass or nature, falling freely in vacuwn, will have the same acceleration, at a given place., This acceleration is called the acceleration due to, it, due, to the gravitational attraction of the body by, is, gravityJ&s, , the earth, towards its centre, (see Chapter VIT). It is denoted by, the better g, and is numerically equal to the force with which a unit, mass is attracted by the earth towards its centre, i.e., equal to the, weight of unit mass*., , The value of g, and the, , the poles, , differs, least, , at, place, being the greatest, Its value,, for all practical, 981 _cms. jsec*., in the C.G.S., F.P.S. system. Due to this, , from place to, , at the equator., , purposes, is however, taken to be, system, and 32^ ft. I sec*., in theT, comparatively large value of g, bodies fall much too quickly to the, surface of the earth, when dropped freely, and hence it becomes, difficult to measure it directly, with any great accuracy. It is,, therefore, determined indirectly with tho, help of a simple or a, or, We shall now proceed, other, methods., by, compound pendulum,, to consider some of these in proper detail., , The Simple Pendulum. A simple ( or, a mathematical ), 60., pendulum is just a heavy particle, (ideally, only a point-mass),, suspended from one end of an inextensible, weightless string, whose, other end is fixed to a rigid support, the point where the string is, fixed to the support bJng known as the point of suspension of the, penduluii. In practice, we usually take a small and a heavy metallic, spherical bob, tied to a fine silk thread., / The motion of ths pendulum is simple harmonic, , and isochronous,, , f, , amplitude of small, is, given by the relation, t, swings, and, 27T\/ Ijg^ where / is the length, of the pendulum, (or the distance between its point of suspension and, i.e.,, , its, , time-psriod, , is, , quite independent for, , its, , =, , *This is so, because the force with which a b->dy of mass m is attracted, by the earth, towards its centre, is equal to its weight mg\ and, therefore, if,, m = 1. i.e., ii' the body be of unit mass, this force of attraction on it, or its weight,, equal to g., For the discovery of this property of the pendulum, we are again, indebted to Galileo, who noticed a swinging lamp in the cathedral at Pisa, and timed its oscillations against his own pulse beats. The time taken for, each swing was found to be the same and, as far as Galileo could judge, quiU, independent of the size of the swing., is, , 160
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161, , ACCELERATION DUB TO GRAVITY, , the centre of gravity of the bob), and, g, the acceleration due to gravity, at the place. This will be clear from the following, :, , S, , be the point of suspension and O, the mean or equilibrium, Then if the bob be given a small, position of the bob, (Fig. 99)., angular displacement 6, in the vertical plane, (or the plane of the, .pendulum itself), so as to occupy the position A, it is clear that it, will be under the action of two forces, viz., (/) its weight mg, acting, vertically downwards, (m being the mass of the bob), and ( ii ) tJie, , Let, , t, , tension, , T, , towards, , of the string acting, along, point of suspension S., , the string,, , its, , Resolving the weight (mg) of the bob,, , two rectangular components, we have, component mg cos 0, acting along, (a), the string, as shown, and, into, , (b), , to, , it, /.e.,, , ~o\, , mg sin 0, at right angles, the, along, tangent to the arc OA,, , component, , in the direction, , AO., , Obviously, the former component (mg cos 0), is just balanced, by the tension (T) of the, string, there being no motion along it either, way; so that, the only force left acting on the, bob is mg sin 0, towards its mean or equilibrium, position O., , m9, , ', , = forcefmass., acceleration of the bob = mg sin, , acceleration, , Now,, , And, therefore,, , Fig. 99., , 6/m, , = g sin, , direction AO, or towards the mean or equilibrium position, bob. And, if 9 be small, we have sin Q, Q,, , =, , acceleration, , Hence,, where, OA, pendulum., , =, , OAJSO, , x, the displacement, , acceleration, , Or,, , putting gjl =, , place,, , /^,, , = gO,, = x/l,, , of the bob, , =, , Again, , of, , of the, , directed towards O., , the bob,, , of the bob, , in the, , [v, and, , = g., , =, , arc /radius., angle, the, /,, length of the, , -=-, , ^, , .x., , a constant, for a given pendulum at a given, , we have, , of the bob = ^ Jt., oc x, and is directed towards O,, Or,, the acceleration of the bob is proportional to its displacement, i.e, from its mean or equilibrium position, and is always directed, towards that position. The bob thus executes a simple harmonic, motion, and its time-period I is given by the relation,, acceleration, , acceleration of the bob, , ,, , t, , =, , Alternative Methods., , Method, oc, , (/)., , Let the bob be given a small angular displacement, , in the vertical plane, so as to occupy the position A, (Fig., , In other words,, , let, , a be the angular amplitude of the bob.
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PROPERTIES OF MATTER, , 02, , f, , Then, clearly, the e.g. of the bob has been raised up through, A vertical distance OC, whon AC is the perpendiculat dropped from, A on to SO., , \, , potential energy of the bob at, is released at A, it starts, , .*., , A~mg. OC., , As the bob, 9, I, , \, , \\, \, , i, , \, *, , \, , !, , ^l\, , \, , {, , \, , I, , potential energy of the bob at, , \, \, , j, , \, , .'., , A, , ?+'''A, , <, , /;, , """"""" ^, , /., , B on, , loss in, , P.E. of the bob, , 1, , in, , moving from position, B=mg.OCmg.OD=mg (OC- OD)., , to posit ion, , =mg[(SO-8C)-(SO-SD)]., Now,, , Fig. 100., , So that,, , B=mg.OD,, , the perpendicular dropped from, , is, , o, , and, , to-, , to SO., , \, , \, , J, , BD, , where, , \, , \, , i, , moving, , wards 0, thus acquiring kinetic energy, (due to ita, motion), at the expense of its potential energy., Consider the bqb to be at B, on its way towards O, and let its angular displacement here be6., Then, clearly,, , SC=^SA, , SO = SA=SB=l,, , cos <x=/ cos a, , loss in P. E., , the length of the pendulum;, cos 0=1 cos 6., , SD=^SB, , and, , of the bob=mg[(l-l cos a)-(/-/ cos, , mgl, , G)], , cos a)., , (cos 9, , This must, therefore, be equal to the gain in K.E. of the bob, Or equal to the K.E. of the bob at B, (since its K E at A was, If / be the moment of ineitia of the bob about the, to, zero)., equal, axis of suspension through S, and oj, its angular velocity at B,, , B, , at, , y, , we have, , K E., , of the bob at, , j?, , = |/co 2, , lja)*=-mgl(cos 6-cos, , And /., , ., , a)., , Differentiating this expression with respect to time, aco, i, aQ, ^^ ~ /77I?/ Sin v ~~, i it}, dt, dt, , .., , .., , (/),, , we have, , (/>, , t, , ', , the angular amplitude (a) of the bob being a constant quantity., dBldt=o) 9 the angular velocity of the bob at B., Now,, , And, , dajjdt is its angular acceleration, here., mgl sin Q.J., Ia>.du)ldt*=, , /., , Thus,, , mgl., , f.da)./dt=, , Or,, , dwjdt, , whence,, , knd, , since 6, , Hence,, , is, , small, sin Q, , =, , \in 6,, , mgl. sin, , 6/1., , = Q(radians),, , acceleration of the bob =, , very nearly., , f, , -, , *0,, , . ., , (//>, , ve sign merely indicating that it is directed towards O, opposite, that in which the angular displacement (0) increases., , the, ro, , For, , ve sign, and putting, Or, neglecting the, wg.///=^, a constant, a given pendulum at a given place, we have, acceleration of the bob = v.$,, , ttid,, , therefore, proportional to, position., , mean or equilibrium, , ,, , its, , angular displacement from it*
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ACCELERATION DUB TO GRAVITY, , The bob thus executes a S.H.M., and, fiT, , =2*, , V, , 163, , time-ppriocUs given by, , its, , = 2", ~mg7,r, , V, , mir*, , be the radius of the bob, we have, by the principle, Now,, 1= 2/wr/5 +m/ J, of parallel axes,, if r, , ., , /Twr*/5 + w/, /=27rA/, ---f, , And, therefore,, , with, , r, ., , Since the bob is a small one, its radius r is negligible compared, the length of the pendulum; so that, we have, , /,, , /, , =, , 27ri/nil'lmgl., , Or,, , t=2iri/ljg~~, , See Borda's pendulum, (next article)., Calculation for g., Squaring the expression for the time-period, of the pendulum, we have, 2, 2, = 47r 2 /// 2, 47r //^,, r, whence,, Thus, knowing /, the length of the pendulum, and f, its timeout the value of g at the given place.*, period, we can easily calculate, , Method, , (//)., , =, , ., , Drawbacks of Simple Pendulum. Though simple in theory, and, value of g is n >t, easy to perform, this method of determining the, numerous, to, its, due, accurate, an, drawbacks, the more, one,, quite, are the following, which, of, ones, important, :, , A simple pendulum just an ideal conception, not realizable, 1., in actual practice, for we can neither have a point -mass, nor a weightless string: so that, the string too has a moment of inertia about the, is, , suspension- ax is., The resistance, 2., , and the buoyancy of air appreciably affect tlte, motion of the bob., The relation for the time-period (/), obtained above, is trite, 3., oscillations, having an infinitely small amplitude., only for, motion, The, 4., of the bob is not, strictly speaking, a motion, of, translation, for it also has a rotatory motion about the axis of suspension pass ing through the point of suspension., The bob has also a relative motion with respect to, 5., its amplitude on either side., suspension-thread at the extremities of, , the, , In this pendulum, the bob is a sphere, 61. Borda's Pendulum., of large radius and, assuming thit it is rigidly fixed to the string, and oscillates only about the axis of suspension, (there being norelative motion between the bob and the string), its time-period, , A///+ V, , 2, , is, , given by, , /, , =, , ^TT, , where / is the length of the pendulum, and, This relation for t may be deduced as foil >ws, , //, >, , r,, , the radius of the bob., , :, , *The earliest determinations of the value of g w^re all made oy means, Thus, Picard, in 1669, used a pencul im in which a, of simple pendulums, copper bob of diameter 1 inch was sispen^ed by an ah? fib>e (\vhich remains, unaffected by moisture) and, in the year 1792, Borda and Castini used one, with, a platinum ball of diameter 1*5 inches, suspended by an iron wire about, J 2-75 ft. long.
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PROPERTIES OF MATTER, , 164, , Let SO be a Borda's pendulum, (Fig. 101), suspended from the, be the mass of the bob, of radius r., point of suspension S and let, , m, , Imagine the bob to oscillate in the plane of the paper, and let it, be displaced from its original position A to the position B, through an, angle 0, at any given instant., , The only, , restoring force on the, , bob in, , this position is its weight mg,, acting vertically downwards, which, has, obviously, a moment about the, , S, , 8, axis of suspension through, perpendicular to the plane of, , and, the, , paper,, , \l, , =, , \, , mg x O'D, , =, , mg.l sin, , 6,, , where / is the length of the pendulum, and .'. O'D = I sin 0., , \, , the restoring, is, then,, bob in the, on, the, aoting, couple,, position B, tending to bring it back, This,, , to its original position., ]f dwjdt be the angular acceleration produced in the bob, and I,, its moment of inertia about the axis, , TTIQ, , Fig. 101., , of suspension through 8, the couple, I.dco/dt, , Or,, dojjdt, where mgl/I =, , =, =, , u. t, , mg, , =, =, , is, , mgJd., , sin 6, , I, , 0.wg//7, , also equal to I.dwjdt., , =, , \, , be small, sin 9, if, 0., the couple due to mg is, clockwise and -'. negative., , v, , [V, , jutf,, , a constant., daj/dt oc 0,, , Or,, , i.e., the angular acceleration ofth3 bob is proportional to 0, its angular, so that, its motion is simple harmonic and, therefore,, displacement, ;, , ;, , the time-period of the pendulum, , is, , given by, "", , 27T, , Or,, , VTt, , I, , -, , lngl/1, /, , =, , ~mgl, , Now,, and, , / =: the, , =, /, diameter, Or,, , M., , I., , of the bob about an axis through, , 2, parallel to the axis of suspension-]- w/, , =, is, , M., , I., , of the bob about, , |mr 2 +/7t/ 2 (v M., , I., , ,, , equal to %, , its, , ., , diameter +ml 2, , of a sphere of mass m,, , about, , mr 2 )., "gl, +, , t, , II, , axes., , ., , 27T, , Or,, , its e.g., , [Principle of, , its
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ACCELERATION DUE TO GRAVITY, , 165, , Thus, as will readily be seen, the time- period of this pendulum, same as that of a simple pendulum of length (/+jjr 2.//) 'which,, for this reason, is called the length of an equivalent simple pendulum,, or the reduced length of the pendulum., Further, if r be equal to zero, i.e., if the bob be just a point, or, the pendulum be a simple pendulum, we have, substituting r =s, in, the relation for t, above,, is, , the, , ., , Or,, , t, , =, , which is the expression for the time-period of a simple pendulum,, given above, ( 60)., This pendulum too cannot give an accurate value of g, as, in, the first place, the string has also a moment of inertia about the axis of, suspension and secondly, there is relative motion between the bob and, the stnng, the bob oscillating about it at each extreme end., , A compound, rigid (or, a physical), just a rigid body, capable of oscillating freely about a horizontal axis passing through it. Its vibrations are also simple harmonic, 62,, , pendulum, , and, , its, , Compound Pendulum., , is, , time-period, , given by the relation,, , is, , *, , "/, ', , mg., , I, , where / is its moment of inertia about the axis of suspension m, its, mass and /, its length (or, the distance between its axis of suspension, and its centre of gravity)., ;, , may be, Let S be the, This, , seen from the following, , :, , point of suspension of the body, (or the pendulum),, through which passes a horizontal axis, perpendicular to the plane of, the paper, about which the body oscillates, its e.g., G, \\ill obviously he vertically below /S>, in its normal position of, rest, (Fig. 102)., , Let the body be displaced through, an angle #, into the dotted position, shown so that, its e.g. is now at G'., Then, the couple acting on the, to its weight mg will, obviousdue, body, it, ly, be mg.l sin 6, tending to bring, back into its original position, (SG', ;, , being equal to /, the length of the pendulum)., If the angular acceleration produced in the body by this couple be, da)jdt thq couple will also be equal to, I.dwjdt, where / is the moment of, Fig. 102., inertia of the body about an axis, through the point of suspension S, and perpendicular to the plane of, the paper., f
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PROPERTIES OF MATTBK, , 166, , So, , =, , that, I.!~, , ~, Or,, , dct), , mg.l, , mg.l0...(A) [v, , =, , mg.l, , =, , -, , =, , sin 6, , _~.0, , M#., , [where, , y, , *, , if, , be small,, , /w^.///, , =, , ~, , sin e, , 0-, , a constant., , A*,, , Thus, the angular acceleration, (dc^/dt), of the body is propor*, tional to its angular displacement, (0). The body, therefore, execute*, a S.H.M., and, t, , its, , A/, , == 2?r, , \, , If /- be the, , is, , time-period, , M., , l, , given by, , = 2v\/ \. =, V mgl I, r, , u,, , /., , of the body criow/ 0H, , and, , parallel to the axis of suspension, principle of parallel axes,, , =, , I, if, , And,, through, , G9, , k be the, , rad>'us, , ", , 27T, , /V/-^r-, , \, , flx/^, , ..., , (B), , /Yj?, , e.g.,, , mgl, , through G.,, , through 5, we have, by the, , /,+/!!/*., , of gyration of the body about the axis, , = mk- so that,, / = mk* +w/, , obviously I g, , :, , 2, , ., , substituting the value of / in relation (B) above,, , .*., , Or,, , t, , i.e.,, , -, , the period of viberation, , of length, , ., , -f/,, , or, , ~t~, , 2, , same as, , the, , is, , which, , ,, , we have, , is,, , that of a simple, , valent simple pendulum, or the reduced length of the, is sometimes denoted by the letter L., , Since k 2, , pendulum, , therefore, the length of an equi-, , pendulum., , It, , always greater than zero, this length of an equivalent, always greater than 1., simple pendulum, Centre of Oscillation. A point 0, on the other side of 0, at a, distance fc 2 // from (7, is called the centre of oscillation, and a horizonis, , is, , tal axis, , $, , is, , through, , known, , GO, , Thus,, , we have, And,, , 2, , .-., , it,, , parallel to the, , axis of suspension,, , as the axis of oscillation of the, , SO, t, , =, =, , & 2 //,, 2, , i, , -, , -J, , Putting, , (Fig. 103)., /2, , pendulum., it, , equal to, , /',, , 2, , = + *-- =, /, , /+/'., , = 2;r, , L, ~g, , K, , /, ', , Interchangeability of the Centres of Suspension, and Oscillation. If the pendulum is inverted and suspended about the axis of os< illation through 0, its time-, , 63., , period of vibration will obviously be given, , by
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167, , ACCELERATION DtJl TO GBAYITY, , And,, , since, , fc, , 2, , /', , =, , we have, , /',, , fe2, , =, , that, the expression for the time-period, , is, , :, , i.e.,, , the, , same as about, , w/, t, , ?, , becomes, , V, , the axis of suspension through S., , Or, the centres of suspension and oscillation are interchangeable,, are reciprocal to each other, a property of the pendulum, first, , discovered by Huyghens., Thus, we get the same values for the time-period and the length, of the equivalent simple pendulum whether the pendulum be suspended, at S or at O, i.e., at a distance I from the e.g., (G), or at a distance, Jc, , 2, , /lfrom, , it., , G, , as centre, and, with, therefore, we draw two circles or arcs,, z, radii equal to / and k /l respectively, they will cut SG produced at S, and P below, G., and, above, and at, If,, , O, , Q, , Then,, , = GP = and GQ =G0 = W\l., = /+/' = SO., QP = GP GQ = /+, SG, , clearly,, , I, , 2, , //, And, therefore,, Thus we have four points, S, Q, O and P, collinear withG, the, which the time-period is the same., e.g. of the pendulum, about, these four points,, If. therefore, we can determine, by experiment,, the equivalent, of, or, L, (/+/'), we can easily find out th* length, of g at the given place, with, the, hence, and, simple pendulum,, y^He, the, of the relation, t = 2 Try^/*, where ' is the time-period of, \-, , help, , the pendulum., Centre of Percussion, Fig. 104 shows a section of a rigid, 64., its, G, body, of mass m, by a vertical plane passing through~ e.g.,, with S, as its poii.t of suspension, the axis of, to, suspension through which is perpendicular, the plane of the paper., Let a force F be applied at O, in the, direction shown, so as to be perpendicular to, and the axis of suspension, both the line, to, through S. Then, this force is equivalent, , SCO, , an equal, and a like parallel force F at G,, the force, (ii) a clockwise couple, formed by, F at O and an equal and opposite force F at G,, the moment of which is clearly equal to Fxl', distance GO., where /', (/), , and, , =, , Now, this force Fat G tends to produce, linear acceleration a, say, in all th^ particles of, the body, including the centre of suspension, , S, , ;, , a, , so that,, , =, , F/m,, , in the direction of the force,, , i.e.,, , from right to, , 104., , left., , the acceleration produced at S by the force F at, the other hand, tends to produce an angular, on, couple,, acceleration a, say, in the body, about a parallel axis through G., [f / be the moment of inertia of the body about this axis, we have, This, then,, , G., , The, , is
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PBOPBBTIES OF MATTER, , 168, , = Fx, , La, where /, , = m&, , ,, , =, , Fxl'/mk*,, whence, a^Fx /'//, being the radius of gyration of the body about, , I',, , 2, (fc, , this axis)., , Now,, , linear acceleration, , = angular acceleration x distance, , from, , the axis., a', , =, , Hence, linear acceleration produced by this couple, , Ixa, , =^, , Fxl'xllmk*,, , in the direction left to right,, , is, , i.e.,, , given by, opposite, , to that of a., , applied at O may produce, a due to force F at G must bo, acceleration a' due to this couple,, equal and opposite to the linear, F x I' X Ijmk*,, In other words, Fjm, a, a'., i.e.,, , F, , In order, therefore, that the force, , no, , effect at S, the linear acceleration, , =, , =, , 2, , -, , -, , 1., Or, /', *//., /'X//, This, therefore, is the distance of the point O from the e.g. of, the body, and the point O is thus the centre of oscillation, (see page, with respect to S., 166), and is here called the centre of percussion,, , whfence,, , It is thus clear that if a body be struck at the centre ofpercusor the centre of oscillation, in a direction perpendicular to its axis, of suspension, it does not move bodily, as a whole, at its point of suspenit., sion, but simply turns about the axis passing through, , 'sion,, , This explains why when a ball strikes against a bat such that, the point where it strikes the latter is the centre of oscillation, or the, centre of percussion, corresponding to the point where it is held in, the hand as the point of suspension, no sting or shock of any kind is, felt., Similarly, a good hammer should be so constructed that its, centre of percussion lies in a line with the driving force., Other points, collinear with the e.g., about which the time65., period is the same., Squaring the expression,, time-period of a compound pendulum, where, radius of gyration about the e.g., we have, , --., , f, /, , =27ry'/, is its, , 2, , 2, , -f/c, , length, , //g">, , for the, , and, , k, its, , Or,, , Or,, , Dividing both sides by, ^, , Or,, , 2, , ~"/, , Thus,, 1, , -, , /, , a, , ./+&*, , 47T, , =, , 2, ,, , 0,, , has two values,, , +, , V, , e, , we have, which, , is, , /, , 2, , 4-&, , 2, , =, , gt*, , ,,-/, , clearly a quadratic equation in, , viz.,, , -* 1 ", , I, , (a, , L, , and, , Therefore, there are two values of / at distances (a-f fe), b) from the e.g., for which the time- period is the same., , and
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ACCELERATION DUE TO GRAVITY, even, , and, , less, , 173, , than one-thousandth of that of the bar, and yet maintaining its strength, The values of g and k may then be determined in the usual manner., , rigidity., , Devised and first used by, 68. Eater's Reversible Pendulum., Captain Kater, in the year 1817, to make the celebrated determination, of the value of g in London, it is a compound pendulum,, consisting of a brass or steel bar with a fixed heavy bob, B and fitted with two adjustable and mutually facing, *, knife-edg^s F l and F2 near its two ends, so that the pendulum may be suspended from either. (Fig. 111). Two, weights, Wi and W^ can be made to slide along the length, of the bar and clamped in the position desired, the smaller, a micrometer screw arrangement for, 2 having, weight, finer adjustment of its position. The position of the e.g. of, the pendulum can be altered by changing the relative, positions of the two ,weights, their positions being, however, so chosen that the e.g. always lies in-between the two, , W, , knife-edges f, , The pendulum is first suspended from the knifeedge F,, and its time-period determined. It is then susK X'-/ B, pended from the knife-edge F2 and its time-period determined again. If there be a divergence in the two values, of the time-period, the heavier weigtit, is moved, l, up, or down and a proper position found for it so that the, time-period is very nearly the same, whether the pendulum be suspsnded from Fi or F2 The smaller weight IV2, is then adjusted by means of the micrometer screw, M, until the, time -periods, in the two cases, are as nearly equal as possible (say,, differing only by -01 sec. or less, i.e., until the number of oscillations, made by the pendulum in 24 hours, in the two cases, differs by just a, fraction of one full oscillation). When this is so, we have, obviously,, one knife-edge at the centre of oscillation of the other. The distance, between the two knife-edges is measured carefully**. This gives the, length L of the equivalent simple pe,idulum$ and the value of g is, 4ir 2 Lit*, (see, then calculated from the relation g, page 171), where, / is the mean of the time-periods about the two, knife-edges, which, Kater determined by the method of coincidences, (see 69)., ,, , W, , T, , ., , =, , The values of L and, and accurately as follows, *It, , made in, , is, , India,, , t, , may, however, be determined more easily, , :, , gratifying to observe that in Kater's own pendulum, these, from a special variety of steel, called *wootz\, , were, , W, , tin fact, the heavier weight, l is there to ensure that this is so., **This is done by means of a travelling microscope. The pendulum is, laid horizontally on a table with Us knife-edges lying alongside, and on a level, with, a standard steel scale, and the positions of the knife-edges read on the, -scale with the help of the microscope., Or, a better method is to use a vertical, comparator (an instrument carrying two microscopes fitted on two massiye stone, slabs), when the cross wires of the microscopes are first focused on the two, knife-edges of the suspended pendulum, and then on a standard steel scale, fixed, vertically in plape of the pendulum. The distance between the positions of the, ttwo cross-wires then gives the distance between the two knife-edges., , and, , jThis deduction from the reciprocal nature of the points of suspension, was first pointed out by Bohnenberger, in the year 1811., , oscillation
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PROPERTIES OF MATTER, , 174, , First a near equility in the time-periods of the pendulum Is, and, obtained about either knife-edge, by adjusting, % as exl, the, above,, time-period, plained, being slightly greater, say about, i.e., when the, F, than about F2, d(BOB DOWN), , W, , W, , ,, , ,, , C(BOB UP), , bob B i* down than when it is up., The weights are now kept fixed in, one of the, their positions and, knife-edges, say F,, moved up or, down a bit to further narrow down, the discrepancy in the two time-, , periods until a position is attained, when a little more displacement, makes the time period, of F,, F2 (with the bob up), about, greater, Fig. 112., than about F,, (with the bob, down), i.e., a reversal in the relative magnitudes of the time-periods, about the two knife-edges takes place. The time-periods of the, pendulum for two slightly different distances between F, and F2 are, noted, just before this happens, and similarly for two slightly different, These four distances, distances between them, after tins happens., the time periods, and, between Ft and, measured, are, accurately, 2, corresponding to them plotted on an exag/eratod scale, as shown in, Fig. 112, where a and b represent the distances and time-periods for, oscillations about F, in the first two cases and c and d in the second, two cases., The coordinates of the point of intersection O of ab and, cdthen give the true length L of the equivalent simple pendulum and, the true time- period / corresponding to it., 69., Kater determined, Eater's Method, of Coincidences., the time-period of his pendulum by what is known as the method, , F, , 1, , in which the oscillations of the experimental pendulum, (Kater's, or any other), are compared with those of a standard, second's pendulum, (i.e., a pendulum of time- period two seconds), which, may be a simple pendulum or a clock pendulum. This gives better, results than those obtained by simply timing the oscillations against, , of coincidences,, , B, , A, , T, Fig., , 113., , a stop watch or clock, the accuracy of which can hardly be expectedf, to go beyond *5 sec.*, , This means, , that the time taken for several thousand of swings will have
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175, , ACCELERATION DUB TO GRAVITY, , The experimental pendulum A (Fig, second's pendulum B, with, , 113) here is suspended in, knife-edge resting on a, lower ends of the two, that, the, to, see, taken, rigid support, care being, level and exactly coincide with each other, the, same, in, lie, pendulums, when viewed from in front, in their mean or equilibrium position., its, , front of the, , A, , marking device*, , suitable, , M, , arranged behind the, , is, , peri*, , dulum 5, such that when the two pendulums are in their mean posiis just covered by their lower ends and is thus not, tion, this mark, visible to the observer viewing them through a telescope T some, , M, , distance away., is to enable the observer, watching the oscillaas possible as to when exactly do theas, accurately, judge, into, two pendulums come, coincidence, i.e., as to when exactly do they, a, reference point, in the same direction., particular, simultaneously pass, And this is perhaps'best done by using a cross- wire in the eye-piece, becomes, of the telescope itself, (in which case the marking device, , The whole idea, , tions, to, , M, , quite unnecessary)., set oscillating, and, if they start, oscillate, time-periods, thpy continue to, 'one', observer, the, to, b&just, pendulum,, they appear, the time. But, just hidden behind pendulum A, all, , The two pendulums, and have identical, , together,, , 'in step', i e.,, , pendulum, , B, , are, , being, , 9, , their time-periods differ, ever so slightly, they soon get 'out of step, and their oscillations are watched carefully until they both simultaneously pass the reference point fixed upon, (say, their lowest positions),, in the same direction. When this happens, a 'coincidence' is said tois just not visible to, behind the, occur, and the mark, if, , ,, , M, , pendulums, , After this, the pendulums again get out of step and, the observer., the next coincidence occurs when one of them gains or loses a whole, swing or oscillation over the other. The oscillations made by both, between two successive coincidences are carefully counted., , made by, , Let n and (n+l) be the oscillations, the experimental pendulums respectively., Then,, , if t', , and, , 1-, , =,'(, -a, , ', , f, , f, , t, , the seconds and*, , be their respsctive time-periods,, , we have, , n, , i, , ^ ver^ nearly,, , and the, , of, , higher powers, neglecting the terms involving the second, n for, with t and t' nearly the same, n is sufficiently large, (about, 500 or more), and these terms become negligibly small., ;, , /, , Now, f'=2, whence, , f,, , *A, mark on, , it,, , sees., so that,, , ?=2, , f, , ^, , 1, , --, , 1, , n, , ., , 'J,, , r pendu, pendulum, B being a ssecond's, L pendulum, , the time* period of the experimental pendulum can be easily, white pafcer pointer or just a black (iron) stand, with a white chalk-, , would do
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PROPERTIES OF MATTER, , 176, , This value of, , calculated out., , then used in the expression for, , / is, , g,, , above., , be easily seen that with all the care taken, it is not, really possible to determine with absolute certainty the particular, oscillations at which the exact coincidence occurs. Luckily; however,, even a difference of a couple of oscillations, this way or that, hardly, For, supposing there ia, matters, ifn be fairly large, as it usually is., an error of two whole oscillations made in judging the point of, It, , will, , coincidence., , Then, clearly,, , =, , 1, , The, out to be, , error, , introduced, , [, , +'0008% and, , {, , 2, , thus, , is, , - -, , 1, , t', , ,,-^2"), , 2/, , ,, , therefore, not of, , is,, , which,, , y^y J, if, , fi=500, worka, , much consequence., , own determination, made at Portland place, the, of a standard clock and the swings of the two pendulums observed by means of a telescope from a distance of 9 feet., In Kater's, , Note., , pendulum was, , set in front, , The standard clock was checked every 24 hours by stellar measurements,, the rate of swing of the pendulum was, in effect, compared with the rate, , io that, , )f rotation, , of the earth, , itself., , Successive coincidences occurred every 530 sees., during which time the, The error thus, reversible pendulum completed 528 swings or half oscillations., vorked out to 1 part in 1,00,000 and the length of seconds pendulum at sea, level, in the latitude of London, cams to 39*13829, inches., , Computed Time Bessel's Contribution. Kater, in his, pondulurn, made the time-period about the two axes, exactly equal, which, as we have seen, is an extremely tedious process., But Bessel showed that it was by no means necessary to make, them exactly equal and that it was enough to make them only nearly, 70., , reversible, , equal., , Thus, suppose the time-periods about the two axes are, respectively, (both being very nearly equal), and that, their respective distances from the e.g. of the pendulum., f, , 2,, , /, , ?,, , and, , I', , and, are, , Then, we have, ~~~, , ', , (*), , fa~, 5, V&2l|L/, ^, and, , ,, , f, , 2, , o, =27r, AA //, , Squaring and re-arranging, /1, , 2, , /g=4-7r, , 2, , 2, (/:, , +/, , 3o that, subtracting, , 2, , ),, , (iv), , ., , from, , /', , 2, , (/), , (///), , (Hi),, , ,..,, , is the radius of, gyration of the pendului, , l-<*but its e.g., , ... (n), , .,, , 2, ., , +, , rwhere k, , and, , (//),, , we have, , and, , we have, , g, (/-/'), , + (*.*-**) (/+/'), , 2(1, , 1'\
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ACCELERATION DUE TO GRAVITY, , 177, , 8, This quantity,, , +-*''-'', , ., , -, , ^, , where T, , L- = T, , l, , *.'+'', , ^~~^, , [>, L, , ,, , ~, , =< 2, , ', , simple pendulum, , computed time of the pendulum., This distance (/+/'), between the two knife-edges, can be, determined accurately, but (//'), the difference between the, distances of the two axes from the c.g cannot bo determined to any, is, , called the, , high degrea of accuracy, because of the difficulty of locating the, Smce, however,, position of the c.g. of the pendulum correctly., 2, 2, 2, 2, /2 ), is much too small a, (/, quantity compared wirh (/ 1 +/, ),, this can only introduce an inappreciable error which does not^, matter., 71., Errors in the Compound Pendulum and their Remedies., Besides the difficulty of adjusting the time- periods to be exactly, the sain? about either knife-edge, and of correctly measuring the, distance between them, there are a number of other sources of error, in a Kater's pendulum (or the compound pendulum, in general) for, which proper corrections must be applied to obtain an accurate result,, the chief among them being the following, :, , The Finite Amplitude of the Fenfluhim. The expression for, (/), the time-period has been deduced on the assumption that the, amplitude of swing of the pendulum is vanishingly small for, then, In actual, alone, will its motion be truly simple harmonic in nature., This reduces its, practice, however, it has always a finite value., The observed, acceleration and thus increases its time-poiiod, time period may be corrected for this error by multiplying it with, are half-swings (in radians) at the, (1, ^.^g/IH), where a, and, 2, beginning and at the end of the experiment, respectively, as can be, seen from the following, ;, , :, , a manner similar to that discussed in connection with a, Proceeding, simple pendulum (Ahernative method (/), page 162), we have, in, , i/w, Or,, , 2, , /G>*, , = mgl (cos 0-cos a)., = 2mgl (cos 9 cos a),, , [Relation, , where a is the angular amplitude of the pendulum and, ment at time /., , Now, So, , that,, , Or,, , Or, , f, , / *=, , mk*+ml 2 = m, , 2, , 2, (A:, , m(k*+ /, , 1, , 2, ), , and, , -!-/ ),, , f, , j, , -, , ^, , <o, , Q, its, , -, , page 162., , angular displace-, , -, , -^, , (/), , ., , 2mgl(cos Q cos, , a).
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OF MATTJfiA, , 178, , to f/4 and, Integrating this expression for the limits, the observed time-period of the pendulum, we have, , -, , ,, , where, , -, , o, , O, , to, , (cos, , Q-cos*Y, , fa, ^, , TT^7, , V2]7", ~~*i Jo, , a, f, , * \ -**\, )*, , Jo, , Or, , Or, , *, , Putting, , j/, , -y-=, , 5//i, , J/V? 5^,, , we have, , J d$ cos -^, , ^, , Substituting the value of dQ in the above expression,, , 2, , sin', , ~ -s, , ^, , we have, , 5W!, , /i, , |*, , s, <f>, , -J//Z, , 2, , cos, , Or,, , -|-, , )*, , = sin, , )*, , we have, , 5/w, , cos, , Or,, , Now,, , "*, , y ^TT . y 1-^4 - (i-, , --, , f, , la
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ACCELERATION DTTB TO GRAVITY, , Or,, , ,, , -, , 2, , */y, , +L, , Now, lTt\l, , imall amplitude., , If a, , be small,, , sin', , ~+, , -., , would be the time-period of the pendulum,, , Denoting, , sin, , -*^[l+i, , 179, , =, , it, , by /,, , ~, , for, , an, , infinitely, , we have, , and we have, , ,, , Since the amplitude (or half swing) does not remain constant but decreases from a, in the beginning to a, at the end, both being small, we replace a 8, , by, , So that, , a^., , <.There are three distinct ways in which the pre(//) Air-effects., sence of air affects the time-period of a pendulum, This tends to reduce the restoring, (a) Buoyancy of (he air., couple acting on the pendulum, due to slight decrease in its weight,, similar to tho one produced in a body immersed in a liquid. For, if, be tho mass of the pendulum and m', that of the air displaced by, to, it*, the restoring couple is reduced from the value my./ 5/72, (ml-m'h).g sin 0, where /?f is the dibtance between the e.g. of the, displaced air and the axis of rotation of the pendulum, and can be, obtained sufficiently accurately from its physical dimensions., The equation of motion of the pendulum thus becomes, :, , m, , m(tf+l*Y, , This, period, , tt, , is, , ^, , g, , sin, , e(ml-m'h)., , obviously the equation of a simple harmonic motion, of time-, , given by, /, , :, , (ml-m'hg), , 2w \/(* +/*)//*, the expression for the timeis clearly greater than/, period in vacuo., The time-period of the pendulum is thus slightly increased due to buoy?, , which, , ancyoftheair., This was -the only correction taken account of by Newtw, followed by, Kater, and it was left to Bessel to show that other corrections due to air-efFectf, were also called fox., , ____,, , *Th>s can easily be obtained from the volume of tfre pendulum and the, density of the air, at the time., t The value of /j may not be the same as that of /, im]es> the .pendulum, hat a uniform density.
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PROPERTIES OP MATTER, , 180, , Some, , (b), , air being, , dragged along with the pendulum, during, , it$, , (Du Buat's Correction). The p3ndulum during its 'to and, fro* motion, ^carries air with it" and this increases it effective mass,, and hence its moment of inertia, making the obssrved time-period, greater than the true one, as will be clear from the following, motion,, , :, , as it oscillate , can bo, ,, by attaching a feather to its bob, in a direction at right angles to its direction of motion. It will be found that the feather, tilts in a direction opposite to that of the mon'on of the bob, showing that the, air surrounding it is at rest. If, however, the feather be sufficiently close to ihe hob t, it is not found to tilt at all, clea^lv indicating that the air in immediate contact, with the bob moves along with it, or that it "carries air with it*., , That the pendulum does, shown by a simple experiment, viz, , 'carry', , ^ormair with, , it, , Let the mass of this air 'carried' by the pendulum be m" and let the distance of its centre of mass from the point of suspension of the pendulum be d., the moment, Then, clearly, the effective moment ot inertia of the pendulum, i.e, of inertia of ths pendulum and the adherent mass of air with it, is equal to, 8, a, w( -f /*)-f/wV, And, therefore, the equation of motion of the pendulum now, ,, , ., , becomes, , " ", , QTt, , _.., , _, , ^1, , /, , The time-period of the pendulum, , -, , t, , in thus given, , ., , by, , t, , 2n^~J^I^j^, , once from the above that, , It follows at, , _A^4/' w^l P-M, +, f, , **, , n, Ur>, , r.'!LZiLC___. e, , 4**, , ml, , /, , 2, *, , tn^h, , 7, , ', , ml, , f Neglecting second, order terms, L, , assuming the time-period here to bs already corrected for the, , ', , finite arc, , of, , its, , swing., , Now, if / t and /., be the distances of the two points of suspension from, the centre of gravity, on cither side, such that the time-periods in the two cases, are nearly the same, then, if h l3 h 2 and */,, d* be the re*pective distances from the, point of suspension of th3 centres of buoyancy and the centres of mass of the air, adhering to the bob, in the two positions, we have, .,, , ...., , '.So, , that, subtracting relation, '.', , Since, , W- W, , -, , t l is, , (//), , from, , (/),, , we have, , (*-*), , 4-, , very nearly equal to, , /a ,, , 4-, , we have k 2, , And, therefore,, , l, , Here,, ing, , it, , by, , T2, , ,, , /, , _/, , we have, , a, , is, , the, , scluare, , of the computed time, (see, , 70)., , Denot-
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181, , AOCELBBATIOH DTJE TO GSAVIfir, , Thus, the obvious method to eliminate this correction is to make, -h9t (i.e., to make the psndulum symmetrical in shape), which will reduce, the two expressions on the right-hand side of relation (///) to zero. This is, precisely what has been done in Repsold's reversible pendulum, (see S 72,, page 188)., Both effects (a) and (6) due to air can, however, be made almost negligible, by arranging to swing the pendulum in reduced pressure, a procedure now being, increasingly adopted for the residual effect in low pressures is found to be a, linear function of the pressure. The required correction can thus be directly, obtained by plotting a graph between pressure and time-period and obtaining the, value of the latter by extrapolating the graph to zero pressure., , ^Aand, , hi, , The viscous drag due to air produces, Viscosity of the air., (c), a damping effect on the pendulum and tends to reduce its amplitude,, thereby increasing its time -period., For, taking the viscosity-drag for small velocities to be proportional to*, motion v ould be of the type, , velocity, the equation of, , Let th, , solution of this equation be, , QAe**, 0., , which, c, So, , is, , ..., that,, , a quadratic equation in, / 7I, , -r\, -, , ^4i*, ----, , 2, , Hence the general solution, , --, , -r, ---y-, , 2, , Then, clearly,, wr+/* = 0,, , r, , 4./A/~I, J, , V, , r, , ~4~, , f", , *, , where 7, , Land, , is, , +7 V/^*/4 1*, , /*, , is, , ., , o> -f, , eo., , -f, , which, , Or,, , Be, , J)-, , |L, , I, , V~^r, , j, , 2, , 2, , /4 I/., , J, , /+;M "" B), , J/n, , V(, , ^"r, , a simple harmonic motion of decaying amplitude, of a time-period, ', , _, , Now,*2rr/v, any viscous drag., , So, , ^, , r, , ,, , the time-psnod of the, , I, , that,, , -, , 1, , +, , pendulum in the absence of, , -, , *.[, , we make use of ths approximate relation f = 2n/^^r for the time2, /a, So that, substituting this value, of, the pendulum, we have /* = 4w /f, period, of f* in the expression for / above, we have, , And,, , if, , ., , 1, , -jjT^J', , very nearly., , This correction due to viscosity is however, ~, the order of 10 9 and is, therefore, usually neglected., , much, , too small,, , being of, , Due to the yielding of the, Non-rigidity of the Support., tends, to, be, the, time-period, support,, greater than, the correct value,, as explained below ;, (///)
Page 190 : PROPERTIES OF MATTER, , 182, , It might, at first sight, appear improbable that the stipport should yield, by the mere swinging of a pendulum suspended from it. This is, however, not, In fact, any ordinary, so, for the simple reason that no support is perfectly rigid, support, we consider to be rigid, would yield under a weight of 100 k.gms. or so., True, a pendulum is seldom as heavy as that, but in view of the fact that we, can measure lengths and time-periods to an accuracy of one in several thousand,, it 1$ only in the fitness of things that we must take into account even this, slight yielding of the support, ii we really aim at a high degiee of precision in, Our work., , Again, it is also true that we can adopt ways and means of eliminating, error altogether (as well as that due to the presence of air) in so far as the, pendulum is concerned, as explained in 72, below, we should, nevertheless,, acquaint ourselves with the method of deducing a proper correction for it should, it become necessary in other similar cases, wfoere its outright elimination is not, feasible or possible., this, , Now,, , on to a, , then, let as pass, , brief consideration of, , it., , We know, , that a vibrating body tends to set into vibration any other body, in contact with it the degree of response of the latter depending upon how nearly, its natural time-period agrees with that of the vibrating body,, the closer this, agreement between the two, the greater the response and vice versa., , a, , In the case of the pendulum, therefore, the support carrying it also yields, vibrations and is forced to oscillate co-penodicaily with it., , little to its, , This oscillation of the support may be resolved into two rectangular com(i) along the vertical and (//) along the horizontal, the latter having a, more pronounced effect on the time-period of the pendulum, than the former., , ponents,, , OQ be, , the mean or equilibrium position of the, of length /, uith the axis of suspension, into, passing through 0, then, as it swings through an angle, the position OG' there are two forces acting upon it. (j) along, the arc of its swing, to which its motion is due and (//') the, other at right angles to it, i.e., along G'O, (the centiipetal, , Thus,, , if, , pendulum, (hig, , 114),, , 9, , force)., , So, , that,, , acceleration of the pendulum along the arc, <Ps, ., d 2Q, ds, d, , where, , ^, , and acceleration along, v, , a, 1, , J", , the length of the, , "*, ds, , /, , ', , pendulum, J", , "*, , *', , v, , = ds ldt, , & /j, , U4., [, , These accelerations, obviously, act, , at the support 0,, , \, , dt, , \, J, , with the component, , in the horizontal plane, , and the component, , So, , that, If 9, , in the vertical, , plane, , be small we have, , horizontal component of the acceleration, , add, , vertical, , component of the acceleration, </e, , (/l)
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DOE TO QRAVttt, , 1OOULLJEHAT10M, , Now, from, , ,/S/j, , mgl, , -, , (cos 8, <*, , ^-^, , jm (&H/, , cos a), , ^/, , ^-cos, (c^, -, , ^, , viz.,, , ~, , 2, ), , (, , ), , p, , whence, expanding cos and cos a and retaining only the, of the small values ot and a, we have, , /, \, I, , de, ir, , dt, , V = gi, }, , /, , 2, (, , ~e 2 ", , ., , ,, , 2, , and, , e4, , *"~~'T"i, , 2, , in, , rJ, , (, , two terms,, , i, , L, , -, , first, , 2, , ~, , cos e, , *", , +r, , Substituting these values of, , above, we, , r, , ), , z, j*2~r>2, , k, , we have, , -* tng't*0,, , pendulum, ^-j/a, , the equation of the, , And, from ths energy equation of the pendulum,, , we have, , 153, , !, , + ~A4:r, !, , relations, , e, , in view, 9, ., , ~^~i, 6, , I, , (/), , and, , (//), , therefore have, horizontal acceleration, , and, , /., , acceleration =*, , vertical, , (-,5,, , ^, , (, , T*, , 6-f /^., , )., , ,5,, , /a, , ^, , ), , 8, , +, , /TTTa, , ^, , ~^T7T, , >, , Or, neglecting the second term as being extremely small, we have, \, , (/+, &*, , and, , vertical acceleration, , And, therefore, horizontal force on, and, , the, , /, (, , #/, , support **(, , /..,, , horizontal force on the support a 9, , and, , vertical, , ,,, , ,,, , \, , ~jj*Tjr )'&*, , vertical force on the support, , ,,, , )'^, , /2, , 2, , =f, , -, , ^ ^, , j.Q, ", , ~~fciZ/r ), , a 68., , 2, , In other words,, Since, 6 is small, O is comparatively very much smaller., the horizontal foice on the support is very much greater than, the vertical force on it and the latter may, therefore, be easily, ignored., Now, as the pendulum moves from O to G' t the point, of suspension moves from O to 0', say, so that its displacement is equivalent to shifting its axis of suspension to Q,, , (Fig 115)., , Then, if P be the displacement of the support per unit, force, in the horizontal direction, when displaced through an angle, a, i, vvil, 6, its displacement due to a horizontal jorce (w#/ /&*-t-/ )0, 2, 2, clearly be equal to (A>/'/ p/P-F/ )G., Since, , Q, , is, , now, , the effective axis of suspension,, , and hence the displacement of the point of suspension, , 00', We,, It, , therefore, have, , =, , OQ.Q., , .e, , -, , 8.0., , --r A, , [Putting, , whence,, , *-, , ~^GQG', OQ =, , -., , follows, therefore, that the effective length of the, , this yielding, , of the support, , is, , S., , pendulum duo to
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184, , MATTER, , PttOPEBTIfiS OF, , And, therefore, the timf-r}eriod of the pendulum, ^, , c, , From, , ^, , now, , is, , V A^w V ^^, /, , A, , ,, , /, , ^(/-f, , this it follows at, , ~, , ), , jk**liW, , A, , =, , Ci+ s i), , -f-, , a+ s 2 ), , +, , 2, , L, , and, , 4^:^, , Since, , /! is, , (/, , ra, , ,, , (/,+/,,, , r-, , 1, , y-(v, , f, V, , _, , i, , i, ^a, , we have, , 2, , + V.-*, , =, , for, , L, , that, putting, , 9, , (/ l /1, , /., , wrnre 5 t and 5 2 are, the corresponding, , \, , y, , additions, , ^, , ^, /, , /A /2 ., , -f,*/1 )/(/1 -/1 ), , =, , to, , the, , two l-rigths of the, L pendulum., , And, therefore,, , ', , j.fcV.., , C.T7, +,,>/r ), , So, , l, for, , which the time-periods, f, , ), , 7l, , ^'J, /a, , very nearly equal to, , (/+*), fWriting, [-W, , ', , once that, , So that, if /! and / 2 be the lengths of the pendulum, fi and / a are v^ry nearly the sams, we have, 2, 2 /, x, /C, st, 4 -7, , given by, , T. (where T, , is, , |v *, , the computed time),, , we have, , P, ', I, , s\', , +", (, , mgp, , (THE PENDULUM), , For, /!^/2, , L, the length of the, , e Q u ^ va ^ent simple, , pendulum., , the weight of the pendulum ;, Clearly, here, mg, f\, S;, so that, //** correction factor wep 75 f/itf displacement of, *, to the weight, j/j^, SUppori due to a horizontal force equal, of the pendulum and can b^ determined directly by suspending the pendulum from a string passing over a, pulley and attached horizontally to the support at O, as, shown, (Fig. 116), when ihc displacement OO'~mg$, can bc read accuratelv bv mcans of a microscope., is, , ^, , Vening Meinesz suggested a method by which, this, , correc-ion, , could bo considerably, , reduced,, , viz.,, , that of using two pendulums, swinging from the same, support but in opposite p ases with each other. This, involves, however, the d fficulfy of having to adjust, Uvir time-periods to very near equality. The correction, is thus bcsi eliminated as explained in, 72, (page 187)., , >y, , Fig. 116., ('V), , rounded)., altered., , The, , knife-edges not being perfectly sharp, (hut, to this also the effective length of the, , Due, , more or less, pendulum i$
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ACCBLEttATIOtf, , DUlfi, , 186, , TO, , is not a mathematical line we have so far, In actuil practice, it has a definite shape, -generally, symmetrical with a finite radius of curvature., Let us, as a first approximation, assume the edge, to be the pai t of a cylinder,, a cylinderical cone, as shown, in Fig. 117., be the centre of nirvaiure of the, Then, if, edge, a line perpendicular to the plane of the paper and, passing through O represents the axis of the c>iindrical, edge., As the pendulum (of length /) swings, the edge, also moves along with it about thi\ ax s through O, so, that the axis of suspension is. in effect, shitted fiom S to, O, i.e., through the distance SO, r, the radius of curvature of the edge, and the effective length of the pendulum, thus becomes (/ 1-r)., , For, the axis of suspension, , tacitly, , assumed, , it, , to be, , Since, 'however, the instantaneous axis of rotation, of the pendulum stiH passes through the bottom of the, knife-edge (S), the moment of inertia (/) is still to be, taken about this axis, In other uords, we still have, / = m.(k*+l 2 ) where m is the mass of the pendulum., t, , The, , equation of motion of the pendulum, , thus, , becomes, -mg(l+r}B., And, therefore,, whence., , its, , is, , time-period, , ., , given by, , Fig. 117., , [, , Q being small., , t, , Or., , =, , /4-r, , we find two lengths of the pndulum, say / t and /, on the two sides of, such that the time-periods ( 1 and r 2 for tnein are nearly the same, (with,, of course ^ not equal to J 2 ), then, if r x and r a be the radii of the two knife-edges, If, , its, , c.g, , ,, , respectively,, , So, , we have, , that, subtracting the, , Since, , t^ is, , second expression from the, , very nearly equal to, , /3 ,, , we have, , k*, , first,, , we have, , And, therefore,, , Or,, , i-W, , (/i-M.), , (Ijzjl)^!-^) ], , [, , Or,, , Again, putting, , ~, , J, , ~], *i, , have, , f, 'a, , -, , =, , T1 ,, , where, , T, , is, , the computed time-period,, , we
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18ft, , OF, , FttOPteBTlfis, , Hcfe, clearly, the correction term, , -!, 'i) tectfffiei, (, (Otay, \ /i*i, , zero, (Wily if, , only if the two knife-edges have the same radius of curvature., Since it is difficult to make the two knife-edges of exactly the same radius, of curvature, the suggestion at once comes to the mind that the same knife-edge, may be used at both the two points of suspension. But this may affect the, position of the e.g.. which might be different for the two positions of the knifeedge. And, then, it would disturb the symmetry of the pendulum, necessitating, the troublesome air-corrections. This difficulty may be tided over by ananging, two knife-edges of the same shape and mass, and by using only one of them for, suspension, i.e.. by interchanging them when we change the sHe of the pendulum., Here too, however, an error may creep in if we do not succeed in replacing one, knife-edge with the other exactly in its true or original position. This difficulty, too may be got over, however, by performing the experiment four-times, first, taking the two observations for / x and /8 with one position of the knife-edges and!, then two, similar observations with the knife-edges inter changed, Thus,, if T! and TJ be the respective computed, the two, cases,, time-periods in, , fi=rt, , ,, , /.*.,, , ., , we have, , So, , that,, , adding the two, we have, *, , v, , 47i*, , correcthe sum of the corre, terms (involving, , {*.andr^-O., , -^ r, , whence,, , 2(/H-/,),, , x, , tion, , -^-J, , 1, , - = (/!+/)., , But, even this correction does not help much. For, a* the pendulum, swings to and fro about its mean position, the edges invariably get chipped off,, resulting in the loss of weight of the pendulum., , The one and only way of eliminating this correction, now being increasingly ustd, is to replace the two knife-edges in the pendulum by just plane, bearings, / <?., by flat plates, and to provide a fixed knife edge on the support,, the latttr being carefully ground to a sharp edge and the foimer being accurately plane or flat and always placed in the same position on the knife-edge., N.B., , In a bid for an extremely high degree of accuracy, the effect, , of, , pendulum was also, , investigated at Potsdam under the, supervision of Helmert, viz., its periodic extension under the varying longitudinal strain and its flexure under the changing bending moment to which, it is subjected as it swings, the latter being the more important of the two and!, resulting in a reduction in the effective length of the pendulum., the elasticity of the, , Thanks, million, , is, , to, , the work of Clark Heyl, or less easily attainable, t, , and Cook, an accuracy of one, , in, , ai, , now more, , of, , This', Temperature during the Experiment., change in the length and hence the timeperiod of the pendulum., A correction for it can, however, be readily applied, if we know, the coefficient of expansion of the material of the pendulum. Or,, the error may be eliminated altogether by using what are called, (v), , Change, , results in a corresponding, , invariable pendulums, (see, , 76)., , We, , have considered above the errors and, (vi), corrections in wo far as they relate to the pendulum itself. To obtain, an accurate value of g at a place, however, certain other corrections, must also be applied, v/z., the corrections (a) for rot at ion of the earth,, Ib) for latitude, (c) for altitude, (d) for elevated masses and (e) fopOther Errors.
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ACCELERATION DUB TO (iRAVIT*, , 1, , of the place, all of which are discussed in, the succ^edin^ chapter., Nevertheless, as a method for determining the value of g, a, compound pendulum, (e.g., the Kater's pendulum), is distinctly, superior to a simple pendulum., For,, the terrain or the topography, , whereas a simple pendulum is just an ideal conception, not, (i), realizable in actual practice, the length of an equivalent simple pendulum, and hence the value of g, can be easily and accurately determined, with, , its, , help, , ;, , vibrate? as a whole, there being no lag between the, (//)*, the string, as in the case of a simple or a Borda's dendulum ;, it, , and, , bob, , the length to be measured here is clearly defined, viz., ttie, (Hi), distance between the two knife- edges, and can thus be easily and, accurately measured whereas, in thn case of a simple pendulum, the, point of suspension and the e.g. of the bob are both more or less, indefinite points, and hence its true length can hardly be expected to, , be determined correctly, , ;, , due to its large mass, t\e compound pendulum keeps on, a fairly long time, thus enabling its time-period to be, determined with accuracy. In a simple pendulum, on thn other hand,, the oscillations die down much too soon due to the comparatively, small mass of the bob, and it becomes difficult to determine its timeperiod to an equivalent degree of accuracy., (iv), , oscillating for, , The one obvious disadvantage in the case of a compound, pendulum, however, is that during its vibrations to and fro, about, its mean position, some air is dragged aloni* with it, as mentioned, above, thus increasing its effective mass and hence its moment of, inertia., But it has been clearly shown by B^ssel that if it be of a, form, symmetrical about the centre of its geometrical shape (which is, not the same thing as its centre of gravity), this error is automatically, eliminated., This explains the symmetrical shapes of various types, of compound pendulum* we use, though, theoretically, a rigid body, of any shape whatever would do., 72., Other Improvements due to, Bcssel., Not only ha, Bessel done away with the trouble and the tedium of having to make, the time- periods about the two axes identical, but he has also, succeeded in removing quite a few other important errors. Thus,, for, , example, (i), , :, , The error due to some air being dragged along with the, is removed by the symmetrical physical form of the instrusuggested and shown by him., , pendulum, ment, as, , The error due to the knife-edges not being perfectly sharp,, (ii), which a correction, proportional to their radii of curvature, would, be necessary, (unless they be of the same radii of curvature), [see 71,, has also been eliminated by him. For, he has shown, (/v), page 184],, that this error would automatically vanish if the two knife-edge,, at the two ends of the pendulum, could be made interchangeable., Tben if f, and f, be the computed times, before and after the in^er-, , for, , f
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188, , fBOPERflfcS OF MATtfitt, , change of knife-edges, the true time-period, , /, , is, , given by the relation, , Bessel, unfortunately, died before he could put his theory, into actual practice but, later, Rep>old did actually, , ^/"ff^, , construct in the year 186), a reversibh pendulum of this, , type and used, , with success., , it, , Repsold's Pendulum, , pendulum but, , is, , is, , svmmetrical, , more, , or, , less, , a Kater-type, , geometrical form^about its, Here, we have a rod 7? fixed on to, mid-point, (Fig 118)., two rings R L and R z at its two ends, which, in their turn,, have two short rods screwed into them, terminating in, inside the rings and carrying two, 2, knife-edges E l and, bobs B l and B2) one solid and the other hollow*., , R, , in, , The time-period of the pendulum can be made nearly, equal about either knife-edge by moving the bobs up and, down and screwing them into the desired position., , With the symmetrical form of the, error, , due to, , air- effects, , is, , peivlulurn, the, automatically eliminated, as, , explained in 71, (ii) above., And, (iii) finally, ths error, due to the yielding of the support, has boon eliminated by D^ffarges, by using two reversible Rep sold, t)p3 p3nlulurm, of the same inns but different lengthy the sime, ratio of I to I' and h.ivin ; a common pair of knife- edges, (to be used, with either of them). He has shown that if L l and L> be the reduced, lengths of tha two reversible pendulums (i.e., the lengths between, 118., , Fig, , 2, 2, the knife-edges) and TJ and T 2 their computed times, then \/(T 1 ~T 2 ), L2 ), as, gives the correct time-period of a pendulum of length (LL, can be seen from the following, ,, , :, , We, , have, , g*i, , (See pages, LI 84 and, , and, 4?r, , where, , /t, , and / 2 are the two lengths on the two sides of the, pendulum and tt and // in that of the other., , that, subtracting th3, , c g. in the, , ', , case of one, , So, , 186., , a, , 83con<l expression, , from the, , first,, , we have, , (i.e., the second term) in this expression, can be made zero if /,:/,::, /,', and this is easily done by adjusting the positions of the bobs of the two pendulums. With this adjust-, , Clearly, ths correcting term,, , V, , :, , ment made, we have, , *This is to ensure that the lengths / t and / t of the pendulum on the two, sides of the e.g. are not very nearly equal, or else the correcting terms for the, c/ror due to air effects, (page 180), will not be small. This is the reason why in, a Kater's pendulum one bob is made smaller than the other.
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ACCELERATION DUB TO GRAVITY, , we use only one, , Or, if, , 71, , explained in, , =, , -^ o, 47T-, , (///),, , knife-edge, (fixed, , 189, , on the support, as, , page 181), we have, , = L +mg S,, , 2, , and, , Li+mg.fi, , J^, 47T a, , L, , Seepage, , 184., , where Z^ and L 2 are the reduced lengths and r l and T 2 the computed, time-periods of the two pendulums respectively., ,, , Thus, subtracting the second expression from the, straightaway,, **, , first,, , we, , have,, , ', , V-T, , \, , )=L -L z, , 2, 2, , i, , ., , This removes at one stroke the errors due to yielding of tke, support and curvature of the knife-edges, as also those due to aireffects,, , and we have, , a, , *., , This, \alue of, , about the most accurate method of, g at a given place., is, , determining the, , 73., Conical Pendulum. A simple conical pendulum is just, a simple pendulum, (ie., a srnill heavy bob attached to a, light, inextensible string), which is given such a, mot on thit the bob describes a horizontal, S, ;, , circle, , and, , the strirg traces out a, , cone., , The, , <, , of the pendulum is the distance, between the point of suspension and the, e.g. of the bob., , *, , length, , Let, , \r, , m, , be the mass of the bob J?;, of the, v, its velocity and r, the radius, circle it describes, (Fig. 119)., Then, its, the, centripetal acceleration towards 0,, centre of the circle, is equal to v 2 /r, and, the centripetal force on it is, therefore,, mv*/r in that direction,, , Let, , /, , /, , r,, , */.-, , ", , '", , -"."." I, , ".-13, , T>^*-'|, , ma, , SO, , be equal to h., Clearly, the forces acting, , nL., , pig, 119^, , bob are (i) its weight, wg, vertically, the string T, in the direction BS., , on the, downwards, and, , (U) the tension, , of, , The weight mg is balanced by the vertical component T cos 0,, of the tension T of the string, and its horizontal, component T sin, 2, provides the centripetal force wv /r towards O, where o is the semivertical angle of the cone, , T sin = mv 2 /r, T sin =, , Thus,, , n, Or,, , =, , tan, , Or,, , Since, , have, whence,, , v2, , v*/rg=~tan 0., ==r 2 .co a , where, , r*.w 2 =r.g. tan, , uP^glh,, , and, , T, , and, mv*lr, L, , ^, , cos, , =, , mg., , v2, , Jl_, , t, , Or,, , w, , ^, ,*., , is, , v*=r.g.tan 0., the angular velocity of the bob,, 2, , r.g,r//j=r g/A,, , [\, , o>, ? 1, , we, , tan 0=r/A.
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PEOPlfiBTIBa OF, , MATTEB, , Now, the time-period of the pendulum, 2*, 27T, _, , =, , is, , given by, , JrTTA /, , g, t, , where, , / is, , =27rA, , [v, , ..(it), , >, , ft, , -/we., , the length of the pendulum., , It will, , thus be noted that the time-period is the same as that,, h. the axial height of the cone., , of a, , simple pendulum of length, , =, , is nearly equal to 1, so that h, If, he very small, cos, /,, /H ctf/zer won/5, the, the time period is almost independent of 0., time-period remains the same whether the bob moves along a circular or, ;, , i.e.,, , a linear path., 74., Steam Engine Governor. It will be seen from relation (/), above, (73), that the angular velocity (co) of the bob of a conical, pendulum varies inversely as the square root of the depth (//) of its, or, conversely, that the depth of, e.g. from the point of suspension, the e.g. of the bob, below its point of suspension varies inversely as the, ;, , ,, , square of its angular velocity,, , made, , use of in the construction of what is called the, a steam engine, which is just a device to maintain the, speed of the engine constant by regulating, or 'governing', the supply, of steam from the boiler to the steam- chest,, , This, , "governor*, , is, is, , In essentials, it is just a combination c*f two similar conical, pendulums, mounted on either side of the vertical shaft (with a common point of suspension), rotated by the engine,, and cons, sts of two rods OP and OQ hinged together at their upper end O to the shaft OS, and, carrying two spherical metallic bobs P and Q at, their lower ends,, , rods connect, , Two other smaller, to a metallic collar C,, , (Fig. 120)., , O/^and, , OQ, , which slides freely along the shaft, thus operating a lever which controls the throttle valve, or, the steam valve, opening it partially or fully,, according as the collar moves up or down the, shaft., , Now, when due, to the cylinder, the, bobs rotate faster,, , to a greater supply of steam, shaft, and, therefore, the, , h propori.e., & increases,, tionately decreases, or the bobs rise up, thus, partially closing the steam valve, thereby partially cutting off the, supply of steam to the cylinder. This automatically results in a, falling off of the speed of rotation (<o) of the shaft or the bobs, and, , when this happens, h increases, i.e. the collar slides down with the, bobs, thus opening the steam valve more fully, allowing more steam, into the cylinder, which then, increases the rate of rotation of the, So that, by proper adjustment, the rate of supply to the, shaft., steam chest or cylinder, and, consequently, the rate of rotation, the shaft, can be maintained at, constant v, 9, , any
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DUB TO, , GRAlfttt, , 19), , The sensitiveness of the device, however, decreases with the, For, we have the relation, increasing speed of the engine., 2, , co, , differentiating which,, , we have, , 2, , whence,, , dft, , by, , (fv), , ~~~dh., (///),, , ~, , h, , g.dh, , as, , =*, , 2cu.rfco, , And, therefore, dividing relation, Zw.da), , ...(/), , -g//f,, , -TT, ^, , X-, , =s, , --, , -, , ,, , Or,, , (', , v), , we have, ., , ,,, , 2rfa>/a>=, , aft/A,, , ~, , 0}, , Or, substituting the value of, , ft,, , from relation, , (Hi), , above,, , we have, , Thus, it is clear that dh oc l/o> 3 i.e., dh decreases as w increases., In other words, the change in the position of the e.g. decreases with, increasing angular velocity of the bobs or the shaft, thus slowing, down the 'up and down motion' of the collar along the shaft or, ,, , decreasing the sensitiveness of the device., 75., Other methods for the determination of <g'. The following, are a few other methods that, iy bs used to determine the value of, g at a place. Although they do not compare favourably with the, , m, , pendulum methods, , in point of accuracy or ease of performance,, , they, , are, nevertheless, valuable laboratory exercises, affording good, trations of tbe various principles employed for the purpose., , illus-, , then, are these different, , methods, , Here,, , :, , We, , The, , Inclined Plane., have seen before, in, 39, (page, acceleration a of a body, rolling down an inclined plane,, (without slipping), is given by the expression,, (1), , 88),, , bow the, , a=*[r*t(k*+r*)]g sin a,, the radius of the body jfc, its radius of gyration about its, axis of rotation a, the angle of inclination of the plane, and, g, the, acceleration due to gravity at the place., , where, , r is, , ;, , ;, , It follows, therefore,*, , *, , thatg=, *, , sin, , a, , So that, knowing r, (by means of a vernier calliper) k, (from the geometrical shape of the body) sin a,, from the height and length of the, plane) and a, (by direct experiment, as explained below), we can, , easily, , calculate out the value of g at the place., , The value of a can be easily and accurately obtained by, noting, the distances covered by the body, down, along the plane, in successive equal intervals of time*, and, plotting the distance-time curve, 2, The equation of the curve being 5, for it., |af (u being zero, because the body starts from rest), it will,, obviously, be parabolic in, *This may be easily done if the angle of inclination, (a) of the plane be, small ; for, then the acceleration of the body will also be small and the time, taken by it in rolling down the place will be, fairly accurately measured by means, , =, , of a stop watch., , ,
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PROPERTIES OP MATTER, , 192, form, with, , its, , axis coinciding with the, , relation, , S, , 2, , \at, , ,, , So that, subon th 3 curve in the, , distance-axis., , any snitabh, , stitu ing tha co-ordinates of, , p'jint, , the value of a can be easily determined., , Let S be the centre of, The Dynamical Spherometer., (2), curvature of a spherical surface, arranged horizontally with its concavity upwards, and, /?, its radius, (Fig., and, and let a steel ball, of mass, 121), radius r 9 be allowed to roll to and fro, , m, , ;, , on, , wilhout slipping., , it,, , Then, , clearly,, , the ball oscillates on the inside of the, surface as though it were a compound, its centre of suspension, centre of oscillation at, t, of the ball ; so that, SO, , pendulum, wiih, , S and, , at, , O, , its, , the centre, , (*-'), Let the angular amplitude of the ball be 0', (i.e., the angle that, it makes at S, when in its extreme positions)., Let it be in the position B at any given instant, such that the angle it now makes at, 9, , s, , is 0., , work done on the, , ball by the force of gravity in, to, weight of the ball x the vertiequal, cal distance through which the ball has fallen down., , Then, clearly,, , bringing, , tial, , down from A, , it, , to, , B, , is, , This must, therefore, be clearly equal to the loss in its poteni e.,, equal to mg x PQ where mg is the weight of the ball., , energy,, , loss in P.E., , Or,, , SQ, , Now,, , And, , =, , =, , and, , (R-r).cos, , SQ-SP =, , .-., , mg(SQ-SP)., , SP, , =, , (Rr).cos 0'., (Rr).(cos 6-cos 0')., , So that, work done by the force of gravity in moving the ball, to B is equal to mg.(R-r) (cos, cos 0') and is equal to the, loss in the potential energy of the ball., This must, clearly, be equal to tha gain in the kinetic energy of, the ball i.e., equal to |/.o/ 2 where / is the moment of inertia of the, ball and a/, its angular velocity about the line of contact., , from, , A, , ,, , So that,, If, , mg, co, , (R-r).(cos B -cos, , 0'), , =, , be the angular velocity of the ball, , 2, , J/o/, about a horizontal axis, ., , ., , _H_ Va,., (jy, .-., , mg.(R-r).(cos 0-cos, , 0'), , whence, differentiating with respect to time,, , ,, , since dd/dt, , =, , w,, , we have, , we have, (J?-r), 0.<o, , da,
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193, , ACCELERATION DXTB TO GRAVIT7, , Or,, , mg.(R-r)jin, , ^, , ~, , ., , ., , ., , wS'r*(R-~ r ) sin 9, 2, , mg.r, , *, .T, , =7, , 6, , .s-m, , ~~, , I(R-rT, , w#.r, , 2, , ., , /n, , p.-, , L, , /.(.R^rp', , 0=6,, , if, , besmal1, , -, , ., , /., , Now,, , ", , is, , Here, clearly, dw\dt, so that,, /.., the, , angular acceleration of the ball, , [v 5g/7(Rr), The, period, , ', , ', , =, is, , T^^V*^*, , proportional to, , 0., , thus executes a simple harmonic motion, and, , ball, , ;, , 5?, , a constant]., , therefore, given, , is,, , 5g_, , the angular acceleration of the ball, , angular acceleration of the ball, , is, , _, , f/, , .~, , i,t, , ., , g, , time-, , by, , o, ^., Squaring this expression, we have, , whence,, , its, , =, , /*, , r, , =, , -, , r, , 287r 2 (/?-r), ~, , o/~, , Thus, knowing the radii of the concave surface and the ball, (with the help of a spherometer and a vernier calliper, respectively),, and noting the time- period of oscillation of the ball, we can easily, calculate the value of g at the given place., , N.B., , Re-arranging the expression for, , 5f, , have, , 2, , =, , 2, , 28w jR-287T, , 2, , r., , Or,, , t, , 2, ,, , obtained above, we, , 287TIR, , =, , 2, , 5g/ +287T, , 2, , r., , R, K, , Or, ur,, , So that knowing, , r,, , the radius of the ball,, , and the value of g, we can, , t,, , its, , period of, , easily calculate the radius, vature (R) of the given spherical suriace., , oscillation, , of, , cur-, , The Atwood's Machine., , In the ribbon-type of machine,, is passed round the flat rim of a, light and frictionless pulley, (running on ball-bearings), with two, and, at its two ends, one of which is initially kept, equal masses, resting on a platform P., (3), , [Fig. 122 (a)],, , a, , strip or ribbon, , M, , An, , M, , ribbon is attached to the lower ends of the two, shown, 90 that when th ayvtem iff set into motion, ap, , identical, 8, , R
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194, , PROPERTIES OF, , additional length of the first ribbon,, passing on to the right side of, the pulley, is exactly balanced, by an equal length of the second, pass, ing on to its left side, thus ensuring thai, no extra mass is transferred from one siJt., of i he pulley to the other., , A, , steel, , or vibrator F,, , strip, , of a, , known, , time-per.od T, is clamped horizontally at one end, and carries a light, or an inked brush B, which just, style,, touches the paper ribbon going round, the pulley., , of mass m is placed, ng on the platform,, to make it slight'y heavier than tho other., Then, with the rna c ses not yet in, motion, the bnibh is moved across the, paper ribbon to mark a horizontal line on, , A, , small rider, , on the mass M,, , it,, , Fig. 122., , r,, , t, , rest, , indicating the starting point., , P is now suddenly made to fall, (by means of, the vibrator simultaneously set vibrating., and, ('trigger releases'), Naturally, the mass loaded with tho rider, moves down and the, And, as the ribbon, other up with a common acceleration, say a., runs, past the brush, a \\avy curve, duo to the transverse vibrations, of K, gets traced OH it, and goes on gradually lengthening out,, [Fig. 122 (/?)], on account of the accelerated motion of the masses,, and hence that of the ribbon., The platform, , Since one wave, , is, , traced out on the ribbon during one vibration, , ofV, the distances occupied by successive waves represent the distances covered by the masses during successive time periods of it., Thus, if S M So, S3 etc., be the distances covered by the masses, in the first, second and third etc., time-periods of K, we have, [/, , and, , S.2, , =, , aT.T+laT*, , S2, , i.e.,, , And,, , So, , similarly,, , that,, , Thus,, , 5,, , S2, , 5, , 7 being known,, Now,, , if v, , =, , =, , 0,, , the masses starting from test., , here u = aT, the velocity, after time T., , aT*+laT*,, , = T0P/2, = 2aT. T., =-, , l, , u, , 53, , S^, , V now u, , c, =, , aT2, , =, , after, , x, say., , 2'T, the velocity, , time 2T., , Or,, , a^x/T*., , the va'ue of a can be easily calculated out., , be tho velocity acquired, , by the masses, when they, , have covered a distance, h we have, gain in K.E. of the pulley and rhe masses ('ogether with (he rider), , = /0ss, , in, , P.E. of the masses and the rider,
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ACCELERATION DUB TO GRAVITY, where /is the moment of inertia of the pulley about, and w, its anguLir velocity at the time., , 195, its axis, , of rota-, , tion, , Now,, , if, , R, , \, , e the radius of the pulley, v, , Hence,, , \Lv-iR-, , Or,, , 4-, , \(*M-\ m).v, , 2, , =, , Rw,, , =, , wg/7., , or, , w, , =, , vjR., , Iv, , Or,, , whence,, , , _v'(//JP, *, , But, , v, , g, , __, ~~ 2flA(//JP, , 2, , +, , =, , 2ah., , 2 A/, , +, , + 2M + m)., 2mA, , ['.', , w), , -, , 2mh, , g(//JP, ~, , + 23f -f, , m~, , u, , =, , and, , w), ', , S-, , A., , -W, , whence, the value of g can be easily calculated., It H, however, 'lesir.ible to eliminate /from this expression,, by, repeating the experiment with the same masses but a different ridtr,, of mass /'., If a' bo no\v the accaleration of the masses,, (determined, as before), we hav e, r, , Re-arranging relations, , mgla, and, So that,, , m'gja', fliibtiMctin^ relation, , mgla, , -, , and, , (/), , =, =, , (///?, 2, , (7//J, , (/V), , (//),, , we have, , + 2Jf + m),, + 2A/ + W)., , from relation, , (///),, , = (m-rn Or, g(Aw/a m m' \*, ,'/ _g = (/H-TH, , we have, , f, , m'%[a', , ,, , ~, , )., , m//'), , ==, , (m-m'),, , ^, , ,, , whence,, , ), , J, , ^, , Thus knowing, , w', ^, , and, , .., , (v ), , the value of g can be easily obtained., A possible source of error, here, is the fi'idional force encountered by the pulley as it rotates about Jts axle, which, obviously,, tends to lower its angular velocity. This may be easily remedied by, placing another auxiliary rider on the loaded mass,,, such that, with, the main rjder (r) removed from it, if an initial velocity be given to, it, to sot the svstem in motion, it continues t-"> move with the same, uniform velocity, i e., (its motion, is neither accelerated nor reta-ded)., Obviously, tlnn, the weight of this auxiliary rider exactly counteracts the retarding force due *r friction., If, therefore, kept on the, miss throughout the expe, ent, it completely eliminates the error, neither its weight nor the fractional, due to friction, and, clearl,, at ions., force need enter into our ca', w?,, , a',, , Plate. A plate of glass, P, smoked by holding, (4) The Dropping, over burning <urnph'>r, is suspended with its plane vertical, by, means of a thread, as shown in Fig 123 (a), and a tuning fork f,, of a known frequency \ is mounted close to it, so that a light aluminium -tyle, (or better still, a hog's b*ii>tle. such as may be obtained, from a discarded hair b ush), attached to one ot its prongs, just touches, the surface of the plate., , it, , f, , case, tjie, , *Or, we could u*e different ma^^c*, shill tuvs g = 2(Af-Af' )/mU/, masses in the second case,, , w:, , M, , a.id, , !/'), , M', , but the *amc rider, in vhich, is the acceleration of, , where a'
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PROPERTIES OF MATTER, , 196, , The fork is set vibrating by lightly drawing a bow across, by simply pinching it strongly, and the thread, supporting the, , it,, , or, , plate,, , burnt or cut simultane-, , Thus, , ously., , the, , plate, , released,, fallstarts, , ing, with an acceleration, equal to the value, , of g at the place, and, the style traces out a, wavy line on it, of the, form shown in Figs. 123, the waves, (b) and (c),, being smaller and closer, together at first, but, lengthening, gradually, out and getting further, apart, due to the accelerated motion of the, plate, though the time, trace each, taken to, wave remains the same,, viz., equal to 1/N, the, time-period of the fork., , (a), , JL., , Y, , Three points D,, , *, , and, ', , F,, , are then, , Fig. 123., , [Fig., , E, , 123 (b)\, , marked on, , this, , DE, , such that, and EF contain the same number of waves, say, n, each Let distances, 'and 7'" be 8 l and S2 respectively, as measured by means of a, travelling microscope, both being covered by the plate in the same, interval of time t, njN, taken by the fork to complete n vibrations., , wavy, , =, , =, , Then, clearly, 5,, ut+\Qt*. Or, 2S t, where u is the velocity, of the plate at D., ", And,, (S,+S? ) = 2w/+|g.(2f) a, because, here, distance = (8 -{-S 9 ) and time, 9, , 80 that, subtracting equation, , Or, , (/), , from, , (//),, , line,, , 2, , 2w/-fgJ, , ,, , ..., , (/), , ., , =, , (2f)., , we have, , >, , whence,, , g, , = iT~LtL, (, , Or, substituting the value ,w/JV for, , g, , -5., , ~, ., , -, , /,, , y as, , Thus, knowing N, n and (S2, value of gat the place., , Sj),, , ...(m), , in relation, , --^, , we can, , (///),, , we have, ..., , (JV), , easily calculate out the, , It will be readily seen ttfat the mass of the style (or the 'hog's, attached to the prong of the foik, together with the friction it encounters, at the plate, will slightly lower its frequency, so that it will actually be somewhat, For greater accuracy, therefore, the frequency of the fork (with the, less than N., style attached to it) must be determined by the method of 'beau\ by sounding it, , N.B., , 9, , bristle, , ),
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to, with another fork of an accurately known frequency. The frequency, thus deterin relation (iv) for g, above., mined, should then replace, , N, , Alternative Calculation. The following is a comparatively more, accurate method of calculating the value of g, because, here, the, possible error in correctly counting n is eliminated., , Three points A, B and C are marked on a portion of the wavy, [Fig. 123 (c)], where the waves are clearly visible and can be, vibrax, distinctly counted. Let there be n l waves (and, therefore,, n, waves, A, and, and, in-between, made, the, fork, B,, (or, 2, by, tions), W 2 vibrations) in between B and C, and let the total distance AC be S., line,, , Then, clearly,, fork in time, , 1, , (, , (tf 1, , waves or vibrations are made by the, 2, So that,, , +w, , +fl 2 )/jV., , ), , ^, , S z=~, , ., , Or,, , V, , = A /"F, Y 4, 2, , "1, ~, , and, ^, , against n i9 therefore,, Plotting, we obtain a straight line, (Fig. 124),, , of slope, , A, , /-?, , /, , N, , 9, , from which the, , value of g can at once be calculated,, without knowing n., Incidentally, the dropping plate, , method also shows that a freely falling body is subjected to a constant, acceleration due to gravity,-*- a fact,, not easy to demonstrate otherwise., of a, (5) Vertical Oscillations, Fig. 124., A spiral spring is, Flat Spiral Spring., just a uniform wire or ribbon, designed to have, in its normal, unstrained condition, the form of a regular helix, such as may be, obtained by winding the wire closely and uniformly round a cylinder,, of a diameter much greater than its own., axis, , If the plane of each coil of the spiral is perpendicular to the, of the cylinder, it is called a flat spiral, but if it be, inclined at a small angle to this axis, it is spoken, of as an inclined spiral. We shall concern ourselves, here only with the flat spiral of a wire of circular, cross-section., , If a small force be applied to a flat spiral, along, along the straight line passing through the, centre of each coil of it), and perpendicular to its plane,, it increases in length a little, but still preserves its, helical form, as will be clear from Figs. 125, its axis, (i.e.,, , (a), , and, , (b)., , Let us consider a flat spiral, of length L and, radius R, (where R is much greater than its pitch),, suspended from a rigid support, with its upper and, lower ends (A and B), bent as shown in Fig. U6 (a), 9, ao as to lie along its
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PEOPEBTIBS OS MATTBB, , 198, , Then, if a mass m b3 suspond^d from its lower free end, & forcd, equal to mg (the weight of the mass) acts vertically downwards, along its axis, producing a statical, extension/ n its length, (/e., an, extension, with the mass m at, rest)., , The, , of this force mg,, of the spiral, axis, the, acting along, is to produce a turning monent,, equal to mg R. at every section of, effect, , [Fig. 12(5, , it,, , And,, , (/?)]., , this, in its, , turn, subjects the wire to a uniform twist 0, say, per unit length, , of, , it., , Now, the, , twisting or torper unit length, , sional couple*, of the wire is, , where n, , is, , the, , 4, equal to rar tf/2,, , coejfif ient, , of, , of the material of the wire, (a;, , radius, , and, , #, its, , rigidity, ;, , r,, , its, , angle (>ftwi>t., , Fig. 126., , This, therefore is the torsional resistance, opposing the turning, For equilibrium, therefore, we, to the weight nig., , moment mg.R due, , mg.R, , have, , =, , ..., , n.ir^d/2., , (/), , per unit length of the wiro corresponds to a, Now, a twist, R.6 per unit length of it. [Fjg 12t> (b)]., extension, or, displacement, Charly, therefore, the extension produced in the \\hole length of the, wire is equal to L.R.6., /, , Or,', , =, , L.R.O, whence,, , 6, , -, , v, , ', ' is the to al, \ tension, L, produced., , IjLR., ., , Substituting this value of 6 in relation, , mg.R, , a=, , Tiw 4, , /, , r~, , ., , j-g, , =, , (/), , alove,, , ', , we have, , Trnr 4 /, .., , -, , p, , ,, , whence,, , The expression Tnr*.l/2LR' thus represents the, , force of elastic, in the length of the spiral, and. therefore,, the elastic reaction per unit incievse in the length of the spiral, as 7tnr*/2LR*, (because /I). Denoting this by K, we have, , reaction for, , an increase, , /, , =K, , mg, , ., , /,, , whence,, , K=, , mg/l., , mass (m) Le now displaced or pulled, vertically, downwards through a distance x and then released, so as to produce, vertical oscillations in the spiral, the restoring force F. acting on the, mass may for small oscillations, be taken to be dirjctly proportional, F = K.x., to its displacement. So that,, If the suspended, , And,, , if, , 2, , d*x/dt, , be the acceleration of the mass at the, given, , See chapter VII, where it is shown that the twisting couple on a cylinder, 4, (or wire) is equal to OTtr G/2/ f where Q i^ the angle of twist and /, its length*
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ACCELERATION DUE TO GRAVTl^, , when, , instant,, is, , its, , t\e mass, , x, the inertial reaction of, , is, , displacement, , l&S, , clearly equal to m.d'x/dt*., , Hence, by Newton's third law of motion, we have, d x/rf/ 2, m.d 2 x/dt 2, whence,, x.K[m., , =, , =Kx,, , Now, since K and m are both constant quantities, we have Kjm = a constant, p,, say., So that,, d z x[dt* oc x, d*xldt* *= - n. x., Or,, , for, , the given, , spiral,, , i.e.,, , the acceleration of the, , ment x, , ;, , and, , mass, , ;, , directly proportional to its displacetherejore, executes a simple harmonic motion, its time-, , it,, , is, , period being given by the expression, , =, , /, , ~, , 2irvT/, , *, , 27TA, , V/, , Or,, , ,, , Kim, , Or,, , =, , t, , = t*^lR =, , 2ir, , v'/7F, , 2* A, , V/ nig, , ...(//), 1, , 1, , ;, , time -period is ihe same as that of a simple pendulum of length, the extension produced in the spiral., i, , e., ///e, , /,, , 2, Squaring and re-arranging this expression for /, we have g = 47T .//f, whence the value of g, at the given place, can be easily calculated out., ?, , ,, , In the above treatment, we have not tak'm into consideration, the mass of the spring, assuming it to be negligible, compared with, the suspended mass m. For gre itcr accuracy, however, it must also, be taken into account. So that, if the effective mass of the spring be, the total mass acting downwards along the axis of the spring, s, becomes, above, for the time-period of, 3 and the expression (//), the spiral, becomes, /, 27t\/(m-\-w s )/K^., , m, , ,, , m+, , =, , m, , This is done, is, however, best to eliminate, s altogether., the, with, two, different, performing, experiment, suspended masses,, It, , by, , m, , 1, , and, , w, , 2., , Then,, in the, , two, , if t l, , and, , cases,, , t%, , be the respective time-periods of oscillation, , we have, , =, , 27rv\w 1 -Fwj7A' and f 2, So that; squaring and subtracting the second from the, tl, , first,, , we have, ....., , ...(in), , Now,, , if, , mabses,, , /,, , and, , /2, , be the statical extensions corresponding to the two, , w e have, r, , mv g =, whence,, , mr g^m, , 2, , .g^Kl, , AT/j, , and, , Kl2, , }, , A, Substituting this value of (m, , have (^-/a, , 1, , =, ), , 47r^-~, , ., , =, , K12, , (w,, , m, , m^g, , Or,, , ,, , 2 ).g, , =, , K(lL, , l^., , g, m.2 )IK in expression, , l, , i?, , ,, , (i/i), , above,, , whence, g, , Thus, observing f, and ^ 2 directly, and measuring /t and /,, by, noting the positions of a light pointer, attached to the spiral, on *
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2UO, , PBOFBRTIBJS OB, , MATtlA, , vertical centimetre scale fixed alongside, , it,, , the value of g can be, , eatfi-, , ly calculated., , The Bifilar Suspension. If a heavy and uniform bar or, (6), cylinder, (or, in fact, any rigid body), be suspended horizontally by, means of two equal, vertical, flexible and inelastic threads, equidistant, from its centre of gravity, the arrangement constitutes what i called, , On, , being displaced a little in its own plane, i.e.,, and then released, the bar or cylinder executes a simple harmonic motion about the vertical axis through its, a, , bifilar suspension., , in the horizontal plane),, , ., , centre of gravity., , Now, two cases, arise, (/) when the two suspension threads are, and (//') when they are not. Let us consider both., , parallel,, , Bifilar Suspension, with Parallel Threads. Let AB [Fig. 127,, (/), represent the original or equilibrium position of a cylinder, of, mass m, and with its e.g. at 0, where its weight nig acts vertically, downwards. Let the two suspension threads PA and, b*, parallel to each other, and distance 2J apart ; and let the length of, each be I., (a)],, , QB, , Now, if the cylinder be displaced a little into the position A'B',, through a small angle 6, about the vertical axis through (9, the suswith, pension threads take up the position PA' and QB' at an angle, their original positions, where, is small., </>, , <f>, , Let, , T be the, , Then, resolving, , it, , tension in in each thread, acting upwards along, two rectangular components, we have, , (), , (W, Fig., , (/), , nd, [Fig., , (//), , it., , into its, , (c), , 127., , the component T cos <f>, acting vertically upwards ;, the component Tsin ^, acting horizontally along B'Band A' A, , 127, , (b)]., , Obviously, the vertical components support th, cylinder., , Hence,, , ?r cos, , J>, , mg., , Or,, , T cos i, , weight of the
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fO GfcAVlTt, , 201, , = 1, very nearly., And, since is small, cos, So that,, T = wg/2., The components, T 5fw <, (acting at ^4' and B')< on the other, hand being equal, opposite and parallel, constitute a couple, tending, to bring the cylinder hack into its original position., And, since A' A, and B B are practically at right angles to A'B', we have, <f>, , <f>, , moment of this, , restoring couple, , =, , BB'IOB, , And, therefore,, , ^, , Now,, , r^ormg, , Hence,, , =, =, =, , C*HJ>&, , =, , T. sin, , #'/rf, , so that,, , ;, , =, , BB'\l, , .y/fl, , l, , 0., , [Fig. 127 (c)., , e.d., , 127 ()., , . 2L^, , f, , But, , . tf ., , i, , 2, , &ldt* 9, , the, , mg.d*, --, , T ' d*8, , n, , V*, , *, , Now, mg.d2 jll, fore,, , .**, , =, , moment of inertia of the cylinder about the vertical, 2, through 0, (its e.g.), and d $/dt*, its angular acceleration., , where, axis, , Ld, , =, , S mall,, , L, , [Fig, , 2&, , =, , /, , Us, , BB', , C, , ', , T.<f>.2d., , 0.<///., , T.'4-'2d, , the restoring couple is' also =?, , =, , </>.2d, , putting, , it, , jf, , J., , a constant quantity, in a given case, and, thereequal to n, we have,, , d^Qldt, , is, , 1, , =, , n.Q., , d*0/dt* oc 0,, , Or,, , /.., f/t angular acceleration of the cylinder /? proportional to its, lar displacement, and is clearly directed towards its mean, , angu-, , The, , position., , cylinder, therefore, executes, , period, , is, , a simple harmonic motion and, , its, , time-, , given by, , T, mg~d*Jll, , Or,, -O, o, mfc 2 , where k is the radius of gyration of the, cylifi^, about the vertical axis through <9, we have, , Or, if, , we put, , /, , =, , x, , s, ri>__, ^7f __^, , M, , T, , whence,, , Now, squaring, , ^, , k/, = 27r.-~7, V, ,, , d, , relation,, , =, =, , (i), , or, , *, , (//),, , s, , --, , ^, , ("), , g, , and re-arranging, we have,, , 2, a, 2, from relation (/),, ..., 47T /.//m.rf .r, g, a, 2, 2, a, 4:r .fc .//d .r, ..., and, from relation (), g, And, thus, the value of g at the given place, ., , ..., , ...(/), , ..., , ..., , (/v), , be easily, , cal-, , may, , culated out., , Non-Parallel Threads. Let the rod or, (ii) Bifilar Suspension with, cylinder AB, [Fig. 128 (#)], be suspended symmetrically by two equal, but non-parallel threads*, each of length /, and let the distance between the threads at the top and at the bottom be 2dlt and 2dt respectively,, , where (d%, , *The threads,, cHicatini the Figure, , dj, , =X, , in this position of the rod, are, , not shown, to avoid, , COJOQ-
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202, , FROFERTIES OF MATTER, , If the cylinder be displaced through an angle 0, in its owri, plane, into the position A' B' t the suspension threads take up positions, PA' and QB'. Then, as befora, tension T acts upwards along each, thread, and may be resolved into two rectangular components, v/z.,, , T cos <, acti'g vertically upwards, as shown, and (ii) T sin <,, (/), acting horizontally, along A'K and B'L [Fig. 128 (0)J. where <f> is, the angle that each thread makes with the vertical, or the perpendiculars, PK and, , QL, from P and Q on, , to, , AB., , Id,, , Ts>n, tfl, , ff-, , '7TT - * "> ^-, , '-', , z, , CP^'>, , T, , S,r>, , (a), , Fig. 128., , The, , vertical, , clearly,, , 2, , LB', , =, , And, c0$ $, , ss, , Now, T, , =, , j>, , ;, , ', , mg., , ...( v ), , [Fig 123 (c), , r, L, , 2,J^dL approximately., ,, , = #,, , (did^), B'R, , 6, , b;i, cing small,, , and, , cos o=l, nearly., , approximately., [Fig 128(6)., , 2, , Zv/, , And, resolving, , and, , cos, , support the weight mg of the, O and, therefore,, , at its e.g.,, , = ^^d^^d^d^sJ., , LB', , L, , Or,, , <f>, , downwards, , 2T, Now,, , T cos, , components, , cylinder, acting vertically, , 5X respectively,, , 2, , From, , ", , -*'//, , (v), , relation, , above., , T, , sin <f>, acting at A' and B', forces, along A'K, into their rectangular components along and at, , right angles to A'B', we have, the components at right angles to A'B', , = T sin, , sh, , <f>., , a,, , Fig. 128 (c)., , [, , Since these two components act in opposite directions at A', and B\ they constitute a couple, tending to rotate the cylinder back, into its original position AB and, clearly,, ;, , moment of this restoring couple, Or,, , = T sin, , restoring torque on the cylinder =* Tsln, *=, , ft, , ,, , 2dr, , mg.l, , -A=, , x, , ., , <f>, , sin a. A'B'., , <f>.sin, , oi.2dt, , ., , C, , ., , r, , r-. sin a., , from, , Fig. 128 (b)., , the sides in a triangle, be ng proportional to the sine* of, the angles opposite to them, we have, from Fig, 128 (c),, , Now,, , ;, , djsin, , *, , LB'Isin, , 9,
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ACCELERATION DTTE TO GRAVITY, ^ ,,,,,., So, that, rf,/5/n a, , =, , ,, , A, And, , ,, , x/0., , =, , ., , sin a, , .-., , n, , 1, , .0., , Hence, restoring torque on the cylinder, , X, , Q, , ,], , =, , also, , is, , restoring torque, , fiut,, , -, , ^.-r-2, , where /is th3 mvnent of inertia of the cylinder about the vortical axis, 2, through its e.g., and d^^jdt its angular acceleration., ,, , ^, , -^ Or,, ', , =, , ?', , where,, 7., , v, , a constant, , cc 0., , Tims,, Or,, , (*., , *', , /, , ///e, , angular acceleration of the cylinder is proportional to its angular, It. therefore, executes a simple haimonic motion and its, , displace'mit., time-period, , is, , given by, , r= 2;, , Or,, , r^., v, , ;, v, ...(v), , .mg, mk* where k is the radium of gyration of, And, if we put /, the cylinder about the vertical axis through its e.g. we have, ^/, 2, , rf,, , 9, , 7 be tin vertical dista ic3 bstwean the two ends of each, we have y = B'R ~ LQ = y/^ #-. [Fig. 128 (6)., thread,, suspension, , Now,, , y, , if, , r=, , ", , So, that,, , =^ AV/, , 27r.--A, i, , J!_., , ..., , (v/ii), , S, , Again, squaring and ro-arran.jiiig relations, , (v//), , and, , (v//7),, , we, , have, i, , /, , j-, , from relation, and, from, , Nation, , /, , ..v, , (v//), , g=, , 47T*.A.\/*, , ~*, , 47T*, (v///),, , g, , =, , -^^, , r/a**m*j.rr", MI, fc, , 2, , V, , ....... -(^), ., , ^, , -,-'1.^., a^.u c .7, , v, , ...... (x), , The value of g, at the place, can thus be calculated from either, of the-e relations, x, , =, , It will be readily seen that if </, =, , d., , =* d,, , and y, , /wo threads are parallel and, original equilibrium position of the c^ Under, we have, 0,, , (tf), , r elation, , i.e.,, , vr/zgfl, , relation, (/),, , //ie, , (v//), , reduced to, , for parallel threads,, , 2*, , *= STT, , -^, , A/, , =, , /,, , so, , that, , vertical in the, , -1,, , the same as
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204, , k-HOPKBTIBS, , relation, , (b), , (v/ff), , reduced to, , fflATTKB, , T=, , 27r.~r-A/, , ,the same ad, , for parallel threads, (see page 201)., (//), Note., It will be clear from the above that the bifilar suspension may, also be used to determine the value of /for the suspended cylinder etc. For,, , relation, , relation, , (///), , and relation, , above,, (ix) 9, , when, , when, , /, , =, , re-arranged, gives /, , =, , re-arranged, gives, , - -., , n 2/, ', , 4n-, , \r, , x", , method is more suitable for determining th, moment of inertia of a body than for determining the value of g., Variation of the value of g'. The value of g at a given, 76., place is affected by a number of factors, viz.,, In, , fact,, , this, , f, , We shall, altitude aiid (Hi) depth., (//, (i) latitude of the place, proceed to study the effuct due to each of these factors a little, ;, , DOW, , in detail., , The effect of latitude on the value, be considered under two headings, v>z., (a) the effect of, the rotation of the earth, and (b) the effect due to the bulge at the, (/), , of g, , Effect due to Latitude., , may, , equator., , Let us consider each separately., , We know, , that the earth is rotating, If it were at rest, and were a, homogeneous sphere, the acceleration due to grav'ty would be the, same for a body at all points on its surface and would be directed, towards its centre. Due to its rotation, however, part of the force, of gravity on the body is used up in overcoming the centripetal force, acting on it, and thus the resultant acceleration on it is different,, the, both in magnitude and direction, at different places,, i.e.,, will, be, clear, in, as, value, different, different, of, is, latitudes,, '#', apparent, , about, , (a) Effect of Rotation., its axis from west to east., , from the following, Let NWSE, (Fig. 129), be a section of the earth, (supposed to, be a perfect sphere), through its polar, yy, diameter NS, and let its radius be r. Then,, if a* be the angular velocity of the earth, about the axis of rotation NS, all points on, its surface rotate about this axis with the, :, , angular velocity o>. The linear velocity of, ach point will depend, however, on its distance from the axis. Thus, the linear velocity, and S will, of a particle at the points E, N,, be r.co, and that at a point P, distant, x from the axis, will be x.eo, where, x is the radius of the circle that the point P, describes as it rotates with the earth., , W, , PM =, , b the latitude in which the point P is situated. Then,, the radius PM, of the circle described by P, is r cos 4>, the, linear velocity of P, r cos <f>.aj ; so that, the centrifugal force acting, QB P, oway from the centre (M) of the circle it describes, and acting, Let, , since, , <f>, , =
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ACCELERATION DUE TO GRAVITY, , 205, , MP, is clearly given by m.r.cos <f>.aP. Let it be represented in, magnitude as well as direction by the straight lino Pp., The force of gravity mg, which would act on the boly if the, earth were at rest, (g being the acceleration due to gravity, with the, earth at rest), would obviously act towards the centre of the earth O., Let it be represented in magnitude as well as direction by the, along, , straight line, , PO., , Thus, there are two forces acting simultaneously at the point, 2, P, viz., (a) the centrifugal force m.r.cos <.eo ahng PF, and (b) the, force mg due to gravity along PO., Completing the triangle of forces, POQ, where PO represents the gravitational force mg, and OQ, the, 2, we have the resultant force at P, centrifugal force m.r.cos <.o>, represented by the third side PQ of the triangle, both in magnitude, ,, , ,, , and, , direction,, , where, , PQ = ^POOQ*~2PO~OQcos>OQ., r.r* cos, , 2, <j>, , to, , 4, , 2m*. g, , r, , [See Appendix, , 1 "7 (2),, , cos^.a)*., , Now, the value of r.w* comes to be about 3-39 cms. I sec*., or, about 1/288 of the value of g; for r, 6378xl0 8 cms., and, of, number, mean solar seconds in, <o, 27T/86164, (where 8(5164 is the, , ^, , =, , one day),, Thus, the expression jn*.r*.cos*<f>-a>* is negligible, compared with, the other terms involving g, and, therefore,, , PQ, , =, , y/mg*, , 2m*.~g.rcos*i~^>*, 9, , mg, PQ, , Or,, , Or,, , mg(\, , 1, , -V, ----, , +, , P, , \/m*(g*^i.f~c(^^)', , frt, , i, , 1, , -- x~, , the resultant force on, if, , .'., , =, , *, , =, , =, , -, , ', , w, , -+ some other negligible terms, , g, , ^, , fOJ, , mg(, , ', , c, , ", , <f>,, , T, '""', , ,,, , we have, , a, , is, , and wo/ towards, , e change in, small., , be g, , obviously smaller than g, and is directed towards, 0, the centre of the earth the angle OPQ, or, direction of the gravitational force is, however, very, , This value, ,, , \, , 1, , acceleration of the point P, in latitude, , ), , /, , ;, , =, , For pointy on the equator, since, 0, and, therefore, cos<j>= 1,, a, r.<o, the value of the centrifugal acceleration, i.e., a maximum., 90, and, therefore,, And, for points on the poles, because ^, cos <^, 0, the value of the centrifugal acceleration is zero, i e., a, <f>, , =, =, , ,, , =, , minimum., It follows, therefore, that the apparent acceleration of a body, the least at the equator, and the $reat$st qt the poles, with \n, , is
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PROPERTIES OP MATTER, , 206, , =, , =, , 3'39'978, Substituting tho valua of r.aj* g, 1/288 in the, expression for g above, (the value of g being 978'03 cms.jsec*. at the, !, , we havy, , equator),, , ,-*( '-!?), _, , This, , is, , a result, not quite in agreement with the experimental, , value., , of the, , The discrepancy may, however, be ascribed (/) to the elliptic ity, earth, its radius increasing as wo proceed from the poles towards, , the equator, so that points in the higher altitudes arj nsa-er to its, centre than those near th^ equator, (//) to the non-homogeneity of its, comyosilio'ii tho density of its different layers b3in-z different, with, the i m,r layers compirjiivelv much denser ih:m (about more than, twice as dense as) the outer ones., ;, , (b), , Effect, , in, , periments, , two, , of Bulge, , 1672, , first, , It was RVier, whose exat the Equator., showed a variation in the value of 'g' at, , different places., , Determining the length of a seconds pendulum at Cavenne, Guiana) and at Paris, he found its length at Paris to be, just over one-tenth of an inch greater than at Cayenne, clearly, showing the value of g to be greater at Paris., Newton soon explained this variation on the assumption that, the earth behaved as though it wera a 'uniformly gravitating fluid, globe' so that, by virtue of its very rotation, it was bound to have, a spheroidal shape, with a bulge or a protuberance at tho equator,, and comparative flattening oh the poles under the influence of the centrifugal force acting on it, tho valua of which varies from zero at tho, In fact, even if the earth \\ero, poles to a maximum at the equator., perfectly rigid, it should have assumed this shape before it actually, cooled down., As a consequence, the equatorial radius is about 13, miles greater than its polar radius. Hence, all bodies in the equatorial, regions are farther from the centre than those in the polar regions,, and the force of attraction due to gravity on the latter is, therefore,, greater than that on the former., (in French, , ;, , is, , It can be shown that the true value of g at a place in latitude A, given by the relation,, (98O61 *025 cos 2 A) cms. /sec*., g, , =, , These changes in the value of g due to latitude are of great, help in determining the figure or the shape of the earth., The correction for altitude we really, (//) Effect of Altitude., owe to Laplace and Stokes, particularly to the latter., , Let g be the value of acceleration due to gravity on the surface, of the earth and g'. its value at a height h above the surface. Then,, if the earth be considered to be a sphere of homogeneous composition,, the acceleration duo to gravity at any point above its nurfaco will, vary inversely as the square of the distance of that point from its, centre so that, ;, , ==, , ,^5, , :, , -.f, , ==:, , f= *, , +, , + pf, , [^radius oftheeartfy
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207, , AOCBLBEATION D0B TO GBAVTTY, , If h b3 small, compared with r the quantity W\r* will ba negligibly small, and we shall, therefore, have, , =, , g/g', , Or, g'lg, , 1+iVi/r., , Or,, , =, , l/(l+?/i/r), , =, , l~2/i/r appro*., , g(l-vA/r),, , g', , the greater the value of A, the smaller the value of g'., Or, the value ofg decreases with altitude., /..,, , The general expression for the acceleration due to gravity at, and in latitude \. thus becomes, , altitude h, , g', , =, , (l-^/r)(9SO-61~-025 cos 2A) cmvJsec*., , The correction term fl, Effect of Elevated Masses, for altitude wo ill only be valid when there is mth'ng but, (ili), , 2/i/r), , spice, , between the surface of tlie earth and the point* /i above, e.g., f<>r, an observer in an aeroplane at height h But if we consider the point, to Ii3 o i the top of a rrn intain, of height A, a complication comes in, due to the effect of the attraction by the mountain., /, , Boug uer suggested the correction, as Bxigier's Rde, where, that of the mountain., , A, , i, , the, , 3, , (, , 3 h p \, , ", I?//, I, , _j_, , mean density of, , '-l-j, , }, , known, , the earth and, , p,, , It is now found, however, that Bouguor somewhat over-estimated, the effect of the mountain and his correction*, therefore, gives the, upper limit, as it were, of its effect, th^ lower limit being that in, which its attraction is neglected altogether. The Board of Trade, have, therefore, adopted the following relation for the combined, effect of latitude and altitude, :, , =, , (9SO-6l--0:55 cos 2x)(l-5/i/4r) cms.lsec*., Again, imagining the earth to be a homo(/v) Effect of Depth., let, a, us, take, g^neous sphere,, body of mass m, inside the earth, at a, depth h below the surface, so that its distance from, the centre of the earth is (r /;), where r is the, radius of the earth. Imagine a sphere of this, radius (r h) to be drawn concentric with the earth,, g', , (Fig., , 1, , 0)., , lies on the surface of, and inside the outer hollow spherical shell, of thickness h., Let g and g' be the, accelerations due to gravity at the surface of the, Fig. 130., earth and at a depth h below it, respectively. And since the force of, attraction on a body inside a hollow shell is zero, the only force of, attraction on the body is that due to the inner solid sphere, of radius, (rh), and is directed towards its centre, its magnitude being clearly, , Then, clearly, the body, , this inner sphere,, , given by, , mg _, ', , where, , G, , is, , mass of the spheres mass of the body, , ^^, , the gravitational constant., , *Thi<? co rection by Bouguor was prompted by the same idea which inspired his Momt.ti'i experiment for the determination of the Gravhationd, Constant G, (Sse p-ge 231), v/z., that the attnction on a mass due to the, mountain cpvrtd sfmply be added, ty tl^at 4ue |Q U*e fajth, (taken to be a,, f, , up
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208, , FKOPEKTIBS OF, , Now, mass of the, , A, , where, , sphere, , =, =, , its, , volume, , x its, 3, , -J.7r.(r-/0, , density,, , xA,, , the density of the earth, supposed uniform., .'. force of attraction on the, body at a depth h inside the surface of, the earth is equal to, is, , whence,, , g', , =, , |.TT., , ..II, , &.G.(rh)., , the body were kept on the surface of the earth, the, force of attraction towards the centre of the earth would be given by, , And,, , mg, , =, , if, , -, , -, , .'~, , 9, , O, , whence, g, , 9, , Dividing relation II by relation, , g'=, , Or,, , g, , IIT,, , =, , ~.7r./\.G.r., , ...Ill, , we have, , (i-^\, , ...IV, , the value of g decreases with depth from the surface of the earth., 7, it follows at once, from relation IV above, that at the centre of, the earth, where h~ /*, the value of g will be zero ; i.e., the accelerai.e.,, , And, , tion due to gravity and, therefore, the weight of a body at the centre of, the earth will be zero., , This correc(v) Effect of Terrain- (Topographical Correction)., tion consists in reducing the result at any given station to that we, would obtain if the laud or the terrain in which it is situated were, just a horizontal plane, instead of its actual form., Obviously, some parts of this terrain would be above and, others below the horizontal plane, so that the former would exert an, upward attraction, thus decreasing the value of g and the latter, a, downward attractive force, thereby increasing tko value of g., It so turns out, however, that this correction is always a positive one., 77. Determination of the value of g at Sea., Until comparatively recently,, the value of g at sea was determined indirectly, because it was not considered, possible to use a pendulum on board a ship. The method, suggested by Hecker, and Duffield, and usually adopted, was to determine the atmospheric pressure in, two different ways, one of which involved g and the other did not, so that, by, equating the two, the value of g could be easily calculated out., Thus, for example, the atmospheric pressure P could be obtained (/) from, a barometer which involved g, because P, ?.g., where // is the height of the, mercury column and p, its density, and (//) from the boiling point of water,, which did not involve g, because it could be calculated from the Tables, giving, the relation between temperature and the saturated vapour pressure of water, vapour or directly from aa aneroid b irometer, (again, without involving g)., Then, equating H.p.g, against P,as obtained from method (), we have, , H, , ;, , f, , ., , P, , ., , '01 cm. /sec*.,, by this method give us an error of about, considerably greater than that given by pendulum methods on land,, the chief source of error being the oscillations or 'bumpings' of the mercury, columns in the barometer, caused by the movement of the vessel,- the ship or the, , The, , results obtained, , which, , is
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AUUBliBKATION, , DUB TO GRAVITY, , Vening Meinesz has shown, however, that pendulums can be used for the, purpose with far greater accuracy, particularly in a submerged submarine. Hii, argument is as follows, A pendulum is subject to four types of disturbances on board a ship, viz.,, (/) the point of suspension having a horizontal acceleration, (//) vertical acceleration, 9, of the support, (Hi) angular movement of the support or 'rocking of the plane of, oscillation, and (iv) slipping or sliding of the knife-edges on their agate planes., Of these, the first disturbance is the most marked, but it can be completely eliminated by simultaneously oscillating two identical half-second pendulums*,, suspended from the same support, oscillating in the same vertical plane, but with, Then, it can, different phases, and noting their angular displacements Q l and 2, be easily shown that (Oi, 2, gives the angular displacement of a pendulum,, altogether unaffected by this disturbance., The vertical acceleration of the support can, however, not be eliminated,, without eliminating g itself, but the disturbance due to this can be greatly minimised by taking the mean of a large nu Tiber of observations. For, the value of, # seems to be affected only by the nmn value of the vertical acceleration during*, the whole period of observation. And, sinee the vertical motion is alternately, up and down the zero position, the mean value of this acceleration becomes, almost inappreciable., The error due to 'rocking* can be easily corrected for, by noting different, values of the rocking angle and computing the necessary correction, which is, :, , ., , ), , usually quite small., , And, with all these errors eliminated, or minimised, to an extent thai, the total angular deviation due to them does not exceed 1, the fourth error, viz.,, the slipping of the knife-edges gsts automatically eliminated., What is done, in actual practice is tint three half-second pendulums are, suspended from the same support and set oscillating, and continuous photograbetween the angular displacements of, 2), phic records of (0 the difference (9 t, the first and the second, and (//) the difference (9283), between those of the, second and the third pendulums obtained on a sensitized paper, by means of a, suitable optical arrangement. Ths value of g is then calculated from each of, these two sets of observations and their mean taken., The whole system is suspended in gimbals, to avoid external disturbances, due to small angular movements of the ship or the submarine ; and, further, to, avoid any possible errors due to any slight change in temperature, the whole, apparatus is kept properly thermally insulated and any small correction, still, necessary, applied. And, finally, to make sure that no magnetic disturbances, affect the result, the pendulums are made, not of invar-steel^, (which would be, so helpful in minimising any temperature corrections), but of brass., The probable error in the value of g thus obtained is claimed to lie with2, *0018 cm./.rec ., obviously, a marked improvement over Duffield's earlier, in, ',, , indirect method., recent and comparatively much more accurate method consists in, measuring the change in the frequency of transverse vibrations of a wire under, tension, due to a weight suspended from it For, whereas, any variations in the valua, of ? produce next to no effeet o i tru d snsity of the wire (density being the ratio, of mass to volume), they dp naturally affect the pull of the earth on the suspended, weight and hence the tension in the wire, resulting in corresponding changes in, These can be easily detected to just a fraction of a, its frequency of vibration, vibration in a frequency of several thousands, by comparison with the vibrations, of a quartz-crystal oscillator by the methods of beats. This explains the high, accuracy of the method, which is obviously equally applicable to the measurement, of the value of g on land, particularly at places where it is difficult or impracticable to use the usual method, as for example, at the bottoms of boreholes etc., , A, , Local and Temporal Changes in the value of g. The value, 78., of g at a point is also affected, to some alight extent, by local causes,, ~"~, "", pendulums, whose full time-period is one second], an alloy of nickel and steel, whose co-efficient of thermal expansion, is exceptionally low, and which is, therefore, used in the construction of what, are called invert able pendulums, i.e., pendulums whose lengths remain practically, unaffected by temperature variations, the name 'invar' for the all% being, suggested by the word 'invariable'., */..,, fit is
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210, , PROPERTIES OF MATTER, , like smalt geological deposits near about, the topography of the region?, or even by masses like buildings etc., in the neighbourhood., It is alsa, affected by time, because deformations in the earth's surface take., place periodically, thus bringing about a change in the equipotoatiall, surface, and hence in the direction of the force of gravity, which isalways perpendicular to this surface. These changes are, however,, much too small to be measured by ordinary pandulum methods, whose accuracy is limited to within 10~ 4 cms. /sec 2 ., or 10- 1 milligals,, , =, , 2, 1000 milligals*). More sensitive methods(where i cm./sec =1 gal, have, therefore, to be used for the purpose. A detailed study of these, is beyond the scope of this volume, and we shall, therefore, deal with, them only briefly here., Small changes in the value of g due to, (/) Local Changes., local causes are measured with the help of (/) what are called, The former, invariable pendulums and (//) gravity -meters or balances., are suitable only for the measurement of place-to-place variationsin the value of g in regions, free of all marked local abnormalities,, and the latter, for changes due to abnormal conditions like irregularities in the density of surface constituents and such other causes., For the most accurate determination of small variations in the value, of g, however, a still more sensitive instrument viz,, the Eotvos gravity, balance must be used., These pendulums are so called, because, (l) The Invariable Pendulums., ., , of their being standardised to such an extent that their time periods (/) vary, $olely due to variations in the value of g and to no other factor., They are usually rigid pendulums of invar-steel, suspended from a massive, tripod in a partially evacuated chamber, with a specified air pressure inside it, to, make all air- corrections constant. And the variations due to temperature already, small on account of the use of invar-steel, (with its negligible coefficient of expansion), are further corrected for by a direct determination of the change in, time- period with temperature., The time-period of such a pendulum is first determined at a chosen base, station, i.e., at a place where the value of g is known and then at the field station,,, Then, clearly, the gravity ratio,, i.e., at thf place where it is to be determined., or the ratio between the values of g at the two stations, will be given by the, inverse ratio of the squares of its time- periods there, since, , The only error possible, after all this standardisation, is that in noting, the time-periods of the pendulum at the two stations, or in the 'timing operation\ as it may be called, and the utmost accuracy is attempted to be secured, here by arranging to have precise time-signals broadcast at frequent intervals., In the ultimate analysis, however, the results obtained will be restricted to the, same order of accuracy to which the time-period of the pendulum and the other, constants involved have been determined at the base station., The use of the time-signals at the field station may be obviated by the, technique used by Bullard in his determination of the value of g in East Africa, one at the base station, (in his, (in 1933), v/z., that of using two pendulums, case, Cambridge) and the other at the field station and recording an agreed, Morse signal, alongside the oscillations of the pendulum, at each station, on a, photographic film, repeating the same an hour or so later. The time-periods of, the pendulums can then be compared with the equal time-intervals given by the, Morse signals, and a high degree of accuracy thus attained in their measurement., , ___, , Next in sensitivity come the gravity meters, various, (2) Gravity Meters., ftorms of which are now in commercial use as prospecting instruments and othershall consider here only a few of them., wise., , We, , *The, , milligal is, , a new unit,, , changes in the value of #., , now, , increasingly being used to express small
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ACCELERATION DUE TO GRAVITY, , 211, , Used, , first, , Pollock, in 1899, it is perhaps the earliest gravity meter, quartz thread AB* (Fig., 131), stretched horizon-, , and, , TheThrelfall and Pollock Gravity Meter., , tally,, , end, , with the, , by Threlfall and, consists of a tine, , A, , and, the end B attached to an, axle which c<m be rotat-, , A, , fixed or 'anchored*, , '/?, , p;, , rig, with the, ed, in line, thread (the latter being thus twisted) by means of a pointer (or a vernier) wljichmoves over a circular scale S A small metal rod R is fused athwart the thread/, near about its midpoint and is so weighted (by a bob or weight w) that its e.g., lies on one side of the thread., *, , ,, , The end B is twisted by means of the pointer, until the rod becomes, horizontal about three full turns of the thread being necessary for the purpose, in which position it is just stable under the balancing forces due to the tension, of the thread and the gravitational pull on itself, / e., when the torsional couple, due to thread just balances that due to the pull of the earth. The position of the, ponter is now read off on the scale, the slightest further movement of it, making the rod lose its precarious balance and turn right over. This is safeguarded against by a suitable stop or arrester, but the veiy fact of this tending, to occur enables its position of approaching instability to be readily determined.*, Thus, with a change in the value of g, the rod will no longer remain horizontal, and the end B of the thread will have to be twisted to restore it to that position., The angular twist thus given to the thread can be read on the scale from the, position of the pointer, and is a measure of the variation in the value of g, the, pressure being kept constant and proper correction for temperature effects (i.e.,, for expansion and change in the rigidity of the thread etc.) being made., The instrument is made direct-reading by first noting the positions of, the pointer at two stations, where the value g is accurately known, its variation, with temperature being determined at one of them. So that, if now the instrument be carried from place to place, the various positions of the pointer indicate the values of g on the scale straightaway., With proper precautions taken,, a, , degree of accuracy., The Boliden Gravity Meter., , this, , simple appliance can yield results of, , fairly high, (//), is, , Fig. 132, , due to Boliden, , (1938),, , A later form of gravity-meter,, , m which two pieces of spring, , M, , ,, , shown, , inr, , S, support a, flat plates, and, S,, , E, , mass, which ends in two, D, above and below, each forming one plate of, the parallel plate condensers AD and BEr, whose other plates A and B are properly insulated from the framework of the instrument, by means of insulating slabs FandG. The, condenser AD above forms part of an oscillatory (or LC) circuit, whose frequency (N), compared with a standard oscillator., i, , Fig., , 1, , 32., , A change Bg in the value of^ brings, about a change in the flexure of the springsand hence a proportionate change Bx in the, air gap (d) between the condenser plates A and, D. This, in its turn, results in a change BO, in the capacity (C) of the condenser, such that, CB/C = Bx/d. And, finally, the change m the, capacity of the condenser is then responsible, for a corresponding change BN in the frequency (AT) of the oscillatory circuit.f So that,,, Bg oc Bx oc SAT,, whence Bg can be easily calculated out., , ", , ~~*For, with the approach of the position of instability, the net couple, acting on the thread varies only slowly with the change in its inclination anfd, hence the time-period of the torsional vibrations of the thread about its equilibrium position goes on increasing., tBecause the frequency of an oscillatory circuit depends upon the capacitance (C) and the inductance (L) included in it.
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PROPERTIES OF MATTER, , 212, , The instrument is calibrated by applying known potential differences to, the plates, and, of ch2 lovvar coadeaser, calculating the force of attraction, between them (and hence on D) and ihs attendant frequsncy change of tho, oscillatory circuit and plotting a graph between the latter two., The sensitivity of the instrument, 1, , cmjr./iec, , .,, , or just, , 1, , is, , rather low, being only about, , 1, , X, , 10-**, , milligal., , (Hi) The Gulf Gravity Meter. This is a more recent (1941) and sensitive, type of gravity- meter and depends uponths same principle as a spring balance,, v/2., that the weight of a b-idy is proportional to the acceleration due to gravity,, so that a mass suspended from a spring will exert a different pull on it for different values of g, the stretch of the spring thus indicating the variations in g,, , The method fails in the case of the ordinary spring balance purely for, want of requisite sensitiveness. In the case of the present instrument, however,, this sensitiveness is well assured, as much by the choice of a suitable type of, spring as by the accuracy of the means of observation., , We, , u^e here a flat, metallic ribbon-spiral spring, fastened to a torsion head,, at the top, and carrying a load at its free end below, including a mirror m, (Fig., *, 133), which untwists the spring by about 8 full revolutions., Any change in the value of g will bring about a, jl, $, proportionate change in the -weight of the suspended mass and, the consequent pull on the spring, resulting in a correspondD, ing rotation of the mirror, which can be measured by the deviation of a beam of light from an illuminated slit, reflected, from it. Th? angle of deviation is magnified by making the, and a fixed reflector and the, beam travel four time* between, image of the slit finally observed by means of a microscope,, fitted with a micrometer eye-piece., The slight changes in, the value of g corresponding to thsse deviations can thus bo, easily determined., , m, , 5, , The sensitivity of the instrument, ~, X 10~ f cms. I sec*., or 5 x 10 2 milligals., , is, , found to be about, , None of the above appli(iv) Eotvos Balance., ances possess the necessary sensitivity to be able to, measure the small change in the value of g due to, neighbouring buildings or small geological deposits, etc. Instruments far more responsive to small variations in the value of g must be employed for these, Fig. 133., delicate measurements and the gravity balance, devised by Baron, Eotros, admirably answers this requirement. It is not only used, for a comparative or an absolute determination, of g, but also for the measurement of other, important quantities connected with the earth's, gravitational field and for purposes of gravitational survey, the accuracy claimed for the, instrument being 10~* cms. I sec*., or 1CT*, milligals., , In essentials, the Eotvos Balance consists, of a rectangular torsion beam B, (Fig. 134),, of aluminium about 40 cms. in length, and, between 3 X 10~4 to 4 x 10~ 4 mms. in diameter,, suspended from a torsion head T, by means of, a fine suspension S, about 60 cms. long, of the, alloy platinum-indium, through an aluminium, rbd R, fixed on to the beam at its c.g O. The, rod carries a small concave mirror C, to enable, the deflections of the beam to be read by the, ,, , Fig. 134.
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213, , ACCELERATION DUE TO GRAVITY, , A, , small, lamp and the scale method, with the help of a telescope., cylindrical weight P, of platinum, gold or silver, of mass about 30, gms. is suspended from one end of the beam, by means of a fine wire, , and a counterpoise weight M, of mass about 25 gms, suspended from, the other end of the beara, as shown., If the instrument be taken to a place, where the value of g, varies from point to point, its suspended system experiences a couple,, producing a twist in the wire and deflecting the beara from the, position, (not known), that it would occupy if the value of g were, (H>) of platinum,, is slid on to, or, , ,, , constant., , Let the beam, in its equilibrium position, make an angle with., the x-axis, (lying along the north-south direction),, i.e-, let 6 be the, and let Sl be the reading on the scale, 'azimuth angle', as it is called,, in this position., Then, if 5 be the scale reading in the (unknown)*, on the, position, in which there would be no gravitational torque, that, shown, it, be, can, beam,, , C, , are the constants of the instrument, 17, the gravitational, the gravitational, 9C//3*, a*7/aj and 3t7/3z, the values of, along the North, the East, and the vertical directions, , where A and, , and, , potential,, , attraction, , one being the value of g). The origin of the, respectively, (this last, three axes along these three directions, is taken to be the mid-point, of the beam,, , O, , ,", , we, , have, from relation (1) above,, A' sin 20+ B' cos, Si-So, , =, , =, , 20+C, , sind+D' cos 0., 0, 60, 120, 180, 240 and 300,, , (2), , Now, taking 6, above, and taking the corresponding values of, , $i>, , S&, , ,, , S*>, , SL, , .(2>, , ., , in equation, , to be, , S19 S2 S9, ,, , >, , we have, , - S + S +S =, =, 2, , A', , 2V3C', 2D', , and, , =, =, , 4, , 6, , 3S, , ,, , S.-S^, , Thus, all the constants of relation (1) being known, the rate of, C'/C), as also that in the eastern, change of g northwards, (given by, direction, (given by D'fC), can be easily determined., its high sensitivity, the balance is used for, And, Shaw and Lancaster Jones, (see 80)., prospecting, geophysical, its, with, out, have successfully mapped, help the local gravitational field, , On, , account of, , in a laboratory.
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214, , PROPERTIES OF MATTER, , The Horizontal Pendulum. As we hare, (ii) Temporal Changes, seen above, a deformation of the earth's surface and its gravitational, equipotential surface results in a change in the direction of the force, of gravity and hence in that of g. Since a plumb line always sets, itself normally to the gravitational equipoential surface of the earth,, it is clear that measuring a change in the direction of, g at a point, is, tantamount to measuring a change in the direction of the plumb line, at that point. These changes, however, are much too small seldom, exceeding 1", to permit of their accurate measurement by means, of a plumb line. The most commonly used device to measure, these is what is called the horizontal pendulum, devised by Hengler, in, the year 1832., This horizontal pendulum essentially consists of a rod AB, (Fig., bob at #, with G as its e.g. It is supported, by moans of two pieces of a light string, AP, and Cg, attached to a rigid support at P and C respectively, such that, the straight line CP, joining the two, meets AB in O, and makes an, with the direction of the forco of gravity., The pendulum, angjle, thus takes up a position in a plane parallel to the force of gravity., On giving the bob a slight lateral displacement (towards or away, from the observer), it begins to oscillate slowly, with a small, amplitude, along an arc with O as its centre and OG as radius. Its, period of vibration is deduced as follows, 135), carrying a knob or, in an inclined position,, , <j>, , :, , B, , /r\, , I, , Fig. 136., , Fig. 135., , Let the bob oscillate along an arc GG', (Fig. 136), which lies in, a plane, making an angle with the normal to the plane of the force, <, , of gravity., in this inclined, If the bob be displaced through an angle, plane, into the position shown, its weight mg acts at its e.g., G' (v G, is now at G') in the direction of the force of gravity., Resolving it, into its two rectangular components, (/) in the inclined plane of its, , rotation, and (ii) perpendicular to, s=, sin <f>, and the latter, , mg, , =, , mg, , //,, , we have the former component, , cos $., , Further, resolving the component, , mg, , sin, , <, , into, , two rectangular, , components, along and perpendicular to 06?', we have, the component along OG', mg sin, , and, , the component perpendicular to, , This latter component (mg sin, , <f>, , OG', , =, =, , mg, , cos, , <f>, , sin, , sin $ ) has, clearly,, , <f>, , Q., , sin 0., , a restoring
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215, , ACCELERATION DUE TO GRAVITY, , = OG, , =, , about the inclined axis, mg sin ^ sin Q.I, (where OG*, =/), tending to bring the pendulum back to its original position., If be small, sin, 9, very nearly. And, therefore,, , =, , =, , moment, , the restoring, , mg, , sin <.#./., , the angular acceleration of the pendulum be da>ldt, and, And,, its moment of inertia about O be 7, we have, if, , restoring, , moment, , So that, neglecting, , (or torque) also, , frictional, , =, , and viscous, , I.dto/dt., , forces,, , we have,, , for, , -equilibrium,, , = mg Lsin +.0, whence, dw/dt = mgl * in *.e., = a constant, we have, putting mg.l.sin, Or,, daj/dt = n.S., </oj/<# oc 0,, Lda>ldt, , Or,, , /i,, , </>/!, , the angular acceleration of the pendulum is proportional to its, .i.e.,, angular displacement., It is, therefore, a case of simple harmonic motion; and its timeperiod Tis given by the relation,, , =, / =, , T, , Or,, , But, , 2*, 7w/c, , 2, ,, , where, , fc, , is, , about O., , Hence, , the radius of gyration of the pendulum, , T = 27rA/ "!^ -,, V ing I sin, , =, , A/, 'V, , 77, n, , $, , in practice, to make Tlargo, ^, , Now,, in this case,, , Hence, whence,, , if, , <f>, , is, , r/r, , =, , 90,, , j/71, , is, , =, , <, , ^/T^inf, sin, , as small as possible., and, therefore, the time-period !F,, , 1,, , <f>, , =, =, , -, , made, , =, , given by the relation T', , =, , k*, ,, , ./.*<, , (l/ 5 f, , T' 2 /T 2, , ^). And, , /., , T^/T'*, , =, , l!sinJ>,, , ., , T, , we can easily calculate sin <f>, and, Thus, knowing T' and, lience ^, which represents the change in the direction of the force oi, gravity, and, therefore, that of g, in the equilibrium plane of the, t, , pendulum., 79, Gravity Survey., , The purpose of a, , gravity, , survey, , is, , two-fold, viz., (i) the main one being to determine the value of the, force of gravity and its direction at various, points of the sea-level, surface of the earth, or the geoid' as it is, called, and (if) a secondary, one being to deduce from it the possible distribution of matter in the, earth, and thus to form an idea about its structure and internal, l, , 9, , -condition., , Now, in any gravity survey, it is found necessary, two new quantities, connected with the earth's, , to introduce, , gravitational field,, These are (1) the gravity gradient, denoted, by the letter O, and (2, the horizontal directive tendency (written as H. D., T., for brevity), denoted by the letter R. Let us try to understand their, meaning.
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PROPERTIES OF MATTER, , 216, (1), , "maximum, , The Gravity Gradient (G). It is a convenient abbreviation for, gradient of g in a horizontal direction, near a point", where ^ is the*, , vertical gravitational intensity at the point, i.e., g =, The gravity gradient, QU/dz, may, therefore, he denoted by tigIds, the rate of variation of g per unit distance, in*, the direction of the maximum rate of change in its value, and it is thus obviously a, , vector quantity., , Now, if the gravity gradient G, (=&e/0s) makes an angle with the axisof x for the north -south direction), and if its components along the axes of x and*, y be Qgld* and dg/Qy respectively, then, clearly,, <j>, , And, since, , So, .'., , g, , -, , = G cos, =, , dgld*, QUIdz,, , and, , <f>, , we have fa fix, , dgfty, , 2, , U xs, , //0Jc0z, , and, t/^ = G cos $, squaring and adding the two, we have, , C/^, , that,, , = G sin, , <f>., , ,, , = G sin, , <f>,, , = V\ Z +V\*. Or, G* (w ^-M/i #) = U*, G = U*xz +U\ zt whence, G = (C/ az -f l/^)*., can be easily determined by means of, {/as and Uv, , G* cos 2, , f+ G 2 sin*, , 2, , a, , Or,, , 2, , f, , 2, , an Eotvof, Now,, Balance, (see page 212), and thus the value of G can be calculated out from theabove expression., It is what is called a 'cur(2) The Horizontal Directive Tendency (R)., vature vector', i.e., a directed quantity, though not a true vector. Its value at a., *, , point, , is, , given by the relation,, , where r and r, stand for the maximum and the minimum radii of curvature of, the level surface, or the gravitational equipotential surface at the point. Its, direction, according to an agreed convention, is taken to be the direction i/r, which the level surface has the least downward curvature and, therefore, the maxi-, , mum radius of curvature., , If the direction of H.D.T. makes an angle, north-south direction, it can be shown that, , R sin, R, , and, , 28, cos 2o, , - 2UXV, , *, , U*xx, , r, , ,, , U* vv, , in, , what are, , where,, , Uxv =, Uy^, , ., , The dimensions of both G and R are, pressed, , with the axis of x, or the, , called Eotvos units,, , [T]~* and, , where, , -=, , fU/dxdy,, cW/0^* and, , they are generally ex-, , one Eotvos unit, , is equal*, to 10- /sec*., In survey maps, the gravity gradient at a point isrepresented, in magnitude and direction, by an arrowhead drawn from the point, whereas the horizontal directive tendency is just represented in magnitude and direction, by a straight line, passing through that point, wimout any arrowhead or feathered tail, as shown in Fig. 137,, where O is the point in question., 9, , Fig. 137., , Further, points, where the value of g is the fame,,, are joined by curves, which are called isogaras, G beingalways directed along the normals to these., , We, , have seen above how, due, 80., Geophysical Prospecting., to the presence of local geological deposits, (i.e., minerals etc.), inside, the earth, small variations are produced in the value of g. Similar, changes are produced, by their presence in the normal values of the, other quantities, magnetic, electrical, seismic* etc., associated with it ;, so that, by measuring these variations, with the help of specially, their presence. This is technically, designed instruments, we can detect, 9, are concerned here only with, called 'geophysical prospecting., the gravitational methods adopted for the purpose, the principle, , We, , *See article on, 'Earthquakes' in the next chapter*
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ACCELERATION DUE TO GRAVITY, , 2 IT, , underlying which is to measure the gravity gradient G and the horizontal directive tendency R, at various points in the region under, survey, with the help of a sensitive instrument, like an Eowos gravity, 78, (iv) the instrument, when so used,, balance, as explained above in, being called a 'gradiometer'., Thus, if & be the angle that R makes with the axis of x, or the, north-south direction at a point, we have, t, , Ssin 28, , =, , 2U^, and, tan 26, , R, , =, , And, therefore,, the two solutions of which,, , cos 26, , =, , U*xx, , 2Uxy/U*xx -U2 yVy, , [/%,., , [, , 79, (2),, , above, , BI and tt t differ by 2/7T and give the, principal axes of curvature of the equipotential, The values of G and, surface, (i.e., the level surface), at the point., at various points are then plotted, the direction of a being such that, 2, secant 2$ and (V 2 XX, VV ) are of opposite signs., , directions of the, , ,, , two, , R, , U, , A, , graphical representation of the variations of g over the, under, examination, is thus obtained, and closed curves or, region,, 79, (2), above], are then clearly marked out on it,, isogams, [see, (which, as we know, are at every place perpendicular to G), so that, we have an isogam chart of the region in question., Interestingly enough, the physical form of these isogams almost, faithfully represents the physical form of the subterranean deposits., Thus, for example, a uniformly monoclwic type of region uould give, isogams which are all parallel and equally spaced, whereas if the subt, , terranean deposits form a dome-like structure, the isogams obtained, also resemble the outline or the contour of a surface dome, as it were., , This method can, however, succeed only in the hands of those, well-trained in the use of the delicate instruments employed and in, the proper interpretation of the results obtained from them., , SOLVED EXAMPLES, A, , 1., , a point on, , its, , its own plane about an axis passing througft, the length of the equivalent simple pendulum ?, , metal disc oscillates in, , What, , edge., , is, , Let the disc of radius r oscillate about an axis through the point, edge, (Fig. 138)., Then, clearly, the time-period of the disc is given, , by the relation,, , where / is, mass and /,, , its, , /, , A/./,, , =, , 2n \/ iiMg, , Pon, , ita, , /,, , about the axis through, , P, , ;, , M,, , its, , its length., , =, , =, , between, 27c\/ /,A/#.r, for /, r, the distance, Or, t, the point of suspension (P) and the e.g. (O) of the disc., , Now,, where, , Ig is the, , through O,, , So, , U. 9, , that,, , And,, , ML, , therefore,, , /, , -, , /=/0+A/r*., of the disc about a parallel axis, l = Mr*\ 2., g, 2, , (Mr /2)+Mr a, t, , -, , 2*, , f, , _3A/r, , A / lMg.r, ', , /2., , Fig. 138., , 2* A, , /, 'y, , the same as that of a simple pendulum of length / = 3r/2., Or, the length of the equivalent simple pendulum is 3/2 times the radius of the disc., Find the period of small oscillations of a rig>'d body, free to turn abou, 2., t fixed horizontal axis, and also find a formula for the length of the equivalent, simple pendulum., I.*.,
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218, , m are fixed to a uniform circular hoop, particles, M and radius, a at the corners of an equilateral, free to, The hoop, Three, , mass, move, , MATTER, , PROFEfcftES OF, of the same mass, , of, , is, , triangle., , in a vertical plane about the point on the circumference, opposite to one of the, masses m. Prove that the equivalent simple pendulum is equal in length to the diameter of the circle., (London Higher School Certificate ; Patna 9 1948), , For first part see 62 (page 165)., Let the three equal masses, m, m and m, be fixed to the hoop, of radius, as shown, (Fig. 139), so as to lie at the corners of an, equilateral triangle., ;, , a,, , Since they are, of the triangle, , from the centre, the e.g., at O, the centre of the circle. The, , all equidistant, is, , whole arrangement, mass (Af-f- 3m), with, , is, , its, , thus equivalent to a hoop of, centre of gravity at its cen-, , tre O., , Clearly, then, the moment of inertia of this, loaded hoop about O, (i.e., about an axis through O, and perpendicular to its plane) == (M>3w)a a, And,, therefore, its moment of inertia about a parallel axis, through the point of suspension P is (by the principle, of parallel axes), given by, ., , I, , Fig. 139., , Now,, , =, , - (M+3m) a +(M+3m)a* =, z, , 2, , Or,, te.,, , 2(M+3m)a*., , the time-period of the hoop about, , /, , _, , 2* A, , =, , 2rr, , P is, , m, , ~, , given by, ., , ', , the same as that of a simple pendulum of length la, the diameter of the hoop., Or, the length of the equivalent simple pendulum is equal to the diameter of, , the circular hoop., 3., How much faster than its present rate should the earth revolve about, axis in order that the weight of a body on the equator, may be zero, and how long, would it take to make one revolution then ?, What would happen if (/) the rotation, became faster still, (//) the rotation were stopped altogether ? (g, 978 cms./sec 2 .), its, , =, , We have seen (page 203),, , that the value of, , tudes, due to the rotation of the earth,, fect sphere,, , and, , 2, , r.c05 ^.o>, , 2, , 8, , Now,, , at the equator,, , ^, , 0,, , and, , .*., , different in different latithat, assuming the earth to be a per-, , g, , is, , \, , Twhere g, , ', , Lin latitude, , cos 2, , ^, , *-, , 1, , ;, , ., , is, , <f>,, , the value of, , V, , (see page 205), , so that,, , 9, , where,, , g is the value of 'g at the equator., With the actual value of g, the value of, , r.w 2, , comes out, , to be 3'39 and,, , ', , therefore,, , we have, , -^ ., , 3 39, , = JL., , 978, 288, g, Thus, in order that the weight of a body may be zero, the value of FO, should be zero, i e., r.^/g should be equal to 1, or the value of r<o 2 should be, 288 times greater than its present value, r, being a constant. It follows, therefore, that <o should be \/288 times, /.<?., 16'97 times greater than its, present value., When this is so, the outward centrifugal force on the body will, obviously,, be just, balanced by the inward force due to gravity., //,, , therefore the earth rotates, 16 97 times, or 17 times, faster than at, preof the body at the equator will be zero., , sent, the weight, , Now, the earth makes one complete revolution in 24 hours, i.e., discnbes an angle of 2* m 24 hours. But, in the case, considered, viz., when the, weight of the body at the equator is zero, it rotates 17 times faster, and will,, therefore, describe an angle 17x2* i n 24, 2* in, hours, or an, hours, or 1-412 hours., , angle, , 24/17
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ACCELERATION DUE TO GRAVITY, , 219, , make one rotation in 1*42 hours., became faster still, i.e., faster than 16*97 or, obviously, all objects kept loose on the equator will, , the earth will then, , .*., , 17 times its, , If the rotation, , normal, , start leaving, fthe surface of the earth ;, for, the increased centrifugal acceleration on them will, be greater than that due to gravity, and, therefore, a resultant force will be act', ing on them outwards, away from the centre of the earth., rate,, , If,, , ped, , on the other hand, the rotation of the earth about its axis were stop=, so that, substituting this value of to in the, we shall have, , altogether,, relation,, o, , ;, , = g(, , \ for the value of V at the equator, we have, , -, , 1, , *o=*U-0)=*., i.e.,, , the value of, , =, , g, , increases by (g, , gQ, , times, , !'<**/, , ), , =, , g., , =, , 2, , =, , 3'39., 1/288 times g, or, /288., [ For r.o>, (3'39/978) times g, Thus, // the motion of the earth were stopped altogether, the value of g, would increase by 1/288 of its normal value., , Or,, , Assuming that the whole variation of the weight of a body with its posithe earth's surface is due to the rotation of the earth, find the difference in, fthe weight of a gram as measured at the equator and at the poles., (Radius of the, earth, 6 '378 x 10 8 cms. ), 4., , tion OH, , =, , We, , have the relation,, r, , g (, , g,, r, , -, , r.o 2, , Since, , 6378x, , 10*, , ', , of, , 3 39 >, , ', , and, , r, , -, , to, , V, , V*, , in latitude, , =, , 97^, , tf., , = 1/288,, , (l-o?s 0/288)., , gj, , =, , is, , 0, so that, cos, , cos, , .'., , 2, , the value of, , 9, , 'g, , =, , I grn., , =, , g, , .*., , cos 2, , $, , =, , 0., , t, , at the poles., , Since the weight of body, weight of, , #(1-0), , =--, , 1., , $, , the value, of '#' at the equator., at the poles, ^ = 90, so that, cos $ and, , gp, is, , and, , <f>, , ^ = #(1-1/288),, , Hence, , and, , for the value, , x, , J^")^, , x(, , And,, where g#, , \, , -?-, , g, , at the equator, ^, , -where gp, , r' os, , 2, , we have, , Now,, Hence, , --, , 1, , \, , is, , mg, where, , at the equator, , 1x^(1-1/288), , w =, , 1, , m is, x#, , its, , mass,, , we have, , ., , = ^(1-1/288),, , =, , (/), , =, , 1 xgp, 1 *xg == g., w', (//), weight at the poles, in, the, the, Hence,, difference, weights of this mass at the poles and at the, , its, , equator, , w'w., , =, , -g{l- 1/288), , = g-g, , #/288, , ^/288, , =, , 978/288 == 3'395 dynes., , Or, the difference in weights of a gram at the poles and at the equator, 3'395 dynes., , train is 100 tons. What will be its weight when, travelling due east, (c) travelling due west, along the equator at, miles per hour ? Radius of the earth is 4000 miles., (Punjab), , The mass of a railway, , 5., , (a) stationary,, , 60, , is, , (/?), , When, , When, , the train is at rest, its apparent, ton, 2240 Ibs.)., t (because 1, When the train travels due east, its, <(b) When the train is moving East., angular velocity about the axis of rotation of the earth increases, because the, earth itself is rotating about the axis from west to east. The centrifugal force, on the train, therefore, increases, (being proportional to r.w 8 ), and hence the, .apparent force of gravity on it and, therefore, the apparent acceleration towards, the centre of earth, i.e., apparent acceleration due to gravity, decreases. And,, mas sx acceleration due to gravity, the apparent weight of the train, since weight, Let us see by how much., decreases.., (a), , weight, , is, , the train, , 100 tons wt., , is, , stationary., , 100 x 2240=224 x 10 3, , Ibs. wt., , =, , =, , =, , = 4000x1 760x3 //., and, there= 2^/24 x 60x60 ft.lsec., since, , 4000 miles, The radius of the earth, r, linear velocity (v) of a point.on the earth, &fce earth makes one complete rotation in 24 hours., fore,
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PROPERTIES OF MATTER, , 220, , A point on the equator will also thus describe a distance 2*r in, its, , velocity, , ==, , ;, , -, , 2*x 4000x1760x3, , 24x60x60, , when, , centrifugal acceleration of the point,, , .*., , is, , -, , 24 hours, , therefore, given by, , is,, , at rest relative to the earth;., , given by, , 1536x1536, , v, , T = Too6TT760x~3, , =, , =, , =, , Since velocity of the train, , 60 m./hr., , the resultant velocity of the train, say,, , -, , And, , 1536488, , --, , =, , 88//./sec.,, , v', , =, , 1624, , ft./ sec., , centrifugal acceleration on the tram moving at 60 m/hr., , /., , v"/r., , 1624x1624, .'., , the centre, , increase in centrifugal acceleration, or decrease in acceleration towards, i.e.. in the acceh ration due to gravity, , of the earth,, , =, .*., , O'1248-O 1116 - 0-0132//./WC*., = mass x decrease in acceleration due to, , decrease in the weight of the train, , gravity., , =, -, , - 100x2240x0-0132/32 Ibs. wt., = 00 xO'Ol 32/32 ton wt. - 0'0412, train = 100-0-0412 = 99'9588 tons wt., , 100x2240x0-0132 poundah, 100x2240x0-0132/32x2240, .'., , apparent weight of the, , 1, , ton wt., , In this case, since the train is moving, (c) When the train is moving West., from east to west, opposite to the ditection oj rotation of the earth, its angular, velocity about the axis of rotation of the earth decreases and, therefore, the, centrifugal acceleration on it also decreases, with the result that the acceleration-, , towards the centre of the earth, i e the acceleration due to gravity increases. The, apparent weight of the train on the equator, therefore, increases. Let us calculate this apparent increase., As before, velocity of a point on the equator, i.e.,, ,, , 2nx4000x 1760x3, , 2nr, , centrifugal acceleration of the point,, , -, , v, , 2, , -, , Hence, "the resultant velocity of the, , And, , when, , at rest relative to the earth, , 1536x1536, 40UOX-17605T, train, say,, , v", , =*, , 1536-88, , =, , 1448, , ft. I sec., , the centritugal acceleration on the train moving at 60 m./hr. is, , .*., , clearly given by, ", , 1448x1448, , v' /a, , T*, , 4000 x 176071, , decrease in cenrtifugal acceleration or increase in acceleration towards the, .'., centre of the earth, i.e., increase in acceleration due to gravity, 0-1116-0-0993, 0-0 123 //./sec 2 ., , -, , And, , .*., , =, , increase in apparent weight of the train, , 100x2240x0-0123^^^/5, 100x2240x0-0123/32x2240, , =, =, , 1 00 x 2240 xO'Ol, 23/32 Ws. wt^, lOOx '0123/32 ton wt., , C '03 844 ton wt., , Thus, the apparent weight of the train, - 100+0-03844 - 100-03884 100'04 tons weight., , EXERCISE VI, What is a simple pendulum ? Is it obtainable in actual practice 7, 1., Deduce an expression for its time-period and show how the value of g maybe, determined with its help. What are the drawbacks of this pendulum ?
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ACCELERATION DUE TO GRAVITY, , 221, , 2., Deduce the formula for the time of vibration of a compound pendulum and show that this is a minimum when the length of the compound, pendulum equals its radius of gyration about a horizontal axis through the centre, of gravity of the compound pendulum., (Punjab, 1951), a, 3., a, and, between, compound, pendulum., simple, Distinguish, For a given compound pendulum, show that the centres of oscillation and, , suspension are interchangeable., How is the value of ** determined with the help of a compound pendu-, , lum, , ?, , (Agra, 1948), , Give the theory of Kater's pendulum and find an expression for the, acceleration due to gravity in terms of two nearly equal periods of oscillation, about the two parallel-knife-edges., 4., , Indicate the sources of error in an experimental determination of ^., (Bombay, 1940-41 ; Punjab, 1948), 5., Borda's pendulum his a bob of radius 12 rwv., which i suspended, by a fine wire, 94 cms. long. Calculate the length of the equivalent simple penduAns. IGO'144 cms., lum., 6., Tf a pendulum beats seconds at a olace where #, 32*2 ft. /see*., how, 32*1 8 ft./*ec z, much would it gain or lose per day at a place where g, Ans. Gains 3 min. 36 sees., , A, , =, , ., , 7., Explain the Dropping Plate method for the determination of tho, value of g. If there be an enor of 1% in, ^asurin r the distance covered by the, plate as also in measuring the frequency of the fork, how would it affect the, result ?, Ans. The remit will b* wrong by 3%., , m, , 8., , The, , length between the knife-edjes of a Kater's, , pendulum, , is, , 89*28, , cms., while the times of oscillation abrjt the txvo edjes ire 1'920 sec*, and 1*933, The e.g. of the pendulum is about 54*4 cwy. from one edge, sees., respectively., What is the value of g 1, Ans. 979 cms. /sec 9, -, , An Atwood's machine has a pulley of radius a and moment of inertia, 9., / ; the masses attached to the ends of the string are each Mand the rider is of, mass m., Prove that the acceleration /of the masses is given by, , assuming that the string does not, , slip, , on the pulley, and neglecting axle, , friction., , (Madron, 1949}, , A uniform rod of length 100 cms can rotate about a horizontal axis, 10., through one end. Find the angular velocity which will enable the rod just to, make a complete rotation., (Madras, 1947), Ans., , 3*83 radians I sec., , A, , solid cylinder, of radius 4 cm?, and mass 250 gms.* rolls down an, inclined plane, with a slope of 1 in 10. Find the acceleration and the total energy, of the cylinder after 5 sees., (Bombav, 1944), 11., , Ans., , 65'4 cms./sec^., , ;, , 4*799 Joules., , A, , cylinder, of mass 100 /6s*. and diameter 12 inches, rolls from rest, down a smooth inclined plane of 1 in 8 and 20 feet long. Calculate the total, kinetic energy and its energy due to rotation, when it reaches the bottom., 12., , Ans., , (/), (it), , (Madras, 1949), ft. poundah., 2'6xlQ*ft.poundals., , 8*0 x 1 0*, , Define 'centre of suspension* and 'centre of oscillation'. Show that, in a compound pendulum they are interchangeable., What is the distance between the centre of suspension and the centre of, oscillation on a uniform cylindrical metal bar used as seconds pendulum ?, (Diameter of the bar=l cm., to density, 8 gms./c.c. and #=978 cmi.lsec*.), 13., , (Allahabad, 1949), , Ans., 14., , 99'1 9 cms., , Obtain an expression for the time-period of a compound pendulum,, , and show that
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PROPERTIES OF MATTER, , 222, , (0 there are four points, collinear with its e.g., about which its time, period is the same., (//) its time-period remains unaffected by the fixing of a small addi, tional mass to it at its centre of suspension., Obtain an expression for the period of vibration of a compound pen, 15., dulum and show that the centre of suspension and the centre of oscillation art*, interchangeable., A thin uniform bar of length 120 cms. is made to oscillate about an axisthrough Us end. Find the period of oscillation and other points about which, it can oscillate with the same period., (Punjab, 7953>, Ans. 1*795 sees. at 40 cms 80 cms., and 120 cms. from the top., ;, , ,, , 16., Derive an expression for the period of oscillation of a circular disc,., supported on a horizontal rail passing through a narrow hole, which is bored, through the disc half-way between the centre and the periphery. (Bombay, 1946), , Ans. T = 2v 3r/2., uniform rectangular sheet of metal is supported by frictionless, hinges, attached to one edge which is horizontal. Determine the period of, oscillation of the sheet if / denotes the length of the side of the rectangle which, (Patna, 1951), hangs downwards., x, , A, , 17., , Ans., , A metal, , T== 2nVT3//L2*., , in turn from two parallel axes on the same, side of its c.^., and its time-periods are four.d to be 1 42 sees in each case. If, the distance of the two axes be 10-8 cms. and 39'2 cms. respectively from the, e.g., calculate the value of g and the radius of gyration of the bar about a, 18., , bar, , is, , suspended, , parallel axis through the e.g., , Ans. g = 979*2 cms.lsec*., and K - 20*58 cms., uniform bar of mass 1000 gms. oscillates about an axis, 40, cms. from the centre, with a period of 7 '48 sees., and about a parallel axis, 10, cms. from the centre, with a period of 1*67 sees. Find the value of g, the, moment of inertia of the bar about its e.g. and the length of the bar., Ans. 990 cms. jsec*., 6'02 x JO 6 gm.-cm z, 85 cms., What is meant by a simple equivalent pendulum ? If the periods of, 20., a Rater's pendulum in the erect and inverted positions are equal, prove that, the distance between the knife-edges is equal to the length of the simple equivalent pendulum., A uniform circular rod, with a radius of 2 cms. oscillates when suspended, from a point on its axis at a distance of 4 cms. from one end. It the length of, the rod is one metre, find the point or points from which, if suspended, the, periodic time would remain unaltered., (Bombay, 1942), Ans. At 31'87 cms. and 68*13 cms., also at 96*0 cms. from the same end., , A narrow, , 19., , ., , ;, , ;, , Define a conical pendulum, and show that, for a small amplitude, its, thU of a "plane" pendulum of the same length. Do simple pendulums exist ? What are the nearest approximations to them ? Why are they, discarded in favour of compound pendulums and what are the main applications, of pendulums ?, (Bombay, 1941}, 22., Describe a conical pendulum and derive an expression for its frequency, Explain how it is used to regulate the speed of steam engines. Show that the, sensitiveness of the pendulum used as a governor increases with diminishing, speed., (Bombay, 1937), 21., , period equals, , ^, , Ans., , n, , =, , IjInV big,, , axial height of the cone described by it, and equal to / cos 0, where, is the length of the pendulum and 0, its angular displacement ; see 74)., , (where h, , is, , 23., , underlying, , What, , is, , a steam engine governor, , its action,, , and discuss, , its, , ?, , /', , Explain clearly the principle-, , limitations., , 24., If the earth were to cease rotating about its axis, what will be the, change in the value of g at a place of latitude 45, assuming the earth to be a*, 8, sphere of radius 6'38 x 10 cms. ?, (Madras, 1947), , Ans., , 1, , -6895 cwj. /,*"-
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ACCELERATION DUE TO GBAVITY, , 223, , Find an expression for the period of swing of a compound pendulum., , 25., , A, , disc of metal, of radius R, with its plane vertical, can be made to swing, about a horizontal axis passing through any one of a series of holes, bored along, a diameter. Show that the minimum period of oscillation is given by, , T=, , (Saugar, 1948), , 2nv/l-414 "Rig., , Give the theory of the compound pendulum and show that the centresof suspension and oscillation are reversible., In a reversible pendulum, the periods about the two knife-edges are t and, The knife-edges are distant / and /' from, (f-f!T), where T is a smaJl quantity., the centre of gravity of the pendulum. Prove that, 26., , /+/', , ., , *Lt+L, , T, , (Madras, 1949), , ., , 27. A heavy uniform rod, 30 cms. long, oscillates in a vertical plane, about a, horizontal axis passing through one end. When a concentrated mass is fixed on to, it at a distance x from its point of suspension, its time-period remains unaffected., Ans 20 cms*, Calculate the value of AT., , 28. Explain how the length of the simple pendulum which has the same, period as a given compound pendulum may be found experimentally., A uniform cube is free to tuin about one edge which is horizontal. Find, in terms of a seconds pendulum, the length of the edge, so that it may execute, a complete oscillation in 2 sees., (Central Welsh Board higher School Certificate], Ans. 3A/2/., 29., body of mass 200 gms. oscillates about a horizontal axis at a distance of 20 cms. from its centre of gravity. If the length of the equivalent simpk, pendulum be 35 cms., find its moment of inertia about the axis of suspension., (Patna, 1954}, Ans. 1 4 x 1 6 gms.-cm*., 30. A pendulum, whose period slightly exceeds 2 sees , is compared with a, standard seconds pendulum by the method of coincidences. Successive coincidences occurred at times, min., 2 nun? 58 sees., 5 wins. 48 sees., 8 mms, 48 sees. Find the exact period of the pendulum., Ans. 2'0224 sees,, A thin rod is suspended bv means of two threads parallel to each, 31., other and tied to its two ends. Compare the time-period of the rod when it, oscillates thus in its own plane with that when it oscillates as a compound, pendulum about a horizontal axis, passing through one of its ends., 1 414., Ans. 1, , A, , :, , 32., , Give the theory of the compound pendulum and show that, , the, , centres of suspension and oscillation are interchangeable., uniform thin rod AB, of mass 100 gms. and length 120 cms., can swing, in a vertical plane about A, as a pendulum., particle of mass 200 gms. is, attached to the rod at a distance x from A. Find x such that the period oi, vibration is a minimum., (Madras 1951}, Ans 2 748 cms,, , A, , A, , ,, , How does g (acceleration due to gravity) vary with latitude anc, Obtain a general relation, assuming the earth to be a Homogeneous, Does the relation agree with observed values ? Give reasons., , 33., , height ?, sphere., , (Punjab, Sept, , ,, , 1955}, , Give the theory of Kater's pendulum and mention the errors, which pendulum experiments are liable. How is the value of g compared, 34., , to, , at, , different places ?, , (Punjab, Sept., 1956], , What, , Describe the constructioc, , are gravity meters and balances ?, and working of one you consider to be the best., 36., Write short explanatory notes on, 35., , :, , (/), , Gravity survey, and (') Geophysical prospecting.
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CHAPTER, , VII, , GRAVITATION, The celestial bodies have been an object of, 81. Historical., interest to scientists all through the ages, and the first astronomical, observations, of which we have any definite knowledge, were perhaps, made by the Chinese, as far back as 2,000 B.C , though the Babylonian astronomers are credited with having mapped out the constelThe first authoritative, to Ptolemy, working in, Alexanderia, about 100 A.D. who formulated his theory on the basis, of the catalogue showing the nightly positions of planets and some, 1000 stars, prepared earlier by the Greek astronomer Hipparchus., Ptolemy's book, the Almagest, enjoyed the authority of the Bible, and reignsd supreme for 1400 years. According to him, the whole of, the heavens, carrying the stars, revolved round the earth, supposed, lations even earlier, near about 2700 B.C., treatise on the subject, however, was due, , The forward and retrograde motion of the planets*, the stars was explained by postulating that the planets, revolved in circles, with their centres revolving in larger circles round, the former circles bsing termed epicycle* and the latter, iihe earth,, ones, deferents. And, it stands to his credit that, with a suitable, choice of radii and velocities, he could explain quite accurately the, /observed facts of the day., stationary., , among, , The Ptolemaic theory was first challenged in 1543, by the, famous Polish monk, Nicolaus Copernicus, in his book, 'Concerning the, Revolutions of the Heavenly bodies^, his geometrical solution being, much neater than that of Ptolemy, involving only thirty four epicycles, as against the eighty of the latter. In it he propounded his heliocentric theory,! according to which the planets moved in perfect, circles round the Sun, which was supposed to be fixed., The theory, was, however, received with reserve and scepticism, being objected, to on the ground that (/) the rotation of the earth should result in, bodies being hurled from its surface, and (//) with greater justification that, no parallax (or relative motion) could be noticed between, stars as was always observed between objects at different distances, from a moving ship. This parallax has since been shown to actually, It is, however,, exist, and was first measured by Bessel, in 1838., *From the Greek word, meaning 'wanderer', because a planet moves, forwards and backwards or 'wanders' about among the stars., tHe hesitated and deferred publishing bis book until he was dying,, dedicating it to the Pope, who, not taking it seriously, expressed himself, pleased with it. And Martin Luther was positively contemptuous towards it, 'Did not Joshua, (m the Bible) command the Sun to stand still and not the, Dearth?', he asked., JTnis was4eally a revival of the theory, first propounded by the Greek, Astronomer Aristarchus, that the earth was not the centre of the universe but, revolved round the Sun, as also did the other planets., , 224
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GRAVITATIOK, , 25, , extremely email, on account of the enormous distances of the stars, from the earth., , As, , is, , belief in, , it,, , was compelled to recant his, and legend has it that blind and helplpss,, he was thrown back into prison for murmuring 'And still it moves',, {E pwr si muove), until he died nine years later, and that Giordano, Bruno was actually mercilessly burnt at the stake for refusing to, , do, , so, , well known, Galileo, , a century, , later,, , so., , Then, appeared on the scene, twenty -five years later, hi 1569,, Tycho Brahe, an imperious nobleman and a brilliant astronomer*,, who rejected the Copernician theory and made careful observations of, the motions of heavenly bodies, on every clear night for thirty long, years, particularly of the motion of the planet Mars, from his, observatory in Denmark, with his celebrated wooden quadrant,, v, , (about 10 ft. in radius), carrying a brass scale. In view of the fact, that the telescope was yet to come, soms forty years later, we cannot, but marvel at the unprecedented accuracy of his observations. No, wonder, they were usad by navigators for centuries together, much in, the manner of the Nautical Almanack tolay. With all his great, , mathematical and experimental skill and his 'infinite capacity for, taking pains', however, Tycho Brahe could not somehow piece his, But later, Keplerf, his, results together into a proper theory., an impocunious but a, assistant at the Royal Observatory at Pragus,, gifted mathematician into whose hinds passed all his data on the, subject, carried on tho work and, accepting the Copernician theory J,, which his chief had rejected, worked on the latter 's figures and finally, succeeded, after twenty- two years of caaaeless work, in evolving the, famous three laws, known after him, the first two in the year 1009, , and the, , third, ten years later, in 1619., , Kepler's Laws. The following are the three laws, formuby Kepler., 1, The path of a planet is an elliptical orbit, with the Sw at one, , 82., , lated, , ., , of its foci., 2., , The, , radius vector, drawn from the, , Sun, , to a planet, , sweeps out, , *He was reputed to be 'an unsurpassed practical astronomer' and made, own instruments for his well-equipped laboratory at Uraniborg, built for, him by Frederick II, King of Denmark. He had, however, a violent temper and, his, , of his nose in a duel, while still young, going about for the rest of his, with this lost part replaced by an artificial one of aa alloy of silver. On the, death of Frederick, he had to flee and seek asylum at Prague, under the patronage of Rudolph //, .Emperor of Bohemia. It was here that Kepler joined him as, lost part, , life, , ,, , his assistant., , fHe actually succeeded Tycho Brahe, who died after a little over one year, of his migration to Prague, under the impressive designation of 'Imperial, Mathematician', at a high salary which was, however, seldom paid, tte Sad,, therefore, to supplement his income by practising astrology, 'the foolish "and, disreputable daughter of astronomy, without which the wise old mother would, He was also the fou tder of Gsomstrical Optics., starve'., {And, for this he had 10 migrate to a Protestant country to save himself, from persecution., He was so filled with ecstasy at his success in enunciating his third law, I will triumph over mankind, that he declared *I will indulge in my sacred fury, by the honest confession that I have stoleil the golden vases of the Egyptians ta, build up a tabernacle for my God.', 1, , 1, , ;
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PROPERTIES OF MATTER, , 226, , equal areas in equal time, i.e., its area! velocity, (or the area swept on, by it per unit time), is a constant., its time-period, or its tim*, a, 3. The, square of planet's year, (i.e.,, is, the, revolution, round, proportional to the cube of the majoi, Sun),, of, axis of, , its orbit., , Unfortunately, Kepler was not aware of the property of inertia, so could not proceed any further. For him, it was necessarj, to suppose a power acting continuously on a body, in order to make, it move. Most of the fellows of the Ro} al Society*, which included, , and, , 7, , men, , Robert Boyle, Edmund Hailey and Somuei, among, Papys, were convinced, by the year 685, that a planet could move, in an elliptical orbit, only if it were attracted by the Sun with a, force, varying inversely as the square of its distance from the Sun,, but they couid not prove it mathematically., Newton, who was also, a member, was at this time / ucesian Professor of Mathematics at, Cambridge and seldom attended the meetings of the society, mostly, Edmund Hailey, therefore, went all the, held at London and Oxford., way to Cambridge to ask him if he could furnish the required prooi, and was simply astonished to learn that he had already done so years, earlier, but had somehow lost his papers., others,, , like, , 1, , Realising that no other member of the Royal Society could, hope to provide the required proof and also that Newton hid really, already achieved something much more than this, Hailey pleaded, with him to reproduce his papers in book form and, though not a, rich, , man, , himself, offered to bear all the cost of publication of the, , same, which ultimately resulted in the appearance of the celebrated, Principia, in the year 1687., , Newton knew that both rest and uniform motion along a, straight line were equally natural and, after a careful study of, Kepler's laws, he showed (i) that it follows from his second law that, only a central force acts on the planet and is directed towards the, Sun, it alone being responsible for keeping the planet in its orbital, path,t (//') that it can be deduced from his first and third laws that, this^force between the planet and the Sun is inversely proportional, to the square of the distance between them f, and (Hi) that it is an*, easy further deduction from the above that this force of attraction., between the two is also directly proportional to the product of their, masses., , He further proceeded to verify these deductions from Kepler's, laws by comparing the value of acceleration of the Moon towards, the Earth, calculated on their basis, with its value obtained experimentally, the two values showing a close agreement with^each other,, a$ will be seen from the following, :, , If g m be the acceleration of the, , Moon towards, , the Earth,, , v,, , its, , Royal Society, the fellowship of Mhich today is consider* d to be, a very high honour, really grew out of informal group meetings of men interested, in natural philosophy, about the year 1645, and received its Royal Charter,, in 1662 from Charles II, who, according to Samuel Papys, 'mightily laughed at, , them, , for spending time only in weighing air*., 9, , fScc 'Note on Newton's deductions from Kepler's laws on page 228.
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227, , GRAVITATION, , linear velocity in ita orbit about the Earth and R, the distance, between the centres of the Earth and the Moon, we have, , gm, where, , <o is, , =, , v*/R, , =, , (wR)*IR,, , =, , <JR,, , the angular velocity of the Moon., , =, , Since to, 27T/7 where T is the time taken, going once round the Sun, we have, , Now,, , T=, R=, , 1, , ,, , =, , 27-3 days, , 60, , ///Her, , r, , 27-3x24x60x60, , he radius of the Earth, 60x40uO/w/feJ., , =, =, , Hence, , gm, , =, , r, , _, , in, , and, , W.T.,, , rv, , 60 X 4000 X 1760 X 3/r.L, x4000x1760x3, nAQAA, , ____?r, , by the Moon, , radius of the, to 4000m?les., , ,, , ,, , ., , -00899/r./^., , Again, if the acceleration due to gravity be g on the surface of, the Earth, its value at the distance of the Moon from it would, in, accordance with deduction (it) above, be equal to g/60 2 ,, , gm, , i.e.,, , =, , /60, , 2, ., , So that, taking the value of g to be 32'2ft./sec 2, of the Earth, we have, , gm, , =, , 32-2/60*, , =, , ., , t, , on the surface, , -00084 ft. I sec*.,, , is practically the same as the one deduced above, thus fully, vindicating the deductions made by Newton, and convincing him of, the existence of a universal and mutual force of attraction between, any two masses., , which, , Not only this, but Newton also put to test his assumption that, so far as the attraction at external points is concerned, both the, Earth and the Moon behave as though their masses were concentrated, at their respective centres. He actually showed that the force of, attraction, exerted at an external point, by a uniform sphere, or by a, sphere consisting of a number of concentric uniform shells, one inside, the other, is the same as that exerted by an equal point-mass, occupying the same position as its centre. In other words, the sphere behaves, as though the whole of its mass were concentrated at its centre., in, , Thus fortified with a clear and complete confirmation of his, deductions and assumptions, Newton announced to the world, in the, year 1687, his celebrated Law of Gravitation in his monumental work,, the Principia*, which the entire scientific world later hailed, in the, words of Langrange, as 'the greatest production of the human, mind'., 'Having lost his papers, as mentioned already, Newton had to iccreate, the whole, step by step, all over again and accomplished the almost superhuman, He used geometrical methods, partly due to his, feat in only 18 months., admiration for ancient geometers and partly to avoid being baited by 'Mttte, imatterers in mathematics*. Perhaps he had Robert Hooke in mind, who- had, claimed priority in the discovery of the Inverse Square Law ; the Royal Society, had, iiowever, sided with Newton.
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PROPERTIES OF MATTER, , 228, , Note on Newton's deductions from Kepler's laws., Let A be the position of a planet, (Fig 140), at a given instant t in, Then, if tho, elliptical path round the sun S, situated at one of its foci., planet moves on to B in a small interval of time, dt, the area swept out by the radius vector SA, in, this interval of time, is equal to the area of, the triangle SA B., 83., I., , its, , equal to i, , i.e.,, , = R, , SA, , because, , =, , SA.AB, , AB, , and, , R.R, , }, , dQ,, , R d$., , area! velocity of the planet, J/?*.</6/df., But, this, according to Kepler's second, Fig. HO., law, must be a constant. Putting it equal to A/2,, h., therefore, we have R*.d$ldt, Now, the fact that the planet moves in a curved path and thus conti.'., , =, , nually changes its direction, means, in accordance with Newton's first law of, motion, that it must be under the action of a force, and must consequently be, possessing an accelerai ion in the direction of the force, Resolving thif acceleration into its two rectangular components, along and at right angles to the, radius vector, we have, (i) component a ly along the radius vector, i.e., the radial acceleration of the, planet, given by, s i, pa result obtained from, simple dynamical conLsideration., , and, , component at at ri%ht angles, acceleration of the planet, given by, (if), , ,, , 1, , --_, , at, , we have, , But,, , ., , d, , seen above that R*., , differential coefficient, , to the radius vector, i.e., the transverse, , must be equil, , elf), , ~, , a constant, , is, , -, , to zero., , It follows,, , and hence, , (h),, , therefore, that a 2, , its, , =, , 0., , In other words, the planet has no transverse acceleration, so that, the only, acceleration it has is the radial one t and, therefore, the only Jorce acting on it is, towards the Sun., ;, , l/R, , 2., , =, , Now, since 7? 4Q Idt =, R = /), we have, 2, , h,, , ., , w, (or, , it, , is, , clear that dQjdt, , dQ/dt, , I, , =, , hu, , =, , hjR, , 1, , Or, putting, , ., , ., , It follows, therefore, that, , = A, , d*, ~df, , d, , 1, , ., , because, , ?, , ., , u*, , ^, , _1, , V, , )**, , \Ju, , = Rn, , -., , l, , (, , dt, , du, ', , ^_, , ', , **L, <to, , d*, ', , -, , ,, , d'tt, , ^, , do, dt, , f,, , d'u, lf, = -""', dv, , Substituting the values of do/dt, , we have, f, , ,, , and, , ', , v, , L, , rf'/f/A' in the, , we have, , Je, </,-, , the equation of the elliptical orbit of the planet be, l + e cos 0., 1 -f e cos 0,, I] R, Or, lu, , =, , =, , / is its latus, , rectum and, , e, its eccentricity., , Differentiating this expression twice, with resp^ect to Q,, ', , T~-, , we have, , =, , h, , ^, , -, , ., , Aw, , ,, , expression for a,, , *--*, where, , ', , dt, , - -*, *, , let, , L, , '~dij, , h., , j-, , ., , d*R, , Now,, , dl, , _/, , dt, , c/0, , ^, 2, , dt, , Differentiating this again with respect to, , above,, , *, , ~u*, , ~di
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229, , GBAVITATION, And, adding, , relations, , (//), , and, , d?u, , lu, , we have, , (///),, , =, , + l.-f, , +e, , 1, , Q-e cos, , cos, , 6, , =, , 1., , whence, Substituting this value of, , fw+^a, , we have, , in relation (0, above,, , -, , - -*V//, , ai, , ), , 1, , ., , ., , for, , *., , [Putting, 2, , we have nx, // by K,, KIR ., 2, a 4 oc -I//?, ... (/v), i.e., the acceleration, and hence he force acting on the planet is inversely proportional to the square of it* distance jrom the Sun, (the, ve sign merely indicating, that the force in question is one of attraction)., , Or,, , denoting the constant A, , 2, , Or,, , ., , /, , the lime-period (T) of the planet (i.e , the time taken by, revolution round the Sun) is given by, , Now,, plete, , one, , its, , to, , it, , com-, , full, , r, , g,, , ___*-^L_, , area, , f tne e [liP se ___, ~~, areal velocity oj the radius vector, , ", , d$, , ., , **D2 *-, , where a and 6 are the semi-major and semi-minor axes of the, , elliptical orbit, , of, , the planet., , And, , r=*fn\2., , Or,, , 2, , Now,, al, , ;, , /a, , clearly,, , =, , .*., , ^. a zb, , T*, , 2, , [vi^, .^at, L, , 42, , 1, , lh\, , the latus rectum of the ellipse, and, therefore,, , /,, , ^, , so that,, , ., , *, , ^, , ?, 3, But, since, in accordance with Kepler's third law, T oc a for every, follows, it, 4x*-,K is a constant, or ihai K is a constant for every, In other words, K is quite independent of (he nature o\ a planet., be the nspcciivc masses of the planet and the, 3., Fin^llv, if AH and, Sun, and F and F', trie force of attraction, exerted by the Sun on the planet,, and the reaction of the planet on the Sun respectively, we have, from relation, (iv) above,, , Um, , planet,, planet., , M, , F-, , where k and, , F', , and, , hnlR*, , K are constants., , = KMIR\, , And, since by Newton's third law of motion, action and reaction are, we have, F F' so that,, K.M., k.m, Or, [k/M - Kim ~ a constant, say, C., k = M.G., So that,, Substituting this value of A- ia the expresssion for Fabove, we have, equal and opposite,, , ;, , mM, P, Fp-.G,, , showing that the force of attraction between the planet, proportional to the product of their ma**es., , Law, , and, , the, , Sun, , is directly, , -This law states that every, universe attracts every other pa> tide with, a force which is directly proportional to the product of their masses, and, inversely proportional to the square of the distance between them., , Newton's, , 84., , particle, , of~~fnatter, , Thus,, apart, and, , if, , m, , C?, , Obviously,, , of Gravitatiaar, , the, , and m' be the masses of two, , particles, distance r, , between them, we have, F = G.w.m'/r 1, , F, the force of attraction, , F, where, , in, , is, , oc, , m, , m'lr*., , Or,, , ,, , a universal constant, called the Gravitational Constant., 1 cm., then, F, G., 1 gm., and r, m', if m, , =, , =, , =
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PBOPERTIES OF MATTER, , 230, , Thus, the Gravitational Constant is equal to the Jorce of attraction, between two unit masses of matter, unit distance apart. Its dimensions, are A/- 1 ! 8 !*- 2 and its latest, accurate value, (as determined by Heyl, The gravitational, in 1930), is taken to be 6-669 x 10~ 8 C.G.S. units., constant is also sometimes referred to as the astronomical unit of, ,, , force., , The law is universal in the sense that it holds good, right from, huge interplanetary distances to the smallest terrestrial ones. The, minimum distance up to which it is valid is probably not yet known, with absolute certainty, but it seems to break down at molecular, 7, We shall discuss latter, in, distances, which are as small as 10~ cm., this chapter, some of the overwhelming evidence in favour of this, law, as well as the small deviations from it and the proper explanation for the-n, on the basis of the new ideas put forth by Einstein., The methods, 85., Determination of the Gravitational Constant., for the determination of the gravitational constant, (and, therefore,, also those for the determination of the density and the mass of the, , may, , earth),, , (/), , be divided into two categories, viz, , ,, , which involve the measurement of the force of attraction exerted by a large natural, mass, like a mountain or the earth's crust*, on a plumb, line suspended on one side of it, which is then compared, with the force of attraction on it due to the earth, as a, Mountains and Mine Methods., , whole., , Laboratory Methods, wh'ch involve the more delicate, measurement of attraction between small masses., We shall deal here only briefly with the former, more for their, historical interest than othorwiss for the results obtained were not,, indeed, they could not be,, very accurate. The latter, i.e., the, laboratory methods, we shall however study in proper detail., (ii), , ;, , ;, , (0 Mountain Methods., 1,, , value of G., , Bouguer was the first to have attempted a determination of the, Wnilc engaged in geographical measurements in the Andes (Peru),, in the year 1740, he suspected a, deflection of his plumb line due to, large mountain-masses. He decided to verify this, and selected a, mountain, Chtmhorazo, 20.000 //., high, (in the Andes) for the purShorn, of, experimental, pose., details, his method was the follow-, , ing, , A, , :, , He chose two stations A, and B[Fig. 141. </) and O/)], the, former due south of the summi t of, the mountain and ch*e to it, and, the latter, in the same latitude, and, at about the same altitude some, distance to its we*t, away from its, influence. At stat ion B, he observed, , r, , Fig. 141, , (/), , a star passing the meridian directly, overhead, so that the plumb line, , *The word is prob tbly a relic of the times when the earth was supposed, be a globe of water, bounded by a solid shell or crust. It. is now used, however, to signify the rigid surface layer of the earth, which is heterogeneous and,, more or less, in a state of permanent stress and strain., to
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231, , GRAVITATION, But at station A, exactly vertically parallel to the telescope., mountain-mass, (ou, that it wa* attracted by the huge, A, He measured this, its nearness)., deflection of the plumb line at A, , he, , hung, , td, , observed, , ^, , and thus compared the horizon-, , ,, , tal pull of the mountain with the, For,, vertical pull of the earth., if F and F' be the forces of gravitational attraction acting on the, plumb line due to the mountain, , and, 4,, , its, , the earth respectively, and, deflection from the vertical,, , (Fig. 141, (*')] we have, tan f, F(F'., , =, , =, , F', dearly,, nig,, the mass of the plumb, the acceleration due to, , Now,, where m, iine and, , is, ,, , So, , gravity., , that,, , F, And,, distance of, , if, , V be the volume and, , its c g., , from the plumb, , p,, , the density of the mountain,, , line,, , V, .G., , Hence, , m.K.p.G/r, , 2, , = mg, , tan $., , and, , r,, , the, , we have, , (Or,, , G, , mass of the mountain, , -, , K.p., , g.r* tan, , Thus, a knowledge of the volume, density and shape of the mountain,, its cen're of gravity, wa* needed to determine the value of G. Bouguer,, for, due to the most, therefore, proceeded 10 do so, but did not quite succeed, adverse conditions of snow and storm under which he had to work, he could not, much, properly survey the mountain, anJ ihe results hs obtained were very, wide of the mark. Thus, for example, he found that his plumb line was drawn, aside by about 8*, and his calculation showed that if the mountain were as, dense as the earth itself, the deflection of the plumb line would have been, twelve times as gr*-at, indicating that the earth was about twelve times at, dense as the mountain. And this, as we know, is very much beyond the truth*, Nevertheless, he had the satisfaction of showing that the attraction due to the, mountain masses did actually exist and thai the method was, therefore, possible., Not only that, but he also deserves the fullest credit for proving conclusively, that the earth was not just a globe of water or a hollow shell, as was fairly widely, supposed at the time., , and hence, , ;, , 2., Maskeiyne, later in the year 1774* repeated, at the request of the, Royal Society, Bousuer's experiment on the mountain Schiehallion, in Perthshire (Scotland). 3547 feet hi eh, an elaborate survey of \vhich *as first made to, determine as accurately as possible, its volume and density (and hence iti mass), and centre of gravity., , Two stations were then chosen at fqual distances from the c g. of the, mountain, on the north-south line (Fig. 142), and the tarn* star was observed,, of telescope, called the, (as in Bou^uer's experiment), by means of a special type, Zenith Sector^, first at the Sduth Station acid, a rmmh later, at the North, StaHon At the former Station, the star which, in the absence of the mountain,, would be directly overlmd, appeared to shift slightly to the north, because the, the zeniih, plumb lins was pulled by the mountain towards the north, (and, *He was, , the Astronomer Royal at the time., , fThe instrument could rotate about a horizontal (Fast ard West) uxfa, , at its object-glass end. pointing upwards, and was provided with a pit rob, so that the, line, suspended from this axis, over a scale, graduated in deuces,, read on it*, be, v, could, crtical, the, from, directly, of, the, distance, telescope, angular
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PROPERTIES OF MATTER, , 232, , thus shifted to the south). At the other Station, on the other hand, the exact, opposite was the case, (the plumb lioe being pulled towards the south, the zenith, thus shifting to the north) and ihe, star, therefore, appearing to shift, eqally to the south- Thus, the, total shift of the star was double, of the deflection of the plumb line, at either station due to its attraction by the mountain. This wa, carefully measured and was found, to be 55". Out of this, a shift of, 43* was calculated lo be due to the, curvature of the earth's surface, so tnat the net shin or deflection, , of the plumb hne, due to the gravitational pull of the mountain, \\as, (55"-43") = 12*. In other words,, the plumb line* at each of the two, stations, was deflected by 6" due to, the mountain-mass The valo-e of, was then calculated, as explained, , G, , above <in Bouguer's experiment),, and was found to be 7*4xlO~, C.G.S units., Further, it was estimated, that if the mountain had the same, density as that ol the earth, the, deflection of the plumb line, due, to its attraction, wouki have been, 9/5 times the observed deflection,, Fig. 142., showing that the earth was 9/5 time*, denser than it. And, since the density of the mountain, determined from, pieces of rocks composing it, was fouiit to be 2 5 gms jc c., the density of the, This was corrected and increased to, earth came to be 9x2'5/5 or 4 5 gms. Ice., 5-0 gms.lc c. after a careful re-survey of the moumain, some thirty years later,, i result nut very much wide of the mark., Since, however, it is almost impossible to determine correctly the mas*, position of the e.g. of a huge natural iruss like a mountain, the value of G,, >b tamed by the above methods, is far from reliable, and is at best only ao, approximation., In these, the time-periods of a pendulum, (say, a, (//) Mine Methods., seconds pendulum), arc compared on the surtace of the earth and at the bottom, >f a mine., It is obviously greater in the latter case, the value of g being less, The change iti its time,here than on tne surface of the earth, {see page 206]., >eriod enables a comparison to be made between the values ol acceleration due, o gravity, and hence between the density of the layers immediately above the, wndulum-bob and that of the rest of the earth, which, in its turn, Teads to a, letermination of Af, the mass of the earth and thus to a calculation of the value, >f G, as will be clear from the following :, , md, , We know, , that the weight of a body at a place is the force with which it, by the earth towards its centre, and is numerically equal to the proiuct of its mass (m) and the acceleration due to gravity at the place., Thus, if g be the acceleration due to gravity on the surface of the earth, nd g', that at the bottom of a mine of dcptn h, we have its weights at the two, laces given respectively by, ', +, ,, w(A/, ,, ~, .G, and, mg, 5, , attracted, , i, , mg~, , M, , is the mass of the earth,, of the earth, of thickness //., , 'here, tie 11, i>, , mM, , ,, , t>that,, , M, , its radius,, , and m' the mass of the outer, , -, , *--Br.0, g', , hence,, , R, , T~, , ,, , and, , M-m', , ST', , t, , g', , f_R_, , (Af-m'>, , -, , V, , \R-itS, , -,, , ="' G>, , '
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GBATITATION, And,, , clearly, if p, , be the density of the outer shell of the earth,, , 233, its, , mass m', , i, , obviously given by the relation,, , m', , = tolP-GR-Wp., , ...(//), , the average value of p was obtained by determining the densities, of the samples of rock at different levels, down to the bottom of the mine, and, thus the mass (m') of the outer shell evaluated. Substituting this value of m' i, relation (/) above, the mass (A/) of the earth was easily determined. And, then,, in the expression for g above, the value of G could be, putting the value of, calculated out straightaway., Airy was the first person to have made a successful attempt of this nature, in the Harton coal pit in Sunderland, in the year 1854, two earlier attempts*, made by him, in a Cornish copper-mine, as early as 1826 and 1828, having come, to naught, due to unfortunate accidents in the mine. Airy's value of G came to, 5'7xlO~ 8 C.G S. units, and that of the density of the earth's surlace, to 6*5, , Now,, , M, , gms./c.c., , Like the earlier Mountain experiments, these experiments by Airy too, gave far from satisfactory results, due mainly to the difficulty of determining;, accurately the density of the outer shell of the earth. His methods, however r, wth improved modifications, now find a wide and useful scope in the branch?, of Geophysics [ 80, (page 216)]., 9, , Laboratory Methods. In these methods, the attraction, between the masses is inevitably feebler clue to their small ness. But, this is more than compensated for by the high degree of accuracy, with which the masses and their sizes can be determined. The first, successful attempt at an accurate method of this type, for the deter*, mination of the Gravitational Constant (G) was made bv Cavendish r, in the year 1798, in which he made use of the Torsion Balance., (///), , It will be of, , some, , interest to recall that Cavendish, , was prob-, , ably also associated with Maskelyne in his Mountain experiment,, performed some twenty five years earl it r. He, however, took his^, cue from Rev. John Michell, who had devised an apparatus almostsimilar to Cavendish's own, but was not destined to use it, due to*, his sudden death., His apparatus fell into the hands of Prof., Wollaston,, , who passed, , it, , on to Cavendish., , The apparatus ussd by Cavendish,, (a) Cavendish's Method., and installed in an outhouse in his garden on Clapham Common, was, as shown in Fig. 143., A long cross bar PQ, about 6 ft. (or about, 180 cms.) long, was suspended from the ceiling of a room and was, free to turn about a vortical axis by means of an arrangement, manipulated from outside. It carried two large and equal lead spheres C, and >, about 8 to 10 inches (or 20 to 25 CMS.) in diameter and weighing about 350 Ibs. each, at the ends of two metal rods attached to, its two ends., Immediately below the mid-point of the cross-bar was a torsion, which could also be worked from outside, from which was, head, suspended a deal-rod RS. (slightly bigger than PQ), by means of a, Two wires (w. vv), fastened, fine torsion wire W, of silvered copper., the ends of the deal-rod to a vertical rod r in the middle, which was, , M, , This increased the strength of the, Two small lead balls, 2, rod, without increasing its moment of inertia., inches (or 5 cms ) in diameter and weighing about a pound and a, , attached to the suspension wire., , half (or 680 gms.) each, were suspended from the two ends of th&, deal-rod RS such that the centres of the four balls lay in the same, horizontal plane, roughly in a horizontal circle of about 3 ft. (or 90
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PROPERTIES OF MATTER, , 534, , cms.) radius. The arrangement was such that, when the line joining, the centres of the large lead spheres was at right angles to the torsion rod, there w&* no twist or torsion in the suspension wire, Each erd of the torsion rod carried a vernier, (of five divisions),, which moved over a fine ivory scale, fixed to vertical stands, and with, each division equal to -05"., , W, , *, , To guard against any changes of temperature, and consequent, air-draughts, which would otherwise mask the gravitational effect,, the room was closed and observations were taken with the help of, telescopes T and T, fixed into the walls of the room, as shown. And,, Further, to avoid the effect of any outside electric charges, the whole, apparatus was enclosed in a gilded glass case, supported on four, levelling screws., The method of procedure was the following, The rod PQ was rotated until the line joining the centres of the, :, , arge spheres was at, , riidit, , angles to the torsion rod,, , in, , i.e.,, , the post', , Fig. 143., , was no, , and the reading on the verniers, attached to the torsion rod at either end, taken., The, large, spheres, were then rotated union in w}iich there, , twist in the suspension wire,, , they lay on oppo, of the torsion rod and near to, , til, , site sides, , the, , small, , either end,, , ** -, , &, , at, , in the, , C and, , D, as, , Fig., , 144,, , positions, in, , shown, , balls, i.e.,, , that thQ lines, joining the centres of, F~~IIJ", g I44<, of near, each, pair, balls were equal in length and perpendicular to the torsion rod. Obviously, then, the forces exerted by the big spheres on the corresponding near small balls were equal and opposite, thus constituting a, This was resisted by the, couple, tend ing to rotate the torsion rod., torsional couple set up in the suspension wire, and equilibrium was, attained when the deflecting couple, due to the forces between the two, *, , such
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GEAVITATION, , 23f, , pain of balls, was just balanced by the restoring torsional couple, set, up in the suspension wire. The position of the verniers was again, noted on the scales by the method of oscillation, as in the case of an, In Cavendish's own experiment, this, ordinary physical balance, distance between each pair of balls was 8" (or 20 cms.), and the, small balls were displace 1 through '7681" (or 1*915 cms.)., The rod, rotated, about, its, vertical, the, was, then, until, axis,, again, large, PQ, r, spheres now occupied the positions C and D' respectively, and the, same adjustment was mad a as before?, viz., that the lines joining the, centres of the two near balls were of the same equal lengths as before, and perpendicular to the torsion rod. The positions of the verniers, were r*?ad on the scales, as before, and their mean taken as the deflection of the torsion rod., , The value of G was then calculated as follows, be the mass of each large sphere, m, that of each small, an' I d tho distance batween each pair of near balls., Then, the, :, , Let, , ball, , M, , between each pair of balls is, clearly, equal to, and, therefore, if/ be the length of the torsion rod, the, deflecting couple formed by this pair of equal, opposite and parallel, force of attraction, , G.Kf.m/d 1, , ;, , ., r, i, ,, forces, is equal to, , M.m, ,, , ., , -G.l., , 2, , [/ IcosQt&l.], , ba tho twist (in radians) in the susp3nsion wire, and, Any], if, C, tho Factoring couple p^r unit (raJia.i) t\yist set up in it, the restorSince the torsion rod is in, ing co'iplo (Itu to torsion is eqpiil to C.Q., equilibrium under these two couples, we have, -, , /, , d2, , - .0./ == C.0., , =, , whence,', , ./M.m, , ,, , .0., , I, , In fwl T to determine the value of, , (7, the torsion rod alone was, vibrations abo'it the suspsnsion wire, and its timeCavendish found it to be 28 minutes in his, period was measured., apparatus., Then, if / b3 tli3 munint of inertia of the torsion rod (together, with the small balls, about tho wire as axis, and /, the time-period of, the rod, we have, , set into torsional, , t, , =, , 27r, , A/, , Substituting this value of, , C, , p, , ,, , in the relation for, , ~~, , "M, Or,, , if, , we ignore the mass of the, /, , So that, substituting, , C =, , whence,, , =, , this valua, , m.7.r 2, , 2/w(//2), , of/, , ., , above, we have, , '', , torsion rod,, 2, , G, , -, , J2, , =, , in the, , we can put, iw/a/2., , above expression, we have, , Corrections and Sources of Error in the Experiment. Corrections were applied for the following, I, (, ) force of attraction between each large sphere and the distant, :, , imall ball, , ;
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PROPERTIES OF MATTER, , 236, , force ofafyrqption between the two large spheres and the tor-, , (2), , sion rod, , ;, , and, , (3) forces, , exerted by the rods carrying the large, , balls., , Tho following are the sources of error in the experiment, (1) The gravitational forces between a pair of balls being small,, the torsion rod had to be made long to increase the deflecting couple., :, , This also minimised the force of attraction between a large sphere, ball, but required a large chamber, and thus, convection currents could not be avoided., In addition to this, the, torsion rod was also disturbed by heavy traffic on a nearby road., (2) For a given deflecting couple, the deflection of the torsion rod, was small, because the suspension wire required a JJarge torque per, , and the distant small, , unit deflection., (3) The torsion wire, being not perfectly elastic, did not return, to its normal position when the applied forces were removed, and, thus the torque was not strictly proportional to the angle of deflection., (4) The distant large spheres decreased the angle of deflection, while the rods carrying them increased it., (5) The method of measuring the angle of deflection was not sen-, , five, , enough., , time- period of the torsion rod, with the attached small, big, i.e., its swings wcro much too sluggish and, impaired, rather than improved, the accuracy of measurement., The value of G obtained by Cavendish, as the> in9an of twenty, nine observations, was 6*754 x lir 8 C.G.S. units *, Many other attempts wera made since Cavendish's time, by, Jolly and Pjynting, amon^ others, to obtain the volue of G moreaccurate ly but the method adopted by Boys, with his newly invented, quartz fibre, used as suspension for the torsion rod, was by far the*, We shall, therefore, discuss that first., best of these., method. Sir Charles Veroon Buys removed, almost a, Boys*, (b), century later, in the year 1895, all the defects of Cavendish's experi(, , rolls, , (6), , 1'he, , was much too, , ment by, (/), , reducing greatly the size of the chamber, thus considerably, minimising convection currents and making it easier to control its temperature, arranging the pairs of balls at different levels, thus making, the attraction between the distant large and small balls, almost negligible, using, for the suspension wire, a quartz-fibre, which required, a comparatively very small torque per unit deflection and, which being almost perfectly elastic, besides being fine and, ;, , (11), , ;, , (ill), , ;, , (iv), , strongf. the angular deflection produced was appreciably, large and also proportional to the torque and, (v) measuring the angular deflection by the telescope and scale, method, which greatly enhanced the accuracy of measure;, , ment., *A musingly enough, Cavendish made a, It, , was, , later pointed out by Baity., fA quartz-fibre is found to, , wane dimensions., li '0125 mm., , slip in calculating this, , mean and, , be much stronger than a steel wire of the, fibres having a diameter as small, , Boys was thus able to use
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GRAVITATION, , 237, , Thus, Boys greatly reduced the size of the apparatus and yet increased its sensitiveness. This ra%y, at first sight, appear to be a, contradiction in terms for, it is commonly b3lieved thit the larger, a piece of apparatus, the greater the degree of accuracy obtained, from it. Boys clearly showed, however, that this was not so, that, the sensitivity of Cavendish's apparatus was quite indep3ndent of its, dimensions and that there was no point, therefore, in attempting a, He argued as follows, larger version of it., ;, , :, , ,,,,, , The, , ., , ', , r, , ., , K/f *, , m, , d*, , ', , deflecting couple, , that,, , I, , ., , =7Restoring couple, (mass, , i.e.'*t, , ', , 2, , (radius)* 13, , oc, , r8 ., , T5, , oc, , i->, , 9, , t, , -, , >, , ', , oc, , oc, restoring, * couple, ^, , Deflecting couvie, , x length of torsion rod, distance he t ween near balls ,, , I, , mass oc volume oc 4*, , Now,, , So, , of the apparatus, , sensitivisy, , Clearly, deflecting couple oc, , ,, , i, , moment of inertia, , *, r - -y, (time of swing?, Now, there is a practical limit to the time of swing which should not, exceed 5 minutes, whatever the size of the apparatus, or else the swings become, very sluggish, thus impairing the accuracy of measurement. This being so, we, , And,, , -, , HT, , -, , have, ., , restoring couple oc, , Or,, , restoring couple oc, , L5, , ., , MK, , 3v, , 2, , ---, , P, , Jl., , <, , And, , .*., , sensitivity, , being, , the, , same, , Lfor any apparatus., , 1, , L6, , In other words, the seniitivity is independent of the size, , (L) of the, apparatus., Thus, if, for example, we double the dimensions of Cavend'sh's apparais increased by (2)',, <O> keeping the time of swing (/) the same, we find that, because it is proportional to (radius)*, d is increased by (2) 8 and / by 2., The net result is that the valuo of d ^MV 2 .G/2*r 2d 2 / remains the same, i.e.,, no advantage is derived by doubling the size of the apparatus, i.e., by increasing, dimensions of all its parts in the same ratio., , M, , sthg, , What Boys, therefore did was to reduce the, dimensions of the different parts of the apparatus, be clear from the, as will, in, different ratios,, following, , :, , A small, , mirror strip S (Fig. 145), about 2-5, cms. long which acts as the torsion bar, is suspended by means of a fine quartz-fibre, from a torsion, head T inside a glass tube, about 4 cms. in diaFrom the two ends of the strip are susmeter., similar quartz-fibres, two small gold*, by, pended,, balls A and B, about 0'5 cm. in diameter and 2-65, fins., in mass each, one ball being about 15 cms., In an outer coaxial, ibove the level of the other., be, the common axis,, rotated, about, which, can, iube,, each about 11*0 cms., jwo large lead balls C and, n diameter, (and about 74 kfgm. in mass), are, t, , D, , Fig. 145., , *On account of the higher density of gold (19 3 gms /c.c.) compared with, hat of lead (l\*3 gms lc.c.) the spheres of gold for the same mass are smaller, ban spheres of lead and thus enable the distance d between the centres of the, arge and small balls to be reduced., t
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PROPERTIES OF MATTER, , 238, , suspended such that the centre of C is in level with that of A, and, that of D, in level with that of B, (to .ensure greater precision in the, measurement of the distance between each pair), the distances between, the centres of the pairs A and C, and B and D, being exactly equal., The deflection is measured by the telescope and scale method, a halfmillimetre scale being placed at a distance of about seven metres, from the strip., , The experiment is performed by rotating the outer cylindrical, tube until the large (lead) balls he on the opposite sides of the twogold balls (but not in a line with the mirror strip*), so as to exert the, maximum moment on the suspended system, i.e., when the angle of, The tube is then rotated, so that the lead*, deflection is the largest.f, balls now lie on the other sides of the gold balls in a similar position,, again exerting the maximum moment, or producing the greatest, The mean of the two is then taken. Let it be Q., deflection., The calculation for the value of G arj then made as follows, Let A, B, C and D, (Fig. 146), be the four balls, when they are, To visualise, in equilibrium, in the position of maximum deflection 0:, , the balls in these positions, we must remember that to start with, the centres of all the, four balls lie in the same vertical plane, there, being no twist in the suspension fibre and, hence no couple acting on the suspended system, If we now rotate the larger balls C and Z>, through a certain angle, the plane containing, their centres will also rotate through the same, with the result that a gravitational, angle, couple now comes into play on the suspended, system, tending to rotate it into a position of, equilibrium in which, once again, the centres, of the small balls come to lie in the same, vertical plane with the centres of the large, balls., this being equally true when the sus;, , Fig. 146., , pended system suffers its maximum deflection., A and B are shown in their initial posito, and the large balls in their final posi0=0, tions, corresponding, tions when they have been rotated into a position BO as to exert the, maximum couple on the suspended system, tending to make the, Here, the small balls, , latter suffer its, , maximum, , deflection 0., , Obviously, equilibrium will, , be attained again only when the centres of the small balls come to, , To bring, the vertical plane of the centres of the large balls., the small balls back to their original positions, (shown in the Figure),, therefore, the torsion head will have to be rotated in a direction, opposite to that in which they have been deflected by this couple., In other words, the deflecting gravitational couple exerted by the larg<, lie in, , ,, , *For, in this position, the gravitational forces due to large balls on the, tmall balls near to them will act in opposite directions along the same straight, line and will thus neutralise each other., is chosen because when the, couple on the, the rate of variation of the couple is small, balls need not be known with any greal, , tThis position of the lead balls, , impended system, and the relative, accuracy., , is, , the, , maximum,, , positions of the
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GBAVITATION, , 239, , on the suspended system, in the position shown, is just balanced, by the restoring torsiona! couple set up in the quartz suspension., Now, let O be the mid-point of the mirror strip, and let / be its, , balls, , half-length, , and, , from, , O, , let, , (i.e.,, , on to, , Then,, , DB, , Let OC=OD=b, AC^BD^a, be the perpendicular drawn, , OA^OB=1)., , AOC~BOD~a., , let, , OE, , Let, , produced., , OBD, we have, , clearly, in the triangle, , BD = \/O~D*+OB, , 2, , ^OD.Utf cos, , a., , [See, , Appendix, , 1, 7, (11);, , rf= ^b*+l*- 2blcosa ~= (b*+l*-2bl cosa)}., k, , Or,, , sin a, -- -=r*, n, sin, , Also,, , EDO, , BD, = OB, 7; D =, , Now,, Or, , d, , in a triangle, the sides are proportional, to the sines of the angles opposite to them*-, , p.*, , /, , L, , /, , [See, , ^-, , OD.sin, , (/), , OED, we have, , EDO, , OE =, , ', , 1, 7, (/;.], , ......, , -, , in the right-angled triangle, , OE =, , Appendix, , a, , BDO =, , sin, , and, therefore,, , ", , =b.sin, b '' S *, , BDO., [From, , d, , (/), , above,, , Obviously, the attraction between the t\\o balls of each of the pairs,, 2, and, is equal to, M.m.G/d , where, A, C and B,, m.GjBD*, are the masses of each large ball and small ball respectively., , M, , D, , M, , =, , m, , These two forces, being equal, opposite and parallel, constitute a, and, quite clearly, the, couple, tending to rotate the mirror strip, ;, , moment of, action, , ^, , ,, , the couple, , EF =, , where, , =, , 2.OE, , ., , ., , Or, the deflecting couple, , _r, , ', , EF = G, , -, , 2.OE,, , ,,, , the perpendicular distance between the lines, , is, , of the two forces., a, ., , '-, , G-, , =, , M.m, , G .--,-., , 2M.m.b.I, , ^, , sin, , b.l sin, , z., , a __, , a, , a, , 2M.m, , r, , Substituting, va j ue, L above., b.l sin a, , -------, , of, ther, , fO, fron?-, , Now, this deflection of the mirror strip is resisted by the torsion, or twist, set up in the suspension-fibre, and the mirror comes to rest,, when the deflecting couple due to the attraction between the twopairs of the balls is just balanced by the restoring torsional coupleIf C be the torsional couple per unit, set up in the suspension fibre., deflection, set up in the suspension-fibre*, the restoring torsiona), couple is equal to C. 6. and, therefore,, b I sm a, 2G., n, , Mm, , w, , u=, , ', , ~*, , w, , be easily calculated out., determined, as in Cavendish's experiment, by oscillating the moving system, in the absence of the lead balls, and noting its time-period., t The value of a is clearly the angle through which the torsion head T, must be rotated to bring the small balls back into their original-positions and canbe easily read on a circular scale attached to it. Since a quartz-fibre is nearly, may be taken to be the same as that of 6., perfectly elastic, the value of, , whence the value of, , ', , 2M.mbJ*ina, , *, , ifhis^js, , Gjmy
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240, , PROPERTIES OF MATTER, , The value of, C.G.S. units., , G, , obtained by Boys was equal to 6-6576, , Alternative Calculation. The value of G, out as follows, using the same symbols as above., , may, , 10~ 8, , alternatively be calculated, , Gravitational pull between each pair of near lar^e, , and small, , the directions, , in, , X, , A, , balls, , C, , to, , =, , and, , -^, B, , *G 9, , to, , D, , respectively, (Fig. 147)., Resolving these, into their two rectangular components, , each,, , and perpendicular to AB, latter components of each, , a'ong, , we have, , t, , the, , equal to, , F^ G, , ^, , cos, , ^^ G Mmp, d*~', , DL ~ p are the perpendicuwhere CK, lars drawn from Cand D respectively on AB, , Fig. 147., , .Now,, , So, , AC*, , BD*, , =, , d2, , =, , 2, /?, , -fa;, , 2, ,, , produced., =^ b sin a and x, , (OK- OA), , where p, , (b cos, , that,, , =, , 6*, , af^cs5, , 1, , t, , Or,, , [/, , And,, , a/)., , *-/)., a-h/ ~26/ cos*., 2, , siri*, , therefore,, , F, , Thus,, , Hence, the, , M.m.b, , G, , deflecting couple on, , =, Now,, twist of, , **, , if, , F.AB, , C be, , =, , AB dse, , to these forces, , =, , G, , M.m.b, , C.Q., , two couples must balance each other, we have, , sin a.2/, 3, , ., , Or,, , M.m.b,21, , r, 1^, , sin a, , the same as expression I in the cas^ above,, obtained., , B., , t, , :, , the torsional couple set up in the suspension fibre per unit, of the suspended system), the total restoring, , Since, for equilibrium, the, , N, , F and F, , M^jA 5iw__a 2/_-, , G, , F.21, , (/e., for unit deflection, storsional couple for a twist a in it, it,, , sin a, , whence the value of, , G, , can be easily, , In case the centres of the neir large and small balls do not lie in the, plane, but a verticil distance h apart, as shown in Fig. 148, then,, , same horizontal, , we have, , if d' be the actual distance between them,, , gravitational couple on the suspended system, , But, clearly, d'*, , *, , (</, , 2, , 4-/r),, , where, , the perpendicular) distance between the, -their centres., , So, , -', , ^, , c/is the horizontal (or, , balls,, , or rather between, , that,, , gravitational couple on the suspended sys'em, , And,, , ', , ==, , therefore, proceeding as above,, , =, , we have, , -2bl cosx+h*)?, M.m.b.U sin a, , -.(ID
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241, , GRAVITATION, , when the centres of the near large and small, same horizontal plane, is reduced to relation (I), above., These methods do not compare in accu(iii) Balance Methods., with, Cavendish's, Boys' or Heyl's methods, but are given here, racy, only for their historical importance., a, (a) Jolly's Method. As early as 1881, Von Jolly had suggested that, , an expression which, when h, , ;, , balls lie in the, , common balance could be used to measure directly the gravitational force, of attraction exerted on a mass, placed in one pan, by a large lead mass, placed, immediately below, , it., , actually performed this experiment in Munich, where he had a common, balance fixed at the top of a tower, 21 metres high, and suspended two long wires, from its two scale pans, carrying two other pans at their lower ends., , He, , Two equal masses were then placed in the two upper scale pans and, One of the masses was then moved down into the, balanced against each other., lower scale pan, on the same side, so that, being now comparatively (about 20, metres) nearer to the centre of the earth than the other mass, its weight increased, a little,, this increase (due to the earth's attraction) being equal to the extra, weights needed in the other scale pan to balance the beam., , A large lead sphere (of known mass) was then placed immediately below the, lower pan carrying the mass, so that due to the additional attraction of it by the, lead sphere, its weight again increased a little. This increase was also determined,, as before, by putting some more weights in the other scale pan. The attraction of, the mass by the lead sphere could thus be compared with its attraction by the, earth., And, since the distance between the centres of gravity of the lead sphere, and the earth was known, the masses of the two could also be compared. Then,, the mass of the lead sphere being known, the mass of earth could be easily calculated out., And, once the mass of the earth was obtained, the value of G could be, deduced as in 85 (*/), page 232., g, , (h), , Poyn ting's Method., , The balance method has perhaps been used, , to the, , best advantage by Prof. J.H. Poynting, whose arrangement was much more elaborate and susceptible of a much higher degree of accuracy. He performed his, experiment in the year 1891, in the basement of the University of Birmingham., , The apparatus used by him, (shown diagrammatically in Fig. 149), conand sensitive bullion type of balance, with a gun-metal beam,, , sisted of a strong, , provided, , with, , steel, , and, , planes., , knife-edges, , The whole apparatus was, fully enclosed in the room, and, , all, , necessary manipu-, , lations were, the outside., , made from, , Two equal spheriA and #, of an, , cal balls,, , alloy, , mony,, , of lead and antiweighing about, , 50 Ibsi each, were susfrom the two, pended, ends of the beam. A, large, , sphere, , S,, , of, , the, , same alloy and weighing, was, Ibs., 350, about, below, on a, arranged, turn-table, which could, its, about, turned, be, , /"", fcea===, , -., , -, , AQ, g. 149., vertical pivot P, so that, or /?, as desired., ball, the, under, the sphere could be brought to lie immediately, To guard against the tilting of the turn-table due to the weight of the sphere S, a, smaller sphere 5', of half the mass of S, was placed on the other side of the pivot, of, at double the distance of S from it, so that, in accordance with the principle, ', , A, , moments, the turn-table was kept, , in equilibrium.
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PROPERTIES OF MATTER, , 242, , To start with, the sphere S was brought to He under the ball A, so that A, was attracted downwards with a force equal to G.M.to/r*, where AT and m ar the, respective masses of 5 and A, and r, the distance between their centres, (which was, about one foot)., The turn-table was next rotated about its pivot until the sphere 5 came, from under A to under B, and the balancing sphere 5' moved on to the other side,, (into the dotted positions shown), so that 5 now exerted a pull on B instead of on, A, resulting in the beam being tilted in the opposite direction to that in the first, now obviously twice* that due to S on A or B. Let, beam be 0., Then, if a be the length of each arm of the balance, (i.e., if 2a be the length, of the beam), we have, , case, the angle of tilt being, this deflection or tilt of the, , change in torque or couple due to the shifting of, (under A) to the second position (under B), , S from, , the, , first, , C, , position, , And, if be the torque or couple required per unit deflection of the, the torque for deflection 9 of the beam is-also equal to C$., , G, , Hence, , m, , .2a*=C.O So, , Thus, knowing, , M, m,, , a, r,, , G, , that,, , C, , and, , beam,, '", , ^/wIL*, , 0,, , value of, , the, , G, , ^, , could be easily, , calculated., , To determine the value of C, a centigram rider was, of the balance and the deflection a of the beam, for a shift, arm, was noted. Then, clearly,, Ca, whence,, , 01, , Substituting this value of, , C in, , relation, , r*j9, , M, , (/), , /, , ems',, , of, , it, , the, , arm, , along the, , C, , above,, , we have, , _-OUr, , ;OU./, , "", , *, , moved along, , 2, , ./, , 2Mm, , ', , ', , a a, a, 2a, whence, G can be easily evaluated., The effect of 5 and S" on the beam was eliminated by repeating the experithe dotted positions shown, and proper, ment with A and B, a foot higher up, corrections were also applied for the cross effect of 5 and S' on A and B., .tn, , m, , Both the angles, 0, and a, being very small, (0 being only about one second), were measured by Kelvin's double suspension mirror method, as illustrated in, is a small mirror, suspended, Fig. 150, where, B, by means of a bi filar suspension (w and >v'), , M, , B, , from two horizontal brackets, , 1, , B and, , B', in level, , with each other, with a small adjustable gap, between them, the former being a movable, one, attached to the pointer (p) of the balance,, and the latter, a fixed one. Thus, when the, beam turned, the wire w also turned with it,, turn about the stationary, makingf the mirror, wire w, This, with the gap BB' suitably adjusted, magnified the deflection of the beam about, 150 times. The scale, graduated in half-millimetres, was arranged about 5 metres in front of, the mirror, and its image in the latter viewed, from the room above by means of a vertical, , M, , ., , telescope, fixed, , up, , in the ceiling., , of the air draughts or currents, was eliminated by using what arc called damping vanes V, suspended from the mirror and, immersed in oil., kept, *""", Poynting obtained the value of G to be, , The, , effect, , 6-6984 x 10~ 8 C.G.S. units., , bSni firsfcbnies, and then gets, tion pf S on #., first tilt, , tilted equally in tfce, , to its'original position from its, direction, due tp the attrac-, , opposite
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243, , GRAVITATION, , The value of G obtained by P.R. Heyl, in, (c) Heyl's Method., year 1930, is taken to be the most accurate one so far. His, is a modification of Braurfs Torsion Balance, experiment,, which, in its turn, was a revised version of Boys' earlier experiment,, referred to above, (page 236)., Heyl performed his experiment in a constant temperature, with the pressure inside reduced to about 2 mms. of, mercury column, in order to minimise, convection currents. The attracting large, masses, used by him, were massive steel, cylinders, each of mass about 66 3 k.gms., suspended from a system, free to rotate, about a vertical axis midway between the, two., The smaller masses, each weighing, 2-44 gms., were balls of gold, platinum, and optical glass, in three different sets, of experiments respectively, and were, suspended from the two ends of a light, aluminium torsion rod R, 28*6 cms. long,, , enclosure*,, , (Fig., , 151),, , supported by a tungsten, (1 metre long and 0-25 mm., , thread, T.W.,, in, , diameter), and two, , inclined, , copper, , wires, (u\ vr), so that almost the whole of, the moment of inertia remains in the balls, He chose as suspension a, themselves., tungsten thread in preference to a quartz-, , t, , Fig. 151., , because the latter is sometimes found to break quite unexpectedly and for no apparent reason, , fibre,, , ., , This suspension system (of the torsion rod and the two small, masses) was made to oscillate in the gravitational field of the two, large masses, which, were arranged once, with the centres of, ^-^, ^^, masses, O, all the four, O, yd, j, , o, ., , (_), , v, , :, , lying along the, horizontal line,, , -d-, , (CL) NEAR POSITION, , (b) DISTANT POSITION, , same, and, , then, with the horizontal line, joining, , their centres,, along, the right bisector of, the torsion rod, the two positions being referred to as the 'near' and, the 'distant' positions respectively, [Figs. 152 (a) and (b)] the gravitational attraction accelerating the oscillations in the first case and, Fig. 152., , Or, as a variation of this, the timeretarding them, in the second., period of the suspension system was first determined, with no other, masses in its neighbourhood and then with the large masses brought, in the near position, shown in Fig. 152 (a)., , To set the system oscillating, bottles of mercury were brought, near the smaller masses for a short while and then removed, when, for, *It was, in fact, the constant-temperature, of Standards, 35 />, below ground level., , room of, , the American Bureau
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244, , OF MATTER, , an angular displacement of, about 20 hours., , the system continued oscillating for, , 4*,, , The usual telescope, lamp and scale method, was employed to, observe the oscillations, and the passages of the lines on the image of, a scale across a vertical cross wire of the telescope were recorded, automatically by a pen on a chronograph, another pen marking down, on it the 'second' signals from a standard clock., , From, , the time-periods of the suspension system, in the two, was then calculated out as indicated in brief, , cases, the value of, , outline below, Let Tl be th time-period of the suspension system, when the large, masses are not yet brought in its neighbourhood. Then, if G be the torsional, couple per unit twist of the suspension wire and /, the moment of inertia of the, system, we have, 7\ =2n\///C., The large masses are then brought into the near position, shown in Fig., 152 (a), such that the distance between the centres of the neighbouring large and, small balls is the same on either side, say, equal to d. This will obviously result, in a gravitational pull Fby each large mass over the corresponding small one,, :, , towards, , whepc, , itself,, , w, , F=, , given by, , G,, , M and m are ths values of each large and small mass respectively., , Considering the gravitational pull between the neighbouring large and, small masses to remain unaltered by any small displacements of the small, ., spheres from their initial or equili", brium positions, the gravitational, ,, pull of each large mass over the, n, ^, small one, when the small masses, C<cc, rf&, *, -***, A and B are deflected a little, through an angle 0, into the positions A' and B' (shown in Pig. 153), will also be equal to, 1, , ',, , M.m, , p=, towards, the, , Fig. 153., , G, , along the line joining, of the two masses., , itself,, , centres, , And, clearly, resolving this gravitational pull on both sides into two, perpendicular to A'B', we have the compo-, , rectangular components, along and, nents perpendicular to A'B' (the line joining the centres of the two small, masses) equal to Fcos a, represented by A' C and B'E respectively, where, , LOA'C, , =, , LO'B'E, , =, , a., , =, , ^, , we have, , F cos, , = LOA'D, , And, therefore,, , LO'B'J - (90-a)., LA'OP = LB'O'P = y, we have, ,., F coi a = M.m.G, r-.sin (0-fy)., , Or,, , F cos, , So, , that,, , where, , Now,, , since, , cos a, , '!?, , a., , d1, , ., , d*, , ., , *sin $,, , (0+y)., , ^, , ,, , M.m.G, ""', , (I), , d*, , and y be small, as they are in actual practice., A' A = B'B = Jy = r$, whence, y, clearly,, this, value, of, Substituting, y in relation I above, we have, , If d, , Now,, , F cos a, , 's2, d, , ', , -f-, , (, , \, 1, , -j~, , d /), , *=, , ---ji', , a*, , r$Id*, , ^, , These two forces acting at A and B' obviously form a couple, tending, to bring the small balls back into their original positions A and B ; and, clearly,
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245, , GRAVITATION, moment of, , M'' G, , -, , this couple, , .(, , 1, , +, , \0 x 2r., , -, , A 'B', , -, , 2r., , [where, , C the torsional couple per unit twist of the suspension wire, the torsional, couple set up in the suspension wire, also tending to bring the small balls back, into their original positions A and B is equal to CQ., , Since, , is, , Thus, the total couple acting on the suspended system of the small balls, , If,, , T, , therefore,, , ..., , /il, , And, thus,, , ~, , Tj, , I, , be the time-period of the suspension system now, we have, , a, , /~, , 2,, -, , C+, i, , Mi, , m \j, , ret, , (, , f, , \, , C >-^, , I, , ), , d,, , ^, , <~VA, , ., , 1, , 4-, , J, , crfj, , T.-r., , T,*, , 2MmG(rdr*), Or,, , Ur, Or, , ', , whence the value of G can be easily calculated., The mean of Heyl's results (for the different small masses, mentioned, a, above) gave the value of G to be 6*670 x 10~ C.G S. units., in, error, this result to be '005., the, So that, the, probable, Birge estimated, best value of, , G, , 005) x 10* dynes, , far is 6 67, , obtained so, , cm 2, , ., , gm~-*., , We, , know that the weight of (or, the, Density of the Earth., force actiiT-Dfbh) a body of mass, t on the surface of the earth, is equal, to mg, where g is the acceleration due to gravity at the place., 86., , m, , Also, if, , M be the mass of the earth and R,, , acting on the mass, , M, , n, , is,, , its radius,, , the force, , by Newton's Law of Gravitation, equal to, , t, , ., , i, , So that,, , wg, , =, , ivi.ni, , /\, , /^, &., , ^2, , Taking the Earth to be a homogeneous, Therefore,, , Hence, , if, , A, g, , be, , =, , its, , density,, , 47r/?, , 3, , 6K, , A, , g, , Or,, , .,, , *G., , =, , ^^^, , sphere, its, , its, , mass, , n, Or, g, , /^, .O., , ^2, , M, , =, , volume F, , =, u, , =, , 4.7r/?, , 3, , .A/3., , 4, o, , .itR.&.G, , whence, the value of A mav bo easily obtained., This gives the value of A t be 55270 gms./c.c., takings the, value ofCto be 6'b'576x 10~ 8 C.G.S. units, (the value obtained by, Boys). And, with Poyntinij's and Heyl's values of G, the values of, , A, , come respactively to 5 4934^/;iy./c.c. and (5-5150'004)^m5./c.c., The most probable value of A is however, taken to be 5'5247, , and, since the density of the upper layers of the earth is, be only 2*7 gms./c.c., it follows that the density of its inner, layers must be very much greater than 5-52 gms./c.c., It is interesting to observe how Newton intuitively made a lucky, guess at the probable density of the earth, placing it so aptly between, gms.jc.c., found to, , ;
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PKOPEETIES Of MATTES, , 246, 5 and, , a truly inspired 'Newtonian' guess, which stands so amply, As to the reasoning that led him to it, we can do, listen reverently to as he puts it himself in his cele-, , 6,, , to-day, no better than to, justified, , !, , brated Principia, , :, , 'But that our globe of earth is of greater density than it would be if the, whole consisted of water only, 1 thus make out. If the whole consisted of water, only, whatever was of less density than water, because of its less specific gravity,, would emerge and float above. And upon this account, if a globe of terrestrial, matter, covered on all sides with water, was less dense than water, it would, emerge somewhere ; and the subsiding water falling back would be gathered to, the opposite side. And such is the condition of our earth, which, in great, measure, is covered with seas. The earth, if it were not for its greater density,, would emerge from the seas, and, according to its degree of levity would be, raised more or less above their surface, the water and the seas flowing backwards, to the opposite side. By the same argument, the spots of the sun which float upon, the lurid matter thereof, are lighter than that matter. And however the planets, have been formed, while they were yet fluid masses all the heavier matter subsided, to the centre., Since, therefore, the common matter of our earth on the surface, thereof is about twice as heavy as watei, and a little lower, in mines, is found to, be three or four or even five times more heavy, it is probable that the quantity of, the whole matter of the earth may be five or six times greater than if it consisted, all of water, especially since I have before showed that the earth is about four, times more dense than Jupiter.', , The attempts made by different workers to determine the values, are tabulated below in chronological order, for the conof G and, venience of the student., , A, , Year, , Name of, Experimenter, , 1775, , Maskelyne, , Mountain method, , 1898, , Cavendish, , Torsion Balance, , 1854, , Airy, , 1881, , Von, , Types of, Experiment, , Mine method, Jolly, , Sensitive, , Boys, Eotvos, , 1901, , Burgess, , 1930, , Heyl, , z, , .gm~*, , 5'0, , A, , gms, , jc.c., , 5'448, 6'5, , 5'692, , 5493, 55270,,, , 6'6576xlO- 8, 6*66xlO- 8, , 5'53, , 664xlO~ 8, Oscillation, , 5'55, , method 6'670x 10~ 8, , Qualities of Gravitation., , 87., , crn, , 6'6984xl()-', , Torsion Balance, , 1895, , 7*4 x 10~* dynes, 6'754xlO5'7xlO-, , Value of, , 6'465xlO~ B, , Poynting, , 1896, , G, , Common, , Balance, 1891, , Value of, , We, , 5'517, , shall, , now proceed, , to see, , whether the gravitational attraction between t\u> Icdii s is in any, way affected by the nature of the intervening medium between them,, by the nature of the masses themselves or by physical conditions, like, temperature, 1., , etc., , Permeability., , From, , the similarity of the formula (F, , for gravitational attraction between two bodies, with these for, electrostatic attraction, it might appear that, like the constants, , the value of, , G might, , also, , =, , magnetic and, M and A there,, , depend upon the nature of the intervening medium, , between them., That this is net so has been clearly shown by Austin and Thwig, who, performed a direct experiment with a modified form of Boys* apparatus, in which, they placed slabs of different materials in between the two attracting masses and, could detect no change whatever in the value of G, within the limits of theii, experiment., , This stands further confirmed by (/) the fact that whereas in the expcri, ments for the determination of G, discussed above, air is the intervening medium, in the case of planets, the intervening medium is just free spoce, and yet thi, astronomical predictions, deduced on the baiii of the same law, come out so sur
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247, , GRAVITATION, , prisingly true, showing clearly that the value of G cannot possibly be very different in the two cases ; and (/*) the very close agreement between the values of G,, obtained by different pendulum experiments, with their bases of different mateWe,, rials, so that different materials lie between the pendulum and theA earth., therefore, conclude that little or no effect is produced in the gravitational attraction, between the masses by the nature of the medium interposed in-between them., , The law simply states that the force of attraction between, 2., Selectivity., two masses depends only upon their magnitude, having nothing to do with their, nature, or their chemical combination, etc. This is amply borne out by the large, volume of experimental evidence in its favour. For, it has been shown by Eotvos, and others, by their experiments with Boys" apparatus, using a laige variety of, materials as the attracting masses that the values of G obtained in the different, all agree admirably, even in the case of radio active substances, thus showing, clearly that gravitational attraction is by no means a selective phenomenon., , cases, , We, , know that in the case of amsotropic* substances, their, 3., Directivity., physical properties, like refractive index, conductivity for heat and electricity etc., depend upon their orientation, i.e., upon the direction of their crystallographic, axes., Or. Mackenzie and a host of other-workers, therefore, tried to investigate, as to whether it had any effect upon the gravitational attraction also, and they all, obtained negative results. An additional obvious proof of the independence of the, value of G of the orientation of crystals is the fact that their weight (which is just, another name for the force of gravitational attraction between them and the earth), is exactly the same whatever the orientation, showing that the phenomenon of gravitational attraction is far from directive., Poynting and Gray confirmed this fact in an ingenious experiment, in, which two quartz crystals or spheres were suspended close to each other, one, enclosed in a case, whereas the other, outside it, being free to rotate. If there, , were even a trace of a directive influence in gravitational attraction, the rotaso that, if their, tion of one crystal would sjt up forced vibrations in the other, time-periods agreed, the enclosed crystal or sphere would be set into appreciably, large resonant or sympathetic oscillations with the outer one., Nothing of the, kind, however, was found to occur., 4., Temperature. Poynting and Phillips, together with a whole lot of, other workers, tried to investigate the effect of temperature on the value of G,, and, once again, the results obtained were absolutely negative. Only Shaw,, experimenting with a Boys-Cavendish type of Torsion Balance, observed that the, value of G, increased slightly with the temperature of the attracting bodiesf, the, value of the coefficient of increase (a) being negligibly small, being only about, l-6xlO-* between (TCand 250C., ;, , All the abovo mass of evidence thus goes to suggest that gravitational attraction is purely a function of the masses of the attracting, bodies and of t lie distance between them, being quite independent of all, other factors., , No wonder, then, that Newton's theory of gravitation held such, an unquestioned sway over the minds of scientists all over the world., And, for the non- scientific people in general, it had equally spectacular, predictions which, when found true, could not but impress them, deeply as to its unerring truth., Thus, for example, Adams, in 1845, predicted, from his calculations, based on the disturbance of the orbit of Uranus, the presence, of a hitherto unknown planet. Unfortunately, Airy, the then Astronomer Royal, did not as much as care to look for this planet, perhaps from sheer scepticism and Challis, the Director of the Cambridge Observatory, though he actually saw a new star, looking like, a, disc did not care to verify whether it was the one predicted by, ;, , *Aniso tropic substances are those whose properties, are different in different, directions,*.?., crystals, in general., G. (l+o/), where G and, t According to Shaw, G, of the Gravitational Constant at, and O'C respectively., , -, , tC, , G9 stand, , for the values
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PROPERTIES OF MATTEB, , made, , A, , Adams., year later, Leverrier, a French mathematician,, similar calculations to those of Adams and communicated his results, to the Berlin Observatory and, lo and behold, the planet we call, Neptune, was there for all to see at the very spot predicted, !, , And, once again, in 1930, the disturbance of the orbit of Neptune itself led to the discovery, by American astronomers, of the, planet Pluto., These two profound discoveries put Newton's theory beyond, the pale of any doubt or scepticism and it came to be looked upon as, Indeed, it continued to enjoy its 'infallible' status until, infallible., the arrival on the scene of that genius of modern times, Albert, Einstein, who showed it to be no better than a close approximation, to the actual law of gravitation propounded by him, as we shall see, in the next article., , 88., , Law, , of Gravitation and the theory of Relativity. Although, as, is found to be, , we have seen above, the Newtonian Law of Gravitation, , valid over a wide range and is supported by a large mass of experimental evidence, there are certain small divergences, not quite in, faith in, conformity with it. But such, indeed, has been the general, the infallibility of the law that any divergences from it were ascribed, to some hitherto undiscovered disturbing influences rather than to, any possible discrepancy or flaw in the law itself. It was only after, Einstein put forward his "Theory of Relativity" that it camo to be, although, realised that Newton's Law was only an approximation, an extremely close one to the true or thfc fundamental Law of, , Gravitation., , A detailed discussion, , of this theory is beyond our present scope,, here only one or two salient points, consider, shall, therefore,, to show how Newt6n's formula lays itself opon to criticism in, , and we, , of it,, the light of these, , :, , One consequence of the theory, fully confirmed experimentwhat may be called the 'inertia of energy viz., that wherever, a change in the energy of a body is brought about, a corresponding, In other words, energy and mass are, change takes place in its mass., mutually convertible, one into the other, the relation between the two, (i), , 1, , ally, is, , ,, , being the following, , Change, , X, , 1C 10, , :, , mass, , (in, , grams), , =, , Change of energy, , (in ergs)jc*,, , velocity (in cms. [sec.) of light in vacuo, (equal to, cms. /sec.)., Further, according to this theory, we have, , where c, 3, , in, , is 'the, , m = mQ l^i~-Ti*Jc*,, , where m is the mass of a body, moving with a velocity w, (called its, moving mass, and m its mass when at rest, (called its rest mass)., Thus, the mass of a body is different when in motion from that, when at rest, i.e., it changes with its velocity, and Newton has not, in his formula for gravitaspecified which one of these is to be used, a noticeable omission., tional attraction,, ,, , of the theory is that the numerical, (ii) Another consequence, value of the distance between two points varies according to the, system of spaoe-timo co-ordinatea chosen, BO that the distance bet-
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249, , GBAVITATIOB, , Ween them changes with the circumstances of the observer, making, the measurement., , The effect of these two discrepancies is, of course, only slight,, but it is there, nevertheless. Einstein took both these factors into, account and was thus led to the formulation of his famous Theory,, which explains satisfactorily the deviations from the Newtonian law,, among which may be mentioned the following, :, , (a), , The precessional motion of, , the perihelion, 10 6 years., , of the, , of the, , orbit, , planet Mercury, with a period 0/ 3 x, On the Newtonian theory of gravitation, it could be explained, only as being due to the influence of another planet, for which the, name Vulcan was chosen but which has never been located., (b) The deflection of a ray of light in a gravitational fieldof light, due to its great velocity, behaves like a material, possessing mass and momentum, etc,, [see (/) above], and it, therefore, suffer deflection in a gravitational field. Thus, a, light from a star would get deflected near the edge of the Sun,, , A, , ray, body,, must,, ray of, , due to, high gravitational field, resulting in an apparent shift in the, Calculated on the basis of both these laws, tho, position of the star., value of this shift, by Einstein's theory, comos out to be twice that by, Newton's law, and actual expoximental observation* fully supports, the former result., its, , (c), , The, , shift in the spectral lines in the solar spectrum., field, the spectral lines in the solar, , the Sun's gravitational, , must have, , different positions, and, therefore, different, , ^\, , Due, , to, , spectrum, , a ve- lengths,, , from those they would have, when emitted by some terrestrial, This has been fully verified in the year 1924, the shift, source., being only very slight, about one-hundredth of an Angstrom Unit, 8, 10- 10 cw.)(or = f^th of 10~ cm., i.e.,, This has been further confirmed by Dr. Adams, who measured, a shift of as much as half an Angstrom Unit in the spectrum of the, dwarf companions of Sirius, due to the greater gravitational field at, the surface of these stars than at that of the Sun., , =, , It is thus clear that whereas the correct Law of Gravitation is, that due to Einstein, the Newtonian Law is a sufficiently close approximation to it for our ordinary experimental purposes, except in, a few rare cases, here and there., , Gravitational FieldIntensity of the Field, 89., The area, round about a body, whithin which its gravitational force of attraction, *This was made on the Island of Principe (on the African coast) aad at, Sobral, in Brazil, on May 29th, 1919, during a solar eclipse, (the stars being not, visible otherwise). Two well-equipped expeditions at these two places obtained, photographs of the portions of the sky, near the Sun, just before the eclipse and, again after the eclipse, when the Sun had shifted away from its earlier position., The stars were found to have been displaced, with respect to the Sun, the respective values of this displacement at the edge of the Sun obtained by the two, expeditions being 1*61" and 1*98*. Their mean, (1*795"), agreed admirably with, the value predicted by Einstein., similar observation was made again at another solar eclipse, three, years later, in 1922, this time at Wallal in West Australia, when the shifts of, as many as eighty stars were observed- The mean shift was found to be 1*74', which wai only about *01' short of the calculated value., , A
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PROPERTIES OF MATTER, , body being near about, , perceptible, (no other, gravitational field., , is, , is, , it),, , called its, , The intensity or strength of a gravitational field at a point is, defined as the force experienced by a unit mass, placed at that point in, the field., I^may also be defined as the rate of change of gravitational, Thus, if /, 91)., potential or the potential gradient at the point, (see, be the intensity of a gravitational field at a point, we have, ,, , ~ ~, /1, , where, , dV is, N B., , dV, >, , dx, , a small change of potential for a small distance dx., The strength or intensity of a field at a point is often spoken of, , merely as the field zl that point,, 90. Gravitational Potential, Potential Energy. Consider a body A, with its gravitational field around it. It will naturally attract any, other body B, placed at any point in its field, in accordance with the, Law of Gravitation, and this force of attraction will decrease with the, increase in the distance of B from A, so that at an infinite distance, from A it will be zero. But, as B is moved away from A, work has, to be done against this force of mutual attraction and, therefore, the, potential energy of B increases, its value depending upon the masses, of A and B arid their distance apart., The work done in moving a, t, , mass from infinity to any point in the gravitational field of body A, called the gravitational potential of that point due to the body A,, and is an important gravitational property of that point. It is, usually denoted by the letter K. Obviously, it will also be the potential energy of the unit mass at that point, with its sign reversed, (for, whereas the potential decreases^ the energy increases, with -the increases in the distance from A)., If we, therefore, replace the unit, , unit, is, , mass by the body B, the potential energy of B will, clearly, be equal, to the product of its mass and the gravitational potential (with its, Thus, the potential energy of a body, sign reversed) at that point., at a point in a gravitational field is equal to the product of the mass, of the body and the gravitational potential (with the sign reversed) at, that point., , Gravitational Potential at a Point distant r from a Body of, be situated at 0, (Fig. 154), and let a unit, mass be situated at P. Then,, the force of attraction on the, ^ ....... x ...... -^, unit mass due to, is clearly, Q^____ /r ____ ,_^, ~~, ....., ........., *, ^, ~, ,.to, G, equal, ,, ,, (7,, d, **, *2, 154., 91., , mass m., , Let a mass, , m, , ^, ', , mxl, , m, = m, , Fig., , where x is the distance of P from O, the force being directed towards, O. Therefore, work done when the unit mass m>ves through a, small distance dx, towards O, is equal to m.G.dx/x*., And, therefore, work done when it moves from B to A, , =, , [Am-., = G.m, -f.G.dx, , [A, , 1, , f-dx.
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251, , GRAVITATION, , t arid r l are the distances of A and B from 0. This, obviously,, the potential difference between the points A and B., If B be at infinity, i.e., if r l, oo, we have, , wtere, is, , =, , potential difference between, , A and, , oo, , =, , G.m f, , --, , _, , = m, , ^, , .G., , difference between A and oo is equal to the potenA, because the gravitational force at oo, duo to m, is equal to, zero and, therefore, the work done in moving a mass about, at oo, is, also zero., In other words, the potential at infinity is zero. Therefore,, the gravitational potential at A duo to the mass, is equal to, , But potential, , tial at, , m, , G.m/r., , Or, denoting the gravitational potential at a point, distant, , m by, , a body of mass, , F, , we have, , K,, , -, , r, , from, , -<7., , It will be noticed that, , whereas the value of gravitational potenmass is zero, it goes on decreasing, as we approach that attracting mass,, i.e., it is an essentially negative, at an infinite distance from a, , tial, , quantity*, its maximum value being zero at infinity,, points, therefore, the potential will be the same., , where,, , at, , all, , We know, , that, ordinarily, a body say, a rifle, to the earth due to the gravitational pull, of the earth on it. Let us see it it is possible to project it with a velocity such, that it will never come back., Obviously, it will be so if it can be given a veloThis velocity, city that will take it beyond the gravitational field of the earth, of the bullet is called the velocity of escape., , Velocity of Escape., , 92., , bullet, projected, , Thus,, , if, , upwards comes down, , m, , =, , .*., , be the mass of the bullet and Af, that of the earth, the force, at a distance x from the centre of the earth is clearly, , on trie bullet, m.M.G./x 2, work done by the, , acting, , ,, , ,., , bullet against the gravitational field,, , a distance dx upwards, *, ,'., , total, , rn.M.G, x*, , ., , ,, , ==, , work done by the, , when, , the body, , moves, , ,, , ., , dx., , bullet escaping, , m M.G, , ", , r, J/, , fa, , x-, , =mM G, R, , ("where, [_of, , R =, , radius, , the earth., , m, , times (i.e., the mass of the bullet times) the, This, as will readily be seen, is, gravitational potential on the surface of the earth., If v, , velocity of the bullet , (i.e., its velocity of escape), its, 2, And this must,, at the time it starts) will be ^wv ., to the work done by the bullet during escape. So that,, , be the initial, energy (t c, , initial kinetic, , ,, , therefore, be equal, , or,, , 9, , which, on substituting the value of A/, , f, , v*, , G and R, , t, , woiks out to, , v= \\'l9xW cms. /sec., Thus,, , velocity, , 93., , 11, , of escape, , Equipotential, , 1, , 9 x 1 0* cms. I we., , Surface., , the gravitational potential, , is, , A, , surface, at points of which, ia called an equipotential, , the same,, , surface., , Thus, if we imagine a hollow sphere, of radius r, with a particle, of mass m at its centre, the potential at each point on it will be the, *Thc, , negative sign, , understood to be there., , is, , often omitted in writing, but, , it, , must always bt
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PROPERTIES OF MATTBB, , 252, same,, , The, , G,w/r., , vfe.,, , surface of the sphere is thus an equipoten-, , tial surface., , since the difference of potential between any two points, is zero, no work is done against the, it., force, in, moving a unit (or any other) mass along, gravitational, In other words, in moving a mass along an equipotential surface, we, must be, it in a direction perpendicular to the gravitational, , Now,, , on an equipotential surface, , moving, , field, every point on it. Or, the direction of the gravitational, an, to, being, is, at every point, p8rp3ndicular, equipotential surface,, directed towards the nearest equipotential surface, having a potential lower than it., , field at, , and Inside a Spherical Shell., a point, distant d from the, centre O of a spherical shell of, radius a, (Fig. 155), and surface, , Potential at a Point Outside, , 94., (a), , At a point outside the, , P be, , Let, , shell., , density,, , (i.e.,, , the surface),, , Join, , mass per, , unit area of, , p., , OP and, , cut out a slice, , in the form of a ring, by, planes close to each other, radius, perpendicular to the, C, and, in, shell, the, Z),, meeting, , CEFD,, two, and, OA,, and, , in, , E, , ^COE =, , and, , F, , respectively., , do., , Clearly, the radius of the ring, that, its, .-., , And, , .-., , =, , circumference, , 2na. sin, , area of the ring or slice, its, , == 2xa. sin, , mass, , EP =, , =, =, , and, , 6,, , the small, , let, , = o sin, = CE = a.dQ., , EK = OE sin, , is, , and, , Q,, , =, , /_EOP, , Let, , width, , its, , circumference, , its, , =, , X its width., , 2x0*. sin O.dO.p., , r from, every point of the slice is at a distance, is, slice, small, this, to, given by, and, therefore, the potential at Pdue, , If, , r,, , _, , mass of slice, , _^*^?-P, , Q=, , in the triangle, , r, , Or,, , 8, , = a*+d*-2n a.d.j, ,, , Differentiating the above expression,, 2r.dr, , Hence, , cos, , 0., , point P., , [See, , 91., , =, , this, , B.dQ^, , OE=, , I,, , 7 (2), , a, the radius, , 2a,d.sin, , a and d being, constants., , __ a.d.sin B.d6, ~~, , dr, , G, , between the, , (d+a), we get, , [Appendix, , we have, , 2a.d.sin 0.d6, , 2, 27ta .sin, , Integrating, , (/), , of lhe she K, , = Q+Q+2a.d.sin d.de =, r, , 0., , i~V, , /i, , 2.dr, , DP =s, , ., , ,, , Substituting this value of r in expression, , r=s, , ., , /*',, , OEP,, , EP 2 = OE* + OP*-20E.OP.cos, /^, , >Cj, , r, , r, , Now,, , BO, , Qxa.dQ, , 2ira. sin, , OXa.ddXP, , ;, , ,, , (/), , above, we have, , __ 2?r a.p.G dr, , = AP =, , limits, r, (d a), and, V, the potential due to the whole shell at the
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GRAVITATION, , Thus,, , Tf, , *, , I, , ,.., , *, , *, , _, , *, , _, , ,, , T., , ](d-a), , I, , _, , _, , d, , d, , </r., , ](d-a), , 4ir.a p is, , 47T0 2 is the surface area of the whole shell, and, therefore,, equal to its mass M., , We thus,, , have, , Now,, 2, , ., , a, , Or, the potential at the point P due to the whole shell is equal to, at O., M.Gjd, i.e., the same as it would be due to a mass, , M, , The mass of the whole shell thus behaves as though, centrated at, , its, , it, , were con-, , centre., , In the above case, if we imagine the point P to be at A, i.e., on, the circumference of the shell itself, we get the potential there by inand r, 20., tegrating the expression for dV', between the limits r, , =, , So that, in, , =, , this case,, 27r, , f*, , -, , a -P- G, , ', , T, , "~d, , Jo, , d, , M.O, ., , Or,, , J, , \_, , M.O, , -, , ,, , ., , [, , d, , ., , here,, , d, , a., , a, , Hence, (b), , At a point, , inside the shell., , Imagine now the point, , P, , to lie, , inside the spherical shell, (Fig. 156)., Proceeding as above, we have, , potential at, , P due, , to the, , slice,, , or ring, , CEFDJ.e.,, , In this case, the limits of, , and (a-f d)., , r are, , So that, we havo, , (ad), , Fig. 156., , 2ir.a.p.(?, , -, , ., , d, Now,, , 47f.a, , 2, f, , p, , =, , -., ", , A/, the, , -., , mass of the, , a, shell,, , dividing by, , a.
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UJf, , V, , Hence, , s=, , .(?,, , a, , the same as at a point on the shell., [See case (a) above], Since the above value for V has been obtained for a point P,, anywhere inside the shell, it follows that the potential at all points, inside a spherical shell is the same, and is numerically equal to the value, of the potential on the surface of the shell itself., i.e.,, , 95. Gravitational Field inside a Spherical Shell or a Hollow, have seen above that the gravitational potential at all, Sphere., points inside a spherical shell is the same., Now, the field at a point is given by the potential gradient (i.e.,, , We, , the rate, , of change of potential with, , Or,, , 7, , i.e., , the field in the interior, , Since, , V, , =, , constant for, , distance), at that point., , -dl'ldx., , =, , 0,, points inside the shell, dVjdx, of the shell, due to the shell, is zero in, other words, there is no gravitational field inside, , is, , all, , ;, , a spherical, , shell., , P, , Let, , Alternative proof., , be any point, , inside a spherical shell, or a hollow sphere, of, surface density, (i.e, mass per unit area), p., Through P, draw straight lines, so as to form, ,, , their apices at P, and, 8 and 5' on the shell,, intercept? ne:, as shown, (Fig. 157), on the opposite sides of, , two small cones, with, , small areas, , XY, , the plane, drawn perpendicular to the, diameter passing through P., Let S and S' be, at distances r and r' from P, and lot the solid, angles* at P be equal to o>, each., , S', Fig, 157., , Then,, to r z .a>, , clearly, the area of the right section of the, 2, 5', equal to r' .o>., , S', , S, , is, , equal, , therefore, a be the angle that the right sections of the cones, make with S and \ respectively, we have, S cos a, r 2 .co and S' cos a, r '2, , =, , =, , r'.o,, , And, , and S', , C OS a, , So that, mass of area, intensity at, , And, , cone, , and of that at, , If,, , S and, , 9, , S, , P due, , intensity at, , *=-'-', cos, , to, , S, , ,, , =, COS a, , and mass of area S', , =, cos a, , 9., , =, , in the direction PS., , ., , cas a, , cos a r*, , P due to S', , ,, , = cos-^,-.0, a.r, , ', , f, , cos a, , in the direction, , These two intensities at, , PS, , 1, , P, being equal and opposite, their resulSimilar is the case for all other pairs of cones on opposite sides of AT, into which the shell may be divided, so that, the, resultant intensity or field at P due to the whole shell is zero. And,, the same is true for any other point inside the shell. In other words,, , tant, , is, , zero., , ;, , there is no gravitational field inside a spherical shell., , _______, , ,, , ,, , _, , t
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255, , QBAVITATION, , Potential and Field Intensity due to a Solid Sphere at a, Inside, the Sphere and (//) Outside the Sphere., (/), Let the point, (a) Potential at a point P, inside the solid sphere., P lie inside a solid sphere of radius a, and of volume density er, at a, distance d from the centre O of the sphere,, 96., , point,, , (Fig. 158),, , The solid sphere may be imagined, to be made up of an inner solid sphere, of radius d surrounded by a number of, hollow spheres, concentric with it, and with, their radii ranging from c/to a. The, potential, at P due to the solid sphere is, therefore,, equal to the sum of the potentials at P due, to the inner solid sphere and all such, spherical shells outside-, , it., , P lies on the, , surface of the solid sphere of radius, inside all the shells of radii greater than d., .*., potential at P due to the sphere of radius d, Clearly,, , = _mas 8 of the, , d and, >, , sphere, , a, , = - 4 ir.d*a.GId -= - 4, 3, , fv, , ., , .TrJ*o.G, , ...(/), , mass of the sphere, , = *</.<,., , I, , j, , To determine the potential at P due to the outer shells, imagine, a shell of radius x, and thickness dx., its volume = area x thickness = 4irx*.dx,, Clearly,, and .-., its mass = <lirx*.dx.G., Now, the potential at any point within a shell is the same as at, any other point on its surface., So that, potential, , at, , P - -, , 47f, , -, , x?- dx -*- G, , Integrating this for the limits, x -=, potential at P due to all the shells., Thus, potential at P due to all the shells, , =, , I, , 4ir.a.G.x.dx, , =, , ___, , 47r, , d and x, , x dx.a.G., , =, , a,, , we, , get the, , x.dx., , 47r.a.G, , }d, , Now, the, , ^, , Jd, , total potential at P duo to the whole solid sphere is, P due to the inner sphere of radius d> plus, , equal to the potential at, potential at, , P due, , to, , all, , the outer shells., , so that, the potential a, *This expression is equal to, In a.G (cPd*), Fdue to all the outer shell = 2n.Q.G(az <**), and, the potential at P due to, the whole solid sphere is, therefore, also, ;
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256, , PROPERTIES, , 4, , _, , MATTER, , P, , total potential at, , *=, , Off, , w, , 3, 4, ~~, , 7r *', , 3, , 4, , -, , But, , ["Multiplying and, 2, by a, , 7T, , (, , -, , 2tf 3, , 3.7T.fl, , potential at, , 3, , the mass of the sphere, , .a is, , P due, , to the sphere, , =, , =, , ., , [^dividing, , M., -, , Potential at a point P, outside a solid sphere. Imagine the, up into a number of thin spherical shells, concentric with the sphere, and of masses, iw lt, Then, as, 2>, 3 etc., (Fig. 159)., we have seen before, the potential at P, due to each spherical shell will be equal, to its mas X G/d, where d is the distance, of the point P from the centre of the, , (/?), , sphere to be broken, , w, , m, , ,, , the same as though the, sphere i.e, mass of each shell were concentrated at, its centre O., So that, the potential at, P duo to the different shells will bo, ,, , Fig., , ~m v Gjd,, , 159, , m, , 3 .Gjd and so on., due to, Therefore, the potential at, the whole solid sphere, , m^G\d,, , f, , P, , =, , all, , such shells,, , /..,, , due to, , m, , /Hi, , _["-, , A/, , <T, where, , h, , ., , ., , ., , Hence potential at P due, , ., , = M,, , the mass of whole solid sphere., , M, , to the solid sphere, , i, , G., , Gravitational field due to a solid sphere at a point inside the, have seen that the potential at a point inside a solid, x from the centre of the sphere, is given by, distant, sphere,, (c), , sphere., , We, , y, , [See above.], , jtf.G--^--", , Now, intensity, or gravitational field, at a point is equal to the, potential gradient, (or the rate of change of potential with distance),, at the point., Therefore, intensity or gravitational field at a point distant x, from *u, the centre, , dV, a, , dx, , =, , d, 5, , dx, , M.G.X*, -, , -Af.G., *, , Taking the value of V, , (see foot-note,, , page, , 255),, , at distance, , we have, , x from, , a*, to, , be, , 2w<j(7/ a*, , intensity or field at distance, , x from, , ^-j, , O
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257, , aRAVITATION, , This shows that the force decreases as x decreases, being zero at, the centre of the sphere. The negative sign only indicates the attractive nature of the field., Gravitational field due to a solid sphere at a point outside the, (d), We know that the potential at a point outside a sphere,, sphere., distant, , x from the centre of the sphere, , above in, , 98, , (6),, , is, , =, , G, as explained, , (page 256)., , And, since intensity or gravitational field at a distance x, equal to the potential gradient at x, we have, gravitational field at distance*, , -, , M.Q. X, , =, , =, , -.-, , M, , =-, , (-lxx-), , ;, , Or,, , M.G. (x~ l ), , ., , =-, , is, , *<?-., , Alternatively, we may get the same result by applying, in this, case also, the assumption, found valid in tli3 case of potential outside, a sphere, viz., that the potential is the same as though the whole mass, of the sphere were concentrated at the centre of the sphere so that, gravitation il intensity or field at a point, distant x from the centre,, ;, , =, , and, , 2, outside the sphere, M.G/x the negative sign merely indifield., the, as, nature, of, the, before,, cating,, ,, , N, , B. This result is of great historical importance in that it enabled, to apply his law of gravitation to the motion of the moon., For, the, radius of the earth not being negligible compared with that of the moon's orbit, , Newton, , around it, there would have been no means, in the absence of the above result,, of determining what correction terms, if any, would be necessary in the equation of the moon's motion, in view of the distribution of matter* inside the earth, and the finite value of its radius., 97., Intensity and Potential of the Gravitational Field at a Point, due to a Circular Disc. Let MN, (Fig. 160), represent a circular disc, , of radius R, with its plane perpendicular to the plane of the, be a point on, paper and let, distant x from its centre, where the intensity and, potential due to its gravitational, its axis,, , C,, , field, , are to be determined., , Imagine the disc to consist, of an infiuito, tric rings,, , mon, , number of concen-, , with, , C, , as their com-, , centre., , Consider one such ring, of radius r and thickness dr., , Join, if, , angle, , O, , AOC, , Pig., , to the extremity A of the radius, be equal to 6, We have, r, , And,, , PQ, , = CA ^ x, , therefore, differentiating, j-, , ifc, , CA, , tan 0., , with respect to 8, , Or,, , 150,, , of the ring., , dr, , 9, , we have, , x.sec*6,d9., , Then,
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PBOPEBTIE3 OF MATTBB, , 258, , AO = x.sec 0., = circumferences thickness, mass of the ring = 2irr.dr.9,, , And,, , Now,, , area of the ring, , And, therefore,, where p is the mass per, , =*2irr,dr, , t, , unit area of the disc., , Considering a very small element A of the ring, we have intenmass of the element x G/AO2 along, sity at O due to this element, , =, , the direction, , ,, , OA., , This can be resolved into two rectangular components,, , OC and, , (//), , at right angles to, , r, , ,,, , ^, , the former component, ,, , ,, , tl, , AA, , ,, , and, the latter component, , So, , it., , along, , (/), , that,, , element, = mass of the, AQ*, , -, , tie element, ~, = mass ofAQ*, ~~~~, , &, ', , ., , *, OC,, , ., , cos, , Q>, , n sm, , *, , along, , n, , *, , ', , v e rtica "y, , upwards., Similarly, for, , an equal element at 5, diametrically opposite, , to A,, , we, , have, intensity at O, resolved into the same two components,, mass, element, ,.,, of, ,, ,, J the -----------.G cos $, along, and, (i), , OC, , --^p, , mass of the element, , ...., , Am, , (j/), , .., , ., , sm, , ', , A\J, , ;, , ,, , 0> v ertica "y, , v/z.,, , ,, , downwards, as shown., , Thus, the two vertical 'components, being equal and opposite,, cancel out, and the components along, alone being effective, are, added up, both acting in the same direction, the same will be the, case with other elements into which the ring may be supposed to be, broken up and, therefore, the intensity at, due to the whole ring, is equal to the sum of the, due to th different, components along, , OC, , OC, , O, OC, , ;, , elements,, , i.e.,, , mass, of the, -, , intensity at, , Or,, , ;, , O, , due to the ring=, , Or, substituting the values of, ^, , intensity at, , ^, O, , ring, , ^ rmg=, =, , j, x, due, to the, , 0., , and AO, we have, _, x tan 6.x sec 2 tf.dQ, -------cosS.O., x 2 sec 2 u, , dr,, , r,, , -, , yUg -'-Gcos, , ., , 2irp.G.sin 8.d6, , t, , along OC., , Therefore, intensity at O due to all the rings into which the, supposed to be divided up, i.e., intensity at O due to the, whole disCj is obtained by integrating the above expression for the, and r=/?, or 6, intensity due to the ring, between the limits, r, , disc, , is, , =, = a, where a is the angle between OC and the, with the extremity M of the radius CM of the disc., and, , Thus, intensity at, I, , Jo, , O, , due to the disc, , torf.Qsin Q.dQ, , =, , 2ir.p.<?, , is, , |, , Jo, , =, , line joining, , given by the expression,, sin
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259, , OBAVITATIOB, , o, , =, , cos, 27r.p.G|, , =, Or, intensity at 0, </we, A, i, And, since cos a, , intensity, ity at 0,, , =, , =, , 27r.p.G.(l, , ~CO, , a+1), coy a), , (0, , ,, , MQ, , due to the disc, , 9, , =, , cos a), Or, again, because 2?r (1, on solid angle below), subtended, from relation (/'), above,, intensity at, , J, , cos, , 27r.p.0.(, , disc, , f, , (cos 0), , cos a, 27r.f>.G.|, , |, , we have, , 27rp.#/l, , is, , ~/^Tj?a, , )', , the solid angle, co, say, (see, disc at the point O, we have,, , by the, , O, due to the disc, , =, , p.G.co., , (KI), , Now, potential at P due to the ring of radius r is equal to the, ZitQ.Gsin O.dQxx, because field, intensity at P due to the ringxx, at P is equal to the potential gradient at P., , =, , Hence, potential at, , =, , P*, , P due, , to the whole disc, , 27T.P.G.X sin Q.dB, , P due, , = 27T.p.G.x|, , sind.dQ,, , JO, , JO, Or, potential at, , is, , to the disc, , ......, Or,, , =, , Or,, , =, , ......, , (iv), , (v), , to give it its proper negative sign., , Note on Solid Angle. Suppose, we have an area PS, (Fig. 161), as the, base of a cone, with its apex at point O. Then, if we draw a sphere, with centre, O and any radius R, so that a surface of area pq of it is cut off by the, cone, then pq is proportional to r*,, where r is the radius of the spherical surface pq ; and, therefore, area, pqlr* is constant for any given cone., This quantity, area pqjr*, is called, the solid angle of the cone, or the, solid angle subtended by the area pq, at O, and is usually denoted by the, letter co., Obviously, it is also equal, to area PS/OP*, or the solid angle, subtended by area PS at O, and its numerical value is equal to area pq, if, 1 cm., r, , Now, suppose the given area be PQ and not PS. Then, if AN be the, normal to it at its centre A, we have area PS - area PQ cos a, where a is the, angle between AO and the normal AN to the surface PQ at A., Thu,.
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PROPERTIES OF MATTER, , 260, , to determine the solid angle o>, subtended by a circular shell or disc MN*, of radius R, at a point O, distant x from its centre C, (Fig. 162), we draw a, , Now,, , t, , sphere, with, , *X, , that, , O as its centre and, , circular face lies, , its, , on, , 2, 2, (R + x ), , (from above),, , \^, , surface area of slice, , \, , as, , radius, such, Then, we have,, , its, , this sphere., , MFN, , Since the area* of a sphere, lying between two parallel, is equal to the area of the circumscribing cylinder, in between these planes, with its axis perpendicular to them,, , /, ', , planes,, , we have, area, , MFN -, , =, , 2, , 2rr(RH* )*xFC, , -* ., , -, , ., , o, , Or,, , where a, , is, , *, , (FO-CO)., , FO - MO,, , CO, , [v, , X(MO-x)., , Hence, , 2n (tff x 1 )* X, , 2n(l- cos, , r, , 2ir, , I, , and, , x,, , _, , a),, , the semi-vertical angle, subtended by the shell or disc at O., , Intensity and Potential of the Gravitational Field at a Point, Infinite Plane., In tho caso of the disc, above,, 97),, becomes i a finite, the disc becomes an infinite, if its radius, In this case, obviously, a becomes 7T/2, so that cos oc, 0,, plane., 98,, , due to an, , (, , R, , and, , =, , = 2?r., , a>, , expression, , (/),, , R =, , we put, , if, , Thus,, , or, , a>, , =, , 2?r in, , is,, , O, , in, , =, , due to an, , Similarly, putting these values of R, cos a, (iv), (v), , =0,, , 2n?G., infinite plane, x, of, the, distance, from it., clearly, quite independent, intensity at, , which, , oo in expression (//), or cos a, expression (///) above, we have, , and, , (vi), , above,, , and, , o>, , in relations,, , we have, , potential at, , due, , to, , an, , plane =2^p.(/ .x., f, , infinite, , 99. Inertia! and Gravitational Mass., We ordinarily define the mass, of a body by the acceleration produced in it by a known force. This is known as, its inert ial mass., , But since, as we have seen, the gravitational field due to body is proportional to its mass, it is also possible to define the mass of a body as proportional, to the gravitational force of attraction it exerts on a standard test body at unit, distance away from it., Thus defined, the mass of the body is called its gravitational mass., , Now, Galileo showed that the acceleration of a falling body was quite independent of its mass and the same is found to be true in the case of pendulums, used for the determination of the value of g t showing that the gravitational, force between a given mass and the earth is proportional to the inertia of the, mass., There appears to be no a priori reason, however, why this should be so ;, in the case of an inclined plane, for example, we have seen how the acceleration of a body loliirg cc\\n the plane depends not only on the mass of the, body but also on the distribution of its mass, ( 39, page 87). The above may, thus be regarded to be only an experimental law., for,, , *MN, , 9, , here, represents the side-view of the shell or disc.
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OEATITATION, , 281, , It is confidently claimed, by some, but equally hotly disputed by others,, that the equality of the inertial and, gravitational mass can be 'predicted' from, the general theory of relativity, so that, nothing very definite can yet be said, on the point. Their eqiulity*, however, is of great, consequence in astronomy., For example, dus to the proportionality of gravitational force to inertial mass,, the orbit of a satellite round a planet /, quite independent of its mass, and we, can thus 'wv/VV, the planet from a mere observation of the orbit of its satellite., And, agiin, we can determine the mass of one component of a double star, by, observing; the orbit of ths othsr ronj thiir CD, centre of gravity, the, required value of G being obtained from terrestrial experiments, dealt with, , -mm, , above., , Earthquakes Seismic Waves and Seismograph. An earthciusod by a portion of the ri*id crustf of the earth giving, way or getting: fractured, soim distanco balow its surface and the, consequent sudden slipping of the resulting portion, or due to 'fault, 80 to speak, it is just a landslide, slipping*, as it technically called., on a largo scale, or a re-adjustmont of the earth's crust, in response, to a change of forces, or more precisely, to, changes of pressure deep, in the earth's crust, down to a distance of 100 w/ev or so, brought, about by a variety of causes like erosion, deposition, tidal forces,, cantrifurral forces, etc etc. An earthquake thus represents the energy, released by this 'relative motion of portions of the earth's crust*., 100., , quake, , is, , The place whore the actual fracture occurs is called the focus of, the earthquake, and it not a geometrical point, but an extended, The point nearest to the foaus, on the surface o the earth,, region., is called the, epicentre., the focus, (which we may, for our purposes here, regard, a, just, point), originate a number of different types of waves,, called, collectively, LQN6 WAVS, \, P, SM L, ,, seismic waves,, which, , From, , as, , spread on to different, points on the surface, of the earth and which, , we fed, tremors'., Fig. 16J,, , as, , "t/l^^^lj, Fig. 163., , 'earthquake, , The general pattern of these seismic waves is as shown in, and thoy consist of the following different types of waves, :, , The Primary or, , P, , The, , arrive at the, (a), the particles, in, which, are, these, waves,, Station,, longitudinal, Observing, of the earth vibrate about their mean position, along the direction of, the waves themselves., , Waves., , first, , to, , If the earth be regarded to be a homogeneous sphere, these, waves, starting from the focus, travel along the chord of a huge, circle of the earth, with a velocity equal to \/ jl~&, vherQ j is what, is called the 'elongational elasticity' $ of the earth and A. its density., ', , These waves arc also variously called as condensational', '/>'rotational', and 'push' waves and their velocity is found to be about 5 miles per, second., *Since, as we have seen, they are proportional to each other, a proper, choice of units can make them equal., fSee foot note on page 230., y (1 -<*)/(! f<*) (1-2<J), where rand a, JThe elongational elasticity j, stand for Young's modulus and Poison's ratio respectively., , =
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262, , PROPERTIES OF MATT1R, , These are transverse waves,, (b) The Secondary or S Waves,, which the particles of the earth vibrate at right angles to the, direction of propagation of the waves, thus having no component, along this direction. Starting from the focus, these waves also, travel along a chord of a huge circle of the earth and are the next to, arrive at the Observing Station, with a velocity equal to \/w/A>, where n and A represent the modulus of rigidity and the density of, the earth respectively. The other names given to these waves are, 'distortional', 'equivoluminal' and 'shake' waves, their velocity being, about 3 miles per second., in, , Discovered by Lord Rayleigh, these waves, (c) Rayleigh Waves., are found to remain confined to a comparatively thin layer in the, and S waves,, Unlike the, close vicinity of the earth's surface., they start from the epicentre and arrive at the Observing Station,, along a huge circle of the earth, the displacement of the particles, at any point on the earth's surface, due to them, being in the vertical, plane containing their direction of propagation. Resolving this, displacement, we have (/) a vertical component and (it) a horizontal, component, along the direction of propagation, there being no horizontal, , P, , component at right angles to it. These waves thus persist over long, distances along the surface of the earth, and are almost unique in, If the earth were a homogeneous sphere, these waves, this respect., also would travel with a constant velocity, but, due to its heterogeneous character, each single wave, starting from the epicentre,, gets split up into a number of different sets of waves, each set, so that, what we, having a different wave-length, velocity etc., receive at Observing Station is a series of oscillations, instead of, one single 'kick' or 'throw' as would be the case if there were no such, splitting up of the original wave, i.e., if the earth were really homo;, , geneous in composition., , The heterogeneity of the layers of the earth, (d) Love Waves., responsible for yet another type of surface waves, known as Love, Waves, in which the displacement of the earth is horizontal, but, The velocity of, transverse to the direction of their pro^apation., these waves is less in the earth's crust than in the matter below., is, , Immediately after an earthquake, oscillations, corresponding to these, waves, can be detected at almost any place on the surface of the, earth., , Unlike P and S waves, which are separately and distinctly, received and recorded at the Observing Station, these waves get, intermingled with Rayleigh waves to form a somewhat complicated, system of waves, (not yet properly understood), called long or L, waves, or the main shock, registering themselves as a long series of, oscillations., , 101., Seismology. The study of the seismic waves constitutes, is called the science of Seismology, and it owes a great deal, to Prof. John Milne, who did almost the whole of the initial pioneering, work on the subject. As early as the year 1883, when he was, residing in Japan, he predicted that 'every large earthquake might, , what, , *
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ORAV1TAT1UB, , 263, , proper appliances, recorded at any point on the land surface, the globe"., And, then, in the year 1889, a curious incident, confirmed his prophetic words., For, a delicate horizontal pendulum,, set up for the measurement of the gravitation- action of the moon,, be, with, , of, , gave recordings, which turned out to be due to an earthquake, with, its origin somewhere in Japan., This started a new era of intensive, researches on the subject, with Prof. Milne in the very forefront, and, in 1895, ho set up his own observatory at Shide, in the Isle of, Wight, which became the centre of a world-wide seismic survey., ;, , By the year 1901, the main facts as to how the tremors travelled through and round the earth were fully established, again, due, in main, to the labours of the eminent Professor himself., His reports to the British Association on Earthquake Phenoin Japan from 1881 to 1895, together with those on Seismological Investigation from 1895 to 1913, (the year of his death), form a, fascinating and a detailed study of the growth and development_of, , mena, , the present-day science of Seismology., 102., Seismographs. A seismograph (or a seismometer), is an, instrument used to record the earth tremors or the seismic waves,, to some dynamical function of which, (like displacement, velocity,, The record of the vibraacceleration, etc.), they respond or react., tions so obtained is called the seismogram., The instruments, responding to displacement, are of the mechanical type and we are,, The following is, in, therefore, concerned here only with those., brief, the theory underlying the mechanical type of seismographs., AH vibrations of the earth may ultimately be resolved into (/), The problem thus reduces, vertical and (//') horizontal components*., , merely recording these vertical and horizontal vibrations. We, our attention here only to the measurement of the, horizontal displacements, accompanying these latter vibrations. There, are 'two types of instruments in use for the purpose, viz., (a) the, itself to, , shall confine, , vertical pendulum, , and, , (b), , the horizontal pendulum type., , A, , vertical pendulum, (a) The Vertical Pendulum Seismographs., just a rigid body, suspended from a stand resting firmly on the, so that, with the horizontal displaceground, merit of the ground and the stand with it, the, also, of, the, of, pendulum, gets, support, point, is, , ;, , displaced horizontally., Thus, if the point of support, vertical, , pendulum,, , (Fig., , 164),, , is, , S, , of the, , displaced, horizontal, , to 5", due to the, displacement of the ground, it can be shown, that a style or pen, attached to its lower end,, reproduces faithfully the movements of the, Fig. 164., support, with precisely the same frequency,, (though on a different scale), it being assumed that the support moves, with a definite frequency and amplitude., , horizontally, , *These components may be along East and West or along North and, South but will be horizontal, nevertheless. These can also be used to measure, the horizontal velocity and acceleration of the earth, or rather of the earth's, crust.
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PROPERTIES OF MATTSK, , 264, , Those vertical pendulum seismographs, however, suffer from, as much as 20 tons, viz., (/) they have to be very heavy,, or more,, if a good magnification of the vibrations be desired, and, , two, (//), , defects,, , their period of vibration, , is, , rather small., , We, , are already, (b) The Horizantal Pendulum Seismographs., familiar with the horizontal pendulum, [see 78 (//), page 214]*., Only some slight additions to it convert it into a sensitive and, a reliable seismograph. With the horizontal movement of the earth,, the supports of the pendulum, which are firmly fixed on to it,, also share its movement, thus setting its stem or 'boom' into motion,, which can then be magnified mechanically or electrically by various, devices., , The best known seismograph of its class is that due to Prince, Boris Galitzin, in which the greatest care has been taken to see that its, indications correspond exactly to the actual movements or vibrations, of the earth. We shall, therefore, discuss in some detail only this, one instrument here., Galitzin's Seismograph., This seismograph measures the, 103., horizontal velocity of the earth's crust, and consists of a horizontal, pendulum, having a boom or stom, 28 cms. lon^. carrying a cylindrical, brass bob, weighing 7 k.gms* ani having its centre at a distance of, 14 cms. from the inner end of the boom. The suspension of the, pendulum is of the Zollner type, (as shown in Fig. 135, page 214),, with a very small inclination of the axes, so that the period of, oscillation of the pendulum is about 24 seconds. The whole pendulum, is built up on a rigid frame-work, firmly secured to the ground, and, consisting of four m3tal pillars, braced together, and arranged, rectangularly on four points on a inetal base or plate, provided with, levelling screws., , The recording of the vibrations or tremors, , at the Observing, done eloctromagnetically, and, for this purp>se, a flat, copper coil is wrapped round a portion of the stern or boom of the, pendulum, extending beyond its cylindrical bob, and connected to a, , Station, , is, , sensitive moving-coil mirror galvanometer., , With the motion of the stem, (caused by the motion of the, ground), the coil moves in the strong magnetic field of a pair of permanent horsa-shoe magnets of tungsten- steel. A current, which is, proportional to the angular velocity ofihe stem, is thus induced in the, coil and produces a deflection in the galvanometer., , A beam of light, reflected from the mirror fixed on to the suspension of the galvanometer coil, is passed through a semi-cylindrical, lens and allowed to fall on a sensitized (i.e., photographic) paper,, wrapped round a rotating drum, worked by a clock-work arrangement and moving uniformly along its axis, with a peripheral speed, of 3 cms. per minute. Time-signals are also similarly recorded on, the paper by cutting off, by means of an accurately- timed shutter,, the beam of light for two seconds at the beginning of each successive, minute. A permanent record of a series of curves, (i.e., the seismoarticle, , The student would do well to refresh his memory by going over this, once again before proceeding further.
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265, , GRAVITATION, , gram), is thus obtained on the sensitized paper, from which the time, of occurrence of any seismic phenomenon can be determined aoou*, rately to within one second., , In order that the horizontal displacement of the earth may be, correctly calculated from the seismogram thus obtained, it must, faithfully correspond to the movements of the earth. To achieve this,, d imping of both the pendulum and the galvanom3t3r is necessary., Or, else, if the period of oscillation of the penduluni'agrees, or-nearly, agrees, with that of tho saismic wave, resonance will occur, producing, largo deflections, which would give an utterly deceptive picture, of the actual movement of the ground. And, if the damping be, made critical, (i.e., dead-beat), the calculations become greatly, simplified., , This damping is produced by attaching to the outer end of the, boom, a horizontal brass plate, which moves in another strong magnetic field, duo to a separate pair of horse-shoe magnets, arranged, above and below it. The eddy currents, thus induced in the plate,, then produce, with proper adjustments, the desired damping effect on, the pendulum., , This seismograph has the additional advantage of great magniarranged, , fication*, as also of enabling the recording apparatus to be, in a separate compartment, away from the pendulum., , N.B. It will be readily understood that for a large or severe earthquake, less sensitive seismographs are more suitable, while, for smaller, local or, nearby earthquakes, the nure sensitive ones or the short period ones, are the, , more, , desirable., , 104., , Determination of the Epicentre and the Focus., , The Epicentre. To determine the epicentre of an earthquake,, we determine what are called the epicentral distances of it from a net-, , work of Observing Stations or Observatories, the epicentral distance, of an earthquake from a given station bei/ig the shortest distance of its, epicentre from the station, measured along the surface of the earth, in, terms of the angle it subtends at the centre of the earth. This is done, with the help of the Tables, compiled by Zoppritz, Turner and others,, which give the relation between the epicentral distances of past, earthquakes and the interval between the first arrivals of the Primary, (P) and the Secondary (S) waves at a station, i.e., which express the, epicentral distances, S P., , as functions of the corresponding time-intervals, , Thus, from the seismogram of an earthquake, obtained at an, Observing Station, we can determine the time-interval S P for it at, that station, and the Tables then give the epicentral distance of the, earthquake from it. This is done at as many stations as possible., Circles are then drawn on a globe, with these different stations as, their respective centres and their epicentral distances as the radii., The point of intersection of these circles then gives the most probable, position of the epicentre of the earthquake in question. Or, the same, may be obtained from the method of least squares., , *Out of a, , set, , city, there are three, , of 8 seismographs at the, , Fordham, , University in, , New York, , which magnify the motion of the ground about 2,000 times.
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266, , FEOPEBTIBS O* MATTttft, , Usually, three seismographs are used for the purpose, one res, ponding to motions of the ground along the East- West direction, the, other along the North-South direction, and the third, having a pendulum suspended by a coiled spring, to respond to the vertical displacements of the ground. Tha information supplied by the three,, when pieced together, enables not only the epicentre, but also the, character, of the earthquake to be determined fairly accurately., , The focus. To determine the position of the focus, imme(ii), diately below the epicentre, we use what is known as Seebach's, method, explained below, , Let, , :, , be the focus of an earthquake, (taken to be, a distance h vertically below the epicentre, and let, O be the position of the Observing, Station, a horizontal distance d from the, , F, (Fig. 165),, , a point here),, , ,, , epicentre., , of the, , Then, assuming the homogeneity, medium in-between the earth's sur-, , and the surface of a sphere, concenwith it and passing through F, the, time t taken by the P waves to travel, face, , tric, , F to O, , from, , t, , *, ', , .., , where, , v is, , (, , clearly given, , by, , =, , whence, (d, , .V, , '., , is, , 2, , +A, , 2, ), , =, , v, , 2, , 2, ./, , the velocity of, , P, , waves in the, , medium and can be determined, , inde-, , pendently by other methods., at, which the earthquake occurs be T O *, and, time, if, the, Now,, the time at which the first P waves arrive at O be T, we have /, T O ), where T naturally varies with distance d., (T, Fig. 165., , =, , Thus, relation, , (i), , above, a, (</, , be put in the form,, , may, , +/l, , 2, ), , =, , V^T-TO)*., , Obtaining the corresponding values of d and t from a number of, different Observing Stations, we plot a graph between d and v/, which, gives a hyperbola, from which h can bo easily calculated out, and, hence the position of the focus determined., , A, , is to calculate, by the method of, most probable values of h and TO, may be of interest to know that the severest earthquakes have, , better method, however,, , least squares, the, , N.B., their foci, , It, , ., , about a hundred kilometres below the earth's surface., , Modern Applications of Seismology. The development of, 105., the modern science of seismology has led to its application in four, important fields, viz., (/) investigation of the nature of the interior of, prospecting for oils and minerals, (Hi) construction of, quake-proof buildings, and (iv) forecasting of the occurrence of, earthquakes., the earth,, , (ii), , It is now almost fully established (according to Jeffreys), (i), that the earth consists of a dense core of a molten mass, mostly of, *Tbis, though not, , known, , to us,, , is, , certainly constant.
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GBAV1TATIOS, , 267, , with some nickel, of a density of about 12*0 gms.jc.c.,, (at the pressure existing there), surrounded by a solid outer shell or, , iron, together, , about 3,000 kilometres thick, the density of which decreases, from about 5'Qgms./c.c. at its innermost layers to about 2-7 gms.jc.c., at the outermost layers, or at the surface of the earth., crust,, , The existence of the dense core is deduced from the observed, refraction of the seismic waves, as they pass through the earth, and, is further confirmed by the, production and propagation of the secondary or shake waves (5) through the core. These waves, as we know,, are transverse in nature and, as such, can only be produced and propagated in media, possessing elasticity of shape or rigidity, viz., in, solids., (/i), Prospecting for, oil, coal and other minerals is now being, increasingly done with the help of seismic waves*, the process being, technically known as 'seismic prospecting'., , Artificial earthquakes as set up in the ground-region to be, surveyed for the purpose, by detonating an explosive, like gun-cotton, , Fig., , 166., , or gelignite, at a point O on the earth's surface, (Fig. 106), and the, time of explosion noted. The time of arrival of the first low frequency longitudinal waves, or the primary waves, thus produced,, is noted, with the help of seismographs, at different stations P, Q, R,, The distances from 0, covered, S, etc., all lying in the same plane., along the chords OP, OQ, OR, OS etc. of the earth are carefully, measured and the mean velocities of the waves calculated along these, different paths or chords., If one of the paths or chords, say, OS, happens to pass through, a mineral deposit, like a salt dome, the value of the mean velocity, along this particular chord will be different from that along the other, chords. The experiment is then repeated along a direction, perpendicular to the first, by exploding a fresh charge of explosives., And,, if this confirms the results of the first experiment, a more elaborate survey determines the positions of the top and the sidesf of the, salt, , dome., , has now been found possible to erect 'quake-proof, in, California,, Japan and other places, frequently visited by, buildings, For,, earthquakes, at a surprisingly low additional cost of just 15%., it has been shown by Prof. Suyehiro that the severest earthquakes of, (Hi) It, , *We have already, , studied the gravitational methods of prospecting, by, etc, (sees, 80, on page 216)., sides of the dome is equally important, because some, almost always found to be there., , means of the Eotvos balance, t The locating of the, mineral oil, , is
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PROPERTIES OF MATTER, , 268, , little damage to buildings, designed to resist a horizontal force, equal to one-tenth of their total weight. The day is thus, not far off when damage to D lildings due to earthquakes will just, become a memory of a dreadful past., , Japan can do but, , And, finally, the prediction of the occurrence of an earth(/v), realm of, quake, a good timo in advanos, i<* also fast coming into the, the, that, established, now, boen, has, it, region,, practical possibility. For,, where an earthquake occurs, exhibits, for quite a few years before, a, rubber tube of a pneumatic, '////', or a gradual rise, very much like the, before, it actually bursts., a, bladder, football, or, swelling, up, tyre, There seems to be but little doubt that much sooner than we, can imagine at the moment, an earthquake forecast will become as, is today., general and universal an affair as the weather forecast, , But even as it is, the loss in buildings etc., due to the severest, confined to a, earthquakes, seldom exceeds 5%, due to their being, The disasone., uninhabited, an, small, area, and, quite often,, very, trous effects of earthquakes have thus been unduly magnified and,, for all we know, they may be for our own good, designed by a benign, Providence, by way of safety devicos to save us from being blown up,, all in a heap., ;, , SOLVED EXAMPLES, 1,, , cms. and, , Given, its, , G, , 6*7xlO~ 8, , =*, , mean, , density,, at the earth's surface., , 55, , c.g, , gms./c, , units, the radius of the earth, , s., , c.,, , = 64xlO, , calculate the acceleration due to gravity, , Imagining the earth to be a perfect sphere, we have, volume of the earth = |..n(6 4x 10V c cs.,, >, , And, , =, , mass, , .*., , 8, , -*.TM6*4xl0 )x5'5 gms., , m, , Consider a mass, gms. on the surface of ths earth. Obviously, the, force with which it is bsing attracted by ths earth towards its centre is, according, to the Law of Gravitation,, , ^|^6^10)2il5, (6'4xl0, this, , must, , g, , whence,, , **m.n.(6-4xl0, , x, , 6*4 x 5 '5, , x, , 6' 1, , =, ~, , 8, , )x5-5x6*7xlO-, , 1, , ', , ., , dynes., , be equal to the weight of ths mass,, nig, , Thus,, , =, , )", , ^rn.it, , /., , <y, , 8, , i.e.,, , mg., , *.7r.mx 6*4x5*5x6*7., , .*.mx6'4x5'5x6*7, , =, , 988'3, , cm, , Or, the acceleration dus to gravity at the earth's surface, , is, , Iscc*., , 988*3 cms. /sec*., , Two lead spheres of 20 cms. and 2 cms. diameter respectively are placed, 2., with their centres 100 cms. apart. Calculate the force of attraction between the, 8, as, spheres, given the radius of the earth as 6*67 x I0 cms. and its mean density, lead, of, 11*5)., gr., (Sp., 533gms,/c.c. ;, , =, , If the lead spheres be replaced, force of attraction be the same ?, ., , ., , Clearly, force of attraction, , -, , by brass spheres of the same, , ., , 3, , *.*.(10)*x* n.(l)B X (ll*5) .G/100, , -, , ., , between the masses, 8, , radii,, , would the, , ~, product, of the masses~ ", -, , ^, , ^/ 5/fl, a, , e \t, , 16n'xlO'x(ll'5) .G/9xlO<., , 8n a x(ll'5) 2 .G/45., 16"*x(ll-5>*xG/90, Now, force on a mass of one gram on the eajth's surface
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269, , a&AVTTATtOH, G.M/R*, , Or,, , G-~-~--X 5-33, , Or,, , -, , g, , 980, taking, , 980 cm*. /we*., , -980., , G.4.7c.*x5'33, , Or,, , -, , 960x3., , G9$Ox3/4.nJ?x5'33., , Or,, , force of attraction between the lead spheres, , ", , *, , x, , 45, , is, , given by, , 980x3, 4.n., , Rx5'33*, , J, , Since the force of attraction between the spheres depends upon their, it will naturally be different in 'he case of brass spheres whose mass wili, be much less than that of the lead spheres, (the density of brass being much less, , masses,, , than that of lead)., Calculate the mass of the earth from the following data :, 3., Radius of the earth -6x10* cms. ; Acceleration due to gravity = 980 cm./sec*., and Gravitational Constant =6 6x 10~ 8 cm.* gm.~ 1 .sec.~, JI, , We know, itself, (/'*.,, , which the earth, , that the force with, , towards, , its, , centre), , = xg, , mass towards, , attracts a unit, , 980 dynes., , 1, , Also, the force of attraction between the mass and the earth is given by, is the mass of the earth, R, it * radius and G, the Gravitational Constant. Clearly, therefore,, M.GJR*^ g., , AfxlxG/K*. where, , M, , M~g.R*IG., G in relation, , Or,, .*., , substituting the values of g,, , R, , and, , ..., , ..., , (/),, , (/), , we have, , A/~ 980x(6xl0 /66xlO- 2, , 8, , ), , the mass of the earth, , Or,, , is, , 53'47xl0 26 gms., 26, equal to 53*47 x 10 gms., , 4., Calculate the mass of the Sun, given that the distance between the Sun,, and the Earth is 1*49 x 10 13 cms., and G * 6 66x 10~ 8 c.g.s. units. Take the year to, consist of 365 days., (Punjab, 1942}, Let the mass of the Sun =, gms. and that of the Earth = m gms., Distance between the two, or the radius of Earth's orbit round the Sun, i.e.,, r = 149 x 10 18 cms., Time of ons revolution of the Earth round the Sun = 365 days., * 365 x 24 x 60 x 60 sees., , M, , Clearly,, , fone of attraction between, , Now,, , G.M.mlr*, , the, , Sun and, , =, , -xfrWx 10- dynes., on the Earth, , centripetal force acting, , And, the distance covered by the Earth, clearly=2nr, , = 2x l'49x 10, , .., , 18, , distance covered by, , it, , in, , 1, , =mv, , revolution,, , a, , /r., , i.e.,, , 2TTX1-49X10, , i, , 1, , sec.,, , in 365 days,, , 1', , v, , or,, , Hence, centripetal force on the Earth, , =, , in, , in its orbit, , cms., , ,u, , ,, , the Earth, , =, , mv*/r., , /2nxl-49xlO l8 \*, "\365 x 24 x 3600 /, , This must, clearly, be equal to the force of attraction between the Sun and, and, therefore,, , the Earth, , ;, , wx MX 6*66 xlQ-' _, ~~ m, J49X10 18 )*, , ', , Or,, , " --, , ., , 8, , *, , _, , f, 6-66xlQ(365x24 x 3600; x6'66xlO"**, M, 19'72xlO gms., the mats of the Sun - 19*72 x 10 M gms., , (365x24 X3600), Or,, , 4**xl 49x1 0", 1, '(365x24x3600), ', , M
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270, , MATTBB, , PttOWBBTiBS 0*, , Assuming that a sphere of mass 40 kilograms is attracted by a second, sphere of mass 80 kilograms, when their centres are 30 cms., apart with a force, equal to the weight of J mg. ; calculate the Constant of Gravitation., Here, force of attraction between the two masses, , 40x1000x80 x 1000 _, , G*, , 30*, , But this, 4, , equal to J mg. wt., , is, , 9, , G *, , r*, , Or,, , *, , 10*, , the value, , *, , 32 x --10*, --G, , G, , ', , 9, , X981, , =, of G ~, , 4600 x32x, , 9xTor, , i.e.,, , "4, , l, , 32 x 10 8, , =, , 6, , ', , 898x, , 6*898 x 10~ C.G.S. units., small balls of mass, each, are suspended side by side by two, equal threads of length /. If the distance between the upper ends of the threads, be a, find through what angle the threads are, pulled out of the vertical by the attraction of the balls., , m, , Two, , 6., , 8, , Let the upper ends of the threads be at, , AB, , distance, , A and, , a., , Due, , B, , B, (Fig., D 167),, " such that the, , to mutual attraction, the balls are, , drawn, , towards each other, say, through a distance x each,, , ~, , 71, , from, , their original positions., , Considering the forces acting on the ball 2,, which keep it in equilibrium, we have, , =, , the weight of ball, , (/), , jy, , mg, acting vertically, , downwards,, (fi), , the force of attraction,, , mx m, , N, , *, , """", , V, , \*, 1, , f, , ~", , F, , ', , **, , f, , v, , m, , (axY, , (a-x), , the tension of the thread T., , (111), , j, , *, , Since the ball is in equilibrium, the three, forces can be represented by the three sides of a, triangle, taken in order., , m9, Fig. 167., , dotted lines show the positions of the threads when the balls are, can be represented in magnitude as well as directhe side BQ, representing the weight mg,, tion by the sides of the triangle, y, the side QN, representing F and the side NB, representing the tension T of the, If the, , in equilibrium, the three forces, , string,, , So, , i/i, , cylic order., , -, , that, clearly,, , tan 9, , whence,, , $, , =, , BQN, , gj, , -, , -, , tan, , 6., , _, , Or,, , -, , tan, , 6., , (, , --, , Or,, , 9, , -, , /*/, , Thus, the threads will be pulled out of the vertical through an angle, tan- 1 mGI(ax)*s., , orbit, r, is 240,000 miles, and the period of, the diameter of the Earth is 8,000 miles and the value of, gravity on its surface is 32 ft./sec*. Verify the statement that the gravitational, force varies inversely as the square of the distance., 7., , revolution, , .., , The radius of the Moon's, , is, , 27 days, , ;, , 2rr x 240000 miles., Here, distance covered by the Moon in 27 days, 760x3, x, 240000x1, 2*, ., f, ~ A, f*-l* ee, velocity of the Moon, v, , ^, , =, , ., , ', , 27x24x60x60, , =, , v'/r. Hence, Now, centripetal acceleration of a body moving in a circle, the centripetal acceleration of the Moon towards the centre of the Earth, is giveo, , by, i, , *m, , "", , [, , 27x24x60x60, , J, , 240000 x 1760x3*
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AVITATIO, , 271, , 4frx 24x1 76x3x10^, , "", , 4frj<I76x3x 10., , ~(27x24x36)xlO, , ~T27x36)x24, , - 0-009189 ft.lsec*., , Then, denoting the acceleration due to gravity on the'surface of the Earth, //. per sec*.), and supposing it to be inversely proportional to the n ih, power of the distance, we have, by #,, , (= 32, , Rg, , where, , is, , and, , the radius of the earth,, , K m9, , the distance of the, , Moon from, , Earth., , ^ /, *, , C'009189, 32, , Or, , Or, taking logarithms,, , 4 4582, , 4000 x|760_x, , 3, , V, , 1, , V2406o6xl760xV, , " /, , V, , 1, , V, , 60, , /, , the, , ', , we have, , -, , n(2-2218)., , Or, n, 2', , 221 o, , Thus, g varies inversely as the second power of the distance, gravitational force varies inversely as the square of the distance., , and hence, , the, , 8., The radius of the earth is 6'37 x 10~ 8 cms., its mean density, 5*5, gms./c.c. and the gravitational constant, 6*66 xlO~- 8 c g s. units., Calculate the, earth's surface potential., , We know, , V, , that potential,, , Now, mass of, , distance,, , G, 6- 66, , GM/x., , the earth,, , =, , x, , M, , r,, , [Taking the earth to be a perfect sphere., , =, , volume x density, , in this case,, , =, , 8 3, *.n(6'37x 10 ) x 5'5,, , 6 37 x 10 8 cms., , 6-66 x 10~ 8 C.G.S. units (given)., , x JO- 8 x 4^(6 37 x "10 8 )x55, , 3x6-37xl0 8, , **, , 6'66x 10 8 x 471(6 37) 8 x5'5, , 3^, , 2'22x 10 8 x4rr(6 37)'x5'5 = 62'27x 10" ergs Igm., Calculate the intensity at a point due to an infinitely long straight wire, , 9., , of line density p., , AC, , be a portion of the wire,, density p, and consider, an element AB of the wire, of length dl., Let AO be a length / of the wire ; and let, P be a point at a distance x from O., Let, , (Fig. 168), of line, , Join PA, and, , Then,, , clearly,, , let, , IAPO =, , tan $, , =, , Differentiating, , 0., , Or, / =-= x.tan 0., with respect to 0,, , IJx., it, , we have, dl, , Therefore,, , And,, /., , .'., , -, , Fig. 168., , x.sec*Q.dQ., , mass of this element AB =, , intensity at, , intensity at, , Fdue, , P due, , to the element, , -, , x, , ', , sec, , 9, '?.G,, , in the direction, , PA., , *j, , to the element, x.sec*e.de.?, , x.jec 8, , G _ P.C.*, x, , along, , it into two rectangular components, along PO and perpendicular, the component along PO, equal to p.G.</0.co$ 0/x, and (ii) the, component at right angles to it, equal to p.G.dQ.sin Qjx., due to an equal element dl and the other end,, Similarly, the intensity at, Resolving it into, at a distance / from O will be p G.dQIx, in the direction PC., two rectangular components, the component p.G.dQ.sin 0/x, being equal and, opposite to that due to the element AB will cancel out and the component, , Resolving, , to, , it,, , we have, , (/;, , P, , ,, , ?.<#= x. sec'Q.dQ.?.
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272, , up., , FHOPKBTilS OF, , MAtflfifl, , ot $lx will act along PO, as before, and the tW6 will, therefore, be added, is true for any two similarly situated equal elements of the wirt., Therefore, considering the whole wire, we have, , The same, , P due, , intensity at, , to the, , =, , whole wire, , p>, , 2, , Or, intensity at a distance, , x due, , to, , an, , ', , &, -, , I, , x, , J, , infinitely, , EXERCISES, , cos, , **, , ., , L x, , long straight wire, , is, , sin, , JO, , 2p.G/x., , VII, , Mention different methods for determining the Constant of universal, and describe one which you consider to be the most accurate., (Punjab, 1940 and 1944), 2., What is meant by 'gravitation constant' ? What are its dimensions ?, Give an account of the experiments of Cavendish and Boys to determine this, 1., , gravitation,, , constant., , 3, , If, , G=, , 6-66, , xlO~ 8, , c.g, , s., , units,, , what, , is, , (Banaras, 1945}, the force between two small, Ans. 2*931 x 10 4 dynes., , spheres weighing 2 k.gms, placed 30 cms. apart., 4., State and explain Newton's law of gravitation and describe an accuWhat celestial evidence led, rate method of measuring the gravitation constant., to the formulation of the law ?, Is this law universally correct ?, Explain your, statement., (Calcutta, 1945), 5., If the earth were a solid sphere of iron, of radius 6 37 million metres,, and of density 7'&6gms /cms 3 ., what would be the value of gravity at its surface,, 8, taking the gravitational constant to be 6 658 x 10~ c.g.s. units ?, , Ans., , 1396 cms./ sec*., , Give the theory of Cavendish experiment, explaining how the density, of the earth is determined. Explain why and how Boys modified the Cavendish, method., (Madras, 1950), 7., Explain how Cavendish determined the value of gravitation constant., Indicate how, from the knowledge of the value of the gravitation constant,, it is a possible to calculate the mass of the earth., (Saugar, 1948), 8., 6*66 xlO~ 8 c g.s units, and the radius of the earth equal to, If G, 6-37 x 10 8 cms., what is the density of the earth ?, Ans. 5 62 gms.lcms*., 9., The earth moves round the Sun in a circle of radius 9*288 x 10 7 miles,, and completes a revolution in 365 days A satellite of Jupiter moves about the, Jupiter in a circle of radius 1*161 x 10* miles, completing one revolution in 16*6, days. Calculate the mass of Jupiter in terms of the mass of the Sun., 6., , Ans., 10., , Assuming the law of gravitation,, , find, , 945xlO~ f, , ., , an expression for the period of, , revolution of a planet., The moon describes a circular orbit of radius 3*8 x 10 5 km. about the earth, in 27 days and the earth describes a circular orbit of radius l*5x 10" km. round, the Sun in 365 days. Determine the mass of the Sun in terms of that of the earth., , (Bombay, 1935), Ans. 3*366x10*., 11., , for, , Define the gravitational constant and describe a laboratory method, , measuring, , it, , accurately., , A, , small satellite revolves round a planet of mean density 10 gmsJc.c., the, radius of its orbit being slightly greater than the radius of the planet. Calculate, 6*66 x 10~ 8 c g.s. unin), the time of revolution of the satellite. (G, (Bombay, 1940), Ans. 1-044 hours., , =, , 12., Define 'Potential' and 'Potential Energy* of a gravitational field., Derive an expression for the potential due to a sphere of uniform density at an, , external point., , The radius of the earth is 6-37 x 10* cms., its mean density 5*5 gmsjcm*., and the gravitation constant, 6 66x 10-*. Calculate the earth's surface potential., (Agra, 1940), , Ans.
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27?, , GRAVITATION, , 13., What is meant by the gravitational potential? How does it vary, with the distance from^tbe centre of the earth ? What initial velocity would be, required to project a body be>ond the attractive force of the earth ? (Radius of, earth is 6*4 x 10 8 cms.), (Cambridge Scholarship), Ans. 1*12 xlO cms.jsec., 14., Explain what you mean by gravitational potential at a point. How, does it differ from other kinds of potential with which you are familiar ?, Find an expression for gravitational potential due to a thin hollow sphere, of uniform density at a point outside it., (Calcutta, 1947), 15., Two balls, each weighing 10 gms are hung side by side by threads, 10, metres long. If the threads are I cm. apart at the upper ends, by how much is the, distance between the centres of the balls less than 1 cm., Ans. l-5xlO- e cmi., ,, , 16., Describe one of the most accurate methods of measuring the constant, of gravita-tion., The star Sirius has a mass of 6*9 x 10 3S gms. and its distance is 8x 10 18 km., The mass of the earth is 6,x 10 27 gms., The tensile strength* of steel is about, Calculate the cross-section of a steel bar which could just with20,000 kg./cm*, stand the gravitational pull between Sirius and the earth. (G, 6'67xlO~ 8, , =, , dyne-cm*. Igm~*., 1, , 7., , (Bombay, 1951), Ans. 2*169x, sq. cm., velocity with which a particle must be projected, , W, , Prove that the, , least, , it may escape, ofji planet of radius R and density p in order that, Calculate the, /?\/8rcO>/3, where G is the gravitational constant., velocity in the case of the moon from the following data, 3'36 gms.jc.c. ;, mean density of earth * 5*52 gms /c.c,, mean density of moon, 1740 km. ;, 638 km., mean radius of moon, mean radius of earth, 980 cms. per sec. per sec., Acceleration of gravity at earth's surface, (Oxford Scholarship), 5, Ans. 2'38 xlO cm. see' 1, , from the surface, , completely, , is, , :, , ;, , ;, , =, , =, , ., , Describe an accurate 'balance -method* for the determination of the, value of G, and write a short note on the 'qualities of gravitation\, 19., What are seismic waves ? Give a brief description of their charac*, teristics., How may they be detected ? Also mention some of the applications of, 18., , tcismology., 20., , What, , is, , an earthquake, , ?, , How, , is, , it, , caused ?, , Describe in brief the, , principle underlying seismographs. Why are they so called ?, 21. Describe in detail Galitzin's seismograph and explain, , how, , the epicentre, , and thefccus of an earthquake may be determined with its help., 22., What is geophysical prospecting ? Write a short descriptive note oa, (/) the gravitational and (11) the seismic methods used for the purpose., , *See next chapter.
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CHAPTER, , VIII, , ELASTICITY, 106., All bodies can, more or less, be deformed, Introductory., by suitably applied forces. The simplest cases of deformation are, those (/) in which a wire, fixed at its upper end, is pulled down by, a weight at its lower end, bringing about a change in its length and, (//) in which an equal compression is applied in all directions, so that, there is a change of volume but no change in shape, or (///) in which a, system of forces may be applied to a body such that, although there, is no motion of the body as a whole,, there is relative displacement, of its continuous layers causing a change in the shape or 'form' of, the body with no change in its volume., In all these cases, the body is, said to be strained or deformed., , When, , the, , deforming forces are removed, the body tends to, , For example, the wire, in the case, original condition, above, tends to come back to its original length when the force due, to the suspended weight is romoved from4t, or, a compressed Volume, of air or gas throws back the piston when it is released, in an attempt, to recover its original volume. This property of a material body to, a, regain its original condition, on th removal of the deforming forces,, recover, , its, , is called elasticity., Bodies, which can recover completely their, original condition, on the removal of the deforming forces, are said, to be perfectly elastic. On the other hand, bodies, which do not show, any tendency to recover their original condition, are said to be plastic., , There are, however, no perfectly elastic or plastic bodies. The nearest, approacli to a porfectly elastic body is a quartz fibre and, to a perBut even the former yields to large, fectly plastic body, is putty., deforming forces and, similarly, the latter recovers from small deformations., Thus, there are only differences of degree, and a body is, more elastic or plastic when compared to another., , We shall consider here only bodies or substances, which are (/), homogeneous and (//) isotropic, i.e., which have the same properties at, all points and in all directions., For, these alone have similar elastic, properties in every direction, (together with other physical properties, like linear expansion, conductivity for heat, , index, class,, , and, , electricity, refractive, , Fluids (i.e., liquids and gases), as a rule, belong to this, but not necessarily all solids, some of which may exhibit, , etc.)., , different properties at different points and in different directions, i.e.,, may be heterogeneous (or non-homogeneous) and anisotropic (or nonExamples of this class of solids are wood, and crystals in, isotropic)., , general, including those metals, which are crystalline in structure., As a class, however, metals, particularly in the form of rods and, wires,, may b3 regarded to be more or less wholly isotropic, in so far, , as their elastic behaviour, , is, , concerned., , As a result of the deforming forces, Stress and Strain., 107., applied to a body, forces of reaction come into play internally in it,, ., , 274
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275, , ELASTICITY, , due to the, , relative displacement of its molecules, tending to restore, to its original Condition., The restoring or recovering force per unit, area set up inside the body is called stress, and is measured by the, deforming force applied per unit area of the body, being equal in, magnitude but opposite in direction to it, until a permanent change, has been brought about in the body, i.e., until its elastic limit has, been reached, (see, If the force be inclined to the sur108, below)., face, its component, perpendicular to the surface, measured per unit, it, , called normal stress* an'l the component acting alon^ the, is called tangential or shearing stress. Further,, the former may be compressive or expansive (i.e., tensile) according as, a decrease or increase in volume is involved, Obviously, being force, per unit area, the units and dimensions of stress are the same as, area,, , is, , surface, per unit area,, , those of pressure,, , viz.,, , ML~, , J, , T~ Z, , ,, , (see, , page, , 5)., , The change produced in the dimensions of a body under a system, of forces or couples, in equilibrium, is called strain, and is measured by, the change per unit length (linear strain), per unit volume, (volume, strain), or the angular deformation, (shear strain, or simply, shear)"\, according as the change takes place in length, volume or shape of the, body. Thus, being just a ratio, (or an angle) it is a dimensionless, , quantity, having no units,, seen that for a perfectly elastic body (/) the, always the same for a given stress, (//) the strain vanishes, completely when the deforming force is removed and (Hi) for maintaining, It, , will be readily, , strain is, , ;, , the strain, the stress, 108., elasticity, , is, , constant., , Hooke's Law. Hooke's law, and states that, provided the, , the fundamental law of, , is, , strain, , is, , small, the stress, , is, , proportional to the strain so that, in such a case, the ratio stress/strain, is a constant, called the modulus of elasticity, (a term first introduced, by Thomas Young), or the coefficient of elasticity., ;, , ,, , Since stress is just pressure, (or tension per unit area), and strain, just a ratio, the units and dimensions of the modulus of elasticity, are the same as those of stress or pressure., is, , When, , the stress is continually increased in the case of a solid, a, reached at which the strain increases more rapidly than is, warranted by Hooke's Law. This point is called the elastic limit,, and if the b'xly happens to be a wire under stretch, it will not regain, its original length on being unloaded, if the elastic limit be passed, as it, On loading it further, a, acquires what is called a 'permanent set'., point is reached when the extension begins to increase still more, rapidly and the wire begins to 'flow down* in spite of the same constant, This point is called the 'yield point', load., and, after a large ex9, In the, tension, it reaches the 'breaking point and the wire snaps., case of plastic substances, like lead, there is a long range between the, yield point and the breaking point., point, , is, , ;, , *The stress is always normal in the case of a change in the length of the, wire, or in the case of a change in the volume of a body, but is tangential in the, case of a change in the shape of a body., tThis will be dealt with more fully later in, , 109, , (3).
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276, , PROPERTIES OF MATTER, , Thus, if we were to plot a graph between the load suspended, from a wire, fixed to a rigid support at its upper end, and the, extension produced thereby, w*, obtain, in general, a curve of, the form shown in Fig 169,, the straight part OA of the curve, showing that the extension produced is directly proportional to, the load applied, or that Hooke's, , law, , obeyed perfectly up to A,, therefore, on being, unloaded at any point between, and A, the wire will come, back to its original condition,, In other, (represented by O)., , and, , in, , question,, , is, , that,, , words, the wire is perfectly elastic, up to A, which thus measures, the elastic limit* of the specimen, the extension here being of the order of 10~ 8 of the, , original length., , On loading the wire beyond the elastic limit, say, up to B, the, curve takes a bend almost vertically upwards, as shown, and, on being, unloaded at any point here, (at B, say), it does not come back to its, original condition but takes the dotted path BC, thus acquiring a 'permanent, , set', , OC., , D, , On, , is reached, where, increasing the load still further, a point, the extension is much greater even for a small increase in the load,, i.e., Hooke's law is obeyed no longer, and, beyond D, the extension, increases continuously, with no addition to the load, the wire starting, For, due to its thinning down, the stresS, 'flowing down', as it were., (or the load per unit area) increases considerably and it cannot, and, if the wire is to be presupport the same load as before, vented from 'snapping', the load applied to it must be decreased., That is why the curve starts turning towards the extension-axi, beyond this point D, which thus represents the yield point of the, wire., And, once the yield point is crossed, the thinning of the, wire no longer remains uniform or even, its cross-section decreasing, more rapidly at some points than at others, resulting in its developing small 'necks or 'waists' at the former points, so that the stress, is greater there than at the latter points, and the wire ultimately, This point on the curve, at which the, 'snaps' at one of these., snapping or the breaking of the wire actually occurs, is called ita, the corresponding stress and strain there being, breaking point,, referred to as the breaking stress (or tensile strength) and the breaking, ;, , ;, , 9, , ;, , strain, respectively., , Note., , The tlastic limit of a material is also sometimes defined as the, maximum reversible or recoverable deformation in it, and may,, , force producing the, , *Jn quite a few cases, Hooke's law is obeyed only up to a point a little, below the elastic limit, represented by A. The portion of the curve from O, to this point (below A), is then**aid to indicate the limit of proportionality, to, distinguish it from the elastic limit. The two are thus not always identical,, though they are generally regarded to be so, in view of the very small difference, between them.
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277, , ELASTICITY, , for a given specimen, be determined by loading and unloading it with a number, of different loads and measuring its length afterVacA unloading, until it acquires, a permanent set. The latter is then plotted against the load, and from the curve, thus obtained, the particular load at which the permanent set just starts, can be, easily estimated., , Even within the elastic limit, however, few solids come back to, their original condition, directly the deforming force is removed., Almost all of them onfy 'creep' back to it, (i.e., take some time to do, This delay in recovering back, so), though they all do so, ultimately., the original condition, on the cessation of the deforming force, is called, clastic-after effect., Glass exhibits this effect to a marked degree, the, few exceptions to this almost general rule being quartz, phosphorbronze, silver and gold, which regain their original condition as soon, as the deforming force ceases to operate. Hence their use in Cavendish's and Boys' experiments for the determination of G, in quadrant, electrometers and moving-coil galvanometers etc. etc., , As a natural consequence of the elastic after-effect the strain in, material, (in glass, for example), tends to persist or lag behind the, stress to which it is subjected, with the, result that during a rapidly changing, stress, the strain is greater for the same, a, , value of stress,, , when, , when, , it is, , decreasing than, , increasing, as is clear from the, curve in Fig. 170. This lag between stress, it is, , and strain, term, , is, , called elastic hysterisis, (the, meaning 'lagging be-, , 'hysterisis\, , The phenomenon is similar in its, )., implications to the familiar magnetic hysterisis, where the magnetic effects tend, to persist or lag behind even after the, magnetising influence is removed, the, curve referred to above may thus be called, hind, , 1, , XTAWOAf, , >, , X, , p| gt 170., , the elastic hysterisis loop. And, exactly, in the same manner the energy, dissipated as heat, during a cycle of, loading and unloading is given by the area enclosed by the loop., There is, however, very little hysterisis in the case of metals or of, quartz., Further, it was shown by Lord Kelvin, during his investigation, of the rate of decay of torsional vibrations of wires, that the vibrations died away much faster in the case of a wire kept vibrating continuously for some time than in that of a fresh wire. The same, happens to any elastic body, subjected to an alternating strain. The, continuously vibrating wire got 'tired* or 'fatigued', as it were, and, found it difficult to continue vibrating. Lord Kelvin fittingly expressed this by the term 'elastic fatigue'., , A, , body, thus subjected to repeated strains beyond its elastic, may break under, within its elastic, This phenomenon is, obviously, of great importance in cases, limit., like those of the piston and the connecting rods in a locomotive,, , limit, has its elastic properties greatly impaired, and, A stress, less than its normal breaking stress even, , which, as we know, are subjected to repeated tensions and compressions during each revolution of the crank shaft.
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278, , PROPERTIES OF MATTER, , It may be mentioned here that all these elastic properties of a material are, linked up with the fine mass of its structure. It is now finally established by careful microscopic examination, that metals are just an aggregation of a large, number of fine crystals, in most cases, arranged in a random or a chaotic, fashion^ i.e., their cleavage planes (or the planes along which their constituent atoms, can easily slide over each other), being distributed haphazardly, in all possible, directions. Now, single crystals, when subjected to deformation, show a, remarkable increase in their hardness. Thus, for example, a single crystal of, silver, on being stretched to a little more than twice its length, is known to, increase to as much as ninety-two times its original strength or stiffness. So, that, operations like hammering and rolling, which help this sort of distribution,, i.e., which break up the crystal grains into smaller units, result in an increase or, extension of their elastic properties whereas, operations like annealing (or heating and then cooling gradually) etc., which tend to produce a uniform pattern of, orientation of the constituent crystals, by orienting them all in one particular, direction and thus forming larger crystal grains, result in a decrease in their clastic properties or an increase in the softness or plasticity of the material., ;, , This is because in the latter case, slipping (or sliding between cleavage, planes), starting at a weak spot proceeds all through the crystal and, in the, former, the slipping is confined to one crystal grain and stops at its boundary, with the adjoining crystal. Indeed, the former may be compared to a small cut,, developing into a regular tear all along a fabric and the latter to the tear stopping, as it reaches a seam in the fabric. Thus, 'paradoxically', as Sir Lawrence Bragg, puts it, */ order to be strong, a metal must be weak,* meaning thereby that metals, with smaller grains are stronger than those with larger ones., , A, , change in the temperature also affects the elastic properties of a, material, a rise in temperature usually decreasing its elasticity and vice versa,, except in certain rare cases, like that of invar steel, whose elasticity remains practically unaffected by any changes in temperature. Thus, for example, lead becomes, quite elastic and rings like steel when struck by a wooden mallet, if it be cooled, in liquid air. And, again, a carbon filament, which is highly elastic at the ordinary, temperature, becomes plastic when heated by the current through it, so much so, that it can be easily distorted by a magnet brought near to it., , 109., , Three Types of Elasticity. Corresponding to the, we have three types of elasticity, v/z.,, , three*, , types of strain,, , (/) linear elasticity, or elasticity of length, called Young's Modulus,, corresponding to linear (or tensile) strain, ;, , (i7), , strain, , ;, , elasticity of volume or Bulk Modulus, corresponding to volume, , and, , of shape, shear modulus, or Modulus of Rigidity,, corresponding to shear strain., When the deforming force is applied to, (1) Young's Modulus., the body only along a particular direction, the change per unit length, in that direction is called longitudinal, linear or elongation strain, and, the force applied per unit area of cross-section is called longitudinal, (Hi) elasticity, , The, , or linear stress., the elastic limit,, the letter Y., , Thus,, , if, , is, , F be, , a, the stress is F/a., , ratio of longitudinal stress to linear *trnin, within, called Young's Modulus, and is usually denoted by, , the force applied normally to a cross-sectional area, And, if there be change / produced in the origi-, , nal length L, the strain, , is, , given by, , Young's Modulus,, , Now,, , if, , In other words,, , if, , L, , 1,, , a, , =, , Y, , =, , So that,, , //L., , -Jijju, , 1, , and, , /, , =, a,, = 1, we have, ., , -., , i, , Y, , F., , a material of unit length and unit area of cross-
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279, , ELASTICITY, , section could be pulled so as to increase in length by unity, i.e., to, double its length, the force applied would measure tbe value of, Young's Modulus for it., is exceeded when the extension, 10~ 8 cm./cm., the material will snap before this much, produced., , Since, however, the elastic limit, , produced, , is, , extension, , is, , In cases, where, elongation produced is not proportional to the, force applied, we can still determine Young's Modulus from the ratio, L.dF/a.dL, where dF/a is the infinitesimal increase in the longitudinal, stress, , and dL/L, the corresponding increase, , in strain., , *'%', dL, , Or,, , a, , N.B. The particular case of rubber may, with advantage, be mentioned, here, which the beginner finds so confusing, when, in ordinary conversational, language, we refer to it as being 'elastic*. For, he knows well enough that it, requires a much smaller force than steel to stretch it, (and that, therefore, its elasIn fact, the value of Young's Modulus for, ticity is much less than that of steel)., rubber is about one-fiftieth of that of steel. What we mean when we say that it is, elastic, therefore, is just that it has a very large range of elasticity, for, whereas a, crystalline body can be stretched to less than even one per cent of its original, length before reaching its elastic limit, rubber can be stretched to about eight, times (or 80%) of its original length., This high extensibility of rubber is due to its molecule containing, on an, some 4,000 molecules of isoprene (C6 8 ), whose 20,OCO carbon atoms,, spreading out in a chain, make it very long and thin, about 1/4000 mm. in length., , #, , average,, , Rubber, in bulk, has thus been rightly compared to an intertwined mass, of long, wriggling snakes, its molecules, like the snakes, tending to uncoil when, stretched and getting coiled up again when the stretching force is removed., , Here, the force is applied normally and uni(2) Bulk Modulus., formly to the whole surface of the body so that, while there is a, change of volume, there is no change of shape. Geometrically speaking,, therefore, we have hero a change in the scale of the coordinates of the, system or the body. The force applied per unit area, (or pressure),, gives the Stress, and the change per unit volume, the Strain, their ratio, It is usually denoted by the, giving the Bulk Modulus for the body., ;, , letter, , K., , F, , be the force applied uniformly and normally on a surThus, if, face area a, the stress, or pressure, is F/a or P, and, if v be the, change in volume produced in an original volume K, the strain is v/K., ;, , and, therefore,, , Bulk Modulus,, , K=, , F, !*, , = Fy ', , a.v, , v/V, If,, , however, the change, , in, , rv Fla, l, , v, , ', , -/., , volume be not proportional to the, , stress or the pressure applied, we consider the infinitesimal change in, volume dV, for the corresponding change in pressure dP so that,, , we have, , K=, , ;, , d, , The Bulk Modulus is sometimes referred to as incompressibility, and hence its reciprocal is called compressibility so that, compressibility of a body is equal to l/#, where K is its Bulk Modulus. It must, thus be quite clear that whereas bulk modulus is stress per unit, ;, , strain, compressibility represents strain per unit stress.
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280, , PBOPHBTIBS OF MATTER, , Since fluids (i.e., liquids and gases) can permanently withstand, or sustain only a hydrostatic, pressure, the only elasticity they possess, is Bulk Modulus, (K), which is, therefore, all that is meant when we, refer to their elasticity., This, however, is of two types, isothermal, and adiabatic., :, , For, when a fluid is compressed, there is always some heat produced. If this heat be removed as fast as it is, produced, the temperature of the fluid remains constant and the, change is said to be, isothermal, but if the heat be allowed to remain in the fluid, its, temperature naturally rises ard the change is then said to be, ;, , adiabatic., , It can be easily, , shown that the isothermal elasticity of a gas (i.e.,, temperature remains constant) is equal to its pressure P, and, adiabatic elasticity equal to yP, where y is the ratio between, C>*, , when, its, , and, , its, , Cy *, , solved, , for the gas in question,, , Example, , 1 (b), , its value being 1*41 for air, [see, at the end of the Chapter.], , It will thus be readily seen that the Bulk Modulus of a, gas, fwhether isothermal or adiabatic) is not a constant quantity, unlike, that of a solid or a liquid., , In this case, while there is a change, (3) Modulus of Rigidity., in the shape of the, body, there is no change in its volume. As indicated already, it takes place by the movement of, contiguous layers of, the body, one over the other, very much in the manner that the cards, would do when a pack of them, placed on the table, is pressed with, the hand and pushed horizontally. Again,, speaking geometrically, we, have, in this case, a change in the inclinations of the coordinate axes, of the system or the body., , 171),, , Consider a rectangular solid cube, whose lower face aDCc,, (Fig., fixed, and to whose upper face a tangential force Fis applied, in the direction shown. The couple, ,, , is, , so produced, , by, , this force, , and an, , equal and opposite force coming, into play on the lower fixed face,, makes the layers, parallel to the, two faces, move over one another,, such that the point A shifts to A', B to B', rf to d' and b to 6', i.e.,, the lines joining the two faces turn, through an angle 0f., t, , F '3- 171., The face A, is then said to, be sheared through an angle 8. This angle, (in radians), through which, a line originally perpendicular to the fixed face is turned, gives the strain, or the shear strain, or the angle of shear, as it is often called. As will, , BCD, , *The symbols Cp and^C*, stand for the specific heats of a gas at constant, pressure and at constant volume respectively, -their ratio r, C>/C, being the, highest (1*67) for a mono-atomic gas, like helium, goes on decreasing with increasing atomicity of the gas but is always greater than 1., fAs a matter of fact, if this were the only couple acting on the body, it, would result in the rotation of the body. This is prevented by another, equal and, opposite couple, formed by the weight of the body (plus any vertical force applied), and the reaction of the surface on which the body rests., , =
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281, , ELASTICITY, , = A A' /DA =, , where /is the displacement AA', or the height of the cube or 9, relative displacement of plane ABbajdistance from the fixed plane, aDCc. So that, if the distance from the fixed plane, i.e., L, 1, we, have 9, /, relative displacement of plane ^4Z?6a., be readily seen, &, , and X, the length of the, , side, , II L,, , AD, , =, , ;, , =, , = =, , Thus, shear strain (or shear) may also be defined as the relative, displacement between two planes unit distance apart., , And, stress or tangential stress is clearly equal to the force F, The ratio of, divided by the area of the face ABbd, i.e., equal to Fja., the tangential stress to the shear strain gives the co-efficient of rigidity, of the material of the body, denoted by n., tangential stress, , Thus,, , And, therefore,, , Fla,, , Co-efficient, , the material of the cube, n, , =, , =, , is, , F/a, , and shear, , strain, , =, , =, , 6, , //L., , of Rigidity, or Modulus of Rigidity of, , given by, , ~~, , Fl a, -, , =, ~~, , ~~L, , (i), , a- 1, , I/L, , to, , This is a relation exactly similar to the one for Young's Moduis the tangential stress, not, with the only difference that, here,, a linear one, and I, a displacement at right angles to L, instead of along, , F, , lus,, , it., , Again, if the shearing strain, or, the shearing stress applied, we have, , shear,, , be not proportional to, , *L, , fl, , where d0ia the increase in the angle of shear for an infinitesimal increase dF/a in the shearing stress., Further, it is clear from relation (/) above, that, F., radian (or 57 18'), we have n, , =, , Q, , =, , 1, , if, , a =, , 1,, , and, , We, , may thus define modulus of rigidity of a material as the shearing stress per unit shear, i.e., a shear of I radian, taking Hooke's law, to be valid even for such a large strain*., 110., , Equivalence of a shear to a compression and an extension, , Consider a, right angles to each other., with the face, fixed, and let the face, A, A, be sheared by a force, applied, in the direction shown,, through an, , at, , DC, , BCD, , cube, , A BCD,, , (Fig. 172),, , R, , j, , 0,, , ^, , ^,, , V", , the position A'B'CD., is inThen, clearly, the diagonal, creased in length to DB', and the diais shortened to A'C., gonal, , angle, , n, , into, , DB, , AC, , The shear, , is really very small in, actual practice, and, therefore, triangles, AFA' and BEE' are isosceles right-angled, -, , triangles, (i.e., right-angled, , 45, , ~, , *, , F, , j, , g, , i 72, , triangles)., , *In the case of metals, however, Hooke's law no longer holds even, shear exceeds 11/200 radian, or '33., , if, , the
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282, , PROPERTIES OF MATTER, , And, therefore,, , v, , EB, , Z.#fi', , 1, , =, , = BB, , 1, , cos, , ., , AB =, , .-,, , BB'E, , =, , =, , 45 and cos 45, , DB =, , =, , If, /, then, clearly,, extension strain along diagonal D2?, __, , "5', , DB, , /-y/2,, , BB', , I, , BB', , e, , -y/2, , /\/2, , 2/, , 2, , Similarly, the compression strain along the diagonal, , '., , cosA'AF, , is, , given, , 45, , AC, [v, , Thus, we see that a simple shear B is equivalent to two equal, an extension and a compression, at right angles to each other., , strains,, , Corollary., , The converse of the, , above follows as a corottary^viz., that, simultaneous equal, "compression and, extension at right angles to each other, are equivalent to a shear, as will be, seen from the following, :, , Let the cube, , ABCD,, , of side /,, be 'compressed along the diagonal AC,, so that the new diagonals become A'C', , and B'D',, Let, , And, Fig., , AA', , OA, , since, , =, , 173., , we have, , (Fig. 173)., , AB., , cos 46, , -=, , AB/\/2, , OA'-OA-AA'-^-a)., OB - OB+BB' -, , ', , ', , and, Clearly,, , = BB' = a., = AB cos BAO, , (A'B'f, , .-., , =, , +, , (, , (OA')*+(pB')*, 2, , I, , In practice, 2a* is very small as compared with, therefore, be neglected., , So that,, , (A'B')*, , a)., , =, , /*., , Or,, , AB' =, , /, , /, , a, ,, , and may,, , = AB., =, , Thus, A'B'C'D' may be rotated through the angle DGD', ^l7^', so that D'C coincides with DC. Then, it is obvious that, A'D' would make an angle 2^4F^4' with AD, so that the angle of shear, is equal to twice the angle AFA', i.e., is equal to 2, , angle, , LAFA, , Or,, , flflgfe, , of shear, , = 2^'/F,, , '., , (/ the angle, , is
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283, , ELASTICITY, , where A'E, , is, , the perpendicular from A' to AF., t, , A'E, , Now,, , =, , EF =, , *Av/2 and, , 7/2., , ~, , ~, , V2, , 2, , Denoting this angle of shear by, , Now, compression, /L4', , we have, , 6, , =, , 2fl\/2//*, , strain along the diagonal ^4C, , ^, , -, , Ad, , 0,, , /, , f, , ~, , fl\/2 __, ~~, , //V2, , is, , *, , 7, , 2, , Or, the compression strain is half the angle of shear,, angle of shear is twice the angle of compression., Similarly,, , it, , can be shown that the extension strain, , is, , i.e.,, , the, , also half the, , angle of shear., , Thus, we see that simultaneous and equal compression and extension at right angles to each other are equivalent to a shear, the direction, of each strain being at an angle of 45, , to the direction of shear., , 111., Shearing stress equivalent to an equal linear tensile stress, and an equal compression stress at right angles to each other. In the, UtVDO, case VJL, if JT, of the, tilt? cube, UULJC above,, l/ilt?, VrClU the, Fwere, itUUVU, 11, J?jC, f" /p\, only force acting on its upper face it, p, ^, would move bodily in the direction of, B, this force., Since, however, the cube is, fixed at its lower face DC, an equal and, ', , \, , opposite force comes into play in the, plane of this face, giving rise to a couple, F./.*, tending to rotate the cube in the, clockwise direction, (Fig. 174)., , F-F, , Again, since the cube does not, is obvious that the plane of, , rotate, it, , DC applies an, , equal and opposite couple, by exerting forces F' and F', along the faces AD and CB, tending to, , F'./,, , rotate, , cube, , ^, "", , say.,, , it, , is, , Fig., , 174., , Thus, because the, under the two couples, we have, , in the anticlockwise direction, as shown., , in equilibrium, , F.I, , =, , F'.l, , Or,, , F, , =, , F',, , a tangential force F applied to the face AB results in an equal tangential force acting along all the other faces of the cube in the directions, shown., i.e.,, , Clearly the resultant of the two forces F and F' or F and F, and CB respectively is F\/ 2 along OB and of those acting, along, is also F\/2 along OD. And, thus, an outward pull, and, along, of the cube at B and, acts on the diagonal, resulting in its, extension, as we have just seen above, (110)., Precisely similarly,, an inward pull acts on the diagonal AC at A and C, thereby bringing, , AB, , AD, , t, , CD, , DB, , about, , its, , D, , compression., , being the length of each edge of the cube and hence the perpendicular, distance between the two forces Fand F., */
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284, , PROPERTIES OP MATTER, , Thus, a tangential force F applied to one face of a cube gives, a force F\/2 outward along one diagonal (BD, in the case, fihown) and an equal force F\/2 inward along the other diagonal (AC), of the cube, resulting in an extension of the former and a compression, of the latter., rise to, , the cube be cut up into two halves, by a plane passing, and perpendicular to the plane of the paper, each face,, 2, /, parallel to the plane, will have an area lxl\/2, \/^ an<* dearly,, the outward force F\/2 along BD will be acting perpendicularly to it., So that, we have, , Now,, , through, , if, , AC, , tensile stress along, , BD =, , F\/'2/l* \/2, , =, , 2, , F//, , ., , we cut the cube, , into its two halves by a plane, and perpendicular to the plane of the paper,, we shall have an inward force Fi/2 along AC acting perpendicularly, to a face on an area / \/2. So that, we have, Similarly,, , passing through, , if, , BD, , compression stress along, 2, , AC =, , F\/2jl *i/2, , the shearing stress over the face, Obviously, F//, which produces the shear $ in it, (see page 281)., , Thus,, , it is, , tensile stress, , is, , =, , 2, , F//, , ., , AB of the, , cube,, , clear that a shearing stress is equivalent to an equal, stress at right angles to each, , and an equal compression, , other., , 112., Work done per unit volume in a strain. In order to deform, a body, work must be done by the applied force. The energy so, , When, spent is stored up in the body and is called the energy strain., the applied forces are removed, the stress disappears and the energy, of strain appears as heat., Let us consider the work done during the three cases of strain., (i), Elongation Strain (stretch of a wire). Let F be the force, applied to a wire, fixed at the upper end. Then, clearly, for a small, increase in length dl of the wire, the work done will be equal to F.dl., And, therefore, during the whole stretch of the wire from to /., , work done, , Now, Young's modulus, , =, , for the material of the wire,, , Y, , =, , L is the original length, /, the increase in length,, ectional area of the wire, and F, the force applied., , where, , And, is, , /., , F=, , i.e.,, , F.L/a.1.,, a,, , the, , L', , t*, , 2, , -, , Y.a.ljL., , Therefore, work done during the stretch of the wire from, given by, , Y.a, , cross, , ", , 1, , Y.a.l, , 2""/r, , ,, , to, , /
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285, , ELASTICITY, , But, , = F, the force applied., F.I =, x stretching force x stretch., , Y.i.llL, , W=, , Henoe, ., , -, , ^, , ,, , unit, , = ^1, , I, , 1, , F, , = UI, , ', , ', , IT, , T-, , Alternatively, the, , as follows, , volume, , ., , work done per, , same, , =, , _, , fV, , 1, , f, , Fix -y~, Lt, a, , L, , 2, , result, , v, of the, volume, , w, , i, , re _., , Lxa, , //L=strain, , may, , ., , also be obtained graphically, , :, , Let a graph OP be plotted between the streitching force applied, and the extension produced in it, within the elastic limit ,, , to the wire, , as, , shown, , in Fig. 175., , Consider a small extension pq of the wire and erect, ordinates at, , p and q to meet the, and q respectively,, 1, , in p', , graph, where pp', to, , is, , very nearly equal, pq being, , (the extension, really small)., <?#'>, , Then, clearly, work done, upon the wire or energy stored, up in it, s=stretching force pp' ^extension, pq., , =pp'Xpq=area of, So, , that,, , strip pp'q'q., , EXTENSION-, , the, , imagining, , whole extension OB = /, of the, wire, to be broken up into small, bits like/N?, , Fig. 175., , and erecting ordinates at their extremities, we have, , total work done upon the wire or total energy stored up in it, sum of the areas of all such strips formed, area of the triangle OBP, \OBxBP $/x/s, / and the stretching force correswhere the total extension OB, , =, =, , =, , =, , =, , ponding to, , Now,, , BP =, L be the, , it is, , F., , if, , original length of the wire, , cross-section, clearly, volume of the wire, .-., , work done, or, , =, , and, , = L x a., , strain energy, per unit volume, 1, , \lFIL.a *=, , v~, A, , F, --, , I, , 1, , -!-=, o, ju, , a, , st r ess, , a, its, , area of, , of the wire, , x strain., , Let p be the stress applied. Then, over, (ii) Volume Strain., an area a the force applied is p.a, and, therefore, the work done for a, small movement dx in the direction of p, is equal to p.a.dx. Now,, a.dx is equal to Jv, the small change produced in volume. Thus,, work done for a change dv is equal to p.dv., And, therefore, total work done for the whole change in volume, from, to v, is given by, t, , W=, Now,, , K, , p.Yjv, , ;, , T p.dv., , so that,, , p, , == K.vjV,
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286, , PROPERTIES OF MATTER, , V is, , where, , And, , .-., , the original volume, and K, the Bulk Modulus., , W - f^V, , ., , =, , dv, , r, , =, , 4-v., Z, , J, , = 9 stress X change in volume., work done per i/mV volume = % p*vjV = \ st ess x strain., ==, , Or,, , ~-, , v^, , -v, , /?.v, , r, , (Hi), , lower face, , Consider a cube (Fig. 176), with its, fixed, and let F be the tangential force applied to its, B', v u PP er face in the plane of ^4, so that, the face A BCD is distorted into the, position A'B'CD, 6r sheared through an, Let tike distance AA' be, angle 0., Strain., , Shearing, , DC, , $, , J, , f v.rfv = J, K, , K, , ;, , 3, , equal to BB'= x\ Then, work done, during a small displacement dx is equal, to F.dx., And, thereiS^, work done for, the whole of the displacement, from, to, x. is given by, , W=, 1, , where L, So that,, .*., , =, , n.a.Q,, , and a, , =L, , F.dx., , I, , 2, ;, , also #, , the length of each edge of the cube., /*..*., F n.L^.xjL, work done during the ivo/^ stretch from to x,, is, , = r, , =, , n.L.x.dx, , =, , volume, , i.e.,, , ", , 2, n.L.x*, , Jo, unit, , x/L,, , =, , =, , work done per, , =, , F'., , 4, , [_the, , V, V, x, = 211n.x.JL, X L ^T', //, I, L, , 7", , ', , Thus, we, , volume, , is, , see that, in, , equal to J stress, , volume of, , cube, , >*, rC*" x, %, , a, , L, , any kind of strain, work done per, , unit, , x strain., , Deformation of a Cube Bulk Modulus. Let A BDCOHEFA, 113., be a unit cube and let forces T x T v and T e act perpendicularly to the, ,, , BEHD, , and AFGC,, and EFGH, and ,4Fand, faces, , ABDC, DHGC, , respectively, as shown, (Fig. 177)., if a be the increase per unit, length per unit tension along the, direction of the force and (3, the, , Then,, , contraction, , produced per unit, length per unit tension, in a direction perpendicular to the force the, elongation produced in the edges, AB, BE and BD, will, obviously,, ,, , be, ly,, , Tx, , T^.OL and TB .a, respectiveand the contractions produced, .<x.,, , perpendicular to them will be TK .$,, Tv .$, and T^. The lengths of the, edges thus become the following, :, , Fig. 177.
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287, , ELASTICITY, , AB = i+arva-2vp-r..p., BE =, BD =, Hence the volume of the cube now becomes, , -2P), neglecting squares and products of a and (3, which are very small, compared with the other quantities involved., Tf, , fn, , /TT, , the volume of the cube becomes, , And, therefore, increase, , =, , in, , /TT, , ___-, , /77, , 1+ (a2p).3T., , the volume of the cube, , l+'3!T(a-2p)-l, , ==, , 3T(a-2(3)., , If, instead of the tension 2 outwards, we apply a pressure P,, compressing the cube, the reduction in its volume will similarly be, 3P(a 2(3), and, therefore, volume strain is equal to 3P(a 2(3)/l, or, r, , equal to 3F(a, , Hence, , volume of the, ["" original, , 2(3)., , Modului,, , Z?w/A:, , K=, , volume strain, , 3P(a, , cube, , =, , 1,, , 2(3), , [, , "* (/), , Or,, , 3(a-2p)', is,, , And, Compressibility, which, therefore, equal to 3(a, 2(3)., , Modulus of, , 114., , is, , the reciprocal of Bulk, , Let the top face, , Rigidity., , (Fig. 178), be 'sheared*, , by a shearing force F,, such that A takes up the position A', , face,, , ABHQ,, , relative, , Modulus,, of a cube, bottom, , to the, , and B the position B\ the angle ADA' being, 6., Then,, equal to the angle BCB', y, , =, , , rr, , ~~, ^ _, , _, , =, L, , where, , is, , /L_, , area of the face, 2, , =, , ABHG, , r, say,, , the length of each edge of the, , cube., , Let the displacement, 5/z^^fr, , Then,, , And, , .-., , coefficient, , .y/r^/n, , =, , A A', //, , =, , BB', , =, , ^., , <=, , /., , Fig. 178., , of rigidity, n =T/0., , Now, extension of the diagonal DB, due to extension along AB, DB.T.a, and that due to contraction* along fA is DB.T,$., Therefore, /o/a/ extension, 5 now becomes, , is, , =, , Jj[^^2fA^22^^, ^^~^^^^3<>^1^, L^.Tfa+p)., , Drop a perpendicular, *, , See, , 117,, , page 288., , [/, , BE from 5 on, , jDJ5, , =, , to DB'.
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PBOPBBTIBS OF MATTER, , 288, Then, increase, , And,, , =, o, , in length, , =, , cos 45, , /, , of DB is practically equal to EB'., BB'. cos BB'E., , =, , EB', , clearly,, , [v <BB'E = 45,, , //<v/2, , ~-~', , ^T _ JL, , ', , *, , =, , i/v2., , T, , n, ', , T =2,%, , And, since TjQ, we have, , ~, , very nearly, and cos 45*, , tv, , -, , '/i, , the coefficient of rigidity of the material of, , n,, , the cube,, , 115., , edge,, , Young's Modulus., , acted upon, , produced, , is a., , Then,, , stress, , Therefore,, , unit, , by, , If, , we now imagine a cube of, , clearly,, 1,, , and, , linear strain, , Young's Modulus,, , Y, , =, , =, , a/1, , =, , a., , I/a., , ...(///>, , Relation connecting the Elastic Constants., 116., relation (/), above,, , a-2p, , =, , And, from relation (77),, a-f fj ==, .*., from, we, have, subtracting (/), (H),, , 30p ~, whence,, , l, , r, , -~, , (//), , 1/3AT., , ...(/), .., , Ij2n., , (//), , **", , by 2 and adding, , to, , (/),, , we have, , ZK+n, , Orr>, l/y from, , [/, , _ ?5 + ", ~ *K+ n 7, A>|, JOi, Rn, = -+, , O f|, , 931, , ,, , whence,, This, then,, , have from, , =, , 11 -*, ZK+n, ~-, , Q, , Y, , 1, -, , We, , 2/j, , p, , Again, multiplying, , unit, , one edge, the extension, , tension along, , is, , (III),, , above., , *, , ...(6), , the relation connecting the three elastic constants., , Poisson's Ratio., It is a commonly observed fact that, stretch a string or a wire, it becomes longer but thinner, i.e.,, the increase in its length is always accompanied by a decrease in its, cross-section (though not sufficient enough to prevent a, slight, increase in its volume)., In other words, a longitudinal or tangential, strain produced in the wire is accompanied by a transverse or a lateral, strain in it., And, of course, what is a true of a wire, is true of all, 117., , when we, , other bodies under strain., , Thus, for example, when a cube, , is, , subject-
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289, , ELASTICITY, , ed to an outward force perpendicular to one pair of its faces, there is, elongation produced along this face, but a contraction in a direction, perpendicular to it, (as we have seen already in 113)., , The ratio between lateral strain (fc) to the tangential strain (a) is, constant* for a body of a given material and is called the Poisson's, ratio for that material\, It is usually denoted by the letter a., p/a., Thus, Poisson's ratio =- lateral stramjtangential strain ; or, a, , =, , It follows, therefore, that if a body under tension suffers no, lateral contraction, the Poisson's ratio (a) for it is zero, and, because, ;, , its, , volume, , increases, is density decreases., , K, , The relations for, and n above, Poisson's ratio, asjollows, , ', , may now, , be put in terms of, , :, , We, , K -=, , have, from relation, , __L_2 _, , a, , above,, , /,, , _ _ I, ~", 3l-*o, 3ai-2a, , ., , l~v, L, , ^, , <C, , Y, , [see (III) above., , a, , =, , Y, , whence,, , ......, , 3^(l-2j),f, Similarly, from relation (II) above, we have, n, , y, , i, , i, , =, , 2(1, , L'a(l+a), , y=2w(l + a)f, , whence,, , Now, from relations, , (/V), , 3A"(l-l>a), , and, , =, , =, , ,, , a, , whence,, , (v), , 2/7(1, , (v), , we have, , + a),, , :iA:~2Ai, , ^^, , which gives the value of Poisson's ratio in terms of, , we, , Similarly, if, , eliminate c from, , y, , -=, , J, , 1, , Limiting values of, , where, , A^ aiifl, , _, , <*,, , a., , We, , ---u, 71, , (iv), ,, , 931, , whence,, , (iv), , A, , -, , and, , (v),, , K and, , n., , we have, , [Sime as relation, , (a),, , above., , [Same as relation, , (b),, , above., , have seen above how, , n arc essentially positive quantities., , Therefore,, , if a be a positive quantity, the right hand expression, and, (/), hence also the left hand expression, must bt> positive, and for this to, be so, 2a<l, or a<| or *5. And,, , Xnis, *i>., the lateral strain is proportional to the longitudinal strain., however, so only when tue latter is small., These relations would not b^ foand ta apply in ths cass of wire speciJ matenals for the simple reason that ths process of wire-drawing brings, about at least a partial alignment of the minute crystals of the substance, which, thus no longer remain oriented at random, with the result that the substance.. loses, is,, , its, , isotropic character,
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f BOFKRTIEB, , 290, , Of MATTBB, , and, if Q be a negative quantity, the left hand expression,, (//), hence also the right hand expression, must be positive, and this is, 1., possible only when a be not less than, 1 and '5. Or, else, as will be, Thus, the limiting values of o are, readily seen from relations (iv) and (v) above, either the bulk modulus, or the modulus of rigidity would become infinite. Further, a negative, value of a would mean that, on being extended, a body should also, expand laterally, and one can hardly expect this to happen, ordinarily., At least, we know of no such substance so far. Similarly, a value of, a, Q-5 would mean that the substance is perfectly incompressible,, and, frankly, we do not know of any such substance either., , =, , In actual practice, the value of a is found to lie between *2 and, although Poisson had a theory that the value of a for all elastic, bodies should be *25, but this is not borne out by any experimental, 4,, , facts., , Determination of Young's Modulus. Young's modulus, as, the ratio between tensil stress (or tangential force applied, per unit area) and elongation strain (or extension per unit length)., The extension produced is rather small and it is difficult to measure, it with any great degree of accuracy., The different methods used are, 118., , we know,, , is, , thus merely attempts at measuring this extension accurately. We, shall consider here only two methods, viz., one for a wire, and the, other for a thick bar., (/), , same, , For a Wire, , material, length, , Searle's, , Two wires, A and B, of the, cross -sect ion, are suspended from, a rigid support and carry, at their, , Method., , and area of, , lower ends, two metal frames,, , shown, , Z>, as, , in Fig. 178,, , C and, , one carryto keep, , W, , ing a constant weight, the wire stretched or taut and the, other, a hanger //, to whic'h slotted, weights can be slipped on, as and, ', , when, , desired., , A, , spirit-level, , L, , rests horizon-, , a point P in frame C, and, on the tip of a micrometer screw (or, spherometei) 5, working through a, nut in frame D., tally at, , The screw, , is, , worked up or, , until the air bubble in the, centre., spirit-level is just in the, , down,, , Weights are now slippedo n to the, moves, hanger, so that the frame, down a little due to the extension, of wire B, and the air bubble shifts, The screw is now, towards P., worked up to restore the bubble, back to its central position. The, distance through which the screw, is moved up is read on the vertical, , D, , 174, , acale., , ar&duated in half- millimetres.
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ELASTICITY, , and fixed alongside the disc of the screw. This gives the increase in, length of wire B. A number of observations are taken by increasing the weight in the hanger by the same equal steps and making the, adjustment for the level for each additional weight. The mean of all, these readings of the screw gives the mean increase in the length of, the wire, for the stretching force due to the given weight. Thus, if, I cms. be the increase in the, length of wire B, and L cms., its original, length,, , we have, elongation strain, , =, , //L., , W, , And if, k.gms. be the weight added each time to the hanger,, the stretching force is equal to IfXlOOO gms. H'/., WxluOOx981, dynes, or equal to F dynes, say., wire,, , So that,, we have, , And,, , /., , if, , a, , 2, sq. cms. be the area of cross-section (irr ), tensile stress =* F/ct., , Young Modulus, , for the material of the wire,, , y~ Ja, , 7, , -L.1, ', , L, , ^ FxL, , of the, , i.e.,, , ., , axl, , The other wire A merely acts as a reference wire, its length, remaining constant throughout, due to the constant weight suspended from it (which need not be known). Any yielding of the support, or change in temperature during the experiment affects both the, wires equally, and the relative increase in the length of, to A) thus remains unaffected by either change., , B, , (with respect, , If a graph be now plotted between the load suspended and the, extension produced, it would be found to be a straight line (just like, OA in Fig. 100), passing through the origin, showing that the extension produced is directly proportional to the load. Hooke's law also, can thus be easily verified., , For a thick BarSwing's Extensometer Method- Ewing's, (n), Extensometer is raorely a device to magnify the small extension of, the bar under test and consists of two metal arms. APS, , and CQD, (Fig. 179), pivoted at P and Q, by means of, pointed screws, on the vertical bar B itself, (the Young s, modulus for the material of, which is to be determined),, 1, , so that they are free to rotate, P and Q. The arm, APS is bent at right angles,, as shown, and carries a micrometer screw S at its lower, , about, , end, and a microscope, fitted, scale,, , M,, , a micrometer, at the end of an arm,, with, , pivoted at, , its, , uppe? end 4-, , Fig., , 179.
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PBOPEBTIES OF MATTER, , 292, , D, , for, The other horizontal arm CQD> has a F-shaped groove at, the micrometer sore v to rest in, and a fine horizontal line marked on, the end C., , The bar B is fixed at its upper end, the two metal arms are, adjusted to be horizontal, by means of the micrometer screw S and, the microscope focused on the, f,, Q, S, JL, horizontal line on C. The bar, 9, , stretched downwards, of, a, testing, (by, machine), so that the horizonis, , now, , means, , tal, , CQD gets tilted a little, D as its fulcrum, the, , arm, , about, , 180., , end C, with the fine mark on, The microthe, and, to, C",, point Q to Q', it,, (Fig. 180),, distance CC' through, scope^is again focused on the mark and the, which it has shifted downwards is measured accurately on the micrometer scale of the eye -piece. Let it be equal to h., , moving down, , obviously, the increase in the length, , Now,, , QQ', , =, , /,, , ., , PQ, , of the rod, , is, , say., , Thon, , ;, , clearly, in the, , we have, , QQ'/CC', , whence,, , /, , Thus, knowing SQ,, high degree of accuracy., , two similar triangles SQQ' and, , =, =, , SQjSO. Or,, , Then, from the length, , SCC\, , SQjSC,, , SQ.h.jSC., , SO and, , and the, , ///i, , =, , PQ, , A,, , we can determine, , /, , to quite a, , its area of cross-section, easily calculate the value, , of the bar,, , stratching; force applied to it,, of Young's Modulus for its material., , we can, , A, , modification of Ewing's Extensometer, as shown in Fig. 181,, Extensometer, in, which there is a vibrating reed R arranged, as shown, the arrangement being such, that as the bar B is stretched by the testing machine, that part of the reed which, touches the micrometer screw M, moves, downwards through a distance five times, the extension of the rod., Thus, by noting, the micrometer screw readings, when the, vibrating reed just touches the micrometer, screw-point both before and, after the rod B has been stretched, we, can directly obtain the increase / in the, 181., Fig., length of the rod., , N.B., , called the, , is, , Cambridge, , Determination of Poisson's Ratio for Rubber. To detervalue, of cr for rubber, we take about a metre-long tube AB, the, mine, of it, (Pig. 182), such, for example, as the tube of an ordinary cycle, with its two ends properly, tyre, and suspend it vertically, as shown,, A glass tube, seccotine*., and, rubber, open, with, bungs, stoppered, at both ends, about half a metre long and about 1 cm. in diameter,, into it through, graduated in cubic centimetres, is fitted vertically, 119., , C, , *atvpeof, , liquid glue.
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293, , BLASTtCITY, , a suitable hole in the stopper at the upper end A, so that a major, part of it projects out., , The rubber tube is completely, with water until the water rises up, in the glass tube to a height of about, 30 cms. from A. A suitable weight, is now suspended from the lower end, B of the tube. This naturally increases, the length as well as the internal, volume of the tube. The increase in, length is read conveniently on a vertical, metre scale M, with the help of a, pointer JP, attached to the suspension, of, and the increase in volume, from, the change in the position of the water, , filled, , C, , W, , W, , \, , ~~(~TF, ., , 1, , 1:, , ,, , column, , in C., , Let the original length, diameter, and volume of the rubber tube be L, D, , and K, , A, , =, , respectively., its, , Then,, , =, , 2, , ir(/)/2), , area of cross-section,, TrD 2 l4,, (/), ., , JA, dA, , whence,, , ., , ., , ., , we have, , differentiating which,, , = -*-dD,, = 2A.dD/D., , ..., , (//), , above,, [From, [_by eliminating TT., (/), , corresponding to a small i7iin the volume of the rubber, tube, the increase hi its length be dL,, and the decrease in its area of cross, section be dA, we have, , Now,, , if, , dV, , crease, , -, , Fig. 182., , v, = (A-dA)(L+dL)., x'\^h, [, = AL+A.dL-dA.L-dA.dL, f where ^.L- V Unoriginal, V+dV = V+A dL-dA.L,, volume of the tube., , "S, , V + dV, , Or,, , ', , [_, , neglecting dA.dL, as a very small quantity, compared with tho other, terms in the expression., , So, , that,, , dV, , = A.dL -, , dA.L, , =, , Substituting the value, , A.dL-^ dD., , from, , Or, dividing both sides by dL, we have, , d?L, , dL, whence, , dD, , = A- 2AL dD, D dL, f, , ., , Or, , dV\ /2AL, , *~, , D, AD, , dV, 'dL, , d, , dL, , dY D, 2AL 'dL 2AL, ', , ,, , dL, ^, , 2L, , D, , dV, ', , 2/il', , ('),, , above.
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P&OPERTiES OF MATTER, , 294, , -, , ', , ~dL, , Now, Poissons, , ratio,, , a, , <,, , Or,, , =_-., , dD, ~, , dDjD, , = -lateral strain, , L, , L dD, = --., dr, dD/dL from, , substituting the value of, , Or,, , ...(in), , A dL, , 2L, , relation, , (///), , above,, , we, , have, n, , *, , L D f., 'D '2L, , I, , 1, , </K\, , X", , 'dZV, , I/,, , ~, , 1-, , 2 ^, , rfK\, , 1, , X dLj*, , in, Thus, knowing the area of cross-section (A) of the tube, the change, calcuwe, can, its, the, in, easily, its volume (dV) and, length (dL),, change, late the value of o for its material., , N.B.An identical method may be used for the determination of the, value of o for glass, but since the change in its volume is comparatively much, too small, we have to use a capillary tube, instead of an ordinary glass tube, to, measure it to an adequate degree of accuracy., By the resilience of an elastic body we understand, , 120. Resilience., , without acquircapacity for resisting a blow or a mechanical shock,, it by the amount of work done, we, measure, and, a, set*, permanent, ing, Let us consider it for, in straining the body up to the elastic limit., its, , a uniform bar of length, , We know, W,, , so that, , it, , that, , L and, , area of cross- sect ion a., , when the bar, , increases in length, , by, , subjected to a stretching force, , is, /,, , we have, , Young's modulus for the material of the bar,, , where, , F denotes the, , WJa, , W_, , l/L, , a, , stress, , L, , .-., , work done, , *, , --, , ', , strain, , Wja., , =, , stress x strain., in elongation strain, \ (stress x strains) x volume., producing extension I, , F, i, , F_, , ^, , I, , Now, work done per unit volume, in, , Y=, , F.-yXvolume, , (V), , ~, , =, , IF, , ~2~~, , 2, , ~T, , ', , V^, , KF* f, '*', , 2Y, , Strai, , " FlY, , ', , I, , Thus, work done, or resilience of the bar,, **, , And, , /., , resilience, , 2Y, , _, , __, ~~~, , ', , 2 xlfoung's modulus, , per unit volume of the bar, ~~, , F*, , 2Y, , "", , (stress)*, , 2x Young s~modulus, }, , Height from which the bar can be dropped without acquiring a permanent set. Since resilience is a measure of the power to resist a, *The meaning attached to the word 'resilience* in our common everyday, that the body comes back to its normal condition wheo, parlance is different, viz.,, (be applied forces are removed.
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295, , ULASTIOITY, , blow or shock without acquiring a permanent set, let us see from, what height the bar can be dropped without taking on a permanent, set., , This height must obviously be one in falling through which, the bar acquires energy equal to its own resilience. Let it be h. Then,, if w be the weight per unit volume of the bar, cJearly,, , =, , energy acquired by the bar in falling through height h, , Vw.h., , this against the resilience of the bar, therefore,, , Equating, , we, , have, , =, , =, , F l2wY., VF*I2Y, whence, h, due, to fall from this much, shock, or, a, a, blow, Thus, the bar can absorb, Vw.h., , 2, , height., , Proof Resilience. The maximum amount of energy per unit, volume that can be stored in a body or a piece of material, without, its acquiring a permanent set, i.e., without its undergoing a permanent strain, is called its proof resilience. Thus, if Fm be the maximum, stress to which a material, in the form of a wire, can be subjected, i.e., if F m be its elastic limit, we have, , =, , proof resilience of the material, , PffiY., , 121. Effect of a suddenly applied load. Suppose we have a uniform, bar of length L and area of cross -section a, suspended vertically, from one end with a collar C provided at the other, and with a weight, in the form of a ring, threaded on to it at a height h from the collar, as shown, , W, , ',, , in Fig. 183., , If we now allow the weight to fall freely so, as to hit the collar, so that the length of the bar, is increased by a small amount /, with the collar, taking up the position C", clearly, the total height, through which the weight has fallen is (h+l)., , W, , =, , .-., potential energy lost by the weight, W(h+l)., This has obviously been utilised in stretching, the bar through / and must, therefore, be equal, to the work done in so stretching it., , F, , If m be the maximum stress in the bar, the, resistance offered by the bar 9 or the restoring force, set up in it ==, m .a., , F, , work done during, And,, we have, X, stretch,, force, ing, since, , work done, , stretch, [see, , Now, as we know,, , Y, , Substituting this value of, , \ stretch-, , in expression, 1, , Fig. 183., , bar, , tensile strain, /, , CL:, , page 285., , = \.Fm .a.l, W(h+l) = J Fm .a.l, tensile stress, - = Fm and, , in stretching the, , And, therefore,, , =, , .-., , 1JL, (j), , -Fm .L, , above,, , we, , /, , = Fm .LlY.
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OF MATTER, , 296, , Or,, , Solving this quadratic equation, we have, , Wh, 2aLIY, , W, a, So that,, , if, , h, , =, , 0,, , aL, we have F m, , Since the zero value of, , Fm, , =, , or, , 2W/a., , has no physical significance, we have, , Fm =, , 2W/a., , W, , is applied to a bar, This clearly shows that when the full load, all at once, the maximum stress is 2Wja, which is clearly twice the, value of the maximum stress W\a* which is set up in the bar when, the load (W) is applied gradually to it, as for example, when the bar, is stretched in a testing machine., , In other words, the effect of a suddenly applied load is to produce, a stress double that produced by a gradually applied one., 122., Twisting Couple on a Cylinder (or Wire). If we have, a cylinder or a wire, clamped at one end, and twist it through an angle, about its axis, it is said to be under tension. Due to the elasticity, of the material of the cylinder or the wire, a restoring couple is set, , up, , in, , it,, , equal and opposite to the twisting couple., , Consider a cylindrical rod of length, of coefficient of rigidity n., , /, , and radius, , r,, , of a material, , Let its upper end be fixed and let a couple be applied, in a plane, perpendicular to its length (with its axis coinciding with that of the, cylinder) twisting it through an angle 6 (radians)., This, incidentally,, , is, , an example of what, , is, , J8, , (), , (6), , called a 'pure' shear,, , 7, , (c), , Fig. 184., , for the twist produces a change neither in the length nor the radius, of the cylinder, the value of the twist for any cross-section of the
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291, , ELASTICITY, , cylinder being quite independent of the direction of the couple, a reversal of which also reverses the direction of twist., , Now,, , in the position, , and opposite, , of equilibrium, the twisting couple is equal, Let us calculate the value of this, , to the restoring couple., , couple., , Imagine the cylinder to consist of a large number of co-axial,, hollow cylinders, and consider one such hollow cylinder of radius X,, and radial thickness dx, [Fig. 184 (a)]. Each radius of the lower end, is turned through the same angle Q, but the, displacement is the greatest, at the rim, decreasing as the centre is approached, where it is reduced to, zero., , Let AB, [Fig. 184, , (6)], be a line, parallel to the axis, before the, twisting, since the point B shifts to B' the, line, takes up the position AB', such that, before twisting, if this, hollow cylinder were to be cub along, and flattened out, it will, form the rectangular plate, A BCD, but, after twisting, it takes the, shape of a parallelogram, AB'C'D, [Fig. 184 (c)]. The angle through, , cylinder, , is, , On, , twisted., , ',, , AB, , AB, , which, Then,, , this hollow cylinder is sheared, , therefore,, , is,, , BAB', , =, , ^, say., , clearly,, , BB', , =, , 14., , Also BB' == x.e., , .-., , =, , </>, , [See Fig. 184, , x.d/i, , (a)., , maximum, , value where x is the, Obviously,, greatest, ie., the maximum strain is on the outermost part of the, In other words, the shearing, cylinder, and the least, on the innermost., stress is not uniform all through., <f>, , will, , have the, , ., , Thus, although the angle of shear is the same for any one hollow, cylinder, it is different for different cylinders,, being the greatest for, the outermost and the least for the innermost cylinder., n, , ., , ., , -- = ~~F, , strain or angle oj shear, , we have, , F=, , Now, face area of, , And, , shearing stress, , ~, , ., , Since, , /,, , n.(f>, , this, , H.x.0/7., , hollow cylinder, , total shearing force on, , =, , =, , this, , =, , 2i:x.dx., , area, , ,, , 27tx.dx, , ,, , <f>, , x, , n.xjf, ,, , B, , =, , ., , 9, , 2irn.-j~.x*.dx., , Therefore, moment of this force about the axis OO' [Fig. 184, of the cylinder is equal to 2vn.0.x*.dx.xll, 2irn.Q x*.dx/L, :, , =, , Intergrating this expression between the limits, x, , =, , and x=r,, , we have, total twisting couple, , Znn.O, ~, , ~-l[r, , /Jo, f, , on the cylinder, , =, , ff, , B, , (&)},, , 8, , 2-nn. ~.-.x .dx.
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298, , OP MATfftft, If 9 SB 1 radian,, , we have, of the cylinder (or wire), , twisting couple per unit twist, , Note. If the cylinder be a hollow one, of inner, rt respectively, we have, twisting couple on the cylinder, , 2rr/i., , 7fr4 /2/., , is also called, , This twisting couple, per unit twist of the wire,, the torsional rigidity of the cylinder or wire., , fa and, , =, , and outer, , radii, equal to, , y.,, , Let us, 123. Variation of stress in a twisted cylinder (or wire)., again imagine a cylinder or wire, of length / and radius n, to consist of a, large number of co-axial, hollow cylinders and, consider both a cylinder of radius x and the outermost cylinder of radius r, (Fig. 185), in which the, lines AB and CD respectively are parallel to, the axis OO', before the cylinder is twisted, and, shift into the positions AB' and CD' after it, hag been twisted, as explained above. Then,, clearly,, , = DOD' =, , LBOE', , =, , <f>, , <f>, , Fig. 185., , DD', , r,, , we have, BB', , r$, , 9, , whence,, , n, , F is the, , And, since, , We,, Or,, , =s=, , c/>, , m, , r, , and, therefore,, , the, , maximum, , F=, , have, , F=, , shearing stress at distance, , t, , i.e., , ,, , .Fm, , xfrom, , n$, , on, , the axis, , jc, , == n,, , from the, , we have, , strain,, , shearing stress on the wire, therefore,, , r, , X maximum strain., , shearing stress at distance, is, , Z)PVr., , 7, , ?, , -, , x, , where, , J?^, , =, , x, , ms, , 5/ra/n fn the cylinder or wire at distance, , Now,, , 57*, , r, , radius of cylinder or wire, , mum, , =, =, , r_, ^, _, /""""/^/'x^x, , So that,, , <j>, , <j>, , OD = OD' =, , Or,, , 0,, , and the angles through which the two cylinders are, sheared are BAB' =, and DCD', m respective= BB'/l and m = DD'/l, this latter, ly, where, being the strain on the surface of the cylinder and,, therefore, the maximum on it., [v DD' = r&., x and, Since OB = OB', , its, , n.^ w, , axis., , = Fw, , surface., , ,, , xfrom the axis, ~, maximum stress., , ,, , the maxl*
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299, In other words, both shearing strain and stress, go on increasing, the axis of the cylinder or wire and acquire, their maximum values on its surface., , as, , we proceed away from, Let us, , and, , now, , see whether there, , is, , any variation of shearing, , strain, , stress along the length of the cylinder or wire also., , Let us, therefore, consider the stress in the plane EO"E' of the, cylinder at a distance a/ from its upper fixed end, where a>0 and, , <1., , Here, clearly,, , =, , .*., , shear strain, , ^EO"E, , f, , EE'lCE, , =, , Now,, So that, shear, , ~.<f>, , m, , =, , as, , >, , I, , r.aO/al, , =, , this, , plane, , rOll., , we have seen above., , strain on the surface, r, , O"E and 0"'=a0., [v O"E = O"E'=r., , between the radii, , EE'jr and .-. EE' = r.ad., on the surface of the wire, in, , of the wire, , in, , plane, , ,, , ,, , the same as in plane DOD', as discussed above. Clearly, therefore,, shearing stress in this plane is also the same as in plane DOD',, i.e.,, , =, , Fm, namely, n<t> m, Thus, we see that the shearing stress at a point in a cylinder, or, a wire, depends only on the distance of the point from the axis, and not, its vertical distance from either end, of the cylinder or the wire., 124. Strain energy in a twisted cylinder (or wire). Let C be the, couple applied to the lower end of a cylinder of length / and radius r,, with its upper end fixed and, y, let B be the angle of twist produced at the former (i.e.,, lower) end. Then, if the limit, of elasticity is not exceeded,, ., , the relation between C and Q, is a linear one and we obtain a, straight line, , graph, , OP, , between, , the two, as shown in Fig 186., So that, for a small increase bC in the value of the, couple, the increase in the, angle of twist is dft, and the, work done on the cylinder, or the, , energy stored up in, fore,, , C.dQ,, , where, , it, is,, , C, , dO, , o, , there-, , is, , ANUE OF TWIST, , the, , >, , Bm, , j|, , Fig. 186., average value of the couple., This is represented by the area of the shaded strip in the Figure., And, therefore, the total work done on the wire, or the total energy, stored up in it for the maximum twist O m (represented by CM), to, which its lower end is subjected, is represented by the whole area, , OAP., This strain energy, , Now,, , is, , obviously equal to, , for a twist 6 in the wire, , is,, , as, , we, , E, , kow, , f, , r*i, =, , I, , C.d0., , equal to m8r*fllt
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300, , of, , p&ottSKTifls, , E-, , And, therefore,, , ". 9 .d9 -, , JJ", , _ _., where, , itnr*0 m /2l is tha value of the, , ing to the, , maximum, , twist, , m, , ;, , i.e.,, , mm, , ^, , ___^_._,, , ,, , maximum couple C m, C m ~ nnr*0 m l2l., , ,, , correspond-, , Or, substituting from this the value of Q m in the expression for, , E, , above, we have, strain energy in the twisted cylinder,, , E=, , =, , Cm Om, , As will be readily seen, this is half the energy (C m .Q m that, would be stored up in the cylinder if the stress in it were to have, the same value throughout, equal to its maximum value on the, surface of the wire, which, as we know, is not actually the case,, the stress increasing from zero at the axis to a maximum on its surface,, ), , (see, , 122 above)., 125., , We know, , Alternative expression for strain energy, that strain energy per unit volume, , = } stressx strain =, , \.F.<f>, , =, , Tp, , J F. n, , in, , =, , terms of, , stress., , IT*, , ---., , ..., , (/), , if we consider an element of the cylinder or wire, defined by, x and x-\-dx the stress will, as we have seen in 123, be constant, at all points in it and its value will be x.Fm fr, where Fm is the maximum value of the stress in the wire on its surface i.e., F = x.Fm /r., , Now,, radii, , y, , ;, , Since the volume of the cylindrical element we are considering, is 2nx.l.dx, we have, from relation (/) above, energy of the cylindrical, element, , _, ~, dE, , Or,, , =, , /, , V, , I*"', , |0r, ^ Jr x*.dx =, , 7T,ljT, , r<, , /., , *, , x*.dx., , E=, , *, , And, , xf, , rfaa, , --, , '*'*, , tri, , I, , If, , <f>, , m bs tho, , mum stress F m, , ,, , maximum, , we have, , nr*, , that,, , E=, , ^, , TJ, , 4, , ~, -= '*'**-?, -v.r-.F,m, , ., , 4rc, , shear strain corresponding to the maxi-, , Fm =, , O m is the angle of twist for the, 1, , ', , -;*- * r, , ., , nr*, , n<f> /n, , == n.r.Bjl, (see Fig. 123),, , maximum, , where, , value of the couple., , SHE, , j-, , i.e., the strain energy is again half the value it would have if all the, elements of the cylinder (or wire) were subjected to the same, , maximum, , stress, , Fm, , ., , A, , Torsional Pendulum., heavy cylindrical rod or disc,, from, of, one, a, fine, end, wire,, (attached to its centre), whose, suspended, 126.
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ELASTICITY, , 301, , constitutes what is called a torsional pendulum, disc is turned in its own (i.e., in the horizontal), or, (Fig. 187)., plane to twist the wire, so that, on being released,, it executes torsional vibrations about the wire as axis., , upper end, , is fixed,, , ,, , The rod, , Let 6 be the angle through which the wire is, twisted. Then, the restoring couple set up in it is equal, tO 7T.W 4 .0/2/, C,0,, where ?ntr*/2/ is the twisting couple per unit (radian) twist, of the wire, usually denoted by the letter C., , =, , This produces an angular acceleration, , dco/df, in, , the rod or the disc., ,\, If 7 be the moment of inertia of the rod, about the wire, we have, Or,, Ldw/dt = -C.O, da>!dt == -C.0/7,, Fig. 187., the, acceleration, i.e.,, angular, (da)[dt) of the disc or the rod is proportional, to its angular displacement (0), and, therefore, its motion is simple, harmonic, Hence, its time-period is given by, t, , c, , V, , Or,, ill., , moment of inertia of the, , ', , __, , disc or rod about the wire, , restoring couple per unit twist of the wire, Determination of the Coefficient of Rigidity (n) for, , a, , Wire., , This method is based on a direct appli(1) Statical Method., cation of the expression for the twisting couple on a wire deduced, in, 122., There are two different types of apparatus used for the, purpose, according as the specimen under test is a rod or a wire. We, shall now consider these in detail., (a), , Horizontal Twisting apparatus for a Rod. Here, a couple^, is applied to a horizontal rod and, equated against the, the, expression for, torsional or twisting, , which can be measured directly,, , 7Tr 4 0/2/,, whence the value of, , couple,, , n for the rod can be, easily calculated., , The arrangement of the apparatus is as shown in, , _^, , ^^^^^^^, ^^Z^^*, , \]r, , Fig. 188, where one, end of the rod, under, test, about 50 cms., i*1, length and of, radius about '25 cm., t, , secured, firmly, to a block B lt with, other end attaits, ched to a steel axle, is, , Fig. 188., , of a large pulley, , J?, , ., , ?
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PROPERTIES OT MA1TER, , 302, , M, , A cord is wound, from, , its, , twist, , it, , round the pulley and has a mass, suspended, lower free end. Thus, a couple acts on the rod, tending to, , about, , its, , own, , axis., , two, pointers p l and /? 2 are clamped on to the rod, at, circuthe, a, over, /, as, to, move, so, known, distance, freely, points,, apart,, lar scales Sj and S2 graduated in degrees, on which the twist produced, in the rod at those two points can be read directly., , Two, , ,, , ,, , Now,, rod,, , if, , R, , be the radius of the pulley, the couple acting on the, is, clearly, equal to Mg.R., , due to the suspended mass (M), , This couple is balanced by the couple due to the torsional reof, action of the rod, equal to mrr* (0 a, 0j)/2/, where r is the radius, the rod and 1 and 2 the angles of twist (in radians*) produced at the, two chosen points, as indicated by the two pointers., ,, , /mr 4 (| 2, , So that,, , 0,)/2/, , =, , Mg.R., , Or,, , n, , =, , -~-JL, , whence, the value of n for the material of the rod can be easily determined., , The apparatus, though quite simple in manipulation,, from two serious drawbacks, viz.,, (i), , an error, to, , it, , suffers, , there being one single pointer moving over the circular scale,, caused due to eccentricity of the axis of the rod with respect, , is, , ,', , there being just one pulley, only one single force is applied to, This, rod, attached to it, thus exerting a side-pull on it., results in friction between the rod and the bearings, thus appreciably, hindering the rod from twisting freely., (ii), , the, , end of the, , (b) Vertical, , Twisting apparatus for a Wire. This was designed, also a couple, which is measured directly, is, applied to the lower end of the vertically, suspended wire, and the twist produced in, it, is, noted. Then, equating this couple, 4, against the expression H7rr 0/2/ for it, the, value of n for the wire can be easily calcu-, , by Barton, and here, , lated., , The wire W, whose, , of rigiclamped at its, 189), and has a heavy, coefficient, , dity is to bo determined, is, , upper end, , T,, , cylinder, , attached to, , C, , (Fig., , it,, , at its lower end., , Two, , pieces of cords are wound round, the cylinder and, leaving it tangentially at, either end, pass over two frictionless pulleys,, as shown, with equal masses, and M,, , M, , M, Fig. 189., , *To convert degrees, , suspended from their free ends., The couple* formed by two masses^, rotates the cylinder about the wire as axis,, , and thus, , twists the wire through, , an angle, , multiply by */}80 f be^apc J80
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303, , ELASTICITY, , (radians), say, which is read directly on the horizontal circular, scale S by the movement over it of the pointer, attached to the wire,, or by the more sensitive lamp and scale method, with the help of a, , telescope., , Then,, , if, , d be, , the diameter of the cylinder,, , we have, , twisting couple applied to the wire, , But, the twisting couple for a twist of, , where, , r is the radius of the wire,, of rigidity for its material., , Clearly, therefore,, , from which n, , 7rr^/2/, , =, , /,, , Mg.d,, , its, , =, , Mg.d., , radians, , is, , also, , length and n, the coefficient, , whence,, , for the material of the wire, , n, , can be easily obtained., , The two sources of error present in the first method are elimiFor, (/) due to the very nature of the arrangement of, the pointer and the scale, the error due to eccentricity of the axis of, the wire does not arise and (//) due to the use of two pulleys, the, side-pull on the wire is also avoided., nated here., , N, , B. It will be noted that the weak point in the above two methods is, the radius r of the rod or or wire, the fourth power of which occurs in the expression for n. It must, therefore, be measured most carefully., , The, Maxwell's Vibrating Needle., (2) Dynamical Method, dynamical method of determining n for the material of a wire, consists in determining, by direct observation, the time period t of, a body, like a disc or a rod, suspended from the wire and executing, torsional vibrations about the wire as axis, i.e., of a torsional, pendulum., , =, , Then, since t, 27T\///C', where 7 is the moment of inertia of, the body about the wire, and (7, the couple per unit (radian) twist of, the wire, we can easily obtain from it the value of C., Equating, this against the expression ?jw 4 /2/ for it, the coefficient of rigidity (n), for the wire can be easily calculated., It is not, however, easy to determine the moment of inertia (/), of the body accurately. Maxwell, therefore, devised a method in, which the necessity of determining it was altogether obviated. Let, us study his method in detail., , Maxwell's Vibrating Needle Method. A hollow tube or cylinder,, open at both ends, is rigidly fastened in the middle to the wire,, the coefficient of rigidity of the material of which is to be determined,, and w}iich is suspended vertically from a support, and has a, small piece of mirror attached to it, as shown, (Fig. 190), to enable, the vibrations of the tube to be observed by the telescope and scale, method., , Two hollow and two solid metal cylinders, of equal lengths and, diameters, can be fitted into the tube such that, put ei4 to eucj,, just fid, , it, , completely.
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304, , MATTER, , PROPERTIES, , The solid cylinders are first put into the inner positions and, the hollow ones in the outer positions, as shown in Fig. 190 (0), and the, tube, so loaded, is then, given a torsional vibratimeand, its, period determined. Let, tion,, it, , )Af, , fr, , H, , H, , /v*., , V/s, , v, , C, , Then,, , the twisting, , is, , couple per unit deflection, or twist of the wire and, 4, is equal to ni:r l'2l, and, , w, Fi s- 19, , t, , where, , H H, , (tv, , be, , Ilf the moment of inertia, of the loaded tube about, , -, , the suspension wire as axis., , The solid and hollow cylinders are then interchanged in position, i.e., the hollow cylinders are now put in the inner positions and, the solid ones in the outer positions, as shown in Fig. 19 ) (ft), and the, time-perioi of the torsional vibration of the loaded tube determined, Let it be t 2, again., ., , Then,, , where 72, , is, , tt, , =, , 2::, , y 7/C,, , ..., , (ii), , now the M.I. of the loaded tube about the suspension, , wire., , Squaring and subtracting, , (/), , from, , we have, , (//),, , C, , ', , .., , {, , (Hi), , Now, let the mass of each hollow cylinder be m and that of, and let the length of the hollow tube be, each solid cylinder,, 2, 20, so that the length of each solid oj hollow cylinder is 20/4, or, , w, , {, , ,, , a/2., , Clearly, then, the centres of mass of the inner and outer cylinders are at distances 0/4 and 30/4 respectively from the axis of, oscillation., , Therefore, the change from the first adjustment, when the solid, cylinders occupy the inner positions to the second adjustment, when they, occupy the outer positions, consists in transferring an extra or excess, mass, from a distance 0/4 to a distance 30/4 from the axis, of oscillation, on either side of it. The moment of inertia of the, , (m^m^, , loaded tube, therefore, increases, and,, principle, , by the application of the, , of parallel axes, we have, , Here, we multiply the mass by, place oj} both t^e, , 2,, , because the change takes
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305, , ELASTICITY, , Or,, , /,, , - M-a^-md x, , =, , /1 4-(W2, , -, , -, , =, , 1, , And .-. (72 /,), (w 2 w^.a .*, wj.fl, Substituting this value of (/a/!) in relation (///) above,, 72, , /.<?.,, , 1, , we, , have, , (tf-V), , 4-7T, , =, , 3, , -, , .(mi-m^a*., , c, , Further, substituting the value of C, we have, 4-7T, , ^, , ,, f, , ,, , U 2 -'i, , Or,, , ox, 2, ), , ?r..a, =, , ,, , r*, , ,, , ,, faa-flM, whence, n, , w, , 47, , ., , %, , %, , -, , n, , 2, , =, , -,, , 7r..., 2, _* i_-i/., , f, and r, the value of n for the, Thus, knowing /, a, Wj,, t, fj, 2, material of the given wire can be easily determined., The vHue of n obtained by the dynamical method is slightly, higher than that obtained by the statical method, because, in most cases, the, twist produced by a torsional couple depends, to some slight extent, upon the, time for which the couple is applied and so, in the dynamical method, where, ,, , ,, , NB, , the time of vibration is rather short, the twist, of the couple than in the statical method., , (o) is, , smaller for the, , same value, , Further, since wires are made by squeezing the molten metal through, outer layers are invariably tougher than the inner, ones, and hence the value of n for a thinner wire needs must be higher than for, a thicker wire of the same material., , holes, (as in a sieve), their, , 128. (a) Determination of Moment of Inertia with the help of, a Torsional Pendulum. The moment of inertia of a body of a regular, geometrical shape can be easily calculated from its mass and dimensions., But, if it be of an irregular shape, it is not possible to do so. In, either case, however, it may be determined by using a torsional, pendulum with a disc or a rod of known moment of inertia / about, the suspension wire and noting the time period (/) for its torsional, vibration., Then, mounting on it the given body, such that the axis, of the moment of inertia of the, two together is again the snme wire,, the time-period (t^) for the torsional vibration of the combination is, determined. Then, if / t be the moment of inertia of the body about, the wire as axis, the moment of inertia of the combination, in the, second case, is clearly equal to /+/,. So that, if C be the torsional, couple per unit twist of the suspension wire, we have, /, , .., , =* 27TV///C, , ..., , squaring and dividing, , (/), , and, , relation, , (//), , t, , by, , =, , l, , (/),, , Zic^T+FJC., we have, , ..., , (ff), , ,2, , *Or, this may easily be deduced as follows, If / be the moment of inertia of the hollow tube about the suspension, wire, and h and /*, those of the solid and the hollow cylinders about the vertical, axes through their respective centres of mass, we have, by the principle of paralle, :, , ., , axes,, , and, So that,, , A*, , /i, , 2, , /+2[/,-fm 8 .(fl/4) ]+2[/A -fm 1 .(3a/4n,, +/M0/4)'] + 2[/f + ntr (3a/4) 2 ]., , /-f 2[/*, , (/i-/i), , (/Wt
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PBOFEBTIBS OF AlATTKB, , 30ft, , =, , ,, , So that,, , f-, , -, , Or,, , t, , fSubtracting the denominator from the numera-, , *, , ^, , Mor, , on, , either side., , whence, /a, , f, , Thus, knowing /, t and f,, we can easily calculate Ilf the, of inertia of the given body., , moment, , If, however, it is simply, (b) Comparison of Moments of Inertia., desired to compare the moments of inertia of two bodies, we first, use one and then the other, as the disc or rod of the torsional pendulum,, t, and t2 respectively for their, i, torsional vibration about the wire as axis. Then, if /j and /g, be, their respective moments of inertia about this axis and C, the torsional, , and determine the time-periods, , couple per unit twist of the wire,, r,, , we have, , =2irVA/C and, , f, , 2, , =, , Zn^TjC-, , So that, squaring and dividing one by the other, we have, f, , and thus, knowing, , may, , t, , l, , and, , t, , z>, , the, , moments of inertia of the two bodies, , be easily compared., , In the above cases, the amplitude of vibration need not be small,, found that the restoring couple continues to be proportional to the, twist B in the wire, up to fairly large values of 0. The assumption made, however,, that even with different bodies suspended from the wire, resulting in a change, in its longitudinal tension, the value of C (or the twisting couple per unit twist of, the wire) remains the same is found to be only approximately true., , Note., , because, , it is, , 129., Bending of Beams Bending Moment. We must first be, about the terms, beam and bending moment., Beam. A beam is a rod of uniform cross-section, circular or, rectangular, whose length is very great compared with its thickness,, so that the shearing stresses over any section are small and may be, clear, , neglected., , Bending Moment. When a beam is fixed at one end and loaded, it bends due to the moment of the load, the plane, of, bending* being the same as that of the couple applied. Restoring forces, are called into play by this deformation of the beam and, in the equi-, , at the other,, , librium state, the restoring or resisting couple is equal and opposite to the, bending couple, both being in the plane of bending., , Irrespective of the manner in which the beam is bent by the, its filaments on the inner or the concave side, get, shortened or compressed, and those, on the outer or the convex side get, , couple applied,, , lengthened or extended, as shown, in Fig. 191., Along a section, in, between these two portions, there, is a layer or surface in which the, 191., , filaments are neither compressed, , *In the case of uniform bending, the longitudinal filaments all get bent, into circular arcs in planes parallel to the plane of symmetry, which is then, known as the plane of bending. And, the straight line, perpendicular to this plane, on which lie the centres of curvature of all these bent filaments, is called the a*ii, , of bending.
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ELASTICITY, , 307, , nor extended. This surface is called the neutral surface and its stetion (EF) by the plane of bending which is, perpendicular to it U, called the neutral axis., , In the unstrained condition of the beam, the neutral surface, becomes a plane surface, and the filament of this unstrained or unst retched, , layer or surface,, , lying in, , the plane of symmetry of the bent, beam, is referred to as the neutral, It passes through the e.g., filament., (or the centroid) of every transverse, section of the beam., , The change in length of any, filament is proportional to its distance, from the neutral surface., Let a small part of the beam be, bent, as shown in Fig. 192, in the, form of a circular arc, subtending an, at the centre of curvature O., angle, Let R be the radius of curvature of, this part of the neutral axis, and let, a'b' be an clement at a distance z, from the neutral axis., Then,, , and, .*., , =, , a'b', , its, , original length, , (R+z).0,, , db == RQ., , =, R.O =, , =, , (R+z).g, , =, , strain, , the strain, , z.e., , original length of the filament, , And, since the, , i.e.,, , ab., , a'b', , increase in length of the filament, , is, , z.e j R.O, , = R.Q,, = z/R,, , we have, , proportional to the distance from the neutral axis., , Since there are no shearing stresses, nor any change of volume,, and extensions oj the filaments are purely due to forces, the, acting along, length of the filaments., , the contractions, , If, , PQRS, , to its length, , p, , (Fig. 193),, , be a section of the beam* at right angles, , and the plane of bending, then, clearly, the forces acting, f, *, on the filaments 'are perpendicular to, t kj g section, and the line AfW lies on the, neutral surface., Let the breadth of the section be, , PQ =, , 6,, , The, , and, , its, , depth,, , QR =, , d., , forces, , producing elongations, and contractions in filaments act perpendicularly to the upper and the lower, halves,, , of the rectangular section, each other., *The section, , is, , PQRS,, , PQNM, , MNRS, , and, respectively,, their directions being opposite to, , shown rectangular purely, , for the sake of convenience.
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308, , PROPERTIES, , Off, , MATTEH, , Consider a small area Sa about a point A, distant 2 from the neuthrough this, , The strain produced in a filament passing, tral surface., area will be z/R, (see above)., , F, , Now,, , =, , stress j strain, , and, , stress, , /., , = Yx strain., , =, , Y xz/R* where, Therefore, stress about the point A, value of Young's Modulus for the material of the beam., And, there fore, force on the area Sa, , =, , and, moment of I his force about the line, , Y, , is, , the, , Sa.Y.z/R, , MN =, , Y.zx$axz/R., , =, , Y.Sa.z*/R., Since the moments of the forces acting on both the upper and, the lower halves of the section aro in the same direction, the total, moment of the forces acting on the filaments in the section PQRS is, , given by, , a.z* is the geometrical moment of inertia (I )* of the secNow,, tion about MN, arid, therefore, equal to ak 2 where a is the whole, area of the surface PQRS and k, its radius of gyration about MN., ,, , Hence, the moment of the forces about, , MN =, , Y, D >ak*, J\, , =, , YI*, ., , i\, , This, then, balances the couple of mo nent M, say, called the, bending moment, acting on the beam due to the load, when the beam, is in equilibrium, for, there is no resultant force acting on the area, PQRS, and the resultant moment about EF, perpendicular to MN,, is also zero., In other words, it is the moment of the stress set up in, the beam or the moment of resistance to bending, as it is usually called, in engineering practice, and is also of the nature of a couple, for, only a couple can balance a couple. Obviously, it acts in the plane of, bending and is equal to the bending moment at the section due to the, load, though, quite frequently, (but, not strictly correctly) it is itself, referred to as the bending moment. This forms tho very basis of the, theory f regarding the bending of beams and is, therefore, a relation, of fundamental importance., ;, , *lt is so called because it is proportional to the mechanical moment of, It is denoted,, inertia of a plane lamina of the same shape as the cross-section., here, by the symbol / 7 , so that, the student may not confuse it with the ordinary, , mechanical, , moment of, , inertia,, , denoted by, , /., , fine theory is subject to the limitations mentioned in 131, (page 313),, which the student would do well to keep in mind., Imagine the section as a rectangular plate of unit mass per unit area,, (Fig. 194)., , Then, area of the strip AB, of length b and breadth, , And,, , p, , therefore, its, , mass, , A, , =, , b.dz.l, , Q, , dz> is equal to b.dz., , b.dz., , Hence, geometric moment of inertia of, b.dz z*, and, therefore,, the strip about, moment of inertia of the whole plate or section about, , MN, , MN, , I*, , m, R, , S, Fig. 194., , 26, , ^, , "124-J, , 12
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ELASTICITY, , The quantity, , YJ, , bending moment, , =, , =, , Y.ak*, , is, , called the flexural rigidity of th, , beam., /., , (Y/R)x geometric moment of, , inertia, , of the, , section., , = flexural rigidity, , /, , R,, , whatever the shape of the cross-section of the beam., For a rectangular cross-section, a = bxd, and, , fc, , For a circular, , section, a, , I9, /..,, .*., , moment, , the same as the, , moment for a, , bending, , =, =, , 7rr, , 0A-2, , and, , =, , A, , =, , 2, , r2, , =, , =, , Hence, bending moment for a rectangular cross-section, a, , 2, , d 2 /12., , Y.b.d*ll2R., , /4., , 4, , 7rr /4,, , of inertia of a disc about a diameter., , circular cross-section, , =, , 4, , 7.irr /4/?., , We, , Note., have seen above how strain in a beam is proportional to the, distance z from Us neutral axis, and is equal to z\R, where R is the radius of, curvature of the poition of the ncutial axis under consideration. So that, if F, be the stress cor res ponding to the strain z//?, we have, , F, If,, , F F, , therefore,, , lf, , 2, , e,c., , ~R, , r', , z, , be the values of stress, , ', , at distances z lt z 2, , ., , from the, , we have, , neutral axis,, , And, , .*., , bending moment, , ^L, , ra, , ., , ^1, , where Zt, , Y, , F, , _, , >-*/*', , /^/^ and, , Z =/, a, , ff, , M=, , y, , ^, -, , /, , /(/, , /z 2, , / ff //?., 2, , 22, , Zi F,, , Jfr, , ^tr, etc., , .//7, , *Z F, 2, , 2, , f, , F, ./i, , Fa, , "~ -r, , 22, , -I, , etc, CIL., , etc.,, , are called the moduli of the section under consi-, , deration., ~, u, , ,, , ., , -, , Ai, Thus, modulus of a section, , Now,, , =-, , of inertia, geometrical, - - moment, ,., ;, ^-., r, ~i, distance from the neutral axis, ., , ., , in the case of a flat bar or, , beam, of rectangular crossbending be small, there is brought about a change in, the shape of th3 section, such that all lines in it, originally perpendicular to the plane of bending, get bent into arcs, which are all concentric and convex to the axis of bending. In other words, the layer of, the beam, which was originally plane and perpendicular to the plane, of bending, and which contained the neutral filament, now gets changed, into what is called an anticlastic surface (Fig. 195), of radius / in, section, if the, , the plane of bending (which, here, coincides with the plane of the, paper), ^nd, of radius R in the plane perpendicular to it. the two, centres of curvature lying on either side of the beam., This is uhat, is to be expected, because a transverse bending must, of necessity,, be associated with a longitudinal bending of the beam, with the curvature of the former opposite to that of the latter. For, the filaments, above the neutral axis, which get extended, must obviously suffer a, lateral contraction a times as great and, similarly, the filaments below, the neutral axis, which get compressed, must suffer a lateral extension., f
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310, Thus, by way of illustration, if a rectangular piece of Indiarubber Le bent longitudinally in the form of an arc, it takes up the, , form shown, fibres, , \A, , a, , in Fig. 196, with its longitudinal, bent so as to be concave with respect to, below, and the transverse fibres, so, , P^, , concave with respect to a point, above, the rubber piece, in the case shown. It is, this bending, which occurs in a plane normal to, as to be, , the, , longitudinal plane,, , that, , gives, , the, , rubber, , piece (or the beam) an anticlastic curvature., , Fig. 195., , And, therefore, as we have seen before, (page 307), the longitudinal and lateral strains in a filament, distant z from the neutral axis, will be given by zjR and zjR' resSo that, Poisson's, , pectively., , the material of the beam,, the expression, , is, , ratio a, for, , given by, , lateral strain, , __ zJR^ _., , longitudinal strain, , zjR, , ~~, , R, , Fig. 196., , R'', , This, then, gives us a method for the determination of cr for the, material of a given boam or bar, the two radii being determined, directly by attaching suitable pointers to the rod and noting the, distances and angles traversed by them, when a known couple is, applied to the beam., , The Cantilever. A cantilever, and loaded at the other., , 130., at one end, (i), , is, , Cantilever loaded at the free end., , a beam fixed horizontally, , Here, two cases, , arise, viz.,, , when the weight of the beam itself produces no bending, and, when it does so. Let us consider both the cases., (a), , (bj, , When, , the weight of the beam is ineffective. Let AB, (Fig. 197), axis of a cantilever, of length L fixed at the end, the, neutral, represent, A, and loaded at B with a, weight W, such that the end B is, deflected or depressed into the, position B' and the neutral axis, takes up the position AB, it, being assumed that the weight, of the beam itself produces no, (a), , bending., , W, , Consider a section / of the, at a distance x from the, fixed end A., , beam, Fig. 197., , the load, , W, , 9, , The moment of the external couple at this section, due to, , or the bending moment acting on, , it
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311, , beam, , Since the, , YI^R, , =, , this must be equal to, the radius of curvature of the neutral, , in equilibrium,, , is, , Y.ak*IR, where, , R, , is, , axis at P., , =, , =, , 7.^ /^-(0, Therefore,, W.(L-x), F./,/^, Since the moment of the load increases as we proceed towards, the fixed end A, the radius of curvature is different at different points, and decreases as we approach the point A. For a point Q, however,, at a small distance dx from P, it is practically the same as at -P., So that,, , PQ =, , R.d6., , Or,, , R, , whence,, , =, , dx, , =, , 2, , iwhere do, , R.d&,, , is, , the L, , POQ-, , dxjdO*, , Substituting the value of R in, , (/), , above,, , we have, , *-, , -, , tangents to the neutral axis at P and Q, meeting the, through BE' in G and D respectively. Then, the angle, subtended by them is also equal to d6 the radii at P and Q being, perpendicular to the tangents there., Now, clearly, the depression of Q below P is equal to CD, equal, , Draw, , vertical line, , t, , to dy, say., rru, , i, , Then, dy, , IT, so = (L-X)W.(L-X).dX, ---= (Lx).dO, ^, , ~, , jTak*~~, , JV(L-x)*.dx, -", Y.ak*, , fFrom, , *, , ......., , ( '7), , above>, , "' V ', , ;, , Therefore, the depression y =, , /?' of the loaded end B below, the fixed end yl, is obtained by integrating the expression for dy, /., and x, between the limits, x, , =, , =, , [Putting back Ig for, , "377"*, Thus, the free end of the cantilever, , is, , 1, ., , depressed by, , _', , "", , ZY.ak*, , oA:, , "377,, , of the beam is elective. In this case, in, at B, the weight of the portion (Lx) of, addition to the weight, the beam is also acting at the mid-point or the e.g. of this portion so, that, if w be weight per unit length of the beam, a weight w(Lx), is acting at a distance (L, x)/2 from the section PQ. And, therefore,, (b), , When, , the weight, , W, , ;, , *See solved example 4, page 332, where it is shown that l/R, d*y/dx*, the, of change of slope. A- mathematical minded student will find the solution, given there with this value R. much neater and also perhaps a trifle easier., rate
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PBOPEBTIES OF MATTEB, , 312, , <, , the total bending moment on the beam, , Since the, , beam, , is, , in equilibrium, this, , And,... rffl==, , dy, , Then,, , And, , =, , must be equal to YLJR or, , _, (L-x).dd, , ==, , y=-, , ..., , Now, , w, , >v.L, , =, , y-+, , WJ2, , WL*, , so that,, , the weight of the beam., , W^j, say,, , ', , "', , ', , 8; 7/, 3, , Or,, i.e.,, , the, , a weight, , beam now behaves as though, , it, , is, , loaded at the end, , W plus 3!8ths of the weight of the beam., , B, , with, , loaded uniformly. Let the uniform load on the, (ii) Cantilever, cantilever be w per unit length. Then, the weight of the portion, , of the beam (Lx), ie., \v(L x) alone produces a bending moment, about the section PQ, there being no weight suspended from the end, B. And, since this weight w(Lx) acts at a distance (L x)/2 from, the section PQ, we have, ding, , moment due, , to, , it, , =, , w(Lx).(Lx)l%., , For equilibrium of the beam, therefore,, , J*, , d0, , whence,, , = --_-.., , Substituting this value of dd, in the relation, ?e, , dy=z(Lx)d9, , have, , Clearly,, , w.L =, , FF, the total load, , on the beam, , t
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818, , BLASTIOITB, , WL*, '-817,', , So that,, , It will be seen that this expression could be obtained directly, in case (/) 6, above, by putting, load at the end, ', the, , W, , from the result, , B equal, , to zero., , 131., In discussing, Limitations of the simple theory of bending., the above simple cases of bending, we have tacitly made the following, , assumptions, (i) That the cross-section of the beam remains unaltered during, This, as we have seen, is not strictly true for, the extenbending., sion of the filaments above the neutral surface brings about their, lateral contraction, a times as great, and the contraction of the, filaments below the neutral surface brings about their lateral extension. So that, the cross-section of a rectangular beam, bent so as to be, concave downwards along its length is convex downwards across its, length., Similarly, a circular cross-section may change into an oval, form., This change in the shape of cross- section of the beam, due to, bending, results in a change in the value of the cross-section and, hence in that of Ig for it. Usually, however, it is much too small to be, :, , ;, , of any practical consequence and may safoly be ignored., That the radius of curvature of the bent beam, or rather that, (/"/'), of its neutral surface, is large compared with its thickness. This is, almost always true for all cases of elastic bending., (///) That the minimum deflection of the beam is small compared, with its length., This, while more or less true for ordinary engineering, problems, is not strictly so in quite a number of cases. Thus, for, example, in the case of a clock spring, the deflection produced is very, We shall, therefore, do well to, large evon within the elastic limit., discuss this particular case here, as representing the more general case, of strongly bent beams., 132. Strongly bent beams When a beam bonds very strongly, its, inclination to its original, unbent or unstrained position, and hence, the tangent of this inclination, is no longer small. Consequently, its, curvature (l/R) can no longer be taken to be equal to the rate of, 2, change of slope, d*yl<Jx as is done in the ordinary cases of small, curvatures (see solved example 4, page 332) but it is now given by, ,, , The problem thus bee >mes, the, , differential equations, solution., , However, there are, , quite complicated in, , obtained, , also, , some, , not being, , many a, , case, with, , amenable to, , easy, , cases which can be investigated in a, , much simpler manner and we, , shall here consider only one of these, v/3.,, that of & flexible cantilever, like a clock-spring, clamped at one point, and loaded at its free end. As we pass or 'pay out' more and more of, , the spring through the clamp, keeping the load constant, its free end, drops further and further down, as a result of the large amount of, bending, until finally it becomes quite vertical. The horizontal distance, between the clamp and the loadod eud of the spring is now the, maximum and any more of the spring 'paid out' through the clamp, , merely hangs vertically.
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314, , *BO*BRIMS Of MATTBB, Thus,, , let, , changed at, , O, , OAB,, , (Fig.198), be the bent position of the spring,, at the end B 9 such that, , and loaded with a weight, , W, , =, , x),, tangent at A, (coordinate x, makes an angle #, and that at B, an angle <, with, tf),, (coordinate x, the axis of x. Then, the bending, ^ ne, , =, , moment, , at, , A, , W(ax)., , -x), , ^, , where, , portion, , is, , equal, , clearly, , to, , So that,, , is, , OA, , = YJ 1, R, ., , g, , ,, , the radius of curvature of, of the spring., , 'Now, -, , =, , d6, , r~Sec, , solved, , Kxample 4,, Lpage 332., , de, , And, , Fig. 198., , YI, YJ, , ', , *'-, , dx, , ds, , Y.Ia cos, ., , Or,, , ds', , dx, , W(a-x).dx== Y.Ig .cos, , 6-, , ~., , dxjds, , ['.-, , =, , cos, , 6., , k, , Q.dB., , x, and, integrating this expression between the limits, we, end, from, have, loaded, the, horizontal, the, of, distance, 0,, a,, .*., , X, , =, , [, , W(a-x).dx, , =, , cos 6.dQ., , Or,, , ~=, , Or,, , Y.I .sin6ff, , Now, when the loaded end becomes vertical, <f>, horizontal distance a becomes the maximum, say, a m, , ., , stituting a, , = am and, , <j>, , =, , in expression / above,, , 90, , = 90, , and the, So that, subtherefore,, , we, , have, H>fl, , w, , 2, , /2 5= Y.* g>, , am, , Or,, , *, whence, a m =, , =, , ^/ZtQW., , It will be seen that the value of, , Y, , ...(II), , for the material of the, , beam can easily be determined from either of the relations /, we know the angle of inclination $ of the loaded end, of the beam with the horizontal, or the maximum horizontal disflexible, , or // above, if, , a or a m ) of the loaded end from the clamp., Transverse Vibrations of a Loaded Cantilever. If the, 133., loaded free end of a cantilever be depressed a little and then released,, it starts moving up and down its original position, i.e., executes, Let us calculate the time-period of these, transverse vibrations., tance,, , (i.e.,, , vibrations.
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Sift, , We have seen, , =, , y, , above,, , 130, pages 311),, , (, , WL*, -s-rrv-, , ,, , whence,, , ,, , how, , W=, TT7, , W, , which gives the load or the force, \ required to maintain the cantilever in equilibrium, with its free end depressed or displaced through, y., This, thus, also measures the elastic reaction of the cantilever,, which is oppositely directed to it., , =, , M, , dzyjdt* be the acceleration of the mass, Now, if a, suspended, from the free end of the cantilever, (i.e., if, be the mass of the load, .a., W), the force of inertial reaction on it is equal to, , M, , M, , and there, , Hence, since the cantilever is in equilibrium, other external force acting on it, we have, .., , M.a, , =, , ,, , 9, , a, , =, , np, , 3.373.YyIg, ---i-' whence, a=--, , Or,, , where 3 YIg [ML*, , is, , fi 9, , =, , u.y,, , a constant for the given cantilever, with the, , given load., , a, , Thus,, , oc y,, , i.e., the acceleration of the mass (or of the free end of the cantilever) is, It is thus a case of simple, directly proportional to its displacement., harmonic motion, and its time-period is, therefore, given by, , t, , cal, , =, , 27T, , A / JL, , V, , =, , 27T, , ft, , A / -_, , X, , yZYIJML, , As can be seen at once, this relation for t gives a good dynamimethod for the determination of the value of Young's modulus, , of a given beam or rod., ( Y) for the material, able for beams like a metre stick etc., Depression of a, , 134, , Beam, , It, , is, , particularly suit-, , supported at the ends., , hen the beam is loaded at the centre. Let a beam be, two knife edges at its two ends A and B, as, on, supported, w/, shown in Fig. 199, and let it be, \y/, A 2, ", * 2, loaded in the middle at C with a, (/) \*, , weight, , W., , The, , each knifebe W/2, in the up', , reaction at, , edge, , will clearly, , ward, , direction., , Pig. 199., , Since the middle part of the beam is horizontal, the beam may, be considered as equivalent to two inverted cantilevers, fixed at C, the, bending being produced by thu loads Wfi, acting upwards, at A, and B., If,, , therefore,, , each cantilever, , L, , be the length of the beam AB, the length of, is L/2 f and the elevation of A or B, , (AC and BC)
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above, , B, , is, , F MATTER, , PROPERTIES, , 316, , C or,, , what, , IB, , the same thing, the depression of, , C below A, , and, , given by, , ., , y, , whence,, , -, , V, ., , ., , [v, , 4, , =, , If the beam be of a circular cross-section, we have ak z, 7rr 4 /4,, where r is the radius of the cross-section so that, for such a beam,, ;, , WL*, , H/L 8, , 4, , ......... CO, , And, if the beam be of a rectangular cross section, of breadth, b and depth d we have ak 2, bd*/l2, and, therefore, for such a beam,, , =, , y, , heara is loaded uniformly., Here, let w be the load, (//) When the, r, per unit length of the beam, so that the total u eight icting downwards at tho c g, is M'L, **, IV,, wliere L i,s the length of the beam., The reaction at each knife-edge is, thus obviously J ivL, acting up*, wards* (Fig 2i)(>) with the beam behaving as a s} stem of two canti<, , =, , T, , W(l-X), , W*urL, , fixed at"C., , s,, 1, , tnsidoi'ing anain, a section, distance .v from the mid-point, C of the beam and taking half-length of the beam equal to /, the, weight of the portion (/Jc) of the beam, /., a weight w(lx) acts, downwards at a distance (/ ,v)/J from tho section PQ., pi, , (, , 200, , PQ, , Thus, the bending momjnt about the section, , For equilibrium, therefore, this must be equal to the moment, of the resistance to bending viz.. Y Ig iR, where R is the radius of, curvature of the neutral axis at PQ., , Y IJR =, , i.e.,, , y./f, , d, ., , page, ,, , C/ J*, , _, , [_, , wl.ence,, , /., , ,,, A, hence, And, , y, , =, , f/, , Jo, , f' Fr f. C, a, , /, , 1 U., j'JO.
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317, , ELASTICITY, , --, , 3 _, , 2, , wL, w, YL, .,-2YI.., , wL, , 1, , ', , Now,, , substituting L/2 for, , w, , L*\, , (L*, , y, , YL, , But, , wL=W, , 6, , '1287, , the total, , 877,,, , we have, , /,, , _, , M', _, , \ 48, t, , J*, , 12, , YT, "384, on, the, beam., weight, Yf, , 384, , Hence, Determination of F by bending of a beam. It will be easily seen, we measure the depression (y) of a beam of known dimensions, supported at the ends and loaded at the centre, as in case (/), above, we can easily determine the value of Y for its material, by, applying relation (/) or (//), as the case may be. In practice, it IB, convenient to use a beam of rectangular cross- section so that, knowing W, L, b d and y Y can be easily calculated from relation (//), above., that, , if, , ;, , }, , t, , is as shown in Figs. 201 (a), and symmetrically, on, horizontally, supported, , The arrangement, , is, , and, two, , (b)., , The beam, , parallel knife-, , Fig. 201., , ., , known, , L, , apart and the load is applied by placing, weights in a scale pan, also supported on a knife-edge, midway between them, as shown. The depi ession y of the mid-point, thus produced, is noted directly with the help of a micrometer screw, [Fig., 201 (0)], or, more accurately, with the help of a microscope, the eyepiece of which is fitted with crossfires, [Fig. 201 (b)]., , edges, a, , distance, , Readings are taken, first with the load increasing, in equal steps,, and then with the load decreasing, in the same equal steps, and their, mean taken. This gives y. Then, if the load were increased (or, decreased) in regular steps of IV each,, , we, , have, as explained above,, , WL*, y, , ", , A, , v, , i%, , wk*
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FBOPBBTIES OF MATTE*, , 318, , where b and d are the breadth and the depth of the beam, and Y, the, Young's modulus for its material., , Y, , Hence, , =, , r-r-ji4y &.d 8, , Now, since the depression of the beam is given by the relation, y = WL*/4Y.b.d3 it is clear that, for a given load, the depression of the, beam is, ,, , (i), (ii), , directly proportional to the cube, , (Hi) inversely proportional to the, (iv) inversely, , may, , of its, , length,, , inversely proportional to its breadth,, , cube of its depth, and, , proportional to the Young's modulus for, , its, , material., , It follows, therefore, that in order that the depression of a beam, be small for a given load, its length should be small, i.e its span, ,, , should be small,, , its, , Young's modulus for, , breadth, its, , and depth should be large and the, , material should also be large., , When, , a girder is supported at its two ends, its middle part is, depressed, and the surfaces above and below the neutral surface are, respectively compressed and extended, the compression being the, utmost at the upper face, and the extension, the maximum at the, lower face, the stresses being the maximum there and decreasing as, we proceed towards the neutral surface from either side. It follows,, therefore, that the upper and the lower faces of the beam should be, much stronger than its middle portions In other wor Is, the middle, portions may be made of a much smaller breadth than the upper and, the lower faces, thus affecting a good deal of saving in the material., It is for this reason that girders are usually manufactured with their, cross-section in the form of the letter I., Stiffness of a beam., The ratio between the maximum deflection, of a beam and its span measures what is called the stiffness of the beam., It is usually denoted by the symbol 1//7., For steel girders of large, span, n should lie between 10UU and 2000 and for those of shorter, And for beams of timber, the value of, spans, between 500 and 700., n should in no case be less than 360., 135. Searle's Method for the Comparison of Young's Modulus, and Coefficient of Rigidity for a given material. A short length of the, wire, the values of Y and n for the material of \*hich are to be compared, is fastened to the middle points of two similar and equal metal, bars AB and CD, (Fig. 202), of circular or rectangular cross-section., The bars are then suspended from a rigid support by means of two, , small vertical lengths of threads, so that, when the wire, the bars are parallel to each other, as shown., , is straight,, , On slightly pulling together the ends A and C of the two bars, symmetrically and through equal distances, the wire is bent into a, circular arc, (Fig. 203)., On releasing the bars, they begin to vibrate, in a horiz >ntal plane from a circular arc on one side to a similar arc, on the other, due to the torque exerted on them by the wire, the midpoint of the bars remaining almost at rest., If /be the length of the wire, , each bar from, , its, , and, , 0,, , the angle of deflection of, , normal position, the angle subtended by the wire at
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319, , ELASTICITY, the centre of curvature of the circular arc into which, clearly 20, such that, / as R.2Q,, , Or,, , where, , R=, , it, , bent;, , is, , R is the radius of the arc., 112$., , Fig. 202., , by, , it is, , Fig. 203., , Now, the bending moment of the wire and the couple exerted, on each bar is, as we know,, , YL, R, , Yirr*, Lfbr the wire., , This couple produces an angular acceleration dwfdt in each bar,, and, therefore,, r, , //, , is*, , I.datjdt, , where 7, , is, , =, , --, , Y.TT.r'.O, , -, , 2[, , ,, , the M./. of each bar about an axis through its mid-point, its length, i.e., about the thread from which it is, , and perpendicular to, suspended., , And, i.e.,, , 7 Trr 4, /., , the acceleration, , is, , proportional to the (angular) displacement., , Therefore, the motion is a simple harmonic one,, time- period of each bar is given by, , whence,, , Y, , =, , ?---, , and hence the, , ., , ..., , (I), , The suspension threads are then removed, and one of the bars, clamped horizontally, so that the other bar hangs vertically below, The suspended bar is then turned, it at the other end of the wire., about the wire in the horizontal plane, so as to twist the wire when,, on being released, it begins to vibrate torsionally. Its time-period f, is, , is, , noted., , Now,, , tl, , -a, , 2wy^/C, , where, , C is the twisting couple, , wire per unit deflection or twist, and, , is, , set, , equal to nnr^fiL, , whence,, , n =*, , ysyr*, , in the, , up, (, , 122)
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PROPERTIES OF MATTE*, , 320, Dividing relation, , (I), , by, , we hav, , relation (II),, , - SnJJ, Zn ~~, , -, , r, , JL1, //*, , - "", r*.tS "87T./.7, , This gives the ratio of Young's modulus and the co-efficient of, of the material of the wire in terras of t l and f ,., , rigidity, , = ~Y, , Poissorfs ratio, a, , Now,, , 1., , >Yl, , So that, substituting the value of Y[2n, we have, /, *, , 2, , o/, ^* 2, , 2, , /, *, , Thus, Poisson's ratio for the material can also be easily determined., will be readily seen, the radius (r) of the wire, the measurement, the chief source of error ;see page 303) has been eliminated altogether., , N.B. As, of which, , is, , 136. Strain energy in a bent, , small portion, , AB of a beam, , is, , beam We have seen before, (page 307), how when a, b^nt into ths form of a circular arc (Fig. 204), subtending an angle 89 at its centre of curvature,, the strain produced in an element of it at a, distance z from the neutral axis is given by, z/R, where R is the radius of curvature of the, neutral axis., , energy per unit volume in any, , Now,, , i stress* strain., , type of strain, , So, , that, energy associated with, the element in, question, J stress X strain X volume of the, g lenient., , =, , if the stress be F, the strain, e,, of cro^s section of the element, normal to the plane of the diagram equal to, <M, and ths length of the neutral axis, Sx,, , Then,, , the, , area, , clearly,, , energy in the element, , Fig. 204., , Now,, , Y=, , =, , stress I strain, , F = Ye =, = f Y. z, ^$A$x., , Fje and, 1, , .'., , =, , Y.zjr., , 2, , So, , that,, , energy, , in the, , element, , Integrating this expression over the whole cross- section,, strain energy in the entire portion AB of the beam, , we have, , Since, as we, section considered., , moment, , Or,, , know, fz*.dA, , strain energy in portion, , =/<,, the geometrical, , AB of the, , beam, , f, f Y1 1, , -~, , 1, , -^, *, , L, , -K, , ., , J, , of inertia of the, , 1, , -^rp.&x., X'ff, , But YlglR -A/, the bending moment of the section., , Hence, , And, , .*., , strain energy in portion, , strain energy, , E, , in the, , AB of the, , beam, , whole beam of length, , =, , L =, , *, , ~-*r, , I, , JO, , ^fjf, , -y^-, , v>yy/^, , ., , &*., , dx>
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ELASTICITY, , So, , 321, , that, substituting the relevant values of the bending, strain energy of the beam in different cases., , moment, we can, , easily, , determine the, , from, , W, , Thus, for example, in the case of a light cantilever with a load, suspended, = W(L x), (see page 310), and, therefore,, we have, , M, , its free end,, , strain energy, in this case, , -, , 1, , 2, , Y[, , Resilience of bent beams., , 137., , f, , Jo, , resilience, is, , .W ZL*., g, , The work done, , in deflecting a horizontal, is equal to, x (deflection at the, the resilience of the beam, as ex-, , W, , beam of whatever type, loaded with a weight, loaded point)., Since this work do.ie measures, plained in, 120, we have, where y, , 1, l, , W*(L-x)*.dx, , of the beam, , =, , \W, , i, , W.y,, , the deflection of the loaded point., , Thus, in the case of a light beam, of length L breadth b, and depth d, sup^, at the centre, we have, ported at the two ends and loaded with a weight, 3, y, WL*/48YIg (page 316) and Ig in this case, is given by bd /\2, (page 308)., So that, we have, y, , ~, , W, , ,, , ,, , 12, resilience, , of such a beam, , ', , 96Y, , bd*, , A long beam of an isotropic, 138. Colums, Pillars and Struts., material used for supporting loads is called a column, a pillar, or a strut., Now, whereas a column or a pillar must always be vertical and, generally fixed rigidly at its ends, a strut may be vertical, horizontal, or inclined and may either have bath its ends fixed rigidly or both connected to the surrounding structure through flexible joints, or it may, have one end rigidly fixed and the other connected to a joint, The, theory underlying the two is, however, the sama, the commonest case, being that in which the load applied is a compressive one, i.e., acts at, one end of the column or, , strut,, , along, , its axis,, , tending to compress, , lengthwise, though, in some cases, there, a lateral load, in addition., Let us take the case of a column or a pillar first,, r, , shorten, , it, , may, , as well be, , Let us take a long and straight, or, wood, 205, of, metal,, arranged in a vertical, [Fig., strip, (a)], position, representing a column, with, both its ends rounded and fitted into, metal sockets, as shown magnified in Fig., 205 (#), so as to allow it freedom to bend, all along its length, and let a load be, applied to it at the top in the form of .a, metal cylinder, containing lead, shot or, mercury, so that its magnitude may be, varied at will, with the cylinder moving, between two parallel guides GO to ensure, 139. Critical load for a long column,, , AR,, , its, , t, , vertical descent., , Now,, cient to, , first,, , bend the, , with the load insuffistrip or the column, we, , apply a lateral force /t at its mid-point, 0, 0,3 shown, to make it bend a little and, we find that, on removing the lateral force,, We, the column straightens itself out., increase the load at the top, apply the, lateral force, as before, and then remove, , (a), Fig. 205., , it., , Perhaps, the column
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322, , FROPEBTIES OP MATTER, , again bends and then straightens itself. In this manner, we go out, repeating the experiment with successively increasing loads at the, top until we find that when the lateral force is applied and then*, removed, the column remains bent. At this stage, we find that, whatever the deflection we produce in the column by the lateral force,, the column continues to retain the same on the removal of this force,, Thisprovided, of course, that the elastic limit has not been exceeded., load which just keeps the column bent, but does not bend it further, z>, called the critical load/br, , it., , we increase the load beyond the critical value and give a, bend to the column, as before, we find that the load now, increases the bending further and the column either acquires a permanent set or collapses due to buckling,*, If, , slight, , Let us see how we may account for this critical load. Let us,, therefore, consider the equilibrium of the column AB, under a vertical load P l and a lateral force, \(P+Q), /! at its mid-point 0, with the, deflection of the column equal, to y v as indicated in Fig. 206, (a)., , Since, for equilibrium, the, lateral force fv must bo balanced, , by two horizontal forces, each, equal to fJ2, acting at A and, B, in the opposite direction tothat of/j, we have, total bending, , O, , = A>, , (b), , (a), , moment about, , be the maximum, jpj, due to bending and Z,, the modulus of cross-section at, O (see note on page 309), the moment about due to stress or the moment of resistance to bending = FX Z.4 So that, for equilibrium,, , And,, , if, , stress, , Fig. 206., , and increase load Pl, If now we decrease the lateral force /i to, to \P (the critical load) so that the deflection of the column remainsthe same y lt clearly, the condition for equilibrium demands that, P. yi, , = ^Z,, , whence,, , P = F Z/y 1, , [v/i, , l, , is, , now, , 0., , the column be in equilibrium when subjected to a verlateral force /2 with its deflection now equal to j a, and we reduce /2 to, and increase P2 to P', with the deflectionremaining unaltered at J 2 we have, proceeding as before, P, where Fa is now the maximum stress due to bending., |, , Again,, , tical load, , P2, , if, , and a, , ,, , f, , ,, , Since the bending, , And, therefore, P', , =P, , is, , proportional to stress,, , we have, , =, , F /y =F2 ly^, 1, , 1, , (the critical load), thus clearly showing that, the column will remain^ equilibrium under the same critical load P for, /.., , bv bending or bulging out.
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323, , 1LASTICITY, , any value of the deflection we choose to give it within the elastic, as we have seen already in the experiment discussed above., , limit,, , however, wo increase the vertical load to a value beyond its, value P, say, to (P+Q), as shown in Fig. 206 (b), the bending moment will clearly increase to (P-\-Q) y^ or to P^i+Qy^ And,, since the moment of resistance to bending, v/z., Fr Z balances only the, portion Pl y l of it (as we have just seen above), the portion Qyl, remains unbalanced, resulting in an increase in the bending or deflection of the beam beyond y 1, In order to keep the deflection at the, same value y v therefore, we shall have to apply a force/, say, at the, mid-point 6 of the column, in the opposite direction this time, so as, to balance the portion Qy of the bending moment. Since/ is supl, posed to be balanced by two equal forces //2 and//2 acting at A and, So that,, B, as shown, its moment about O =s (//2).(i/2), /.L/4., to prevent the column from bending further (beyond y\) we must, If,, , critical, , ., , <, , =, , have/L/4, , =, , Q.y r, , Now, within the elastic limit, the moment of resistance to, bending is proportional to the stress, i.e., FZ oc F, and hence also to, the deflection of the column (because, then, y oc F)., But, once the, elastic limit is exceeded, the column acquires a permanent set, though, it is also possible that, due to the moment of resistance due to bending now increasing more rapidly (as it always does beyond the elastic, limit), the column may acquire a new position of equilibrium under, the additional load Q. But, if this does not happen, the column will, continue to bend further and further and finally collapse., 140., , Filler's, , Theory of Long Columns., , Let, (/) When the two ends of the column are rounded or hinged., represent a long and initially straight column of an isotropic, material, of length L and of a uniform cross-section, and uniform elasticity, with rounded or hinged, ends so as to be free to bend throughout its length., ~, Further, let the critical load P act axially upon it,, f, i.e., in a line with its axis in its straight unloaded, ;, position, and let it be given a slight bend by the, 2, application of a lateral force for an instant, (Fig., , AB, , ^, , i, , 207)., , !, , Now, consider a point C in the column, at a, distance x from its mid-point O. If the deflection, here be y, clearly, the bending moment here due to, , P=, , *~, , But, if the radius of curvature of the, be R, the moment of the resistance to, bending there is YJgiIR. And, clearly, therefore,, Y.I /R, P.y, whence, IjR, P.ylY.Ig ., P.y, , bend at, , t, , C, , =, , =, , =, , ve sign, d yldx 2 (the, But, as we know, l/R, to, make, jR, being given, positive, for dyjdx decreases, as y increases). We thus have the differential equation, 2, , ,, , Ylg, , Fig. 207.
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PROPERTIES OF MATTER, , 324, , The, , solution of this equation gives, , = A sin, , A x+ B cos A x l9, and, are constants to be determined., B, and, A, <\/ P\Y.Ig, we, have, Differentiating this,, , y, , (!), , =, , where A, , = A\ cos \x~~- B\ sin pc., x = 0, rfy/rfx = and, obviously,, , dy/dx, , sin Ax =, Now, when, = 0. From equation I, therefore, we have 7 = B cos \x., and .*. 05- AX = 1, But when x = 0, we have j> = ; 1? Ax --=, and sin Ax = 0. So that, from equation I, we have y = B., ==, And .-. B cos A^/2=0., (b) When x, L/2, (i.e., at 4), >> = 0., (a), , so that,, , ;, , A, , J, , [From, , This means that either, , have seen, , [in (a) above],, , cos, , 7\, , B =, , = y\., = 0, or., , B, , L/2, , L/2, , ="-, , or cos AL/2, , =, , (a), , above., , But, as, , 0., , we, , It follows, therefore, that, , that A, , w /2, , /2, , =, , ir/2,, , [for> A, , =, , 9 87 YI g, , whence,, , L*, , therefore, the value of the critical load, or the load which, can just keep the column bent at the initial curvature given to it An, addition to this makes the column collapse., , This, , is,, , It will be clear from this expression for the critical load that for, the same values of Y and /,, the smaller the length of the column, the, greater the critical load for it., (ii), , and B, , When, , the two ends of the column are fixed., Let the ends A, now fixed, as shown in Fig. 208 (a), so that, when it gets bent or deflected, the tan-, , of the column bo, , gents to, vertical,, , it, , at points A,, , O, , and B are, , all, , with the line of action of the, , load now no longer passing, through the centres of its end-points. It, passes, instead, between the initial unbent position AB of the column, and its, mid-point O in the bent position, cutting, At, the bent column in points C and D., these points, therefore, there is no bendresultant, , ing moment, RO that they are points of, opposite flexure., , considering the portions CA, of the bent column, we observe, that the deflections at certain points in, the two curves (as measured from the, At, vertical line through C ) are equal., all such points in the two curves, therefore, the bending moments must bo, equal, and hence also the radii of curvature there must be the same, in view of, , Now,, , and, , (I)), , Fig. 208., , CO
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325, , ELASTICITY, , the fact that the column is of a uniform cross- section. Further, the, two curves (GA and CO) have clearly the same slope at C and also, at A and O, the tangents at all these points being vertical. Obviously f, The same is also*, therefore, the two curves are equal and similar., true of the curves, thus divide, and DO. The points (7, O and, the whole column into four equal parts and the length of the portion, of the column is, therefore, equal to half its total length, i.e.,, , D, , DB, , COD, COD =, , L/2., , COD, , of the column, the whole of which i, cases considered above, i.e., like a, column of length L/2, with its ends rounded or hinged and carrying, an axial load P at C, So that, proceeding as in case (/) above, we, Clearly, this portion, bent, behaves exactly as the, , have, , P, , --, , L*, , the critical load (P) for the column, in this case,, that in case (/)., i.e.,, , is, , four times, , Thus, a column with its ends fixed, has four times the strength (to, it, will have with its ends rounded or hinged., Or,, putting it differently, ci column, with its ends fixed, can support,, without bonding, the same load as one of half the length, with itsends rounded or hinged, would do., resist thrust) that, , (Hi) When one end of the column is fixed and the other loaded,, This is an easy deduction from case (i) above. For, suppose we havea column AS, with rounded or hinged ends, and of length L as shown in, Fig, 209 (a), with P as the critical, load on it., Then, the tangent to, it, at, its, mid-point O is ver1, , ',, , we clamp it, therefore,, If,, tightly at O, without disturbing the, direction of the tangent at that point,, the lower half, of the column might, as well be removed, without in any, way affecting the upper half OA., So that, the upper half then behaves, as an independent column, of length, L s= L'fi, fixed at its lower ends and, loaded at the top, as indicated in, , tical., , OB, , (b), , Fig. 209., , Fig. 209(&)., , All that we have to do to calculate the critical load, in thisto consider the column of length L and fixed at one end, as, 2L, with both its ends roundequivalent to a column of length L', ed or hinged. Therefore, proceeding in the same manner as in case, case,, , (i),, , is, , =, , we have the, , critical load, , P, which, , is, , L, with, , given by, , =, , clearly one-fourth of the critical load for a, its, , ends rounded or hinged., , column of length
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PROPERTIES OF MATTER, , 326, , Thus, we find that a column, hinged at one end and loaded at the, of the same column when hinged, , other, has only one-fourth the strength, at both ends., , N.B. Exactly a similar treatment to that in cases (i) and (//) applies to, the corresponding cases of a strut, arranged horizontally, so long as the strut is, , Fig. 210., , toaded axially, or along its axis, like the columns incases (/) and (//). Thus, if, the two ends of the strut be rounded or hinged, so that the whole of it can, bend,, we can represent its behaviour as in Fig. 210 (a), and, when its two ends are fixed,, as in Fig. 210 (b). It will be noted that these are essentially the same, figures as in, cases (i) and (//), respectively, but are rotated, as it were, through an, angle of 90,, so that instead of a vertical load we now have a horizontal load. The method of, calculation for the critical load, therefore, remains the same., 141. Elastic Waves. When a system of stresses, to which a body or a medium is being subjected, is suddenly altered, we have (/) a corresponding motion, of the body or the medium itself and (11) propagation through it of the changes in, i\\t two occurring, stress,, simultaneously and constituting what is called the propagation of an elastic or a stress wave., as, we, Now,, know, even in the case of an iso tropic medium, a deformation, in one direction is invariably, accompanied by deformations in two other direcat, tions,, right angles to the first, (the familiar case of the deformation of a cube),, so that the theory of elastic waves is, and this, really quite a complicated one, complication is further aggravated in the case of bodies like the earth, for, example, where the elastic properties vary with depth, which explains at once, the complicated pattern of the seismic waves ( 100)., ;, , In general, however, we have three types of elementary elastic waves in the, case of a uniform, isotropic medium, viz., (i) compressional, (//) shear and (///), shall only briefly touch upon them here., fltxural waves., , We, , Gompressional Waves. These waves are produced when we give an, axial blow to a long bar, i.e., strike it along its axis, and, assuming the sides of the, barJo have freedom of movement, their velocity is given by the relation v, Y l9, where Y is the value of Young's modulus for the material of the bar and p,, its density., But, in case the sides of the bar too are fixed, Y is to be replaced by, Y(l a)/(l + a)(l-2(j), where a is the value of Poisson's ratio for the material, of the bar. This expression takes many forms, the simplest among them being, and n are the coefficients of bulk and rigidity modulii for, K+4nfi, where, the material of the bar. Thus, we have different types of compressional waves, all, of which, however, have the common features that (i) the vibrations occur along the, direction of propagation of the wave, i.e., the wave is of the longitudinal type, and, <//) the velocity of the wave is given by, modulus of elasticity, ~~, (/), , V, , K, , aentity, , Thus, in the case of a liquid or a gas, n, , =, , and we,, , therefore,, , have the
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HYDBOSTATICS, .Pascal discussed this result in his 'Treatise on the equilibrium oj, liquids', published in the year 1663, with reference to vessels of different shapet r, known as- Pascal's vases, (Fig. 213), all having equal bases and containing wafer, upto the same vertical height h, so that the pressure on the base of each vessel was*, equal to hgms. w/./rmV, and, therefore, the thrust on, a is its area., , it, , was, , h.a., , gms* wt., where^, , He was perhaps the first person to have pointed out the paradoxical truth, that even if vessel (i) contains 100 Ibs of water and vessel (v) only 1 oz. of it r, the thrusts on the bases of, both is the same. Aptly,, therefore, it is called the, hydrostatic paradox., Strange as it may, seem, but if the water in, vessel (v>) be frozen into ice, and detached from its, sides, the thrust exerted by, this ice on its base will be only, , ELB, , 1 oz. >v/., but once this ice is melted back intowater, the thrust again increases to 100 Ibs. wt. The explanation of this seeming paradox is, however, simple. The ice does not exert any upward thtttittyl, the part of the vessel opposite to the base and the latter, therefore, exerts, tify, But the water does exert an upward thrust O$, squal and opposite thrust on it., it and hence receives back an, equal and opposite downward thrust from itt, , In case of vessel, of the water on it., , (/),, , the thrust, , on the base, , is, , equal to the entire weight, , In vessel (//), the upward component of the thrust due to the left side of, the vessel supports the weight of water in it, between the left side and the dotted, line A, while the downward component of the thrust due to the right side of the, vessel exerts a downward thrust on it, equal to the weight of the water inbetween the right side of the vessel and the dotted line B ; so that, the thrust or*, the base is the same as due to a vertical column h of water., In vessel (///), the upward components of the thrusts due to both the, and the right side* of the vessel support the extra weight of the water,, between the two sides and the dotted lines C and D, and, again, therefore,, , left, , inthe, equal to that due to the cylindrical column h of water in-bet-, , thrust on the base is, ween the dot ted lines C and D., , And, similarly, in vessel (/v), the downward components of the thrust, due to the two sides of the vessel exert an extra thrust on the base, equal to the, and F;, weights of the water contained between either side and the dotted lines, so that, once again, the total thrust on the base is the same as that due to a cylindrical water column, , of, , above fact by supporting, by means of, vases of the above shapes, one by one, on a large-, , Pascal ex t, a separate stand,, , .verified the, , disc,, , D, , (Fig., , 214),, , suspended from, , the shorter pan of a hydrostatic, balance and kept pressed against, their bases by placing a heavy weight, in the longer pan, and pouring water, into the vessel. The disc just got detached from its base as the water, reached the same level in each case r, thus clearly demonstrating the equivalence of the thrust on the disc in*, each rase and fully vindicating hisdeductions., Fig. 214.
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PEOPBBTIES OF MATTER, , 328, , mum values being, and, , respectively 1, in the case of perfectly elastic bodies,, zero, in the case of perfectly plastic ones., , Thus, if M t and u 2 be the velocities of two bodies before the impact and v l and v 2 after the impact, we have, ,, , v a/), , (v, , (, , where, , w 2 ) and, , (U L, , e, , u t), , (U l, , Or,, , =, , ve sign of thethe, (-"neglecting, relative velocity after the im-, , ,, , L, , -v = <?K-w, Vl, a), , v2 ), , (v l, , pact., ..., , 2 ),, , are their relative velocities, before, , (i>, , and, , after impact., , =, , It will thus be seen that if e, elastic,, , v2 ), , (Vj, , =, , w 2 ),, , (i^, , 1, i.e., if the bodies be perfectly, relative velocities of the bodies are, , i.e., //*e, , the same before and after the impact, (suffering only a reversal of, direction, in the latter case)., , But, , =v, , or, vl, , if e, , =, , 2 , i.e.,, , 0,, 0, i.e., if the bodies be perfectly plastic, (v^v^), the two bodies move with the same common velocity r, , after the impact., , What happens is that when one elastic body, of mass m v moving, with a velocity u v collides against another elastic body, of mass m zr, w 2 ), the surface between then*, moving with velocity u^ (where u l, , >, , compressed and when this compression or pressure reachesits maximum value, their relative velocity becomes zero., Thereafter,, the elastic stress between them makes them recede from each other,, the compression is released, and the two bodies move away with, gets, , v, , different velocities, say v 1 and v 2, Clearly, then, in accordance, ., , momentum, we have, sum total of momenta, , with the law of conservation of, , after the impact, , = sum total of momenta before the impact., Now, sum total of momenta after the impact = /W 1 v 1 +/w a v, and sum total of momenta before the impact = m^+m^., So that,, m jVi+ AW 2v 2 = m u1 -{-m 2 u 2, ... (//'), From relations (/) and ('), we can easily calculate the values of, ., , 1, , and, , vt, , v2, , For, multiplying relation, , "Vi w 2 v =, a, , And, adding, , relations, , 'WiVi+/w a v 1, , (//), , a 1, , v^, , 2, , 2, , vl, , =, , 1, , 2, , ,, , we have, ...(/>, , ,)., , m^ i-m u +m e.(u -u, 2, , ^, ll99, , x, , 2,, , we have, , ^, , -, , ,, , whence,, , m, , w^K, , (///),, , mv =, , +m v, , Or,, , and, , by, , (i), , 2, , v, , 2, , 2, , 2, , 9, a/, ., , t, l, , ., ,, , Similarly, multiplying relation, relation (ff), we have, , (i), , by, , m, , lt, , it, , can also be shown that, , _, , """, , Impulse during, , 2 )., , ,..(/v, v, , and subtracting from, , (m 1+ m 2 ), Further,, , l, , restitution, , Impulse during compression
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329, , ELASTICITY, , The value of, , found to diminish with increase in the, , is, , nowever,, , e,, , velocities of the approaching bodies, and vice versa, and it has been, shown by Sir C.V. Raman, that its value is very nearly equal to 1, if, , the collision be very weak., 143. Loss of Kinetic Energy on Impact., the above relations, that, *, , i, , i, , ,, , *, , i, , >, , where J^v^+Jw^Vj 1, , ,, , 9, , i, , It, , E2 =, E, , Or, , an, , toss, , ,, , 2, , 1, , ,-, , ., , 2, l, , -E ~, , ( 1, , 2, , r, , f, o/ energy on impact, , /, , i.e.,, , 1, , w2 ) a, , the total kinetic energy E2 of the colliding, and Jw 1 w 1 2 +|m 2 w 2 2 their total kinetic, , is, , bodies, after the impact,, energy E1 before the impact., , Thus, , e^.ttiiWVfW!, , (1, , ^, , can be shown from, , =, , ^-', , >, , J, , expression, with always a positive value, showing that there, loss of energy on impact between two bodies., , is, , always a, , Now, the following, , special cases arise, 1, i.e., when the colliding bodies are perfectly, (i), In this case, (EL, elastic., 0, i.e., there is no loss of energy on', 2), impact of perfectly elastic bodies., , When, , When, , (ii), , plastic., , e, , =, , e, , =, , Here (E1, , :, , E =, , 0,, , E2, , ), , i.e.,, , when the colliding bodies, , has the, 1, *, , (/W1, , maximum, , m |.(w, 2, , 1, , value,, , *are perfectly, , viz.,, , w2 ) 2, , 2, , (Wi+wtaj, of energy on impact of plastic bodies., w 2 i.e., when the bodies have the same velocity,, (Hi) When u^, (in magnitude as well as direction). In this case, the relative velocity, of one body, witli respect to the other, is zero, so that no impact takes, 0, or, place at all between the two bodies, and, therefore, (El -r-E2 ), again, there is no loss of energy., i.e.,, , there, , is, , maximum, , =, , loss, ,, , =, , The question now arises as to what happens to this loss oi, energy on impact ? Until very recently, it was supposed that the, energy lost during impact was converted into (i) sound, (ii) heat, or, (7) vibration or rotation of the colliding bodies., Sir C. V. Raman's experiments have shown, however, that the, production of sound is in no way related to the energy of impact,, being solely due to the impulse set up in the air during the reversal, of the motion of the colliding bodies, after impact., , The change, , in temperature too is almost always very small and, appears that an appreciable portion of this energy lost during, impact is used up in bringing about a re-distribution of the molecules, in the surface layers of the colliding bodies. Indeed, it has been shown, by Hertz that impact produces a definite flattening of the point of, contact of the colliding bodies, with a finite common area between, , hence, , it
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PROPERTIES OF MATTER, , 33U, , them, each body being compressed in its neighbourhood, the compression increasing at first to a maximum, (which is proportional to, the two-fifth power of the velocity), then diminishing and finally, vanishing altogether, when the bodies get separated from each other., Prof. Deodhar has also verified this molecular displacement in, the surface layers of the impinging bodies, by making different bodies, ('similar and dissimilar') impinge upon each other, with 'extremely low, velocities' and measuring their velocities before and after the impact., Prom his experiments, he has come to the following conclusions, :, , great velocities of the colliding or impinging bodies, the changes, in their surface are 'vivid*, and a greater portion of their energy of impact is used, jp in producing these deformations., (/), , With, , other hand,, (//) With very small velocities of the colliding bodies, on the, -he value of e increases, in the limit, to unity, the increase of e with the 'minimal, velocities' being independent of the nature of their material., (HI ) The rate of change of e, the impact takes place., (iv) The duration of impact, , qui te independent of the, , is, , medium, , in, , which, , is observed to be greater in water than in air,, depends upon the density of the medium., (v) A distinct change in the structure of the impinging bodies is noticeable, under the microscope, though no trace of it is visible to the naked eye., i.e., it, , He estimates from this that energy, of the order of 1000, Jc.gms.lcm*. is used up 'in displacing the molecular aggregates'. Further,, bodies, when strained, take time to recover their original condition,, and a rapid rise and fall in the stress may result in the dissipation of, some energy, provided the, varying forces,, , is, , elastic limit of the bodies, for gradually, , not exceeded.*, , Relative masses of colliding bodies., If, in the above, two, one be at rest, so, of, bodies, or, the, second, example, balls,, colliding, as to have no kinetic energy, we have, 144., , = Jm, !, i, loss of energy during impact =, , total kinetic energy before impact, A, , if, , i, , And,, , So, , that., , ,, , -, , a, ., , (, v, , !?"f"*W, total energy, , loss, , Or,, , -, , 1 .w 1, , of energy, , =, , -^, , m, , 1, , I "h, Clearly, therefore, the loss of energy will be small if, , and, , mjm 3, , be large,, , vice versa., , order to minimise loss of energy, the ratio mj/n, must be, the mass of the striking body must be much greater, than that of the body struck. Hence it is that a slow-moving heavy, hammer is more suitable for imparting momentum to a body than a, , Thus,, , made, , large,, , in, , i.e.,, , *This should not, however, be understood to mean that an exceeding OF, is necessary for a loss of energy to occur., , 'overlapping* of the elastic limit
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331, , ELASTICITY, quick- moving, 'momentum., , lighter one,, , even though the two, , may, , possess the, , same, , On the other hand, if the loss o energy is to be converted inta, be much, must be small, i.e.,, useful work, the ratio, 2 must, 1 /m 2, greater than, r That is why while forging instruments etc., we, (must have heavy anvils underneath them., , w, , m, , w, , SOLVED EXAMPLES, Show, , that (a) a small and uniform strain v is equivalent to three linear, strains v/3, in any three perpendicular directions;, (h) the bulk modulus for a gas (/) at constant temperature (i.e., under, Isothermal conditions) is equal to its pressure and (/*) when the temperature is not, constant, (i.e , when the conditions are adiabatic), it is equal to r times its pressure,, where y is the ratio c p /c v for it., 1., , (a) Imagine a unit cube to be compressed equally and uniformly, ides, so that the length of each edge is decreased by a length /, i.e.., , clearly, decrease in volume, , Then,, , ., , v, 2, , l-(l-/), , and, , =, , of the cube,, , =, , i-i + 3/_3/4-/, , from all, becomes, , i.e.,, , =, , 3/, i.e.. I, , v/3,, , the value of/ being small., Thus, a small uniform volume strain is equal to three linear strains, each, equal to v/3, in three perpendicular directions., neglecting, , /, , /*,, , (b) (/) Let P be the pressure and K, the volams of a gas, and let it be, compressed isothsrmally* by increasing the pressure to (P+dp), so that the, volume is reduced by dv and becomes (K dv)., = dp, and volume, Then, clearly, stress = force per unit area, pressure applied, strain = change in volume/ original volume = d\\V., =, , .-., , Bulk modulus for the gas,, , =, , K=, , i.e.., , -j?., , *, , y, , .K., , Since the temperature of the gas remains constant, Boyle's law holds, good, and we, therefore, have, , PV = (pdp}x(V-dv] = PV-P.dvdp.V-dp.dv., PV = PY-P.dv+dp.V. Or, P.dv - dp.V., , Or,, , whence,, , V.dpjdv, , =, , = P., Since, V.dp/dv = K, we have K, Or, the Bulk Modulus for a gas, at constant temperature,, ticity, is, , equal to, , (//) If,, , its, , [neglecting dp.dv., , P., , i.e., its, , isothermal elas-, , pressure., , on the other hand, the change, , in the, , volume, , is, , brought about, , we have, , =, Diffrentiating this,, , PyKr, Or,, , a constant., , we have, , t, , Vvf V r dp =, , 0., , Or,, , fa, -V~rV =, , (", , rP- j, , JT-yr., , thus, the adiabatic elasticity of a gas is equal to, , The ve sign merely indicates that dv and dp arc, of opposite signs., , |, y times, , its, , pressure,, , i.e., is, , y, , times its isothermal elasticity., , This may be done by using a cylinder and a piston of a perfectly conducting material, so that the heat H conducted out into the surrounding air as, soon as it is generated and the temperature of the gas remains the same as, before., fin this case, the cylinder and piston are of a perfectly non-conducting, material or the cylinder is placed on a perfect insulator, so that the heat generated on compression of the gas cannot escape out but remains inside the gas, itself, thus raising its temperature a little.
Page 340 : 332, , PROPERTIES OF MATTER, 2., , ratio, , We, , a, , Show, , that the Bulk modulus, , A' -, , have, , =, -^, , Now,, , K-, , 3., , relation,, , We, , n, , K=, , Y/3(l, , p/a, , =, , n, , =, , I/2(a+, , But, , I/a, , -, , Delhi. 1947}, , [See pages 288-8^., , a., , 7/3(1-2(7)., , Y, , that lhe rigidity n, and Young's modulus, r/2(l -fo), where <r is Poisson's ratio., , have, , ;, , [See page 28 7 ., , ., , Show, , =, , and the Poisson'ff, , 2<*)., , (Punjab, 1940, , ^^, , l/a=y, and, , Therefore,, , Y, , K, Young's modulus, , are connected together by the relation,, , are connected by the, (Punjab, 1938), , *, , [See page 289., , ., , zaii-t-p/a), , Y, and p/a, , or., , 4., Obtain an expression for the radius of curvature of a flat curve in term*, of the slope of the curve, and use the result to find the value of deflection in the case, of a bar fixed horizontally at one end and loaded at the other., (Bombay, 1928)*, , Let, , APQ, , be a, , flat, , curve (Fig. 21 1), and let P and Q be two points on it, Draw tangents to the curve at Pand Q, and let O be, the centre and /?, the radius of curvature of the portion PQ of the curve., , small distance $x apart., , Then,, , if, , LPOQ =, Sx, , Or,, , Now,, , 6, gents at, , =, P, , we have PQ, , 9,, , -, , R.Q., , R.e., , ('), , difference in slope, , of, , the tan-, , and Q., , And, , since slope of the tangents at a, point is measured by dy\dx at the point,, , we have, , the rate of change of slope is giventhe second, coefficient r, differential, , Now,, Fig. 211., f, , Or,, , by, , change in slope from P to Q, = $v.d*yldx\ And, /. 8x, z, , =, , *x d*y/dx*., , =, =, , '*x.R.<Pyldx\, , [From, , (/), , above., , 1., R.d*yldx, Or,, l/R, d*y/dx* -= rate of change of slope at P., Since in the case of bent rods, or beams, the curve of the neutral axis isvery slight, the relation \/R, d*yjdx* gives the radius of curvature of the axisat any given point., Now, for a bar fixed horizontally at one end and loaded at the other,., (/.*., in the case of a cantilever), we have, [See page 310., W.(L-x) = Y.Ig !R,, *he axis of x being taken along the horizontal and the axis of y, vertically down-, , t.e. 9, , vards., , Here, L is the length of the bar from the fixed to the free end, x, the distance of the section PQ from^the fixed end, and, the weight applied at the9, free end., , W, , Therefore, substituting the value of, , ~, -A, x), Integrating this,, , we have, f- Y.I, J, , Or,, , j', , 1, , /R 9 from above, we have, , d'y, , Y, J, Or, ur,, r./flr.-., , YJff, , d *y, , F-'*, ., , y\, (Lx)., , =* (T
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333, , ELASTICITY, where, , Q, , is, , a constant of integration, to be determined from the conditions of the, , experiment., , dy/dx, , Clearly,, , is, , zero at A,, , Substituting these values of, , and x, , J, , Again, integrating this expression, , y/, , 0,, , when, , x0., , Q =0., , we have, , 2, , -, , ', , 2, , ,... v, (l, , ">, , we have, , T Ya, , a; 8, , 2, , 6, , TT->"-, , where Ct, , (//),, , (//),, , x, , r ~, = LX, , dy, , W-d*, , Or., , in, , =, , dy/dx, , i.e.,, , - (/V), , + C", , another constant of integration., , is, , To determine, , this constant, we observe that the depression y of the rod, the end A ; so that, j=0, when x=0., Putting these values of y and x in (iv), we have Ca = 0., us zero at, , z, , r./^ v, .y, , ^, , Hence, , TU, , Tben,, , x9, , 2, , 6, , =, , y./r*, , ,, , ^, , clearly,, , .;>, , LJT', , -, , 2, , y, , Or,, , =*, , 8, , f, i, j, for a rectangular rod,, , 3, , =, , 6, , ', , 3, , W L*l3Y.Ig, , the rod be of rectangular cross-section,, breadth and depth respectively ; so that,, if, , its, , ', , to obtain the deflection of the loaded end, let us put x =- L., , Now,, ., , - Lx~-, , ., , bd3 /l2, where ^ and, , k, , arc, , y, , the rod be of a circular cross-section,, And,, rcr/4, where r is its radius so that,, if, , ia, , </, , if it, , (i.e.,, , be cylindrical),, , ;, , for a cylindrical rod,, , ^, , W, , =, , -, , /, , 3, , 4H^, , /, , 3, ', , A brass bar 1 cm. square in cross-section is supported on two knife5., edges 100 cms. apart. A load of 1 k.gm. at the centre of the bar depresses that, point by 2*51 mm. What is Young's modulus for brass ?, We know that the depression of the mid-point of the bar is given by, y, , Now,, , =, , jF/ 3 /48, , Yf, , [See page 3 1 6., , ., , ff, , for a bar of rectangular cross-section,, , jg, , b.d*, , ,, , s, , [See foot-note page 308., , b.cl l\2., , r, , ., , =, /, , TMrcfort., , =, , d = 1 cm., becauss the bar is 1 cm. square in cross* section., = 1 k. gm. wt. - 1000x981 dynes., 1 x 1 - 1, = 100 cms. and y = 2*51 mm. "25 i cm., \2Wl*, -Wl*, - --Wl*, , Here, b, , ;, , W, , -, , x, Or, the value of Young's Modulus, , ,, , Or,, , 1, , . , 77x ou, , _, for brass, , ,, , is, , 9 77, , x, , 10, , 11, , dynes/cm, , 9, ., , Establish an expression for the work done in stretching a wire through, J cms. assuming Hooke's law to hold., 6., , Find the work done in Joules in stretching a wire of cross-section 1 sq. mm., length 2 metres through 01 mm., if Young's modulus for the material of the, 18, 1, ^vire is.2x 10, (London Inter. Science), dynes /cm, , and, , ., , 112 (i), where, see, In stretching the wire *= J stretching force* the stretch., , For answer to, , first part,, , it, , is, , shown, , that work done
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PROPERTIES OF MATTER, , 334, work done, , r,, , ., , x-01, 200, , 3, , Y.al, , 1, , =, , ., , work done, , 18, , '01 sq., , 10, , 10*, , SOO, , 2, , c, , --., , =, , ./, , ^, , - 5, ^5- 5000, 10?, ro4, , ._ n, 5, , is, , ----, , c, ,A, - 5x10-, , ., , x 10~ 4 Joules., , YIN=, , that for a homogeneous isotropic substance,, the Young's modulus, A , the simple rigidity, and a, the, r, , is, , cm.,, , 2xl0 12 x-01-x Olx-01, , \, , ., , -, , ^, , Show, , 1., , Y, , the stretching forces, , 1, , .,, , Thus, work done in stretching the wire, where, , wnereFis, , .,, , 2, , ;, , -, , 10, , F.l=-^, , l>, , /, , Therefore,, , ', , '-', , --, , Y = 2x 10 dynes /c/w a = sq. mm. = 1/100 =, '1 TW/W. = 01 cm., and L =2 metres ~ 200 aws-, , Numerical. Here,, -., , =, , 2(a-hl),, , PoissonV, , ratio., , A gold wire 32 mm. in diameter, elongates by 1 mm., when stretched by, a force of 330 gm. wt., and twists through 1 radian, when equal and opposite torques*, of 145 dyne-cm, are applied at its ends. Find the value of Poisson's ratio for, gold., , An isotropic substance is such that two equal, similar portions cut, with any orientation, arc exactly like and indistinguishable., For proof of the relation,, (page 332), where, , it is, , shown, , that, , Y/N =, , N=, , 2(crH-l), see solved, , Y/2(cr-H), whence,, , example, , Y/N =, , 3,, , from, , it,, , above,, , 2(cr-h 1)., , Y = FyLr, , Now,, , a <l, , Here, F=330x 981 dynes; 1=1 ww.=-l cm., 2, [because radius, 032/2 = *016 cm., and a = nr ]., , =, , and a=rcx('016) 2, , ;, , s<?., , cms.,, , v ~ 330x981xL, Y, WX -016- xl, , Since couple acting on the wire ~ 145 dynes-cm., and angle of twist, we have, couple per unit twist, 145/1 *= 145 dynes-cm., , =, , This must be equal to N.nr*l2L, where, , Mir/2I, , Thus,, , 7, Ur>, , -, , 3^0x98 lxl, , __, "", , 2, , ^-, , =, , 2(a-fl),, , N-, , 1, , 145X2L*, , 330x911 x(, , :, , Since, , radian ^, , _, , -l, , 330x981x(016), "*, 2x lxl45, , 1, , the radius of the wire., , whence,, , 145,, , ^, , r is, , =, , 2, , 016), , 29, , we have, , 2(<j, , --^^., , + l)=2'858., , - 1'429. And,, (a-fl) = 2-858/2, Or,, 0'429., Hence, Poisson's ratio for gold, , =, , a = r429-l, , .-., , =, , '429., , A square metal bar of 2*51 cms. side, 37*95 cms long, and weighing 826*, 8., suspended by a wire 37 85 cms long and 0501 cm. radius. It is observed, What is the rigidity coefficient of the, to make 50 complete swings in 335 7 sees., wire ?, gms., , is, , -, , =, , 6'714 sees., 335'7/50, Here, time-period of the bar, i.e., t, Now, time-period of a body executing a torsional vibration, , f=2TrV i\Ct where /, , is its, , C, the twisting couple per, Here,, , 7, , -, , moment of inertia about, , n, , mass, 826., , (^"'*', 1440, , ^, , is, , given, , by, , the suspension wire as axis an(T, twit deflection or twist of the wire., , "'", ), , 63, 6301, , ^, , 2, 99540 gm. cm, , ), ., , .'., , ., , =, , 826., , ^, 3, , 826 x, , 6-714, , =, , ^J-, , -, , 826x120-5., , ^, 2*W 99540
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33 1, , ELASTICITY, , we have, , squaring which,, , ^TUM, , A, , (6-714)', , Now,, , C is also, , =, , 9954, , 2, , ^, Or,, , -4*'x-c-., , ^, C-, , .wr 4 /2/,, , where the symbols have, 4, = 4Tt*x 99540, a, 2x37-85, (6'714), 87CX 99540x37'85, , 4Tr*x99540, , their usual meanings., , 77Xnx(-0501), , whence,/i, , -, , Or, the rigidity coefficient of the wire, , ', , 3'357x 10 11 dynesjcm*., , is, , 9. A disc of 10 cms. radius and mass 1 k. gm. is suspended in a horizontal, plane by a vertical wire attached to its centre. If the diameter of the wire is 1 mm., and its length is 1*5 metres and the period of torsiona? vibration of the disc i:, 5 sees,, find the rigidity of the material of the wire. Prove the equation you employ., , For proof of the formula, see, , 126, (pages, , (Bombay, 1931}, 300-301), which gives the, , relation,, , 5 sec., and /, for the disc, about the axis of susHere, in this case, t, 2, is the mass of the disc, and r, its radius,, pension, is given by A/> /2, where, Since, 1 k.gm., and r -= 10 cms we have, 1000 gms, , M=, , =, 100, / -, , M, , :, , X, , 102, , =, , 2, , -, , r, , Therefore,, , _, , ,, , Since, , C is, , =, , 4Ti, , C, , whence,, , 5, , 2, , 2*^/5 x~10, , x5xl0 =, ~~, 4, , also equal to nnr*!2l,, , substituting the values of r, , /., , ,, , 4, , 4n3, , /,, , ., , --, , -, , 25, , Or,, , /C., , X- 10 4, , we have, , and, , = 5x 10* gms. cw a, , 500 x 100, , -, , WTTA**, -, , =, , 4?^ X 10 4, , we have, 2, , nxnx('05)*_, , 2x150, r == '5 771/w., , (V, , whence,, , *05, , -, , /i, , r/77., , and, , /, , " 4* xlO^, ^, """', 5, ~ 1*5 metres =, , -, , 0777.5.), , -, , the rigidity of the material of the wire, , .'., , 150, , 4, __ _4_xj, 4^ x 1 x 2 x 14 50"~, X7TX ( 5), Ifo, , is, , l*206x 10 ia dynes/ cm 2, , ., , M, , 10. An elastic string has a mass, suspended at its lower end, the uppei, being fixed to a support. The mass is pulled down over a short distance and let go, Explain the motion that ensues and find an expression for the time of oscillation., If a, , mass, , 777, , is, , added to the mass, , M,, , the time, , is, , altered in the ratio of 5, , :, , 4,, , Compare the masses 777 and M., (Bombay, 1936], For first part, see solved example 10, Chapter IV, (page 143), where it i<, shown that if / be the extension produced in the string in the equilibrium positior, due to the mass, the string executes a S.H.M.,, , its, , time period being given by, , M, , Let the time-period in the first case, when the mass, is suspended frorr, the end of the string be r 2 , and, in the second case, when the mass (77* -hM) is susThen, if /! and lt be the respective extensions produced by the twc, pended, be t t, masses in the equilibrium position, we have, ., , tl, , So, , = 2*V, , / i/, , r, , 8, , that,, , Now,, , Mg, , and, 2, , -, , /, , /i//,., , 2"VQg-, , sees, and f t = 5 sees., we have 16/25, /V/V, and / are "directly proportional to the stretching force applied, and (M-f m)g respectively ; for,, , if /!, , v/i.,, , /i //,, , 4, , /!
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PROPERTIES OF MATTER, , 336, , ***, , Y, , in the 1st case,, , and, , Y-, , 2nd case, , in the, , fl./i, , a.li, , -whence,, , where L, , /,., , is, , =, , Mg.L/ya and, , /f, , the original length of the string, a,, its material., , =, , its, , (M+m)g.LIY.a,, area 0/ cross-section and y, the, , Young's modulus for, , Hence, , M+m =, , M, , ', , 25, , M-hm-M, , ., , 16', , Whence, , 25-16, , M, , ', , -"16", , ., , ''", , m =, , M, , 9, ', , 16, , Thus, the two masses, m, and A/, are in the ratio 9:16., , The breaking stress of Aluminium is 7*5 X 10 8 dynes cm."* and of Cop11., 8, 2, per, 22 xlO dynes/cm.- . Find the greatest lengths of the two wires that could, 2*7 gms./c.c. and of, hang vertically without breaking. Density of Aluminium, 8 '9 gms, Copper, , =, , =, , /c.c., , cms. be the aica of cross-section of the wires and /x and, their lengths respectively that could hang vertically without breaking., Let a, , (i), , sq., , Case of the Aluminium, its, , weight, , Now,, , .'., , /j, , This must be equal to the, , maximum, , breaking stress, , /1, , whence,, ( ii, , ), , 9, , 7 5, , x 10 s dynes /cm a, , -75, , = -~r^t =, , Ci ^ of coppe r wire, , x ax 2 7 gms.,, ., , ., , can be applied to the wire, without breaking, , XflX2 7x981, 7<; v 10 B, , I,, , /j, , d>wi, , force the wire can withstand., , =, , Therefore, total force that, 8, is equal to 7 5 x 10 x a dynes., , Or,, , cms.,, , wire., , = lxac cs., and its mass, = ^ x a x 2 7 gms. wt. = x a x 2 '7 x 981, , Here, volume of aluminium wire, , and, , lt, , it,, , x'lO'xo,, , 283,100 cms.* 2'831 kilometres., , ., , Proceeding as above, we have, in this case,, , /,Xflx8'9x981, 22* y 10 8, whence,, , /,, , =, , Q, , ^, , =, , -, , ,, , 22xl0 8 Xfl,, , 252,000 cms., , =, , 252, , kilometres., , Thus the required lengths of the aluminium and copper wires are 2*831 and, 2 52 kilometres respectively., , A copper wire 3 metres long for which Young's modulus is 12*5 x 10 11, 12., If a weight of 10 k. gms. is attached, dynes per. sq. cm., has a diameter of I mm., to one end, what extension is produced ? If Poisson's ratio is 0*26, what lateral, compression is produced ?, original length of the wire (L) = 3 metres = 300 cms.,, Here,, Young's modulus for the wire (Y), radius of the wire, , and, , ."., , its, , 12*5x 10 U dynes. cm.~ a ,, '5 mm. = '05 cm., mm., , J, , 1, , area of cross-section, , wr*, 7ix('05) 5^. cms., force applied (F), 10 x 1000 1 gms. wt, 10 k. gm. wt., 10 x 100 x 981 dynes ** 981 X 10* dynes., , =, , =, , And, , =, Now, we have, , (r), , =, **, , the relation,, , v, , - FxL, , ,, , ., , whence,, , /-, , 98 1 x 10* x 300, , '5xlO n, 981_x3 __, n x (-05) xT2r5 x, , W*, , F.L, , t
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337, , ELASTICITY, , ., , '2997 cm., , extension produced, , Or,, ., , _, , ,, , Again, we have, , ., , Poisson, , ,, , a, , s ratio,, , lateral strain, , ,, , JA, 26, , ong ifu(, inal, , strain, , ', , l,L, , x //L, , '26, , lateral strain, , Or,, , '26, , x -2997/300=, , 2'598, , x, , 10-*., , This, therefore, gives the value of lateral strain, i.e., d\D, where, decrease in diameter and D, the original diameter of the wire., , Hence, Since, , D=, dl'l, , Hence,, , d, , is, , the, , d\D ==2-598x10-*., 1, , mm., , -, , lateral, , -= -1, cm-,, , 2-598, , x, , we have, , 10-*., , -, , d, , Or,, , 2*598 x 10-*, , 2'598, , compression produced, , x, , 2'598, , '1, , x 10~ 5, , ., , 6, , x 10~ cms, , uniform glass tube is hung from a support and stretched by a weight., found that one metre of the tube stretches by '08 cm but that a column of, water 1 metre long contained within the tube lengthens by only 0'4 cm. Find POH13., , It, , A, , is, , ,, , son's ratio for glass., , We know that Poissorfs ratio,, Now,, and, , let, , = &/., ~ P dynesjcm*., = r cms., , a, , the stress be, , internal radius of the tube, , =P, , Then, increase per unit length of the tube, And, decrease per unit^radius of the tube, , cms., , =, , P.p cms., 100.P, s. increase in 1 metre or 100 cms length of the tube, and, P.p r. cms., decrease in the radius of the tube, , So, , that, the radius of the tube is, , =, , now, , =, , increase in length of the tube, , Now,, , (r~P.p.r, -06 cm., , ), , cms., , cms., , = r(l-P p), , cms., , a = 06/100 P., whence,, = *r* sq. cms.,, tube, the, of, And clearly, initial cross-section, a, -=, )], ^. rmv., final cross-section of the tube, "[/-(I -P, and,, = nr 2 X 1 -2P ft 4- (P )*] ^. cms., , =, , lOO.P.oc, , -06,, , 1, , 7rr, , a, , x[l, , 2P./S], , ^., , c/W5.,, , 2, neglecting (P-jS) as a very small quantity., , = 100 rcr c cs., Therefore, volume of water column initially, And volume of water column finally = 103 4x *r\\ -2P./3) c, length of the water column is now 100-I--04 cms., 2, , c^.,, , v, , Since volume of the water column remains the same, we have, , 100-4-100-04x2P=100., , Or,, , whence,, , ft, , =, , 2Px TdOW, 04, ==, , a, , ~, Hence,, 14., , A, , 2Px, , -, , 100-04, , 4 ______, ~~, , :, , 2 x 1 00 04 x '06, , X, , 100P, , -04x100, 2xlOO-04x-06, , *06, 1, , =, , 50-02 x '06, , Poisson's ratio for glass, steel, , 100 04 x2Pj8=100'04- 100-00 ='04,, , Or,, , 0*3332., , mm. in diameter is just stretched between two fixed, 20C. Determine its tension when the temperature falls, , wire* 2, , points at a temperature of, to 10C., [Coefficient of linear expansion of steel, 12, for steel is 2'lx 10 dynes per sq. cm.], , Let the length of the wire be, , /, , is, , 000011 and Young's modulus, , cms., , Then, on a fall in its temperature, from, crease by an amount =/x '000011 x 10 cms., , 20C to, , 10C,, , its, , length will de-
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PROPERTIES OF MATTER, , And, , .%, , strain, , Let, , V, Now,, , produced in, , =, , it, , T dynes be the tension, , 10//, , =, , 5, 000011 x 10=11 x 10~, , Then,, stress=Tlnr***TjT*('\)*=TlKX '01 sq. cm., '1 cm., and /., radius of the wire, r--=\ mm., =irr 8 = n x(*l) 2 sq. cms., , Young's modulus, (Y), , r~, , r=, , whence,, , /x -000011 x, , So, , stress/strain., , X, , ., , in the wire., , T-, , Or, , its, , area of cross-section*, , that,, , ', , ic, , 2'lxl0 12 X7rx-ll xlO~ B =2 Ix 10 7 xnx'll=7 257x10', , dynes., , Therefore, tension of the wire = 7'257x 10 dynes., If one body impinges on another which is at rest, find the relation, 15., between (a) momenta, (b) the kinetic energies of the system before and after, impact., , A steel ball, ,,,, , ., , We know, , through a height of 64 cms. on a plate of steel. The, rebounds is 36 cms. Calculate the coefficient of restitution., , is let fall, , height through which, .., , it, , ., , that, , e, , =, , 77, , Here, relative velocity after impact, say,, , and, e, , Therefore,, , t', , relative velocity after impact, -------- -, , r, , /, , -, , lelative velocity before impact, , v= \/2..36,, , ['', , [v here,, before impact, say, u = \/2.g.64., = V^-36/2 g 64= V36/64 - v/9; 1^-3/4 -'75., , n ~ 36 c ms, , -, , /i= 64 cms., , Thus, the coefficient of restitution =-75., , EXERCISE, 1., , Define Young's modulus., , VIII, , Show, , that a shear, , is, , equivalent to a com-, , pression and an extension., , Find an expression for the work done in stretching a wire and hence deduce an expression for the energy per unit volume of the wire., (Madras B.A., 1947), , A, , wire 300 cms. long and 0*625 sq. cm. in cross-section is found to, of 1200 kilogrammes. What is the Young's, modulus of the material of the wire ?, (A.M I.E., 1961), Ans. 2'3x 10 1 * dynes I sq cm., 2., , stretch 0*3 cm. under a tension, , 3., Explain the terms stress, strain, Young's modulus, Poisson's ratio,, bulk and rigidity moduli. Show that the value of Poisson's ratio must lie between - 1 and +1/2., (Calcutta), :, , ,, , IT, , Define Young's modulus, Bulk modulus and modulus of Rigidity. If, 4., and n represent these moduli respectively, prove the relation E=9nKj3K+n., (Allahabad, 1943), , A, , solid ball 330 cms. in diameter is submerged in a lake at such a, 2, Find the change, depth that the pressure exerted by water is 1-00 k. gm. wt Icm, 7, 1 '00 x 10, in volume of the ball., (K for the material, dynes/cm*.), 5., , ., , =, , (Bombay, 1959), 1 386 c.cs,, rigidly fixed at both, Ans., , While at 0C., a square steel bar of 1 cm. side is, 6., ends so that it cannot expand. Its temperature is then raised to 20C. What, lf, force does it exert on the clamps ?, (Young's modulus for steel = 2x 10, 000011)., dynes/ sq. cm. and coefficient of expansion of steel =, Ans. 448 k. g m., Find the formula for the work done in stretching a wire, and apply it, 7., to find the elastic energy stored up in a wire, originally 5 metres long and 1 mm., in diameter, which has been stretched by 3/10 mm. due to a load of 10 k. gm. Take, g - 300 w., (Bombay)', Ans. 4-5Tcxl0 4 er,s,, bar of iron, 0'4 sq. /.- in cross-section is heated to 100C. It is then, 8., Hxed at both ends and cooled to 15C Calculate the force exerted by the bar on the, , A
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339, , ELASTICITY, , Young's modulus for iron is 30,00,000 Ib./sq. in. The coefficient of, is 0-0000121C., (Institute of Civil and Electrical Engineering), , fixings., , linear expansion of iron, , Ans., , 5-464 tons., , A, , wire of length 50 cms., and diameter 9 mms. was fixed at the upper, end while a wheel of 10 cms diameter was fastened to the lower end. Two threads, were wrapped round the wheel and passed horizontally over pulleys ; each, thread supported a scale pan., On placing a weight of 230 gms. on each pan, the lower end of the wire was twisted through 45C'. What is the rigidity coefficient of the material of the wire ?, Ans. 7-96 x 10 11 dynes tmr*, 9., , n radians = 180.], Convert degrees into radians, Explain what you understand by 'shearing strain*. What are its, dimensions ? Deduce an expression for the moment of the couple required to, twist the lower end of a rod of circular cross-section by 90, the upper end being, clamped., (Agra, 1945), 2, Ans. Couple w .nr4 /4/., [Hint,, e ~90 ~ Tt/2 radians.], [Hint., , 10., , What, , couple must be applied to a wire, 1 metre long, 1 mm. diameter,, it through 90, the other end remaining fixed ? The, 2, 11, rigidity modulus is 2 8x 10, dynes cm", 90, Ans. 4'3x 10 6 dynes cm.~*, [Hint., 7t/2 radians.], 11., , in order to twist, , one end of, , Explain what is meant by 'modulus of rigidity' and find out its dimenDescribe one method of finding experimentally the modulus of rigidity of, a v/ire and give the theory oi the method. Find the force necessary to stretch by, 1 mm. a rod of iron 1 metie, long and 2 mms in diameter. Also calculate the, U C.G.S., energy stored in the stretched rod, [Young's Modulus for Iron =2x 12, units ], (Patna, 1949), Ans. (/) 64 k gm wt (//) TCX 10 6 ergs., 13., Find the relation between the bending moment and the curvature of, the neutral axis at any point in a bar., A vertical rod of circular section of radius 1 cm- is rigidly fixed in the, earth and its upper end is 3 metres from the ground level. A thick string which, can stand a maximum tension of 2 A gm. is tied at the upper end of the rod and, pulled horizontally. Find how much will the top be deflected before the string, U>, C.G 5. units, g = 1000 C.G.S. units)., snaps. (Y for steel = 2x 10, 12., , sions., , (Saugar. 1948), Ans. J 1 '47 cms., cm. square in section, is clamped fiimly in a horizontal position at a point, 100 <v//5. from one end, and a weight of one k. em. is, 9-78 x, applied at the end, what depression would be produced ? (Y for bra^s, IQ 11 dynes cm.-*)., Ans. 4-01 cms., 14., , If, , a brass bar,, , 1, , A, , 15., uniform beam is clamped horizontally at one end and loaded at, the othei. Obtain the relation between the load and the depression at the, loaded end., , Compare loads required to produce equal depression for two beams,, of the same material and having the same length and weight, with the only, difference that while one has a circular cross-section, the cross-section of the, other is a square., (Saugar, 1950), Ans. 3 TT., A weight is suspended from the free end of a uniform cantilever., 16., Find the equation of the curve into which the cantilever is bent. The weight of, the cantilever may be neglected., made, , :, , A uniform rigid rod 120 cms. long is clamped horizontally at one end., weight of 100 gms. is attached to the free end. Calculate the depression of a, point 90 cms distant from the clamped end. The diameter of the rod is 2 cms., 11, Young's modulus of the mateiial of the rod is l'013x 10 dynes per sq. cm. and, (Bombay, 1940}, g =='980 cm.lsec*., Ans. 2*834 mmx., A light beam of circular cross -section is clamped horizontally at one, 17., end and a heavy mass is attached at the other end. Find the depression at the, loaded end., , A, , If the mass is pressed down a little and then released, show that it will, form simple harmonic motion. Explain how from a knowlege of the period
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PROPERTIES OF MATTER, , 340, , of oscillation, the mass and the dimensions of the bar, the value of Young's, for the material of the bar may be determined., (Madras), A vertical wire is loaded (within the limits of Hooke's Law) by, 18., weights, which, produce a total extension of 3 mms. and 5 mms. respectively., Compare the amounts of work necessary to produce these extensions., , modulus, , Ans., , 9, , :, , 25., , A sphere, , of mass 800 gins, and radius 3 cms. is suspended from a, wire of length 100 cms- and radius 0*5 mm. If the period of torsional vibration, is T23 sees , calculate the'rigidity of the material of the wire., (Bombay), Ans. 7'654 x 10 11 dynes cm~*, 19., , 20., A bar, one metre long, 5 mmi. square in section, supported horizontally at its ends and loaded at the middle, is depressed T96 mm. by a load of, 100 zms. Calculate Young's modulus for the material of the bar., , 980 cm.} sec*.), , (Take g, , Ans., , 19-99, , x 10 U dynes cm.~*, , Calculate the time of vertical oscillation of amass of 1 k. gm. hang= 2x 10 11, ing by a steel wire 3 metres long and *5 mm. in diameter. (Y for steel, Ans. '05 sec., C.G.S. units)., 21., , [Hint., , W,page, , Find extension, , I, , Then,, , produced, , t, , (See solved example, , 2-K^ljg., , 143J., , Prove that Young's modulus Y, the bulk modulus K, the modulus of, and Poisson's ratio cr satisfy the relations, , 22., , rigidity n, , (/), , :, , ", , 1+ff, , 2n, , ( ">, , ;, , = 1 - 2ff, , IK, , 3nd <"V >, , ;, , 4, , K, , =, , I, , ', , 23., Define Poissoa's ratio, and show mathematically, from first princi,*, that it must be, 1., Calculate, than 0'5 and cannot be less than, Poisson's ratio for, anc. :.'w rigidity of silver from the following data, , ples,, , :, , Young's modulus for silver wire, Bulk modulus for silver wire, , 7-25, , =, n =, , x 10 11 dynes cw.~ a, , x 10 U dynes cm.~*, Ans., 2 607 x 10 11 dynes cm.-*, and cr = 0'39., = 2n(l i <*)., [Hint,, (i) From (/) and (ii) above (Ex. 22>, we have 3K (/-2or), !., Since K and n are both -i-ive, G cannot be more than *5 and less than, (ii) See, 116, page 288, whence, it can be shown that, FAT, , 3, , n ~~, , (9K~ Y), AI, Also, , 2, , 3^=, , A, and, , a -, , Or,, , directly, , A, , -, , 2n, ~, , 1, , v, , from, , ., , (/), , nY, , ^, , 9, , 1 1, , v~~, , A Y, and, , (9/i"-3 K), 1, , K, , ^, whence,, , ^, Or,, , v, F, , =, , (Ex. 22),, , 3Y~2n, -, , p, , 9AT/I, , 1, , -, , a, 2, , --1., , metre in length and 1 mm. in diameter is stretched by 0', gms M/. and is twisted through 70 by a force of 5 gms., wt. applied to each end of a 20 cm. rod soldered at its mid-point to the end of, the wire. Calculate (1) Young's modulus, (2) Shear modulus, and (3) Bulk, modulus of the wire., Ans, Y = 20-81 x 10 11 dynes cmr* n = 8-268 X 10 11 dynes cm~ 2, and K = 14 35 = 10 11 dynes cm.= mtQr*!2l = 5 x 981 x 20 dyne-cm.,, [Hint., (/) Twisting couple, 24., , wire, , mm. by a load of, , 1, , 10 k., , ;, , and, , 9, (ii), , A, , K, , =, , 70 x, , TT/ 1, , 80 radians., , nYI(9n-3Y)., , [see Ex. 23, Hint, , (//)), , block of soft rubber, 5" square, has one face fixed, while the opposite face is sheared through a distance *5* parallel to the fixed face by a tangential force of 39 Ibs. wt., How much work is done per unit volume of the cube to, Ans. 8'64/r. Ibs., do this?, 25.
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ELASTICITY, , 341, , 26., Calculate the depression at the free end of a thin light beam, clamped horizontally at one end and loaded at the other., For the same mass per unit length, show that a beam of square section, is stiffer than one of circular section, the deflections being in the ratio 3/w., (Bombay, 1949), 27. A rectangular bar of iron is supported at its two ends on knife-edges, .and a load is applied at the middle point. Calculate the depression of the, middle point., How can this be utilized to determine Young's modulus of iron ?, , (Allahabad, 1947), , Find the value of Young's modulus for copper. In an experiment,, the diameter of the rod was 1-26 cms. and the distance between the knife-edges, 70 cms. On putting a load of 900 gms. at the middle point, the depression was, 0*025 cm. Calculate the Young's modulus of the substance., (Agra, 1948), 28., , Ans. 20-42 X 10 U dynes [cm*., Define Poisson's ratio and describe a method for its determinationDerive the formula used., (Agra, 1947), 30. Derive the expression for the bending of a tube supported at the two, ends and loaded in the middle., (Banaras, 1947), 29., , 31., How do you differentiate between a column and a strut ? Obtain an, expression for the critical load for a long column with its ends rounded or, hinged., 32. Discuss Eulefs theory of Ions columns for the case (/) when both ends, of a column are rounded or hinged, (ii) when both ends of the column are fixed., , Show that (/) a column, with its ends fixed, has four limes the strength, a thrust than a similar column, with its ends rounded or hinged and, (11) a column, hinged at one end and loaded at the other has only one-fourth the, strength of the same column when hinged at both ends., 34., Two steel balls of masses 1 and 10 k.g. respectively are moving, each towards the other with a relative velocity of 4 metres per second. Find the, loss of energy after impact and state the reason thereof., (Bombay, 1932), Ans. 50290 ergs., A sphere of mass 3 Ibs., moving with a velocity of 7 ft.jsec., impin35., ges directly on another sphere, of mass 5 Ihs., at rest after the impact, the velocities of the spheres are in the ratio of 2, 3., Find the velocities after impact, and the loss of kinetic energy., (London University), Ans. (i) 2ft.jsec. and 3ft.jscc. (ii) 45 ft. poundals*, 36., resilience and stiffness of a beam. What, Explain briefly the terms, 33., , to resist, , ;, , :, , :, , is, , proof resilience ?, 37., Write a brief note on, , clastic waves.
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CHAPTER IX, , HYDROSTATICS, 145., Fluids Liquids and Gases. Hydrostatics deals witn tne, mechanics of fluids in equilibrium and our first step, therefore, is to, 9, understand clearly as to what exactly do we mean by a 'fluid, ., , Unlike a solid, in which, as we have seen, the strain set up, under a shearing stress lasts throughout the p3riod of application of, the stress, a fluid may be defined as that state of matter which cannot, In fact,, indefinitely or permanently oppose or resist a shearing stress., it constantly and continuously yields to it, though the yield may be, rapid in some cases and slow in others. In the former case, the, liquid is said to be mobile, (like water, alcohol etc.) and in the latter,, In either case, however, & fluid has, viscous, (like honey, treacle etc.)., no definite shap 3 of its own and assumes ultimately the shape of the, containing vessel., , And, , yet, with all this seemingly clear-cut distinction, , between, , a solid and a fluid, it is not quite so easy, two in many a border-line case. Thus,, , to distinguish between the, for example, pitch, which, looks so much like a solid that it has to be hammered in order to be, broken, is essentially a fluid, for, when subjected to the shearing, stress of its own weight, by putting a piece of it in a funnel, or by, putting a barrel of it on its side, it does begin to yield or flow,, ;, , although intinitely slowly. On the other hand, metal wires, which, Are obviously solids, when subjected to an excessive tension, begin to, flow in the manner of fluids, and, indeed, may be considered to be, Once the yield is over, however,, so, for the duration of the yield, , they behave, , like solids, , they in fact are., , Then, again, we have, on the one hand, highly elastic solids,, in which no change of shapo is discernible even in, millions of years, as is evidenced by the sharpness of its crystals, which look as though they head just been formed, and, on the other,, fluids, like water, the rapid flow of which almost instantaneously, does away with any sharpness of its edges and which, in small, quantities, assumes a spherical form, with no sharp edges or corners, whatever., like quartz,, , The fundamental distinction between the two nevertheless, remains, and we declare a substance to be a fluid or a solid according, as it does or does not yield to a shearing stress applied to it over a, long enough period., Now, fluids too are further divided into two classes, viz., (i), liquids, , and, , (ii), , gases., , A liquid is a fluid which, although it has no shape of its own,, occupies a definite volume, which cannot be altered, however great the, 342
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HYDROSTATICS, , 343, , force applied to it*. In other words, a liquid is a fluid which is quite, incompressible and has a free surface of its own, as, for example,, water, alcohol, ether, honey, treacle etc., ;, , <4* A gas, on the other hand, is a fluid, which cannot only be easily, compressed when subjected to pressure but, which, with a progressive, reduction of the pressure on it, can also be made to expand indefinitely,, occupying all the space made available to it. Thus, the whole of the, gas will escape out from a vessel, if there be the tiniest aperture in, it somewhere., , a free, , Summarizing then, a gas is a fluid which has neither a shape nor, surface of its own ; as, for example, oxygen, hydrogen, carbon, , dioxide, air (a mixture of gases) etc., , We, , shall consider first the case, , of liquids., 146., Hydrostatic Pressure. Since a liquid possesses weight, it, exerts force on all bodies in contact with it, e.g., on the bottom and, the walls of the vessel containing it,, the force duo to it being always, , And, if this force be uniformly distributed, spread over an areaf, over tho whole area, i.e be the same on each small equal element of, the surface, its value per unit area is called pressure or hydrostatic, pressure of the liquid,, meaning pressure due to the liquid at rest%., And, if the force be not uniform, the ratio between the small force, SFand the area BA on which it acts gives the pressure., ., , ,, , Thus,, , pressure, , So that, when S A, , is, , =, , 8F/BA., , progressively diminishing, we have, , pressure at the point, , -, , Or, denoting pressure at the point, , by, , ,, , ., , Limit force, , ^, , p,, , we, , -, , have, in mathematical, , notation,, , p, , =, , Tho, , total force exerted by a liquid column on the whole of the area, in contact with it is called thrust., , thrust, , Thus,, , = pressure X area., , That a, , liquid, at rest, always exerts a thrust normally to the surface, in contact with it is obvious., For, if it were not so, there would be, , a component of the thrust along, or parallel to, tho bounding surface,, and an equal and opposite thrust on it due to the reaction of the, surface would cause it to flow, since it must, by its very nature,, , a tangential force. It follows, therefore, that since the, at, rest, the thrust due to it must be perpendicular to the, liquid, bounding surface at every point., yield to, us, , *Strictly speaking, all liquids do get compressed a little, when subjected, The compression is, however, almost negligible. Thus,, to very high pressures., water, when subjected to a pressure of about 200 atmospheres, undergoes a reducI, tion of only a hundredth part of its original volume., when., for, is, a, same, of, the, force, exerted, on, true, as,, example,, fThe, liquid,, we press the piston down in a cylinder containing the liquid., , }It, , is, , also sometimes called the pressur* in a liquid due to gravity.
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PBOPEaTIBS OF MATTER, , 344, , In other words, the free surface of a liquid at rest must, be at, Thus, when the only, force, right angles to the forces acting on it., acting on it is duo to gravity, its surface remains horizontal*, being, blowperpendicular to the force of gravity, but, with a steady wind, to the, ing, it is slightly inclined, again, however, at right angles, resultant of the forces due to gravity and the wind., , it, , Further, since every layer of a liquid at rest is in equilibrium,, follows that the downward thrust on it, due to the liquid column, , above, , to the, just balanced by an equal upward thrust due, In other words, at any given level, in a liquid, it., downward thrust due the liquid column is equal to jhe, , is, , it,, , liquid column below, at rest,, , upward, , tfte, , thrust on, , it., , Let us now, 147., Hydrostatic Pressure due to a Liquid Column., calculate the hydrostatic pressure due to a liquid column A., Imagine, a narrow metal cylinder, of area of cross section a and fitted with a frictionless pis^ ne gl*e*k]e weight, to be supported, ^ on>, in a liquid of density, , p,, , (Fig. 212)., , the upthrust on the piston, due to the water below it be F, obviously, an equal and opposite force F has to be, , Then,, , if, , exerted on the pistor^to keep it in position,, Hence, if the piston be moved down, through a distance x, work done on it i&, clearly equal to F.x., 2l2, , This downward motion of the piston*, will obviously expel a volume x.a of the liquid out of the tube, itsmass being x.a.9 and its weight, equal to x.fl.p.g., -, , Since the level of the liquid in the containing vessel, it is tantamount to this weight of the liquid, rising up through a vertical distance h up to the liquid surface., slightly raised,, , is, , thus, , x.fl.p.g., , In other words,, x.a.p.g.h., , And,, , equal to the work, F.x, , -, , increase, in potential energy of the liquid, this gain in potential energy will obviously be, done on the piston. So that. we have, , x.a.p.g.h., , Or,, , f\a, , h^jf, , i.e., the hydrostatic pressure due to a liquid ronfigi, the surface is equal to h.p.g., (v Fja, force /area, , =, , '", , ^, t, , depth h from, , pressure)., , N.B. The argument remains the same even if the metal tube is inclined, and not vertical, so long as the vertical depth of the piston remains the same. It, will thus be seen that the pressure due to a liquid column depends only upon its, depth and density, and not to any other factor like the surface area of the containing vessel etc., 148., The Hydrostatic Paradox. A remarkable fact follows from the, above, viz,, that so long as the vertical height of the column of a liquid remains, the same, the pressure exerted by it remains the same, 'irrespective of its actual, mass or weight., , *In the case of a large expanse of water, the surface is spherical and, thus again perpendicular to the direction of gravity at every point.
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HYDBOSTATICS, .Pascal discussed this result in his 'Trefttise on the equilibrium oj, published in the year 1663, with reference to vessels of different shapet r, known as Pascal's vases, (Fig. 213), all having equal bases and containing water, upto the same vertical height h, so that the pressure on the base of each vessel was, equal to hgms. w/./rmV, and, therefore, the thrust on it was h.a. gms* wt. t where*, liquids',, , a, , is its, , area., , He was perhaps the first person to have pointed out the paradoxical truth, that even if vessel (0 contains 100 Ibs. of water and vessel (v) only 1 oz. of \\ r, the thrusts on the bases of, both is the same. Aptly,, therefore, it is called the, hydrostatic paradox., Strange as it may, the water in, vessel (v) be frozen into ice, , seem,, , and, , EO3, W, , t>ut if, , detached, , from, , its, , sides, the thrust exerted, this ice on its base will, , by, be only, , 1 oz. \vt., but once this ice is melted back intowater, the thrust again increases to 100 Ibs. wt. The explanation of this seeming paradox is, however, simple. The ice does not exert any upward thwft$fl, the part of the vessel opposite to the base and the latter, therefore, exerts, mjj, But the water does exert an upward thrtist Oil*, squal and opposite thrust on it., it and hence receives back an, equal and opposite downward thrust from it., , In case of vessel, of the water on it., , (/),, , the thrust, , on, , the base, , is, , equal to the entire weight, , In vessel (//), the upward component of the thrust due to the left side of, the vessel supports the weight of water in it, between the left side and the dotted, line A, while the downward component of the thrust due to the right side of the, vessel exerts a downward thrust on it, equal to the weight of the water inbetween the right side of the vessel and the dotted line, so that, the thrust or*, ;, the base is the same as due to a vertical column h of water., , B, , In vessel (///), the upward components of the thrusts due to both the left, and the right side* of the vessel support the extra weight of the water, inbetween the two sides and the dotted lines C and D, .and, again, therefore, the, thrust on the base is equal to that due to the cylindrical column h of water in~between the dot ted lines CandD., , And, similarly, in vessel (/v), the downward components of the thrust, due to the two sides of the vessel exert an extra thrust on the base, equal to the, weights of the water contained between either side and the dotted lines E and F;, so that, once again, the total thrust on the base is the same as that due. to a cylindri*, cal water column of htigtefa, Pascal, ex^lm^t|i||, verified the above fact by supporting, by means of, a separate stand, bottoMS* vases of the above shapes, one by one, on a large, t, , disc,, , ., , Fig. 214., , D, , (Fig., , 214),, , suspended from, , the shorter pan of a hydrostatic, balance and kept pressed against, their bases by placing a heavy weight, in the longer pan, and pouring water, into the vessel. The disc just got detached from its base as the water, reached the same level in each case*, thus clearly demonstrating the equivalence of the thrust on the disc ir>, ach case and fully vindicating hisdeductions.
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346, , PROPERTIES OF MATTER, , A, , 149., Pascal's Law., , transmits pressure equally in all directionsSince we do not have any, in, boundary demarcated, x, the interior of a liquid,, , liquid, , we may define pressure, there as the force exerted, per unit area across any, plane in it, and it can, be easily shown that this, pressure, equally in, , is, , all, , exerted, directions, , in the liquid., , Thus, let us consia portion of the, liquid, in the form of a, faces ABC and A'B'C' vertider, , (/)] with its, edges AA' BB' and CC' horizontal., , triangular prism, [Fig. 215, cal, , and, , its, , ,, , This triangular liquid prism is obviously in equilibrium under, the action of the forces acting on its different faces. Let us study, the inter-relation of all these forces., It is clear that due to its small size, every part of this prism, can be taken to be at the same depth from the liquid surface and also, the pressure on each face of it to be uniform*., , Now, the forces on the two end- faces are equal and opposite,, thus neutralising each other's effect and may, therefore, be ignored in, our discussion. Hence, if Plt P2 and P3 be the pressures on the faces, BCC'B', CAA'C' and ABB' A! respectively, and /, the length of the, , we have, , prism,, , Fl on, force F2, , force, , and, , force F,, , face, , = /^xarea BCC'B' = Prl.BC,, CAA'C' = P xarea CAA'C' = P .lCA,, ABB' A' = P xarea ABB' A' = P^.lAB., BCC'B', , 2, , 2, , ,,, , 3, , Since these three forces keep the liquid-prism in equilibrium,, be represented by the three sides of a triangle, taken in, can, they, Let PQR, [Fig. 215 (/7)], be this triangle of forces, with its, order., and RP representing Fv F2 and F respectively. Then,, sides PQ,, , QR, , 3, , clearly,, , sin, , a, , Pr l.BC, , Or,, , sin, , sin, , sin, , [Lame's theorem., , 7, , P .l.CA, 2, , a, , sin, , sin, , y, , PV BC, , Or,, , sin a, , sin, , B, , ..(0, , sin, , B, , since angles A,, and C, of the triangle ABC, are respecand, to, and, a,, y (the sides PQ,, angles, p, tively equal, being, and, y, perpendicular to, respectively, we have, , Now,, , f, , BC GA, , *This, , is so,, , AB, , because of the small dimensions of, , QR, , its faces., , RP
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", , HYDROSTATICS, , EC, CA, -*L, = _fBA~, ^ _^L., , sin, , From, , a, , sin, , relations, , sin, , fi, , (/), , and, , (f/),, , ., , ., , y, , ("The sides of a triangle being, proportion to the sines of the, Bangles opposite to them., , (ft), , therefore,, , we have, , =P =, 2, , v.e.>, , the pressures on the three faces are equal, , Further, since the same relation holds good in whatever position, the prism may be rotated, it follows that a liquid exerts pressure, equally in all directions within itself., N.B. It also follows from the above that if we replace the liquid prism, would, by a solid ons, of the sans size and weight, the forces acting on the latter, .also be the same and hence it would also be in equilibrium., Thrust on an Immersed Plane. If we have a plane hori, 150., , zontal surface of area A, immersed in a liquid of density P, the pressure, jPon it is uniform, since ail points on it are at the same depth h, say,, from the liquid surface, which, as we know, is also horizontal and,, therefore, parallel to the immersed surface., , So, , .that,, , Now, force or, , thrust,, , i.e.,, , P = h.p.g., F = pressure xarea =, , P. A, , =, , h.p.g.A., , the plane of the immersed surface be, If, on the other hand,, inclined' at an an^le 6 to the liquid surface, (Fig. 216), we must first, determine the thrust on a, , dA of the surface, integrate its value over, , small area, , and, , the entire surface., , Let h be the depth of, element of area dA., Then, the thrust dF on this, area is clearly equal to, , this, , Ji, , p.g.d.A., , Fig. 216., Now, if x be the distance of this element from, the line OF, in which the plane of the immersed surface meets the, , liquid surface,, , we have, sin 6, , =, dF, , So. that,, , Or,, , h\x., , =, , h, , And, therefore, the thrust on entire area, , F, , =s, , fp.g.sin e.x.d, , =x sin, , Q., , p.g.sin b.x.dA., , A =, , A, , of the surface, , clearly the moment of the element, face about the lino OF, and, therefore, J" x.dA is the, whole area A about CF, i.e.,, , Now, x.dA, , is, , given by, , p.g.sin d J x.dA., , is, , $x.dA, , =, , ., , .(/), , dA of the surmoment of the, , A.X,, , where ^.is the distance of the centroid*, , G of the, , area from CF., , "The term 'Centroid' or 'Centre of mass* is ordinarily used synonymously with "Centre of gravity'* and, in a uniform gravitational field, the two are, one and the sams point. Bat, in a non-uniform field, the weights of the particles, are not proportional to their misses. In such a case, therefore, the weights may, not form a system of parallel forces, reducing to a single resultant force, but, may form a couple, instead, varying with the different orientations of the body,, whereas the centre of mass is quite independent of the gravitational field., -
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PROPERTIES OF MATTER, So that, substituting the value of fx.dA in relation, we have, F = p.g.sin 0.A.X., Again,, , if, , H, , be the depth of the centroid, , surface from the liquid surface,, , And,, , therefore,, , F=, , we have, , *.g.~.A.X.=, A., , sin 9, , =, , G, , (i), , for, , F, , above,, , of the immersed, , H/X., , H.P.g.A., , ., , ., , (fl), , Clearly, H.p.g is equal to the pressure at the centroid or the centre, area. So that, we have, resultant thrust on the, , It, is, , quite, surface., , immersed plane, , =, , of, , pressure at centroid or centre, of area x area of the plane., , should be carefully noted that the thrust on the immersed plane, independent of its angle of inclination (0) with the liquid, , 151. Centre of Pressure. Having obtained the value of the, resultant thrust on the immersed plane, our next step obviously is to, determine the point of the plane through which this resultant acts,, this point be ing known as the centre of pressure., , We know that the liquid pressure acts normally at every point, of the immersed plane. So that, the forces h.p.g.d.A acting on elementary areas of the plane, (like dA), are so many like parallel forces. We, ma y, therefore, determine the centre of these parallel forces (i.e., the, point through which their resultant acts) b}^ an application of the, principle of moments, viz., that the algebraic sum of the moments of a, system of parallel forces, about a given axis, is equal to the moment of, the resultant about the same axis., Now,, , clearly,, , about, , CF, , t, , moments of the thrust, , (Fig. 216), , =, , h.p.g.dA.x, , (or force) h.p.g.dA acting, , = p.g.x sin 0.dA.x., = p.g sin0.x*.dA., f, , Therefore, total, , areas, , dA of the plane, , moment of, about, , CF =, , =, , ., , dA, , on area, , r ..., n, L, , x, , ., , sin, , cr-, , the forces on such like elementary, 2, J p.g.sin 9.x .dA,, p.g. sin 6 J, , x 2 .dA,, , ("'), , where the integration extends over the entire surface of the plane., 2, Now, J x .dA is the geometrical moment of inertia Ig of the area, A of the plane about OF. So that,, total, , moment about, , CF =, , And, since Ig = Ak where k, A about CF, we have, 2, , ,, , area, , total, , moment about, , is, , CF =, , p.g. sin 8.1ff ., , the radius of gyration of the, p.g.sin, , Q.Ak 2, , ., , Again, if A" be the distance of the point P through which thfr, resultant thrust F acts on the plane, i.e., the distance of the centre of, , CF, we have, moment of the resultant, , pressure from, , thrust about, , CF = F.X, , And, therefore, by the principle of moments, we have, , =, , whence, X,, , p.g. sin, , H.Q.g., , .
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349, , HYDROSTATICS, , Or,, If, , through, , fc, , its, , be the rqdius of gyration of the area about a parallel axis, centroid G, we hav, by the principle of parallel axes, 7, *, , Ak 2, , Or,, , And,, , B= XsinB., , ..(iv), , =, , f where 7 is its, of inertia about, L through G., , IQ-^A.X*,, , |, , = Ak*+A X\, , whence, k*, , ---, , moment, to axis, , f, , k, , therefore,, , whence,, , Z, , (v), , may be, , easily determined., , Alternatively, equating F.XQ against expression (Hi) above, for, moment about GF due to tho thrusts on elementary area dA, wo, have, F.XQ = p.g.sw J x*.dA, (vi), , total, , ., , H, , ., , be tho distance of the centre of pressure from, Q, the liquid surface, we have 7/ /A", s/Vz, and, therefore,, , And,, , clearly, if, , =, , ;, , A", , ffjsin 0., Putting this value of A^ in relation (v/) above,, , F'Hn, , jw /, , ^, , Pig, , ', , 5w, , ^, , * ^, , %, , we have, , ,, , h, , Or,, , .dA., sin, 2, , R.g.j//i, , Or,, , -, , j oin, , ~, , x, , \ and.'., '\, L, , sin 0., /f, , J//I, , -dA, , ,, , (7, , TT*, , the value of the integral J A .<i4, like ths expression J, depending upon the shape of the immersed plane., We thus see that whereas the distance of the centre of pressure, from the liquid surface is quite independent of the density of the liquid, it, depends upon the shape of the plane., 152. Particular Cases of Centre of Pressure., Let us now consider some simple cases of centre of pressure on surface of a definite, geometrical shape., (/) Centre of Pressure on a Rectangular, Lamina. Suppose we have a rectangular, lamina of length I and breadth b, immersed, vertically in a liquid to a depth h below its, 2, , free surface, (Fig. 217)., , Then, since for a rectangular lamina7 = A.kQ* = /.6 3 /12, (see page 308),, where 7 is its geometrical moment of inertia, about an axis through its centre of area (or, centroid) and parallel to its length, and k, its radius of gyration about this axis, we have, ,, , Fig. 217.
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350, , PROPERTIES OF MATTER, A.k Q *=l.b.b*II2=Ab*l\2, , p.. ixb=A, the area of the, , t, , lamina., , And,, , clearly,, , X == h + ~, , ., , 2i, , b2, , * =, , So that,, , V2, , --V, , --12, == ", , /, , ft, , + (//+,, , i, , ,, , "&, , ~, , 2, , whence, the position of the centre of pressure for the rectangular, lamina may be easily calculated for any depth h., the liquid,, Clearly, therefore, if the lamina be just submerged in, its upper edge just lying in the liquid surface, A, 0, and we, , with, , i.e.,, , =, , =, , have, X$, |fr,, lb*/b, the centre of pressure, in this particular case, lies two-thirds below, , therefore,, i.e.,, , the top, , of, , the lamina., , Centre of Pressure on a Circular Lamina. Let the centroid, from the free surof a circular lamina, ot radius r, lie at a depth, face of the liquid., 2, 2, Then, since & for a circular lamina is r /4, we have, (ii), , X, , r*, , X =, , +x*, , =, , X, , X, , =, , 4X, , r2, , Or, the centre of pressure lies at//*,, , -, , below the liquid, ^ +/Y, , however, the lamina be submerged, , just touching the liquid surface,, , ^~X, , +r =, +, 4r, , r, , r2, , 4, , in the liquid,, , X=, +r ~, , we have, , surface., , with, , its, , edge, , and, therefore,, , r,, , 5, , ^~, , r, , ', , Or, the centre of pressure, in this case, lies at 5r/4, , from the liquid, , surface., , Centre of Pressure on a Triangular Lamina. Hero, two cases, when the lamina is immersed upright into the liquid, i e.,, with its vertex up and base down and (b) when it is immersed upside, down, with its base up and vertex down., (iii), , arise, , viz., (a), , Let h be the height, (a) When the Lamina is immersed Upright., of the lamina and let its apex^be at a depth d below the free surface, of the liquid. Then, clearly,, , k*, , =, , /i, , 2, , and, , /18, , X=, , d+\h., , Its, , [, , centroid lying, , [below the apex,, *, -, , And, therefore,, , 8, -A>=, ", , _,, , =, , 2/J/3-
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HYDBOSTATICS, If,, , of the, , however, the vertex of the lamina just touches the free surface', , liquid,, , we have d, , = 0., X =, , In, 3A, , Q, , i.e.,, , 3511, , the centre of pressure, , now, , this case, therefore,, 2, , /4/z, , lies, , =, , f h., , $ths down, , the height of the lamina., , (b) When the Lamina is immersed Upside Down., Again, let thedepth of the base of the lamina be d from the free surface of the*, liquid., , Then, we have, , And, therefore,, , And,, , i/, , Me, , Jr, , =, , 2, , /z, , 2, , /18, , X =, , (, , X=, , lamina be just submerged in the liquid, with its base, we lave, d, 0., In that case, therefore,, , in, , the, , =, , surface of the liquid,, , /..,, , and, , the centre of pressure, , lies, , a distance \h below the liquid surface., , 153., Change of Depth of Centre of Pressure. Let a plane, lamina of area A be immersed in a liquid such that its centroid, is*, at a depth X from the liquid surface CD,, (Fig. 218),, y, , Then, if k Q be the radius of gyration- -X, of the lamina about the axis AB passing ^J^v, , ^^j, , >, N, , _ ./_ V., , through G, in the pJane of the lamina, snd, parallel to the liquid surface CD, itBradiu8^..^i, of g} ration about a parallel axis, lying in -^f^E7 ^-^!^, the liquid surface, will clearJy be k l9 ^uc, that, , kf, , =, , r, , k *+X*., , Now, let the level of the liquid surface be raised through a distance h by, adding some more liquid to that already, , Fig., , Then, clearly,, present., radius of gyration k z of the lamina about a, parallel axis to AB, and', is given by kJ, lying in this elevated liquid surface, /:</, , CD, , Subtracting one from the other,, , k*-k* =, &a2, , Or,, If, , XQ, , &j, , =, , 1, , ,, , we have, , [k, 2 =, , be the depth of the centre of pressure of the original, liquid from its free surface, we have, , amount of the, , X =, Q, , kf\X,, , [, , See relation, , So that, after the addition of a liquid layer of thickness, becomes, jrt, , y, , i, , L, , _, , (/v),, , h,, , Page 349v, , its, , depth*
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PROPERTIES OF MATTER, , 352, , =, , - X.XQ ~. Or, k* = X.XJ-Xh X(XQ '-h)., be the depth of the new centre of pressure from the, we, raised surface of the liquid, after the addition of the liquid layer,, , whence,, And, if, , k,*+Xh, , X", , jfaave, , 80, , *'-(*, =, , X '(X+h)-X(X9 '-h)., , k^-W, , that,, , From relations, , and, , (/), , (//),, , therefore,, , Or,, , ...(), , we have, , XJ(X+h)-X(XJ-h) = h(2X+h)., X (X+h)- X.X + Xh = 2Xh+h 2, XQ "(X+h) H, , Or,, , *'-*<*+>, , --, , ', , ., , a, , X, , Or,, , "(X+h), , Dividing throughout, , &nd, therefore, the, liven by, , A, , Or,, , O, , =, , by (X+h). we have, , shift in the position, , '-~AY', , X, , ", , Q, , h, , *.(*, , +, , +fc), , of the centre of pressure, , is, , --, , be easily seen that the distance between the depths, centre of pressure and the centroid of the lamina is given by, , It will also, -of, , Or,, , the, , new, , X, , ", , Q, , which approaches zero as h approaches infinity., Thus, the greater the depth of the liquid, the nearer does the, centre of pressure come to the centroid of the lamina so that, at an, coincide with each other., infinite distance, the two must just, 154., Imagine a body ABCD to be, Principle of Archimedes., a vertical line GA to travel, and, a, in, immersed, liquid, (Fig. 219),, round it, touching it along the line, AECF and meeting the liquid surface, in the curve GHKL., ;, , Then, clearly, the resultant up-, , ward thrust Fl on the surface ABCEA, and the resultant downward thrust F2, on the surface ADCEA are given by, the weights of the liquid that would, , occupy the spaces, , ABCKG, , and, , ADOKO, , respectively, acting through their resSo that,, pective centres of gravity., the resultant thrust on the whole body, Fig. 219., , is, , given by
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363, , HYDROSTATICS, , Fa = 'weight, , F, , l, , of liquid occupying space, , ABCKG, , minus, , weight of, , ADCKO., space ABCD., , liquid occupying space, , =, , weight of liquid occupying, , =, , =, , weight of liquid equal in volume to that of the immersed body,, weight of liquid displaced by the body., , In other words, when a body is immersed in a liquid, it experiences an upward thrust equal to the weight of the liquid displaced by, it., It can easily be shown that the same is also true for/ a body, which is only partially immersed in the liquid, the upward thrust oa, it being, equal to the weight of the liquid displaced by its immersed, We may, therefore, generalise and state that, part., , when a body is wholly or partly immersed in a liquid, it experiences an upthrust equal to the weight of the liquid displaced by it, (i.e.,, by, , its, , who, , immersed part}., , This is known as the Principle of Archimedes*, for, first enunciated it., , it, , was he, , The point where this upthrust acts is obviously the e.g. of the, displaced liquid, which is called the centre of buoyancy, the upthrust, being referred to as the force of buoyancy., N. B. The applications of Archimedes Principle are many and. -various, gives us the method of determining ipecific gravities or densities of liquids, as well as the instruments, kn^wn as Hydrometers, with which the Degree, students are no doubt already familiar., It, , 155. Equilibrium of Floating bodies. A body, immersed wholly, or partly in a liquid, is subject to two forces, viz., (i) its own weight, v, W> acting vertically downwards at its, e.g., G, and (il) the upihrust W*', acting vertically upwards at its centre of, ,, , buoyancy B, (Fig. 220)., If these, tion,, , two points of, , applica-, , (G and B), of the two forces, , respectively, coincide or lie in the, vertical line, called the centre, the, line,, body sinks, just remains sus-, , ~, :, , game, , pended (or float ing), or, , W, , ing as, less than, , is, , -_}--~\, , rises up, accord- i ji~-r-J- I, , _ -.!*-'_", , greater than, equal to, or, , W., , W>W', , Fig. 220., , the body sinks further down, disFor, obviously, if, placing more and more of the liquid and thereby increasing the, upthrust until the two balance each other, and the body just stays, there,, , i.e., is, , If, opjtosite, , in equilibrium., , W=, and, , t, , W,, , then obviously, the two forces on it are equal and, their line of action being the same, they just neutralise, , *, Archimedes, (287212 B.C.), was a Greek philosopher. He was asked, by King Heiro, at Syracuse, to test the gold-content of a crown. Engaged on, this problem, he suddenly discovered the law of upthtust, while taking a bath,, w hich enabled bun to determine the specific gravity, and hence the quantity of, gold in the crown, without in any way damaging it, Overjoyed at his success,, he ran home, with the triumphant cry 'Eureka', 'Eureka'. 'I have found it, 1, have found it.*
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354, , PROPERTIES OF MATTER, , each other and the body remains suspended or floating in the liquid., And finally, if, the body rises up, so that a lesser volume of, it is under the liquid, i.e., it, displaces a smaller volume of the liquid,, and the upthrust on it is now less. This rise of the body continues, until the upthrust is just equal to the weight of the body, and the, body then continues to float in that very position., Thus, two conditions are necessary for the equilibrium of a floating body, viz., (/) its weight must be equal to the weight of the displaced liquid*, and (//*) the e.g. of the body and the centre of buoyancy of the displaced liquid must either coincide with each other-f or lie in, , W <W, , ',, , the, , same, , vertical line., , If the floating body be tilted a, Stability of Equilibrium., to one side or the other from its original equilibrium position,, through a small angle 6, (Fig. 221),, so that the weight of the displaced, liquid, or the upthrust on it, remains, pj the same, then, since the shape of the, its centre, -jz displaced liquid changes,, of buoyancy shifts a little, say, into the position B' so that, the vertical line drawn through the new, centre of buoyancy B', meets the, old centre line in M. This point, is called the metacentre of the body, and the distance MG', where G' is, its centre of gravity, is called its, Fig. 221., metacentric height., , 156., little, , ;, , M, , Now, whether the meta-centre (M) coincides with, lies above, or, below, the shifted position of the centre of gravity (G') of the body,, depends upon the shapo of the body and determines whether the body, will be in neutral, stable or unstable equilibrium., (/), , Thus, in the case of a sphere, [Fig. 222 (/')], a tilt this way, no change in the shape of the displaced liquid,, , or that brings about, , (/), , Fig 222., , the metaoentre coinciding with the e.g. of the, , (//), , body, , all, , the time., , *lt, for ihis reason that the weight of a ship or boat is often referred to, the weight of the water displace J by it being equal to its, is {^displacement^, own weight, | As happens in the case of a spherical body., is
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355, , HYDROSTATICS, It, , and continues to, , therefore, in neutral equilibrium, , is,, , float in ail, , positions., (ft"), , In the case of a rectangular body, floating in a liquid, as, , M, , the position of the e.g. of, lies above G', shown in Fig. 222 (ft*),, and W\, the body in its tilted position, and a couple is formed by, the two equal, opposite, parallel and non-collinear forces acting at G', and B', which represent the shifted posi-tions of the e.g. of the body and the centre, of buoyancy of the displaced liquid respectively. This couple tends to rotate the body, back into its original position, thus making, , W, , ',, , equilibrium a stable one., Obviously, the moment of this restorW.G'M sin 0. It has been, ing couple, appropriately called the 'righting moment',, (particularly in the case of ships and other, floating vessels), because it tends to bring, the body, or to 'right it' into its original, its, , =, , Fig., , 222., , (///), , position., , la the case of a rectangular body, floating in the manner,, and the body is, therefore,, lies below G', (///'),, and, tends, in unstable equilibrium., For, the couple formed by, here to rotate the body further in the same direction in which it has, There is thus no prospect of its coming back (or, -been tilted already., (//'/), , shown, , M, , in Fig. 222, , being righted) into, , its, , ',, , W, , original position., , Let us now discuss the problem in a little, particular reference to a floating vessel or ship., Tihip, , </, , ;, , W, , more, , detail,, , with, , Identical consideration to the above applies also to a floating, so that, when the ship is 'on an even keel\ its centre of gravity, the centre of buoyancy B, of tho displaced water, he in the, vertical, its plane of symmetry (W'} is, vertical line, i e., , md, , same, , 9, , (Fig. 220)., If,, , however, the ship, , (Fig. 221), its plane of, , rolls or gots tilted, , symmetry (VV'), , is, , through an angle, , no longer, , vertical, and,, , 0,, , al-, , though this roiling or tilting does not alter its. e.g. with respect to the, shifts to B\ giving rise to the righting, ship, the centra of buoyancy, or W.h sin 0, where h denotes the metacentric, sin, moment, G'M. If $ be small, so that sin 9, 8, this righting moment is, , W.G'M, , height, equal to W.h, , =, , 9., , It will thus be clear that the greater the value of h, the metacenof a ship (or a floating body, in general), the greater the sta-, , tric height, bility, , of its equilibrium. It, , is, , heavy cargo is stowprovided with a leaden, , for tiis reason that, , ed as low as possible in a ship or that, , it is, , keel, to lower its e.g. or to increase h*., ", , N.B The lowering of the c g. is, however, not quite so desirable beyond a, certain point., For, due to the waves ia the sea, the ship is subject to lateral forbe quite considerces in different directions and the moment of thsse forces can, able if the e.g. of the ship is very low down, resulting in its being tossed about, broad at the, *The~stabiHty may also to increased by making the ship quite, it is just touched by the water-surface., which, along, lins, ths, i.e.., line,
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PROPERTIES^ OF MATTER, , 856, , this way and that, which is obviously most unpleasant and annoying to those on, board. Judicious care must, therefore, be taken to lower the e.g. of the ship within reasonable limits., , W, , Rolling and Pitching of a Ship- The righting moment, h.O, acting, it is tilted through a small angle, results in its oscillation (or, as, we, have, if, T, seen, its, of, and, be, natural, oscillation, we, rolling),, above;, period, 157., , on the ship when, have, , T=, , V, , 2, , W, , where /is the moment of inertia of the ship and, , M,, , [See pages 300-301, , the turning, , moment on, , it, , ,, , per, , unit (radian) deflection., for, if, , Obviously, the turning moment per unit deflection is also equal to W.h, So that,, 1, the value of the righting or turning moment becomes W.h., ;, , =, , W.h, , substituting, , for, , M, we have, , It is thus clear that the period of rolling (T) of a ship is inversely proportional to the, square root of its metacentric height (h)., ship, with a small metacentnc height,, It is for this reason that large ocean liners are, is, therefore, less liable to rolling., designed to have a comparatively small metacentric height of just a few metres, for, small displacements, which obviously makes them much steadier. At the same, time, however, to avoid the danger of the ship turning over or capsizing, if the, deflections be large, the designing is such that the metacentric height increases for, , A, , large deflections*., , Similarly, to avoid 'pitching', or tilting of a ship in the direction of its, length, its metacentric height in this direction also is suitably adjusted., , 158., Determination of Metacentric Height. The displacement, of the ship through an angle B causes a wedge -shaped portion of the, ship, (shown shaded in Fig. 221), to be immersed on the right hand, gide and an equal wedge-shaped portion of it to rise out of the water,, on the left hand side. Let these wedge-shaped portions be divided, into a number of elementary vertical prisms, by planes perpendicular, to the water surface, on either side, and consider one such prif m, of, height //, at a dt&tance x from O, where the plane of symmetry meet&, the water- surface., , Then, clearly,, , H *=, , x tan, , Since, small., , Q =3 x.0., , prism be dA, we have, volume of the prism, its mass, x.Q.dA, an, where P is the density of water., , is, , supposed, , If the base area of the, , =, , Clearly, therefore, weight of the prism, or the weight, placed,, , =, , x.O.dA.p,, , of the water, , dis, , i.e.,, , the upthrust on the prism, , and, , its, , moment about, , O, , =, , x.Q.dA.p.g.,, , =-x0 dA.p.g.x, , =, , p.g.0.x*.dA., , Similarly, considering the equal wedeje-shaped portion on the lefthand side, we find that there is a loss of upthrust due to its rising out, of the water, whose moment about O is, obviously, also equal toz, in either case., Hence,, fi.g.Q.x .dA> the direction being anticlockwise, , the, , moment of the, , general,, , couple acting on the ship (or a floating body, in, displacement 0, is given by, 2, p g.OAk*, P.g.0. jx*.dA, J P.g.0.x .J4, (/), , due to, , The, Gyrocompass, , its, , =, , =, , rolling motion of a ship can be greately, (See page 98)., , ......., , minimised by the use of a
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367, , HYDROSTATICS, , =, , -where / x*.dA, Ig =.y4fc 2 the geometrical moment of inertia of the, k being its, surface-plane of the liquid about the axig through O,, radius of gyration about this axis., This displacement (0) of the ship being small, the volume V of, water displaced remains unaffected by it, and the upthrust p g.F, due, to this displaced water, acts through its new centre of buoyancy after, the displacement 0. The floating body or ship is thus acted upon by, a couple equal to, ,, , ...(ii), p.g.V.BM sin 6 = p.g.V.BM.O., Equating the two values of the couple, we have, , p.g.V.BM.O, , V.BM =, , Or,, , Ak*., , =, , Or,, , [Q, , 9 .g.0.Ak*., , EM =, , So that, the metacentric height h of the ship or the vessel, , <}'M=BM--BG' =, , Ak 2, , being small., , BG' and may thus be, , is, , given by, , easily determined., , Alternatively, in the case of a ship, its metacentric height, be easily determined by moving a known weight w from, point A to another point B across, the deck, say through a distance */,, , may, , AB =, , i.e.,, , Now,, , this, , d., , shift, , of weight, , w, , to B is equivalent to an, ,upward force w at B and a downward force w at A, (Fig. 224), thus, , from, , A, , constituting a couple, of moment, w.d cos 6. For equilibrium, therethe, fore, this must be equal to, of the, couple due to the weight, , W, , ship and an equivalent upward, thrust at the new centre of buoyancy B', i.e.. equal to couple of mo-, , ment W.G'M sin, , **>, , [See above], , =, , W.G'M sin, , Or,, s\, , 9., , GM =, rntr, , COS 6, , sine-, , G'M =, , Or, Thus knowing, , W, , W.d, , w,, , W, d and, , 0,, , rr, ^^"^-^^^3p~^^^, ~-r-_:r-_r-jr-~ ?j^, ~, T, , -JT, , Fig. 224., , w.d cos, , W.d, , W, , ', , Wt(i, , C, , 8., , n, , te, , W.d, , 1, , -^tanO, T, , $ being, , smaU, , ', , W.6, L rfl/l ~ 0., we can easily calculate the metacentrio, , height of the ship., 159., Pressure due to a Compressible Fluid or a Gas., A gas, differs from a liquid in that, unlike the letter, it is highly, compressible* and, therefore, also highly expansible, tending to, expand, perpetually and indefinitely., , *An idea of the high compressibility of a gas, compared with that of a, uquid, can be had from the fact that whereas the density of sea- water at a deptb, 3f 5 miles is about the same as that of the surface layer, the density of the, atmosphere at the same height above sea -level is reduced to one quarter of that at, the latter., , 1
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PROPERTJ&S OF MATTER, The pressure exerted by a gas, is thus fundamentally different in, the nature of its cause from that of a liquid and cannot be taken to be, proportional to the height of the gaseous column, for the simple, reason that, except in the case of a small volume of a gas, the, density goes on progressively increasing as we go further down the, column, due to the layers above pressing down upon the layers, below, thus giving riso to a well defined pressure gradient ail along the, column., , So that, whereas a liquid everts pressure only under the action, i.e., due to its weight, or due to an external force applied, to it, as, for example, when it is pressed down by a piston, the pressure due to a gas is entirely a consequence of the incessant mobility and, the kinetic energy of its molecules, or due to what Boyle called the 'spring, of the gas., The mad and random motion of the gaseous molecules results, in their colliding not only agairst eech other but also against the, walls of the containing vessel, and it is this bombardment of the, walls by the fast and haphazardly moving molecules that causes the, pressure., (See Chapter XV)., We are here concerned mainly with the pressure exerted by the, gaseous mantle or envelope, surrounding us over land and sea alike, and in all latitudes, which we call 'air' or 'atmosphtre' and which, as, we know, is a mixture of a number of gases and vapours, In, of gravity,, , pursuance of its inherent property of indefinite expansion, this air or, atmosphere should expand to an infinite distance above the earth,, but the earth's gravitational attraction on a huge mass like it sets a, limit to its expansion., Even so, it has been known to exist up to a, of, 300, from, the surface of the earth, although even at, miles, height, 25 miles or so, its density and pressure start falling oif so rapidly, that at altitudes above 300 miles, it may be said to be as good as, non- existent, \*ith just a void or a vacuum beyond., Now the atmosphere can be divided into two very distinct, regions, viz., (i) a lower region, called the troposphere or the convective, zone, and (//) an upper region, called the stratosphere or the advective, zone, the surface ol separation of the t\*o being known as the tropopause, which varies with the latitude and falls from a height of about, 14 kilometres at the equator to about 8 or 10 kilometres at the poles,, and is found to be higher in summer than in winter., This extends to a height of about 6 miles at the, (0 The Troposphere, poles and about 10 miles at the equator, with a vertical distribution of temperature, as its chief characteristic, the temperature falling off rapidly with altitude, there, being a vertical temperature gradient or a lapse rate* of, per 500 feet rise in, , 1C, , altitude., , This temperature gradient is probably due to a variety of causes. Lord, Kelvin attributes it to the atmosphere being in a state of C3nvective equilibrium,,, which is brought about, on the one hand, by the earth getting hcaied by the, solar radiation Jailing on Uf, and then warming up the layers above it, by direct, , *UsuaUy, a vertical temperature gradient is taken 10 bd the fall in temperature per 100 metres rise in altitude and the lapse rate, as the fall in temperature per one kilometre rise in altitude., t Little or no heat is absorbed by the air during the passage of the solar, radiation to the earth through it, and whatever little is, is distributed over too, large a mass to be able to produce any appreciable rise in its temperature, this, absorption being the same at all altitudes,
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HYDROSTATICS, , 359, , contact and by emitting out radiations which are absorbed by them, and, on, the other, by the lower atmosphere getting cooled by radiation due to its, emitting out more energy than it absorbs at the ordinary temperature. The, two processes, going on side by side, produce changes in the density of the air,, conducive to the setting ip of vertical convection currents, the lower warmer, air rising up and getting cooled by adiabatic expansion and the upper coldei, air coming down and getting heated up by adiabatic compression., vertical, temperature gradient is thus established and maintained throughout this region, of vertical convection. Hence the name, 'convective zone* also given to it., This seems to be amply borne out by the fact that the lapse rate for, dry air, calculated on this assumption, comes out to be, per 1000 ft., which,,, though appreciably higher than the observed value, is quite understandable,, considering that the air is really never 'dry' and the moisture present in it, inevitably tends to lower the lapse rate., Also known as the a&dctive zone, it is the regidb, (//) The Stratosphere., above the troposphere, where the vertical convection, relerred to above,, becomes much too feeble, with the temperature falling to such an extent that, the heat radiated out is equal to the heat absorbed from radiations from the, earth and the solar radiation parsing through it, there being set up a radiative, equilibrium in the region, the temperature remaining constant at about 55C,, hence the name, 'isothermal layer* also given to it., It will thus be readily seen that the stratosphere is a direct consequence, of, and is characterised by, the cessation of vertical convection and the setting, c, up of a radiative equilibrium, with the temperature constant at 55 Cuptoa, height of 300 miles or so, after vihich it probably shoots up to 700C or thereabouts., , A, , 3C, , ,, , 160., Measurement of Atmospheric Pressure. The instruments, used to measure the atmospheric pressure are known as barometers,, one of the bast forms of which is the cistern-type Foriin's barometer., Another hand}7 and portable type of barometer is the, s no, Aneroid barometer, (from 'a, without, and 'neros' liquid), or, used, is, it., studied, other, in, We, have, mercury, any, already, liquid, these in good detail in the junior classes and shall not, therefore,, repeat them here. Instead, we shall pass on to a consideration of the, corrections that must be applied to the readings obtained from them., 9, , Correction of Barometric Reading. Although the Fortirfs, 161., barometer is quite an efficient instrument, a few corrections Lave to, We shall consider, be applied to its readings for greater accuracy., here only t\vo important ones of them, v/z.,, the expansion of the brass scale, on which the, (/) correction for, reading is taken, and which is usually calibrated at, , 0C, , correction for expansion of mercury,, , (//), , of, , ;, , and consequent lowering, , its density., (/), , and, , let, , Let the temperature, at which the reading is taken, be /C,, is, be the observed reading at this temperature. Then,, t, , H, , H, , t, , in fact just the value of the divisions of the scale, correct only at 0C., If, therefore, a be the coefficient of linear expansion of brass, the, correct length at tC, is given by, t (l+at) cms., , H= H, , Again if v and P O be the volume and density of a certain, of mercury at 0C, and v t and P/, its volume and density resv t?f, v .P, pectively, at tC, we have, is the cofficient of, whe r e, V, P, ^, JL* r, - (l+70, n, ur,, Or,, of mercury, expanssion, ^, -^, ^cubical, (ii), , mass, , m, , _, , =, , Or,, , =, , m=, , po/p,, , '>', , ., , =, , 1+yt, whence,, , Po, , =, , p,(l+70-
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360, , PROPERTIES OF MATTER, , Now,, , H, , clearly,, , H, #, , where, Or,, , is, ., , Po, , Q .? Q .g, , = tf.p,.g,, , the true barometric height at 0<7., , =, , H.? t, , =, , //O .p/(l+70, , Or,, , ., , H.? t, , =, , //,(l+a/). P< ,, , #, , whence,, , neglecting squares and higher powers of a and J., , Or, , # =, = -00018, = fli[l - (-00018, , >, , #,!l-(y-<*)']-, , For mercury, y, , and, , for brass, a, , =-000019., , #o, -000019X]. Or, #,=jf7,(l-'0001610,, whence, the barometric height at, can be easily calculated., Other errors, due to pressure of mercury vapour and, capillarity, etc., are much too small for tubes of, reasonably wide bores, and are,, , 0C, , therefore, usually neglected., 162., , A, , B, , and, , B, ,, , x, , *, , T, j, , x, , _, , Fig. 225., , ~, , Or, , distance, , Since the density of air and, therefore, its pressure, decreases with altitude, for a pressure p at A, that at B will, be, If, therefore, p be the density of air betsay., ween A and B, and g, the acceleration due to gravity, we, , pdp,, , have, , dp, , p.g.dx,, , ...(/), , with height., If the temperature of the air be constant,, T v l/^, rn, P, , >, , where, , =, , ve sign being used, because the pressure decreases, , the, , I, , i, , Change of Pressure with Altitude. Consider two points, dx apart, vertically below each other, in, If A be at a height jc above the ground,, air, (Fig. 225)., the height of B from the ground is obviously (x+dx)., , 9, , = K'P>, , K is a constant, equal to, -, , K.p.g.dx., , Integrating, , this,, , p., , p//?., , Or,, , -, , dptp+K.g.dx, , Or,, , A (Boyle's law), , and l\V oc, , [, , relation, Substituting this value of p in, , -dp, , P oc p., , (/), , above,, , -2-, , =, , we have, , = K.g.dx., , 0., , we have, loge p+K.g.x, , s= a constant C., , ...(#), , Kfow, if the pressure at heights h and I/be p and P respectively,, we have, from (ii) above,, = C ...(/v), = C,, ...(/ii) and log, P+K.g.H, log, p+K.g.h., .-., subtracting (iv) from (in). w h av, log,, , Or,, , Thus,, , p-log,, , P = K.g.H-K.g.h =, ^K.g.(H-h)., , log,(^-), , (//-/(, , ), , =, , ~-, , K.g.(H-h)., ...(v), , ...(v)
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361, , HYDROSTATICS, substituting the value of, , Or,, , K, , 9, , in, , i.e., p//>,, , we have, , (v/),, , .P, , (H-h), , P-, , p.g, , Thus, knowing p 9 P,, determined., , and, , p, , g, the altitude, , (H-h) can be, , easily, , In the above treatment, it has been assumed that the temperature of the air, or the atmosphere, remains the same throughout., the case. Nevertheless, the result, This, as we know, is far from being, small heights., is accurate enough for the determination of, A, of, number, a, heights, 1? A 3 A 3 etc., in arithmeIf now we have, tical progression then (A a, (63^2) and s6 on, A,), at these heights, we have, etc., be the pressures, And, if p L9 /> 2r, ,, , =, , -, , A, , from, , (v), , above,, , log,, , Since, , (pjp2 ), , Aj, , (A 2, , = K.g.fa-hJ, and, = (A, A we have, , log,, , (pM = K.g.fa-hJ., , a ),, , log, (ft/A), , A /A, , Or,, , A A etc, , == lo S, , (A/ft)-, , = A/A-, , S eometrlcat P r, , Sression., increases in arithmetical, altitude, or, Thus, we see that as the height, in, progression., geometrical, decreases, progression, the pressure, , f, , Pv, , ^->, , Note., , o the base, , -, , are in, , into, convert logarithm* to the base e, 2'302., multiply the former by, , To, , 10),, , common, , logarithms,, , (i.e.,, , SOLVED EXAMPLES, , with water, r ft long and 5 ft. wide is filled, r ct ng u lar clster, B 62'5 Ibs., find the magnitude, A !u, f, l ., Af water to weigh, a depth, of 3 ft. Taking one cu. ft. of, 2fton, nd position of the resultant fluid thrust, ach side., -, , a, , (a), , A, , ^, , ., , J, , ., , ., , l, , ,, , w, , Here, clearly, (Fig 226),, depth of water, , 3 ft., , centre of area for each side of the cistern, , ., , =, -*., , V, , 3/2, , =, , 1-5 ft., , = h.p.g., = I'5x62'5x32poundals., = 1-5X62-5 lb. Wl, , pressure at centre of area, , 7, , ., , Now,, , Fig 226, area of each longer side in contact with water, 9 sq. ft., 3 x3, and area of each smaller side in contact with water = 2x3=6 sq.ft., Since thrust, pressure at centroid or centre of area X area,, , we have, , =, , and,, , pressure on each longer side, pressure on each smaller side, , And,, , centre of pressure, , =, , ~, , -, , =, =, , x depth, , 1'5, 1'5, , =, , -, , x 62, x 62, -, , 5, 5, , x3, , x9, X6, ==, , =, , =, =, , 843*7, , Ibs., , 562*6, , Ibs. wt., , wt., , 2/r., , 2., Find the position of the centre of pressure of a triangular plate immersed in a liquid with its plane vertical and one side in the surface., ABC is a vertical triangular door in the side of a ship, AB is horizontal,, C below AB, and the triangle equilateral of side 5 ft. The door is hinged along, AB, and kept shut against the pressure of the water by a fastening at C. If the
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362, , PBOPEETIES OF MATTER, , water rises to the, weighs 62-5 Ibs.), , of, , level, , AB,, , find the force, , on the fastening. (One, , cu. ft., , of water, , (Liv. Inter.), , The centre ofpressure of the triangular plate, with one of its sides in the, plane of the liquid surface will be at a depth /z/2, from the liquid surface,, where h is the height of the plate, (see page 351)., , ABC be the triangular door hinged, and having a fastening at C wheie, , Let, , along, , AB, , ABlies in the plane of the water surface, (Fig., 227)., , Obviously, height h of the triangular door, 75 = 4 329 ft., I, , Since the centre of a triangular lamina isits height below the vertex, its depth below AB, or the water surface, is $rd ot its height,., ,e., equal to Jx4329=r443/r., \rds of, , Fig. 227., , And, , /., , Therefore, pressure at the ccntroid=, 1 '443x62 5, thrust on the door, , =, , =, , 1, , -443, , x 62-5 x, , i, , Now, centre of pressure, , moment, , /., , Ibs wt., , =, , 1-443x62-5, , Ibs. wt., , X area of the door., = 1 "443 x 62 5 x i x 5 x 4-329 Ibs. wt., = 1-443 x 62-5 x 2-5 x 4 329 Ibs. wt., , base x altitude, , Ibs., , of the triangular door lies at i h,, 2 164ft., ix4329, , i.e.,, , at, , =, , of the thrust about, , [See above-, , AB, , =-~t'nrustx, , depth of centre of pressure., 5 X 4-329x2- J 64 /6s wt., , = 1-443x62 5 x2, And,, , if, , Fbe, , the magnitude of the force on the hinge,, , its, , moment about, , Ibs. wt., , Clearly, therefore,, , Fx4 329=1-443x62 5 x2'5x 4 329x2, , Or,, , 164., , 443x625x25x2-164-487-5, , F-=l, , Thus, the force on the fastening at, , C=, , Ibs., , wt., , 487 5 Ibs wt., , Find the centre of pressure of a rectangular sheet 'a' in, long and 'b' in., 3,, of unifoim density, with one side, wide, of uniform thickness, immersed in a liquid, of length V?' in. in the surface, the plane of the rectangle being inclined at a, to the vertical., angle, If the rectangular sheet remains in the same position with respect to the, \essel containing the liquid, and the depth of the liquid be increased by h in , find, (London Higher School Certificate), the new position of the centre of pressure., , its, , a, , Let^BCDbe the rectangle, immersed, AB = b in the liquid suiface F, and, , side, , tan angle, , 0,, , (Fig. 228), , Then, since the, angle, , is, , vertical depth of the rect-, , BK^BC, , clearly, , troid lies, , in the liquid, of density p, with, plane loclincd to the vertical, , its, , cos, , =, , a cos, , 9, its, , cen-, , at a vertical depth -r- cos 0, from the, , liquid surface., , And, hence, proceeding as, , in, , 152 [case, , page 349], we have, depth X ef the centre of pressure, the surface of the liquid, clearly given by, (i),, , ., , X =, , |-, , x vertical depth, , =, , a cos, , Q., , P, , from, Fig. 228., , Now, let a column of liquid EE'F'F, h in. thick, be added on to the top, of the liquid surface to increase its depth by h in., and let P' be the new centre, of pressure of the rectangle, whose position is otherwise unchanged with respect, to the vessel; at a distance XJ from the new surface E'F'.
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363, , HYDROSTATICS, Then,, , clearly, thrust on the rectangle due to the original liquid column, area of the rectangle X depth ofcentroid G x density oj the liquid*, , =ab^, and, , So, , ., , x p xg, , cos e, , =, , -r-, , a*b cos 6, , x p xg, , g, , ;, , on the rectangle due to the new layer h of the liquid added', of rectangle x depth of new liquid column added x density of liquid* g., , increase in thrust, , that, the total thrust on the rectangle, , =, , \a*b cos 9 X P, , x g-}- ab X h x p X g., , ~ab$g(\a cos 0-f/z)., Clearly, therefore, the moment of this thrust about the new liquid surface E'F', ..., ..... (i), ~ab9g(\a<osQ + h).XQ '., Again, the distances of the new centre of pressure P' and the centroid of the, -h h) and, rectangle from the new liquid surface E'F' = (X ^fi) = (| a cos, (J a cos 0-f-/z) respectively., , And, therefore, the moment about the new liquid surface E'F', , Equating, , (/), , ( a- cos-, , Or,, Or,, the, , *, , or the depth of the, liquid surface, _, , (, , K, , '=, , A'o',, , new, , is, , also equal to, , -f h) + ab.h p-g. ($ a cos 6 +h),, ka*b>cos 6>p-g(% a cos, (//), =abpg[$ a cos B (| a cos e+A)+Mi a cosQ + h)]., and (//), therefore, we have, cos e + /0 Xo'=a.b.p.g[k a cos 0(*. a cos 6-h/iH /*(i a cos Q+h)], , cos* o + lah cos, , <j, , + $ah, , new centre of pressure of, , cos, , the rectangle from, , ., , ro5^, 2, , +, , 2/z\, , ), , Neglecting atmospheric pressure, find the depth of the centre of pressure, of a circular lamina just completely immersed with its plane vertical in an incom4., , pressible liquid., , A circular door in the vertical side of a tank is 'hinged' at the top and, opens inwards, and the tank contains water to a height just sufficient to cover, the door. If the diameter of the door is 2 ft , find the magnitude of the force that, must be applied normally to the centre of the door in order just to open the door., Find also the reaction at the hinge when this force is being applied., (Cambridge Higher School Certificate), The centre of pressure of the vertical circular lamina, just immersed in an, 152 (//;, (page 350)., incompressible liquid, is equal to 5r|4, as explained in, Here, obviously, the centroid of the circular lamina is at its centre, at a, depth equal to its radius from the water surface, / e. 9 at a depth 2 ft. 12 or 1 ft., below it, (the lamina bemgyw^/ immersed in water)., = // p.# = 1 x 62 Sxgpoundals = 62'5 Ibs. wt., pressure at the lamina, and, area of the circular lamina = nr 2 = TT x 1 = TT sq. ft., So that, thrust on the lamina = pressure on the centroid x area of the lamina, , =, , 62*5 XTT /fo. wt., , =, , 196-3 Ibs. wt., , This thrust acts at the centre of pressure of the lamina, whose depth from, the water surface, as we know,, , =, , .'., , moment of this, , =, , If, , we have, , F Ib., , /., , =, , 5/4, wt. be the force applied to the centre of the door,, just to open, , moment, , And, , =, , 5r/4, 5x1/4= 5/4 ft., thrust about the liquid surface, or about the hinge, 62-5 XTTX, 245-5 Ibs. wt., , of, , F above the hinge = Fx r = Fx 1, Fx 1 - 245*5., FOr,, , Ib, ft., , 245*5, , Ibs. wt., , it,
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PBOPERTIBS OF MATTER, , 364, The, , force required to be applied to the centre of the door to just open it is thus, equal to 245*5 Ibs. wt., Thus, the forces acting on the two sides of the door are respectively 245 '5, Ibs. wt. and 196*3 Ibs. wt. ; and, therefore, reaction at the hinge, 245 5-196-3 = 49 '2 Ibs. wt., , =, , 5., , If a load of, , 20 tons, moved 50, , Ions displacement, causes the ship to, , tilt, , across the deck of a ship of 20,000, through i, vhat is its metacentric, , ft., , height ?, , 20 tons be moved across the deck of a ship from A to B, Let a load w, through a distance of 50//. (Fig. 229;, and let the ship be tilted through an angle, i into the position shown. This, as we, _r, , -_~^.-^_-_~, , __-_-__--_ --_i, , know,, , -, , equivalent to the application, , is, , of a couple, of moment 20x50 ton-feet,, tending to turn the ship clockwise, [see, 158 (alternative treatment), page 357}., , And, the couple, formed by the, weight of the ship acting vertically at, /X2 its c g., G' and the equal weight of disB, Jj-JE placed water at its centre of buoyancy, 1, , ,, , -^= tends to restore the ship back into, , its, , original position, the moment of the, is the, couple being WxMP. where, on, perpendicular from the metacentre, , MP, , to the vertical line through, , Since the ship, , Or,, , in, , M, , ., , equilibrium, , under the action of these two opposite, couples, they must obviously be balancing each other. So that,, , Fig. 229., , =, , is, , G, , 1, , 20,000 x MP = 20 x 50., =, -20x50/20,000, 1/20 /V. = -05 ft., MG' sinV '= AfG'X'0087,, , 20x50., , Or,, , MP^, , Now,, , MG', , whence,, , 0087, , =, , =, , 5-748, , ft., , =, , 5 '748 ft., Thus, the metacentric height of the ship, 6., State the theorem of Archimedes, and explain what you understand by, the terms "force of buoyancy", "centre of buoyancy". A cylinder of radius 1 cm., and length 4 cms., made of material of specific gravity 0*75 is floated in water with, its axis vertical., It is then pushed vertically downwards so as to be just immersed., Find (a) the work done, (b) the reduction in the force on the bottom of the containing vessel when the cylinder is subsequently taken out of the water,, (Oxford and Cambridge Higher School Certificate}, is clearly equal to m*l, n X 4 x 0*75, 3rc gms. wt., weight, floats vertically in the water, we have, 3 n gms. wt., weight of displaced water also, , Here, the volume of the cylinder, , And, , .'., , Since, , it, , its, , =, , =, , TT., , 1., , 4, , c-cs., , =, , =, , Let length of the cylinder inside water be = x cms., Then, the volume of the immersed part of the cylinder, i.e.,, , And, , /., , Hence,, i.e.,, , = IT. 1.x =* *x c.cs., volume of water displaced by the cylinder = *x c.cs., the weight of this displaced volume of water = it.x.l gms wt., *x = 3^. Or, x, 3 cms., , =, , 3 cms., length of the cylinder inside water, length of the cylinder outside water, , 43, , =, = 1 cm., Thus, to immerse the whole of the cylinder just inside the water, we have, to simply push it down through 1 cm., Obviously, the volume of the displaced water or upthrust on the cylinder, TT x 1 x4x 1 = 4*, will be, gms. wt., , And,, , therefore,, , Hence, increase in the upthrust on the cylinder will be, , =, , 4rc, , 3*, , =, , n gms., , wt.
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HYDROSTATICS, , 365, , Since this increase in the upthrust takes place gradually from, to w, a*, the cylinder is pushed steadily down into the water from its initial position, we, may take the average value of the upthrust against which we work in pushing the, cylinder down through 1 cm. to be (0-j-7i)/2 or ir/2 gms- wt., , And, therefoie, work done in pushing the cylinder down through 1 cm ,, against this average increase in the upward thrust, will clearly be equal to, average thrust x the distance through which the cylinder is pushed down., i.e ,, work done, 1-571 gms. wt. cm., (^12) x 1, ?i/2, , =, , =, , Now, with the whole of the cylinder immersed in the water, the weight of, water displaced, , =, , 4n gm. wt., , =, , 12-57 gm. wt., , This must also, therefore, be the downward thrust ori the bottom of the, containing vessel. So that, when the cylinder is removed out of the water, the, reduction in the thrust on the bottom of the containing vessel will also be the, same, viz., 12-57 gm. wt., Calculate the metacentric height and determine the necessary condition, 7., for the stable equilibrium of a cylinder of length /, radius r, and density p, floating, ', , vertically in water., , Then,, , And, , .*., , Let a portion x of the cylinder be inside water., n r z .x, volume of water displaced by the cylinder, i.e., v, weight of water displaced or upthrust on the cylinder nr*.x.l, , c.cs., , m x., z, , This must, for equilibrium, be equal to the weight of the cylinder,, 2, equal to w.r logins- wt., , i.e.,, , ., , =, , 2, , 2, , *r .x, 7tr ./.p gms. wt., / p., Therefore, x, centre of buoyancy of the displaced liquid must, therefore, be at a, height xj2 = /p/2 from the bottom of the cylinder., Now, as we know, the distance between the centre of buoyancy of the, z, 2, displaced liquid and the metacentre is Ak jv t (see page 357), where Ak is th(, moment of inertia of the surface-plane of the cylinder about its diameter. Sc, , Or,, , The, , 2, /c, =/ 2 /4,, that,, k being the radius of gyration of the plane about the surface-line or the diamete, , of the cylinder., , =, , 2, , iir x, therefore, we have, Substituting the value of v, distance between the centres of buoyancy and metacentre, , rtr*, , Now,, , x, , distance of the e.g. of the cylinder, 1, , ""2, , _, , *, , from the centre of buoyancy, , - JL ~ JP, ~~, , .JLLiP), , 2""~, , 2, , 2", , 2, , of the body and the metacentre, or the metacentric height, h, of the cylinder = distance between the centre o, buoyancy and the metacentre minus distance between the e.g. of the body an<, the centre of buoyancy., , And, therefore, distance between the, , a, , e.g., , 2, , a, , 2, , ~2/p/(l-p), _r -2/ p(l-~p), ^-p, 4/p, Now, for stable equilibrium of the cylinder, the metacentre should be, above the e.g. of the body, i.e., h should have a positive value., 2, 2, only when r >2/ p<l p). This is, therefore,, And, obviously, this is, n, , Or,, , -, , r, , ^-, , /(1-p)2, , _, -, , r, , possib^, , the necessary condition requirea,, , EXERCISE IX, , Define pressure at a point in a fluid. Find the total thrust on the, sides and vertical ends of a V-shaped trough, 1 ft. deep, 2 ft. wide at the top, and 4ft. long, when nlled with water, density 62*5 Ib.jcu. ft., (Oxford and Cambridge Higher School Certificate), 1., , m, , Ans, 9, , for, , (/), , Ibs., , wt.;2Q'Zlbs, , wt., , Determine its position, Define clearly the term 'Centre of pressure, a circular lamina of radius r just immersed vertically, and (11) a triangular, 2., , ., , t
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PROPEREIES OF MATTER, , 366, , immersed, , apex, and (b) its base in the, 5r/4 ; () (a) 3H/4 ; (b) h/2., A square lamina with its sides 3ft. long is just immersed vertically, 3., a water with an edge in the surface and is then lowered 10 ft. Find the dis,ance of the centre of pressure in the new position from the centre of the square, ;Neglect the pressure of the atmosphere in each case)., (Joint Matriculation Board and H-S. Certificate), Ans. 0-0303 //., A circular area of radius a ft is immeised in water, with its plane, 4., The surface of water rises from 2a ft above the centre of the circle to, vertical., Neglecting atmospheric pressure, prove that the centre of prts[aft- above it., (London Higher School Certificate), ,ure rises through a distance a\ 16 ft, State the Principle of Archimedes and define clearly the terms (i), 5, Centre of buoyancy, (it) Metacentre, and (///) Me tacentric Height. Discuss in, ^detail the conditions for the stable equilibrium of a floating body, with particular reference to a floating ship., lamina of height h,, mrface., , vertically with, , (a) its, , Ans., , (/), , Show that if a floating body be given a small rotational displacement, 6., plane of symmetry, the distance between the centre of buoyancy of the, displaced liquid and the metacentre is Ak^\V\ where A is the area of the surface, plane of the body, k, the radius of gyration about the surface-line and V, the, volume of the displaced liquid, Discuss the conditions necessary for a hollow cylinder of height h, and, density p, open at both ends, with i\ and r a as its internal and external aradii, to, Ans. r^-f r 2 2 >2/z .p(l p), float vertically in a liquid in stable equilibrium., Discuss how the atmospheric pressure changes with altitude above, 7., the surface of the earth, the temperature remaining constant, and show how if, the altitude increases in arithmetical progression, the pressure decreases in, in its, , geometrical progress ion., , A mercury barometer is known to be defective and to contain a small, 8., quantity of air in the space above the mercury. When an accurate barometer, the defective one reacts 760 mm- and when the accurate one, reads 770 mm, What is the true atmospheric, reads 750 mm., the defective one reads 742 mm, pressure, when the defective barometer reads 750 mm. ?, (Cambridge Scholarship} Ans. 758'8 mm., A simple barometer has the glass tube attached to a spring balance., 9., What weight does the balance record when the open end of the tube is just dipping under the surface of the mercury in the reservoir, and what changes occur, when the tube is lowered so that m:>re of it dtps under the mercury ?, (Oxford Higher School Certificate), Ans. (/) The balance records the weight of the tub and the mercury column., (//) A progressive decrease in the weights, due to buoyancy of the tube, un,, , ,, , til, , when finally, , the fatter is full,, 10., , A, , it, , decreases to zero., , sealed spherical cellophane balloon has a diameter of 5 metres, , and, , the apparatus it carries, 1 k.gm* It contains one-tenth of the, volume of hydrogen required to fill it at atmospheric pressure. The balJoon is, illowcd to ascend if the cellophane does not expand and if the temperature of, ,he atmosphere is assumed to be constant at 0<7 at all heights, calculate at, vhat height the envelope becomes full and the height to which the balloon rises., The pressure p at height h (km.) is related to that at the ground (p Q ) by the reation/i = 20 log lo ip n lp). (Densities of air and hydrogen atO'O and atmos09 gm per litre respectively)., )heric pressure are 1'29 and, (Oxford and Cambridge Higher School Certificate), weighs, with, , ;, , Ans, , (/), , 20 km. and, , (), , 34'5, , km.
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CHAPTER X, , FLYING MACHINES, Jet Planes, Rockets and Satellites, Flying Machines. There are two types of flying-machines,, machines, or Air ships, (//) heavier-than*air, , 163, mz.,, , light er-than-air, , (/), , machines, or Airplanes., The Airship. An airship, , is based on the principle of Archimedes., The, weight of the air displaced by it is greater than its own weight, i.e., the upward, -thrust on it, due to the displaced air, is greater than the downward thrust,, (due", to its weight), and hence it rises up., 1, , An, , airship, , is "in, , fact a big cigar-shaped, , balloon of a light material, like, , aluminium or its alloy, covered with a specially treated water- proof linen or silk, and divided up into a number of compartments contain iag bags filled with a, 'light g is like hydrogen or helium (preferably the latter, due to its non-inflammable nature) from which it derives its buoyancy, i.e., which makes the total weight, of the airship less than the weight of the air displaced by it, or the upward thrust, on it greater than its weight. This excess of upward thrust that it possesses over, jts weight is called its liftm ; power, and gives the maximum extra load it can be, , made, , to carry., , For steering purposes,, for horizontal motion,, ful engines., , it is, , it is fitted with rudders or other suitable devices and, provided with propsllers, worked by light and power-, , The, , Kite. Before dealing with the airplane, it will be helpthe, This will be, study, principle underlying the ordinary kite., understood from the following, , 164., , ful to, , :, , Let AB,, , (Fig. 230) represent the mid-line of the kite., Then,, the different forces ou it are (/) its weight W, acting vertically downwards at its e.g., G. (//) the tension, , T, , acting along the, (///) the pressure, due to the wind, acting along the, direction of the wind, all along the, undersurface of the kite., , of the, , string,, , shown,, , string, as, , of, , the, , wind may be resolved into, rectangular, components at, , t\\o, , Now,, , points,, , (/), , this, , pressure, , perpendicular to, , all, , the, , plane of the kite, and (b) along the, plane of the kite. These latter components play no part in supporting, the kite and may thus be ignored, and the former components p, p..., ;, , being so, , many, , like parallel forces,, , have a resultant P, equal to their sum, called the effective pressure of, the wind, acting at the point C., Thus, the three forces acting on the, kite are, (/), , W>, , acting vertically downwards at G, the e.g. of the kite., , 367
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PROPERTIES OF MATTER, , 368, , T, acting along the string at E, and, , (//), , (Hi) P, acting perpendicularly to the plane, , of the, , kite at C., , Condition for the Kite to be in Equilibrium. In order that the, kite may be in equilibrium, these three forces acting on it must be*, in, represented by the three sides of a triangle, taken in order. And, order that this may t>e so, they must all meet in a point, say, at O., It will easily be seen that this can be possible only when G, the e.g., of the kite, lies below the point C, where the effective pressure (P) of, the wind acts. In other words, the e.g. of the kite must be pretty low, down for it to be in equilibrium. It is for this reason that the lower, part of the kite is made slightly heavier, and that a small paper tail, is sometimes attached to it, which, in addition to bringing its e.g., down, also makes it look more attractive., , P, , Condition for the Kite to rise up. The effective pressure, may also bo resolved into two rectangular components,, , of the wind, viz.,, (/), , D, along the direction of the wind, called the, , drift,, , or the-, , drag., , L, upwards, perpendicularly to //, called the lift., be the angle that the kite makes with the direction of the;, , (ii), , If, , wind, we have, , = P cos (90 0) = P sin 0,, = P sin (90 -0) = P cos, , Drift (D), , and, , Lift (L), , 6., , Similarly the tension (T) of the string may be resolved into two, rectangular components, (/) along the horizontal and (ii) downwards,*, along the vertical, (shown dotted)., Now, clearly, the only force tending to make the kite rise upcos 0, and the forces tending to make it fall, wards is the lift L, downwards are (/) its weight, arid (//') the downward component of, , =P, , W, , the tension of the string., The moment, therefore, that the lift (L) is4- the downward component of T, the kite rises upwards., greater than, , W, , Thus, to make the kito rise up, we must increase the lift, i.e^, This can be done by increasing P, i e. by running against, cos 6., the wind, and by decreasing 0, by giving small jerks, (Tanka) to th&, , P, , t, , If, however, the drift (P sin 0) be greater than the, string., (P cos 0), the kite drifts along in the direction ol the wind., , lift, , An airplane is a heavier-than-air machine, underlying it is in main the same as that of, the kite. Obviously, however, there is no tension of the string,, here, pulling it downwards, so that the only force ttnding totake it up is the lift and the force tending to take it down is its, 165., , and the, , The, , Airplane., , principle, , weight., air, , Farther, as the propeller- blades rotate rapidly, they throw the, in front of the plane, and its reaction is a thrust, , backwards from, , R, forwards., Let us now consider the relation between these different forces-', on the plane in the different phases of its flight, viz., (i) when itflie&, level, (ii), , when, , it, , climbs, (Hi), , when, , it, , dives, , and, , (/>), , when, , it, , glides.
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FLYING MACHINES, (/) Level Flight., said to be flying level., , An, , 36, , aeroplane flying along the horizontal, , HORIZONTAL, ATTITUDE, , I, , P<, , is-, , W, , LINE OF FLIGHT, , w, , W, , (c), , Fig. 231., , Fig. 231 (a)* shows an aeroplane in level flight, from right to left,, with a constant speed K. This is tantamount to wind blowing from, left to right with velocity V and striking the undersurface of the, plane so that, proceeding, as explained above, we have, Reaction or Thrust R, forwards, the drift or drag, backwards,, both acting practically along the horizontal, ;, , =, , D, , y, , ;, , W, , =, , of the plane downwards, and, weight, acting along the vertical, [Fig. 231 (b)], , the, , lift, , upwards, both, , if,, , L=W,, , Or,, , (/), , R =, , ..., ..., D,, ...(//), form a closed polygon, [Fig. 231 (c)],, , and,, , which, represented vectorially,, the plane being in equilibrium., , It will at once be clear from relation (//) that, for level flight, the, forward thrust R must just be balanced by (i.e., must be equal to) the, backward drag D, at that particular speed of flight., , Further, if the speed, , falls,, , the, , lift, , decreases, , and the, , so that, a minimum speed (about, starts losing altitude, essential to keep the plane at a certain height., ;, , (ii), , Climbing., , If an aeroplane, , flies, , 'plane, , 50 m./hr.), , obliquely upwards,, , it, , is, , is, , said to be doing a <climb\, , *, HORIZONTAL _, ANGLE OF AT JACK, , (WOW*), , *, , ^, , W, \f, , *&,, , %, , W, , (W, , (c), , Fig. 232., , We, , shall, for the sake of simplicity, take the line, , of, , flight dur-, , *For simplicity, the student may simply show these forces acting on the wing, or the aerofoil, instead of sketching the whole plane., fine lift is not necessarily vertical. It is just the component perpendicular, io the current, , of air.
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PBOPEETIBS Of MATTEB, , 370, , ing the climb to coincide with the direction of thrust R due to the, propeller, or with the attitude of the plane.*, Since the relative velocity of the wind now makes an angle a, with the horizontal, the lift (L) no longer acts in a vertical line with, the weight (W) of the plane and is, therefore, balanced by the comcos a of the weight [Fig 232 (a) and (b)] and, similarly,, ponent, the thrust R of the propeller, by the drift or drag, (D) along the line, of flight plus the component, sin a of the weight, i.e., now, , W, , ;, , W, , = W cos a, R = D+W sin a., L, , and, , ...(/), (*>), , R, , It will thus be clear that, must be greater than, for a climb, the thrust, the drag (D) by the factor, sin a and that it increases with the angle, , W, , &, , or the steepness of the climb., It follows, therefore, that if a, and cos fl=l ; so, 0, sin a, that, equations (///) and (/v) reduce to (/) and (//) respectively. In othei*, words, the plane then flies level with a constant velocity, without a, climb., , =, , =, , The forces in equilibrium, during the climb of the 'plane, represented vectorially, give a closed polygon [Fig. 232 (c)], which, in the, case of level flight, reduces to a rectangle, [Fig. 231 (r )], with R and, , D, , equal and horizontal and L and, When a plane, (Hi) Diving., to be making a dive., , equal and vertical., , W,, flies, , obliquely downwards,, , it is, , said, , Again, taking the speed of the plane to be constant and its line, of flight coincident with its attitude, the different forces on the 'plane, are as shown in Fig. 233 (a) and (fe)., VERTICAL, , COMPONENTS, Of, , L&D, , ~, , L/, , (a), , (b), , (c), , Fig. 233., , Since the relative wind velocity (V) makes an angle a with the, horizontal, the lift here also does not act along the vertical line with, W\ and, since it makes an acute angle with the downward vertical line,, the vertical components of both the lift and the drag act upwards,, , thus opposing W. A& will be readily seen from Fig. 233 (a) and (b)., cos a of the weight (W), the lift is balanced here by the component, sin a of which acts along the, of the 'plane, the other component, same direction as the thrust (R) so that, for equilibrium, we now, , W, , W, , ;, , have, , can, jicts, , *lt is by no means necessary that it should always be so. The line of flight, fact it often does make an angle with the thrust (R), which, of course, along the attitude of the 'plane., , &, , .
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FLYING MACHINES, , R = D - W sin, showing that,/0r a, , dive, the thrust, , R, , 37), , a,, , . ., , must be, , .(v/), , less than the, , drag (D) by, the factor, sin a and, therefore, it decreases with a or the, steepness, of the dive,, necessitating the throttling down of the engine., And, once again, if a=0, sin a =0 and cos a, 1, so that reiations (v) and (v/) reduce to (i) and (//), respectively, the 'plane flying, level with a constant velocity, without a dive., , W, , =, , Representing the different forces vectorially, we again obtain *, closed polygon. [Fig. 233 (c)], which with a, 0, becomes a rectangle,, , =, , with, , R and D, , equal and opposite, and acting along the horizontal,, and L and W equal and opposite, and acting alohg the vertical., (iv) Gliding. With the engine not functioning, i.e., with R = 0,, a,8 the 'plane descends down, it is said to be, gliding., In this case, obviously, the lift and the drag are balanced by, the components, cos a and, sin a of the weight (W) of the 'plant, respectively, and we have, for equilibrium,, ',, , W, , W, , L = Wcosa,, , D =, , and, so that, with the, decreases., , W sin a, , increase of a,, , ...(vff), ...(viff), , ;, , the, , drag increases and the, , 166. Different parts of an Airplane and their functions., different parts of an airplane, respective functions., , The following are the, , and, , lift, , their, , The wings or the aerofoils, as they are techni(i) The Wings., cally called, are, appropriately, the most important part of an, airplane (a flying machine) and much research has gone into perfecting their design, in order to obtain the maximum lift for the 'plane., In fact, the lift due to them accounts for as much as about two -thirds, of the total available lift., To minimise the fractional force to, , its, , smooth air-flow along its surfaces, the wing, lines of the air through which, , motion and to ensure a, shaped to the stream*,, , is, , a gradual taperof its thickness from its, front or leading edge to its rear, or trailing edge, with the upper, surface more curved than the, lower, as shown in Fig. 234., The axis of wing (shown dotted), is called the chord and the angle, Fig. 234., that the chord makes with the, direction of the wind is called the angle of attack., The air, moving more rapidly over the upper than the lower, surface, brings about a difference of pressure on the two surfaces,, In accordance with Bernoulli's principle, (see, and, ft, , passes, with, , ing, , ff, , Chapter XII), , This lift on the, gives the wing an upward lift., consists of (/) an upward thrust on its lower surface and, this, , effect, , on, , its, , upper surface., , wing, (it), , really, , a suction, , For, as the leading edge of the, , wing
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PROPEBTIES OF MATTER, , 372, , tend, air, it parts the air stream into two parts, which, The upper stream, to flow as close to the two surfaces as possible., curved shoulder of its upper, is, however, deflected upwards by the, surface and its return back to that surface is retarded due to its, and a conseinertia, resulting in an area of partial vacuum above it, suction., to, due, on, it, quent upward pull, For a given wind- speed, the lift increases with the angle of, attack up to a certain limit, beyond which it begins to decrease and, This limiting value of the angle of attack is, the 'plane begins to sink., called the stalling angle and its value varies from about 15 to 20., The ratio lift/drag is, however, the maximum when the angle of attack, is about 4., Hence we have the maximum efficiency in flight at this, , moves through, , angle of attack., important consideration in the structure of the wing is to combine, and it, therefore usually consists of two main spars of, wood or metal, running all along its length, with light girders of the same, material, set perpendicularly to them at suitable intervals, the whole framework being covered with a 'skin' of sheet metal or thin plywood, having a tightly, stretched fabric over it, well coated with a liquid solution, called 'dope', which, not only shrinks the fabric and makes it taut like the skin of a drum, but also, serves to increase its strength and to make it water and air-proof., , An, , lightness with strength,, , found that the force or effective pressure (P) due to the, is called, depends (/) directly upon the area A of the, the wind,, aerofoil, (ii) directly upon the square of the velocity (V) of, and (i/f) directly upon the density (p) of the air at the height of the, Thus,, 'plane., It is, , -wind, as, , it, , P, , K is, , oc A.?.V*., , P=K.A. 9 .V 2, , Or,, , ,, , a constant, depending upon the shape of the aerofoil and, the angle of attack., the drag D =* P sin 0,, P cos 0,, and, Since the lift, L, , where, , =, , we have, , L, , =, , K.A. ? V*.cos 9, , and, , D=, , K.A.p.V, , Or, multiplying and dividing each expression by, , 2, , 2,, , ., , sin 6., , we have, , D = 2K sin, and, 2K cos 6 \A$V*, The factor 2K cos is called the Lift coefficient and, , 2K, , L, , =, , sin, , 0,, , the factor, the symbols, , the Drag coefficient, usually denoted by, , CL and C^ respectively. So that,, and, L = CL } ApV*, in, Ib., where L and D are expressed, , D = CD, , ., , wt., , ;, , A, in, , .\ A?V\, , sq. ft., , ;, , p,, , in slugs* per, , sec., c.ft. and V, in ft. per, The Lift and Drag coefficients increase with the angle of attack,, the former having its maximum value 1-2 at about 16, when the, value of CD is about 20. The ratio of the two coefficients i.e., CL \CD, or the ratio Lift /Drag also varies with the angle of attack, and has, its maximum value (12) at about 4, at which value of the angle of, in flight., Norattack, therefore, we have the maximum efficiency, is arranged to lie between 3 to 6., of, attack, the, angle, mally,, Further, it will be clear from the expression for L above that a, certain minimum wind speed is essential for the lift to be large enough, to make the 'plane rise up against the force of gravity. It is for this, , *Mass, , in slugs is equal to, , weight in pounds weight, divided by 32.
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FLYING MACHINES, reason that the plane must, it can take off., , first, , be, , made, , 373, , to run on the ground before, , () The Propeller or the Air-Screw. It is a large fan- like, structure, carried right in front* of the plane and rotated rapidly, about a horizontal axis by an internal combustion engine. Its tw<?, (or more) blades are set at an, in a central hub, a shown in, angle, , Fig. 235, and may be made of, wood or metal, consisting, in, , the former case, of a number, of layers firmly glued toFig- 235, gether, with their edges tipped with metal, and their surface provided, with a suitable protective covering of fabric or cellulose., A propeller blade is in fact a small wing and functions precisely, as such. For, just as a moving wing, meeting the air at an angle,, experiences an upward thrust in a direction almost at right angles to, that of its motion, so also does a revolving propeller blade experience, a thrust at right angles to its direction of motion, i.e., along the, horizontal, for the very air which it sweeps from in front of it and, throws backwards, pushes it forwards., its way through air, much, way through wood or metal, it, , Thus, because the propeller cuts, the, , manner of a screw cutting, , its, , on that analogy, also referred to as the, , in, is,, , air-screw., , from the two most important parts of an airplane, discussed, above, there are others which make for its stability and easy manageability, These together constitute what, In any desired position and direction in the air., are called the surface controls of the airplane and we shall now deal briefly with, Apart, , these., , Carried at the rear end of the airplane, it consists, (Hi) The Tail Unit., of two sets of surfaces, (/) vertical and (//) horizontal, each being made up of two, parts, one fixed and the other movable, viz., the/z/j and the rudder ; and the tail, plane (or stabilizer) and the elevator respectively., It is the fixed or the front part of the vertical surface of, (a) The Fin., the tail unit and takes the form of a vertical plate, arranged at a small angle with, , RUQDEP, , ELEVATOR, , LEFT, WING, Fig. 236., the central line, , of the fuselage or the body of the, , 'plane, (Fig. 236)., , Its, , function, , is, , *This is the most usual position of the propeller in most 'planes, such, to the tractor type, because of their being pulled through ail, planes belonging, by the action of the propeller. In what are called the pusher type of 'planes, the, propeller is carried behind the line of the wings, so that it exerts a pushing action, OB them.
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PROPERTIES OF MATTER, , 474, , to give directional stability lo the 'plane, (very much in the manner of the featheis, and, tipping a dart), making for its straight-line flight in the horizontal plane, it to, tending to bring it back to its original course, should a disturbance cause, turn to one side or the other., be thrown to the left, the fin will deflect the air to the, Thus, if the, 'plane, , which would then push, course and vice versa., , right,, , it, , back to the, , left,, , to resume, , its, , original straight, , The student may perhaps wonder as to why the/w, with its avowed purpose, of keeping the 'plane along its straight-line course, should be offset a few, degrees from the central line of the fuselage. The reason is that the air stream,, blown back by the propeller, (and called the slip stream), shares with the latter, its corkscrew motion and would strike the fin at an angle, were it set along the, central line, producing precisely the opposite of the desired effect ; for, it would, The, result in turning the plane rather than keeping it along its straight course., small inclination of the fin to the central line just counteracts this turning effect due, to the slip stream., It is the rear portion of the vertical surface, (Fig. 236),, (b) The Rudder., hinged on to the front portion or the fin, and has freedom of lateral movement in, the vertical plane., Its function is very much similar to the rudder on a boat and, it enables the 'plane, in level flight, to be steered to the right or to the left ID, the horizontal plane., Connected by means of cables to the rudder bar, pivoted horizontally OD, a central vertical pin in the cockpit*, it is operated by the pressure of the pilot's, a pressure with the right foot (i.e., on the right-hand end ol, feet, (see Fig. 236), the bar) makes it swing out of the central line and turns the plane to the right,, and a pressure with the left foot similarly turns the plane to the left., (c) The Tail Plane or the Stabilizer. This is the fixed pat t of the horllontal surface of the tail unit, (Fig. 236), and its function is identical with that, of the fin, but in the up and down direction, i.e., it serves to give the airplam, Mobility in the vertical plane, or the 'fore and aft* stability, as it is called., , (d) The Elevator. It is the movable part of the horizontal surface, of the tail unit and controls the vertical motion of the 'plane, i.e.,, its, , climbing and gliding movements., Lying normally in level with the, , tail plane or the stabilizer, it, controlled by the central column, or the, central stick, (or, simply the stick, as it is usually called), which is, connected to it by cables and is arranged conveniently in front of the, backward or inward pull on the, pilot'i seat, (see Fig. 236, above)., stick raises the elevator up above the level of the tail plane and the, air, rushing over the surface of the plane, strikes against it, tending, to blow it down to its original position, in level with the tail plane,, , up and down movement, , is, , A, , thus exerting a downward pressure on the tail of the 'plane, as a, whole, with the result that its nose is pushed upwards and it climbs, up., Similarly, a forward or outward push on the stick lowers the, elevator below the level of the tail plan and the air thrust on it now, pushes the tail up, which is the same thing as pushing the nose down,, and the 'plane, therefore, now glides down., These are hinged flaps, free to move up and down ai, (/v) The Ailerons., the rear or the trailing edges of the two wings, extending from the tip of eacl, wing to almost its mid-point, (Big. 236), their up and down movement being con, trolled by the side-ways pull on the stick-f, to which they are connected by meani, , *The Cockpit is a closed or open well, in the front portion of the aer*, plane in which the pilot takes his seat, (Fig. 236), with different controls and 10, it rumen ts arranged in front of him ., fin the larger type of aircraft, the aileron is controlled not by the sticl, but by what looks like an incomplete steering wheel of a motor ear, fitted on t<, 4he top of the stick.
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FLYING MACHINES, , 375, , of cables., The arrangement is such that as the stick is pulled to one side, it, simultaneously causes one aileron to be raised above, and the other to be lower*, ed below, the undersurface of the corresponding wing, with the result that the, lift on one, wing increases and that on the other decreases, making the plane, 'bank' or heel over to one side,, a pull on the stick towards the left making the, 'plane bank to the left and a pu)l towards the right, making it bank to the right., It will thus be seen that the stick and the rudder bar, between themselves, either singly or in combination with each other enable the 'plane to bf, manoeuvred into any desired position and to perform all sorts of aerobatics., , The Tail Trim. If an airplane continues to fly level, even when the, hold on the stick for a while, it is said to be 'flying trimmed *, This ideal state of affairs may however be easily disturbed by the entry or exit, of a passenger or two, the plane becoming 'nose heavy' or 'tail heavy' and thui, starting to fall down or to rise up. This puts an undue strain on tbe pilot, always alert to exert an inward or an outward pull on th;s stick., The tail trim is just the device to prevent all this and to enable the 'plane, to fly trimmed even with different loads in it, by automatically adjusting the inward or outward pressure on the stick, to suit the load. Of immense help to the, pilot during 'take offs' and 'landings', it just consists of a lever on one side of the, cock pit which, working on a quadrant, suitably alters the tension of a spring, Attached to the lower end of the stick*, always exerting the requisite pull, com(v), , 9, , pilot releases his, , mensurate with the load in the 'plane., It is that part of the airplane, (vi) The Undercarriage or the Chassis., behind the engine and at the base of the fuselage, which serves as a carriage for, the 'plane to run on the ground and includes the wheels and a shock-absorbing, mechanism (the oleomechanism) to take up the unavoidable impact on landing or, the bumps on uneven ground, which may otherwise severely strain the fuselage, even to the extent of damaging it., To minimise the air resistance to the flight of the airplane, the undercarriage is now almost universally made retractable (except perhaps in the case, of very small aircraft) ; so that, it can be drawn up into the fuselage once the, 'plane is up in the air, and lowered again when about to land, there being a, case his, suitable device to warn the pilot in time, when preparing to land,, undercarriage remains retracted., The undercarriage is supported on twof wheels (ex(v//) The Wheels., cluding the one at the tail end), fitted with wide-track pneumatic rubber tyres, inflated at low pressure., These, besides enabling the 'plane to run on the ground, before a take off also absorb part of the shock of impact, on landing, passing on, , m, , the rest to the oleomechanism., , In modern aircraft, we have also wheel brakes fitted more or less in the, manner of our motor car brakes, which (a) keep the plane stationary during the, running of the engine on the ground and (b) also shorten its run on landing. In, addition, they enable more pressure to be applied to one wheel than to the, ther, thereby greatly facilitating the steering and the manoeuvring of the plane,, while still on the ground., The rear of an airplane is supported either on a, (v///) The Tail Skid., small wheel or a spar-like structure, called the (ail skid- When the two front, wheels and this spar, or small wheel, touch the ground simultaneously on land-, , supposed to have made a perfect 'three point landing'., Slot., Oftentimes, when an aeroplane climbs too steeply, or, when it is about to land, and in fact, when for any reason, the speed of the, 'plane falls below a certain minimum, the lift on the wings becomes insufficient, Not only, to keep the 'plane flying and there 15 every possibility of its 'stalling'., that, but with an insufficient air-fbw, the other controls, and particularly the, ailerons cease to function properly and the 'plane starts dropping in a dive., ing, the plane is, a, , (tx), , The, , ^Sometimes, the lever is replaced by a wheel, whose movement suitably, adjusts the position of the tail plane instead of acting on the stick., tin some cases, we have a three-wheeled or a 'tricycle' undercarriage, the, third wheel being arranged well ahead of the other two. This not only prevents, the 'plane tipping on its nose, thus greatly reducing the possibility of accidents, on landing or manoeuvring the plane on the ground, but also greatly simplifies, both take-offs and landings.
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PROPERTIES OF MATTER, , "376, , An, , ingenious saftcy device, known as the Handley Page Slot, or, simply, therefore, used to avert this danger of a 'stair. It is just a small gap, between the upper surface of the wing and another miniature wing-like structure, 9, the 'slat arranged over its leading edge*., Without the slat, if the airplane were to stall, the air-flow ceases along, the upper surface of the wing and breaks up to form a series of eddies, as shown, , the Slot,, , is,, , THE SLOT, , (b), , (a), , Fig. 237., , thus depriving the 'plane of about 60% of its lifting power, the slat is fitted to the 'plane, it opens up as shown in Fig. 237, ait, (6), and forms t small passage or slot between itself and the wings and the, stream is directed through it on to the wing surface, instead of breaking up into, eddies. The lift on the wing is thus avoided and the danger of a stall averted., In Fig., , 237, , a),, , When, however,, , If, however, the wing be tilted too steeply, a stall may eventually occur,, but the 'plane recovers from it much sooner than would be possible without the, , slat., , Among these, the main 01, (x) Engine Controls and Other Instruments., the important ones are the following, The Throttle. This corresponds to the accelerator of the car and controls, the speed of the 'plane. Operated by the throttle lever on one side of the cockpit, it differs from the car accelerator in that it stays in the position in which it, is set, without spring ng back when the pressure is released on it, thus enabling, 'plane to fly at the desired constant speed. There is no gear changing or slowing, down for negotiating corners, for which, indeed, it must fly a little faster., :, , As its very name indicates, it is an instrument to, (xi) The Altimeter., indicate the altitude of the airplane. It is, in fact, a modification of the aneroid, barometer and is calibrated to indicate height or altitude in terms of 'thousands', of feet. Thus, if the pointer be at 5, it indicates a height of 5000 ft and so on., Since, ho ^ever, the altimeter really measures variations of pressure at ground, level, which can occur due to changes of weather, it may indicate different height, even at one fixed point on the ground, and its readings may thus be highly misleading and may prove dangerous. To obviate this risk, therefore, it is so, arranged that the pilot sets it at zero altitude before taking off, so that its readings later indicate the heights above this starting point, and not the absolute height, above the ground at any given moment. Thus, even if it indicates a height of 5000, //., it may well be within a couple of hundred feet from a mountain top., Improved instruments to indicate the absolute height of the 'plane above the, ground at a given moment (instead of from the starting point) are however well, in the offing and would greatly reduce the hazard of an airplane flight in fogg>, weather., This enables the pilot to feel the, (*//) The 'Engine' Revolution Counter, of the engine, as it were, telling him all about the condition of the engine,, pulse, Including its undue vibrations and uneven running etc. Further, should there be, an unexpected or unaccountable drop in the revolutions of the engine, it is a, warning to the pilot that trouble is jmminent. The revolutions are measured in, terms of hundreds per minute., , The Oil Pressure Gauge. It is a small but vitally important instru, indicates the pressure (in pounds) under which the oil is pumped round, to the different parts of the aeroengine,, an operation about just as essential to, (xiii), , ment and, , *Sometimes the slot is also arranged close to the aileron flap, when, helps to maintain the requisite air flow over the aileron surface, thus enabling, to function effectively even at low speeds of the 'plane., , it, it
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FLYINQ MACHINES, , 377, , Us, , life as the blood supply to ths various parts of our body., this pressure forewarns the pilot of a coming serious trouble, take remedial measures in time., , is, a, machine., , It, , 2., , It, , 1, , 'lighter-than-air, , 1., , flying, , on the principle of, and its lifting power is, provided by the buoyancy of the air, displaced by it., is, , and, , drop, , in, , him, , to, , alerts, , Airplane, , Airship, , I., , A sudden, , is, a, It, machine., , 'heavier-than-air', , flying, , the, lifting power is due, thrust, produced by a, created wind, strong artificially, and the characteristic shape of its, , based, , Here,, , floatation, , the, , to, , wings., It rises vertically, , upwards, directly, , from the ground., , !*, , 3., , I, , must first be made to, on the ground before it can, It, , run, 'take, , off*., , 4., , It is, , very, , much, , bigger in size than, , an airplane., , 4., , It, , is, , comparatively, , smaller, , ID, , size., , though the air-ship arid the airplane are based on entirely, different principles, they have in common (/) an upward motion against, the action of gravity and (//) propulsion through air., Thus,, , We, , are all familiar with the meaning of the, 167. Jet Propulsion,, which is just the term applied to a high velocity stream oj, fluid (liquid or gas) issuing out of a nozzle, as for example, a 'jet of, water' or a 'jet of steam' etc., And, therefore, jet-propulsion is, obviously the method of driving or propelling a body or a machine, forwards through the agency of a jet, the body or the machine thus, driven being said to be jet propelled., , word, , 'jet*, , That a jet possesses such a motive or tractive force can be, easily seen from a number of facts of every day life, if only we care, to stop a while and analyse them., Thus, for example, when a bullet, is, , forced out of the barrel of a, , rifle by the exploding mixture of gases, the rifle suddenly moves or 'kicks' back in a direction oppoto that of the bullet and the exploded gases., , inside, site, , it,, , So that, if we continuously fire a rifle fastened to the rear of a, boat, with its barrel facing outwards, we shall find that the boat, continues to move forwards with a jerky motion so long as the firing, each bullet fired producing a push forwards. We, therefore,, continues,, In fact, even when we ply the oais,, the action is similar. For, what we do is simply to push some volume, of water backwards and the boat, as a consequence, moves forwards,, , have here a jet-propelled boat, , !, , Indeed, if we did nothing else but simply sit quietly in the boat, and throw stones into the water, with our face towards the stern of, the boat, the boat will still move forwards, direction in which the stones are thrown)., , (i.e.,, , opposite to the, , All these examples are, as the student is no doubt already, aware, a consequence of the well known Newton's third law of motion,, according to which action and reaction are equal and opposite, or what, follows from it, viz., the Jaw of conservation of momentum, which
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PEOPERTIES OF MATTBB, , 378, , lost with the momentum, cases, in, the, So, the, above, moves oppositely as a, gained., boat,, that,, result of the reaction to the motion of the bullet (and the gases) 01, , demands the equality of the momentum, , the stones, or because the momentum lost by the bullets or the stones, is, equal to the momentum gained by the boat. A force such as the one, experienced by the boat is called the reactive force and, in the case, of a jet, sometimes alo the jet-force., ;, , Now, does it surprise the student when he is told that even the, usual type of airplane, in which we use the ordinary reciprocating, for(i.e., the piston- type) engine makes use of a jet for its propulsion, wards* ? For, the propeller, as it whirls round at a high speed,, throws a jet of air (or in the case of a ship, a jet of water) backwards,, as a reaction to which the plane (or the ship) is pushed forwards, The question,, against the viscous resistance of tiie air (or water)., therefore, naturally arises as to why then do we not call them jetpropelled planes. The answer is that, technically speaking, the, narrower the cross-section of the high- velocity fluid stream, the more, nearly does it come up to the definition of a jet, and the term jetpropelled planes is, therefore, reserved for planes in which the jet is a, narrow one, about one foot in diameter, as compared with ten feet of, more in the ca.se of the ordinary airplane., , Again, it must not be inferred from wheat has been said above, that a jet must necessarily consist of hot gases., No, it may just as, well be of cold air, as in the case of what are called the ducted-fan, type of planesf, or as was the case with perhaps the earliest jetthe jet in this, in Italy,, propelled plane, constructed by, , CampM, , latter case, being, , produced by a compressor, driven by the ordinary, , reciprocating type of engine., 168. Thrust supplied by the jet. Let us now calculate the thrust, supplied to an aircraft by the jet produced by the power unit, inside it., , Suppose we have an aircraft travelling with a speed V and fitted, with a power-unit which produces a jet of fluid, of velocity w, relative, to the aircraft, where u is higher than F,, the velocity u of the jet, a, little, measured, a, in, it, at, being, away from the nozzle, where, point, the static pressure is the same as that in the surrounding air. Then,, a velocity V on the aircraft in the opposite direction to, own, the aircraft comes to rest, with the air streaming past it with, velocity V. So that, if a be the area of cross-section of the jet at the, point where its velocity is u, the volume of the fluid flowing per second, if, , we impose, , its, , In the jet is clearly a.u., , If,, , therefore, p be the density of the fluid,, , have, mass-flow of the fluid per second in the jet =a.u.p, , =, , we, , m, say, , And, therefore, momentum of this fluid in the jet = m.u., If this mass (m) of the fluid finally emerges out from the aircraft, *The same being the case with a ship., tin these planes, air is sucked in through two holes or ducts, by two fans,, the latter thrusting the air away with considerrotating in opposite directions,, This sucking action also helps to buoy, able force, propelling the plane forward., the plane up. Further, the gyroscopic effect, produced by the oppositely rotatini, fans greatly helps in enhancing and assuring the stability of the plane.
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PLYING MAOHINBS, , 3713, , ., , with a velocity F, its momentum is clearly reduced to m.V. It thus, a loss of momentum, equal to (mu mV) or m(uV)pcr second, I.e., its rate of change of momentum, m(uV)., And this, therefore, in accordance with Newton's second law oj, motion, must be the force, or the thrust, F supplied to the aircraft bj, the jet in the direction opposite., So that,, , suffers, , t, , =, , F = m(uV) =, , 0wp, , (ti, , V), , 169. Efficiency of the jet., If we consider the exact state oj, affairs in the case above, viz., that the aircraft is really not at rest, as, fre had, imagined, but is moving with velocity V, then, the velocity ot, the final jet, moving in the opposite direction clearly becomes (w, V), , and, therefore,, , = m, , =, , K.E. of the final jet, \, (u-V)*, \F(u-V) [-.- m(u-~V)-P}, Also, the aircraft does FV amount of work per second against, ihe air resistance'as it moves forwards. So that,, , must be supplied by the jet propulsion unit,, by the power-unit) = FV+\F(u V)., Of this, obviously, the portion usefully employed is only FV, the, being simply a waste, creating a disturbance behind the aircraft., total energy that, (i.e.,, , rest, , So that,, efficiency, , of the, , jet, or, , the Froude efficiency, as, , _, , converted into useful work, ~~ energy, ~~, total energy supplied, , ., , _, , ~, , it is, , FV, , commonly, , _, , FV~+\F(u^V)', , 2V, , u+V, , ', , Note. Clearly, the efficiency will have the maximum value 1, when, /.., when the initial jet velocity is equal to the flight velocity of the aircraft, for, then, the energy wasted in the form of K.E. of the final jet [\F(u, V)], will also become zero., But, then, the thrust on the aircraft [m(u-V)} will also, become zero., condition of maximum efficiency is, therefore, not a practia, , V, , y, , .This, cable proposition, just as, , it is not in any other type of machine also., 170. Effect of smaller cross-section of the jet. As indicated earlier,, the cross-section of the jet in a jet-propelled plane should be narrow,, Let us see what advantage is to bs gained by it., Apparently, from the relation F, m(u V) for the thrust, supplied to the aircraft by the jet, we find that a reduction in its, cross-section will mean a diminution in the value of the mass flow of, the fluid, m, so that, to obtain the same thrust F, as before, (u V), , =, , will have to be correspondingly greater., This will naturally mean a, higher value of \(u, K)*. the K.E. of the final jet, which, as we have, seen, is a mere waste of energy. Not only that, but, as a natural, oonsequence, the efficiency of the jet FF/FK+|F(wF), will also fall, below its previous value. It would thus appear that a decrease in the, cross-section of the jet, far from improving matters, does just, the reverse, viz., increases the loss of energy and decreases the, In what manner, then, is jet-propulsion a, fficiency for propulsion,, better mode of propulsion ?, , The answer, (i), , Initially,, , is, , manifold, , when, , bout the year 1940,, , it, , :, , jet-propulsion was just introduced round, was intended to render auxiliary support to
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PROPERTIES OF MATTER, , 380, , the then prevalent gas turbine engine. The materials of the gas, turbine could not function satisfactorily at the temperatures obtaining, in the earlier reciprocating type of engines and the products of combustion required to be diluted with a large excess of air. This seeming, difficulty was actually turned into an advantage by the enginedesigners, who used this necessary excess of air as a narrow jet, to supply the entire thrust required to be given to the aircraft, thus, eliminating the necessity of the propeller and quite a few other, accessories., The jet was made to escape through a small turbine, which then supplied the necessary power to the generator, the fuel, pumps and the compressor etc. Thus, although the introduction of the, jet inevitably entailed a loss in efficiency, with the fuel -consumption, rate rising higher, it gave the distinct advantage of reducing the weight, of the whole unit for the same value of power. In view of this smaller, weight but higher rate of fuel-consumption, the turbo-jet engineg, ae, these engines were aptly -christened, came to be considered more, suitable for flights of shorter durations, say, of less than 2 hours in, those early days when the highest speed was only 400 miles per hour., of a narrow jet is, (//) It was found that although the efficiency, rather low at moderate flight speeds, it increases rapidly with the, In fact, if we take into consideration also the other, flight speed., that, advantages, go with higli speed, (e.g., assistance given to, the compress ion- process in the engine, etc.), the over-all result is that, the po^er output (FF) increases directly with flight speed with only a, comparatively very small increase in fuel consumption, i.e., FV oc V., Clearly, therefore, F remains practically constant for varying flight, epeeds.*, , This linear increase in power (FK) with speed (K), with practically a constant fuel-con sumption rate, necessarily implies that if the, flight speed bo high, the turbo-jet unit will also be about as economical as the ordinary propeller-engine and will, in addition, possess the, , advantage of (a) having less weight and (b) capacity of packing large, power in a smaller space., In fact, both the turbo-jet and the propeller engine will have, the same efficiency, i.e., their power output for the same fuelconsumption will be the same, at a speed of 700 m.p.h., provided the, propeller engine had a constant power-output upto this speed. And this, is the point where the jet-unit scores over the propeller unit. For, the, power output of the propeller engine does not really remain constant, with speed but falls steeply as the flight speed approaches the speed, of sound, v/j., 762 m.p.h. at ground level and 660 m.p.h. at altitudes, above 3600 ft. This is so, because a propeller may be regarded, essentially as a wing, with the difference that whereas the latter provides a lifting force to the aircraft against the force of gravity,, the former supplies a similar force in the form of a thrust in the direction of its motion, for which purpose it is rotated in a plane perpendicular to the direction of flight,, the lifting force in the case of, the wing and the forward thrust in the case of the propeller being, always roughly perpendicular to the direction of their respective, motions through air, both experiencing an air-resistance or 'drag*, and not, , "That is why, in the case of a turbo-jet unit, only, its power (FV)., , its thrust, , (F) is indicated
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FLYING MACHINES, , 381, , opposing their motion, with some power used up in overcoming the, same.* So that, despite all improvements made in the designsof propellers (such as making their sections near the tips very thin,, etc.) their propulsive power falls greatly at high flight speeds, whereaa, that of a jet-unit rises equally greatly. This comes about because the, actual velocity of the propeller blade is the resultant of its velocity of, rotation and translation and, as such, is higher than the flight speed, of the aircraft itself, even a non-rotating wing experiencing a large, increase in the drag on it much before the speed of sound is attained., Thus, from the point of view of over-all efficiency, a jet unit is, certainly superior to a propeller- unit at speeds of 600 tn.p.h. and, above., , Then, again, another advantage that a jet unit possesses over, the propeller- unit is that there being no accessories and profcubrancea, like radiators, oil-cookers etc., the drag is comparatively less., And, the absence of the propeller which makes for a smoother flow over the, entire surface of the plane, cuts down the drag over these surfaces by, as much as 20 to 30 per cent., 171. Rocket Planes. The small fire-work rockets, rising pretty, high up in the air, to the amusement of on-lookers, are a common enough, sight everywhere and they, are obviously jet-propelled, on their own small scale,, rocket plane is merely a large, scale version of the same, , A, , phenomenon., , PROPELLED,, , NOZZLE, , It possesses a, , higher speed and can rise to, a much greater height than, , even a turbo-jet plane., fact,, OXIDI-, , it is, , '(HOTGASES), , In, , a turbo-jet plane,, , (0, in, , r, , SER foil, , which the technique, , jet- propulsion, stage further,, , <s, JTmtm, , (COMBUSTIONJ, ?, , X, , CHAMBER!, , t/F, , ,, , ., , Fs, , ', , 2 8, , PROPELLER, NOZZLE, , ., , (HOT6ASZS), , is, , carried, , with the, , of, , a, jet, , still, , narrower in cross-section, , and, , its, , velocity higher., this difference in, degree, then, the only factor, that distinguishes it from a, Of course not;, jet-plane?, for, the essential difference, of the jet. In a jetIs, , between the two lie* in the method of production, is carried on the aircraft, with the oxygen, propulsion unit, the fuel alone, drawn from the surrounding air, [Fig,, combustion, its, being, necessary for, 238 (/)] only a fraction of which is usually consumed, the rest, together, with the considerable larger quantity of nitrogen 'swallowed', merely, the jet-propulsion, serving to keep the temperature down throughout, , _____, , ~~, , ^^, , comprised iairki front of the aircraft is of little, moves away with the, consequence at speeds below that of sound, for it simply, that of sound,, loeed of sound. But when the speed of the aircraft is higher than, with the result, the condensed air in front can oniy move sideways but not forward, witn a, that the nose of the aircraft has to carry along a bulk of compressed air,, on, it., in, the, increase, drag, large, consequent, ^
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PROPERTIES OF MATTEB, , 382, , In the case of a rocket-propulsion unit, on the other hand,, the fuel as well as the oxidising agent, required for its combustion, are, together carried on the aircraft and no air has to be drawn in from tht, It is thus a self-contained unit in itself, ['Fig. 238 (//)]., surroundings., tinit., , This is a point of great importance in that it makes the working, of the motor quite independent of the presence or absence of any, surrounding air. Thus, whereas a jet plane can only attain a height, of 50 to 80 thousand feet up to which it can have its supply of, air from the surroundings, there is no such limit to the height, of a rocket-plane, which alone is capable of rising up to higher altitudes beyond the earth's atmosphere, where there is obviously no air, to be drawn in. So that, its being self-contained, with its own supply, of the oxidising agent, while it may be a comparative disadvantage at, lower altitudes, is clearly a tremendous advantage at higher altitudes, Calculating the thrust (F) given to the aircraft by the issuing, jet in the same manner as in the case of the jet propulsion unit, 168, page 378), we have Fin.u, where m is the mass-flow of the flui<3, (, per second in the jet and u, its final velocity, its initial velocity hert, being zero, since all the constituents are carried on the aircraft itself., 2, Its efficiency thus works out to 2Fw/(w 2, which again, as in the, ),, V., ase of jet propulsion, will have the maximum value 1, when u, , +F, , =, , Further, the power-unit, in the case of the rocket plane, is also, much lighter. Thus, while, for supplying a thrust of 1 Ib. at ground, level, or of 0-2 Ib. at an altitude of 50,000 ft. a turbo-jet unit weighs, about 0*3 Ib., a rocket-unit, weighing only 0-1 Ib. can supply the, same thrust of 1 Ib. at all altitudes. It follows, therefore, that a, rocket plane is the more suitable for use only for flights of short, duration or at very high speeds., , The rocket-unit, in which both the fuel and the, amount, of, oxygen for its consumption are carried on the, required, aircraft itself, is much simpler than that other one which requires the, 172. Rocket Fuel., , compression of large quantities of air. For in this case, the only, problems with which we are concerned are those of the combustion, chamber and the propelling jet., , The oxygen may be carried either in the liquid form, or in the, form of oxidisers rich in oxygen, like hydrogen peroxide, (H2 O t ) or, nitric acid (HNO Z ). In the latter case, the remaining part of the oxidiser, going into the propelling jet, merely serves to cool the jet and the, combustion chamber., , Now,, , if the fuel, , system, , contains, , own, , its, , and the, , oxidiser are carried in separate conas a bipropellant rocket but if the fuel, oxidiser with it and is carried in a simple container,, , tainers, the, , is, , known, , ',, , is broughl, It is known as a monopropellant and its decomposition, about either by the application of heat or through the agency of, Obviously, a monopropellant must be some sort of an ex, catalyst., Quite a commoi!, plosive and, therefore, requires careful handling., one being hydrogen peroxide, which decomposes as shown by thi, , equation, , 2H2Oa, , 2HiO+O a +69Q C.H.U., , *1 C.H.U. (Centigrade heat unit), tore of, , 1 Ib., , of water through re., , is, , Ib*, , the heat required to raise the tempera
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FLYING MACHINES, , 383, , The products of decomposition of a monopropellant substance, are sometimes themselves rich in oxygen, as we can see in this case, of /JgOj, and the substance can, therefore, also be used as one of the, components of a bipropellant. Thus, for example, H^O^ used with, methyl alcohol (CH$OH) would react as shown, :, , releasing heat at a much higher rate and hence resulting in a much, higher exit velocity of the gases through the exhaust nozzle or the, *enturi, as it is called, and consequently a much higher thrust*, , The propellant may be injected into the combustion chamber in, (/) by exerting pressure by a compressed gas, like, nitrogen or air on the propellant tanks or (') by means of a pumping mechanism, usually a turbine. The former method admits of no, one of the two ways, , :, , variation or control of thrust and is, therefore, suitable only for, short-duration flights or remotely controlled missiles, and the latter, Is the one, commonly used for rocket-propelled air-crafts., 173. Specific Impulse., The performance of a rocket motor is, measured in terms of what is called the specific impulse or the specific, pull, /, which is the thrust generated by unit rate offuel-consumption, i.e.,, , _ F, , thrust (Ibs,), , ~~, rate, , offuel consumption, , ~~, (Ibs. /sec.), , nig', , So that, the dimensions of / are the same as those of time. Phytime for which a unit thrust can be generated, by a unit weight offuel., Now, as we have seen, the thrust in the case of a rocket is, equal to mu, where m is the mass-flow through the nozzle and u, the, exhaust velocity of the gases. So that,, , sically, therefore, it is the, , /, , =, , mujmg, , =, , ujg., , And, therefore, the higher the jet-velocity, the higher the specific impulse and the smaller the fuel-consumption for a given thrust., Besides fuel-consumption and thrust, there are quite or few, other factors which determine the suitability of various fuels, e.g., the, weight of the engine, the temperature in the combustion chamber etc.,, In the modern rocket motors, the total weight of the pump,, etc., control and installation etc. must be about one-tenth of the maximum thrust developed. In short, the performance of a rocket depends, chiefly upon three factors, (/) jet velocity, (ii) density of the propellant, and (Hi) weight of the power plant, which includes that of the propeliant tanks and the fuel-supply system etc., into details of which we, need not enter in an elementary discussion of the type we are concerned with here., 174. Shape of the Rocket. During an upward flight, particularly,, through the denser layers of the atmosphere, the components of the, rocket are subjected to intense air pressure, and also a lot of heat, Both these factors, is produced due to viscous friction of the air., are taken into account while designing a rocket. Its frame is accordingly made of a heat-resisting material and its velocity during the, first part of its flight, through the denser layers of the air, kept suffiFurther, it is so designed as to reduce the air pressure, eiently low.
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PBOPBETIBS OF MATTBB, , 384, , each individual part to the very minimum, its over-all shape, or less like that of a cigar., more, being, 175. The Multi-stage Rocket. If a rocket is hurled into space beearth's gravitational field then, supposing that its acceleration, the, yond, takes place in tho latter region, where the value of g is 32 ft. /sec*.,, the velocity F that it must acquire to escape from the earth's gravias it is called, is given by the, tational field or the 'escape velocity, of, 2 =, relation V, 92,, page 251) from which the value, 2MG/P, (see, s, K works out to about lM9x 10 cms. [sec. or about 36000 ft.jsec., , on, , its, , 7, , ,, , Now, at the present stage of rocket development, no single rocket can achieve this velocity. To tide over the difficulty, therefore,, we make use of what is called a ww///-, , -3RD STAGE, , ENGINE Of, STAGE, , 3RD., , ROCKET, , 2ND., , stage rocket, which is just a combination, of rockets, either (i) joined consecutively, or in series, as it were, or (//') one inside, the other or (///) with the rear port oj, one inside the nozzle of the other, as indicated diagrammatically in Fig. 239. In, , these three types, the first stage, rocket is the largest in both dimensions and weight, and the last stage, one, the smallest., all, , Naturally, the, , STAGE, it, , first, , stage rocket, , is, , and when it has done its job,, and IB discarded, with, detached, gets, , used, , first, , the second stage rocket taking over the, task of producing further acceleration., ENGINE Of, Then, this too is discarded and the, 2ND. STAGE, third stage rocket takes over and so on., ROCKET, The velocity thus goes on increasing at, each stage by the same amount as it, in a single stage rocket and each, does, 1ST. STAGE, stage has its own propulsion and control system., Obviously enough, the, ENGINE OF fuel-consumption and the thrust for, 1ST. STAGE, the first stage rocket are the highest, XOCKET, of all, say about a hundred times the, corresponding values for the third stage, and the fuel-stock too in the, rocket, that, first stage is about sixty times, in the third stage, the same being the, ;, , Fig. 239., , ratio of the total weights carried by, the former to that by the latter., , Considerations of both weight and cost demand that the number, of stages should not be large and that, therefore, the pay-load of, each stage (which includes, in addition to the useful load of the, final stage, the weights of the intervening stage rockets to be discarded later) be limited to about 20% of its own weight. Clearly, the, useful pay load of the final stage thus works out to be a very small, fraction of the initial over-all weight., Thus, for example, if there be, , n stages, , in, , all,, , this fraction is just l/5, , n, , of the, , initial total, , weight. Or,
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FLYING MACHINES, , 385, , to give a more concrete example, a space ship of the size of the wellknown V2. designed by Dr. Verner Von Braun, would be about just, sufficient to land a match box or a packet of cigarettes, by means of, a purachute, on the planet Mars., , Each individual rocket of the multi-stage, , rocket, has its, , own, , independent design and basic characteristics, with its function, These characteristics include the, correlated with those of the others., following, , :, , Net weight. The net weight for a single stage rocket includes also, the weight of the instruments and appliances or the weight of ammunition,, if any, etc. And, in the case of a multi-stage rocket, obviously, the total weight, of the second stage is the net weight of the first stage and the total weight, of the third stage, the net weight of the second and so on, the ratio between the, two being usually for 3 1 for each stage., Steering Equipment. This is necessary to steer the course of the, (ii), rocket during its flight during the other stages except the first which only serves, as a sort of runway for the rocket, as it were., (i), , :, , (///), , dual rockets,, , Design. This includes the frame of the rocket or of the indivithe case of a multi-stage rocket, with its fortifications and fasten-, , m, , ings etc., (iV), Rocket-length. This obviously means the height of the rocket or, that of the individual rockets of the multi-stage one. This is an important factor in as much as the very stability of the rocket in its trajectory depends upon the, ratio between its length and in mean diameter (ie. the mean diameter of the, whole rocket or of each one of the stage- rockets), y, , Number of Motors. Each stage rocket has its own separate motors., stage rocket, naturally, in view of the highest total weight it has to, carry and the greatest resistance of the lower denser layers of air it has to, overcome, has more thin one motor and the last stage rocket, because of its, lightest load and the least resistance to be overcome, is provided with only one, (r), , The, , first, , motor., Apar* from these, there are also other characteristics of a rocket, like, fuel-consumption, thrust, specific pull or impulse, time of combustion (in seconds),, acceleration, lift or range etc., \\.%, , 176. Take off of the rocket. This is perhaps the most important, part in the flight of a rocket and must be fully ensured to be correct., The slightest error in the timing or the accuracy of firing makes all, the difference between the rocket returning back in this generation or, the next or perhaps not at all., , Salvaging the various stage rockets. Let us wind up our, elementary study of a rocket flight with a word about salvaging the, various stage rockets which are discarded after they have performed, This problem cannot yet be said to have, their respective functions., , 177, , however, being made with, if they succeed,, it will mean a tremendous economy in cost., And for all one knows,, the ideal solution may turn out to be the utilisation of the material, of the stage used up as fuel for the next stage., been satisfactorily solved. Experiments, , are., , various systems of parachutes and other devices and,, , Satellites. Among celestial bodies, a satellite is what may, 178., be called a minor or a junior member of the solar system revolvinground one of the major planets in its own prescribed orbit. Till recently,, it was not thought possible that anything man-made could also be so, placed round the earth or any other major planet to revolve in a given, orbit. But, then, with the development of jet-propulsion (in the year, 1940), followed by that of high speed rockets, man began to dream of
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PBOPBBTIES OF MATTER, , 386, , flight into space an'd of inter-planetary travel,, , when,, , all, , of a sudden,, , on October 4, 1957, the Russian scientists made the whole world gasp, with wonder and surprise by launching their first 'sputnik' or artificial, This Sputnik /, of the form of a ball, 58 cms. in diameter, and weighing 83-6 kilogrammes (roughly 185 Ibs.) was placed into an, exactly like a celestial satellite, elliptical trajectory round the earth, satellite., , making, in its initial phase, one full revolution in 96 2 minutes and, attaining a speed of 8 km. or nearly 5 miles/'sec. at a distance of 950, km. from the earth., , The progress of this latest wonder was watched with dumb admiration by scientists all over the globe and the radio signals sent out, by it listened to attentively as long as its source of power lasted. It, existed as a satellite for full 58 days, during which it made 1400, revolutions of the earth, thus covering a distance of 39 million kiloIts existence, however, continued for 92 days and the entire, metres., distance covered by it totalled up to the enormous figure of 60 million, kilometres, when, finally, on January 4, 1958, it entered the denser, layers of the atmosphere and got burnt out due to the intense heat, produced by friction., This artificial satellite was obviously an automatic rocket, hurled, into its pre-determinod and well-calculated orbit by a multi-stage, rocket. Indeed, the rocket carrier too continued to revolve round the, earth at about the same height as the sputnik but at a distance of, about a thousand kilometres from it, and, then, while descending, of, the, denser, it also began to burn, the, layers, atmosphere,, through, out, with fragments from it falling somewhere in Alaska and North, America., After almost exactly a month, on November 3, 1957, the, Russians put their socond artificial satellite 'Sputnik IT into orbit, ;, , round the earth, containing, , scientific equipment for exploratory purposes, as well as the first space traveller, the dog 'Laika , in a sealed, The, cabin, which they successfully retrieved back, safe and sound., total weight of the Sputnik was this time much greater, being 508'3, kgms. or 1126 Ibs. (including the dog). Its distance from the earth, was al-o greater, 1700 kms. f its period of revolution, 102 sees.,, with the angle of tilt of its orbit roughly 65 from the equatorial, 9, , plane., first American artificial satellite, 'The, January 31, 1958, though of a comparatively much, smaller weight and size., These sensational events brought still more sensational and, breath-taking ones in thoir wake, with the Russians putting the first, cosmonaut of the immortalised name, Major Yuri Gagarin, into space, in a much larger space-vehicle or space-ship and retrieving him back,, with the Americans later repeating the performance. The race in still, on in right earnest and who knows what greater wonders yet are in, , This wis followed by the, , Explorer', on, , store for us., , Let us try to understand the basic principles underlying this, , phenomenon., 179. Conditions for a satellite to be placed in orbit. It is obvious, artificial satellite goes round the earth exactly as a celestial, , that an
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387, , SATELLITES, , satellite goes round a planet, as the moon, which, for all practical purposes, is a satellite of the earth, goes round it, or as the earth and the, , other planets go round the sun, i.e., in accordance with the laws, first, enunciated by Kepler, leading to Newton's celebrated Law of Gravitation, which forms the basis of the entire celestial mechanics., The student is quite familiar with the whirling motion of, -a stone, tied to one end of a string, the other end of which is held in, the hand. Precisely similar is the case with a planet going round the, sun or an artificial satellite going round the earfch, with the force of, gravitational attraction replacing the tension in the string. There is,, however, one fundamental difference between the two, viz., that whereas the tension in the string is, within limits, a variable quantiof the stone, the attractive, ty, permitting a lower or a higher velocity, a, in, onihe, the, satellite, earth, specific, quantity and thus perforce of, mils only a specific velocity for the satellite, if it is to remain in orbit,, this velocity for a satellite close to the earth being, as mentioned already, about 8 kms. or 5 miles per second. Since, however, the, gravitational force decreases with increase of distance from the centre, of the earth, a satellite further away from the earth will need, a smaller velocity to remain in its orbit than the one nearer to the, earth, though up to about a 1000 kms. above the earth's surface, this, reduction in velocity is only nominal. This is clear from the fact that, the moon, which is roughly 38000 kms. away from the earth and,, therefore, moves in a much larger orbit, has only a velocity of about, 1 km. /sec., which is about one-eighth of a satellite close to tho earth, so that, whereas the moon makes only one revolution of the earth in, one month, the satellite makes as many as 15 revolutions in one day., Now, the question is how to have the satellite with such a high, into arbit around it., velocity away from the earth, to enable it to go, As can be seen, not only has the opposing gravitational force to be, overcome but also the very considerable air resistance, particularly, in the lower denser part of it. As we have seen above, the least velo5 miles j'sec. called the first, city for the purpose is about 8 km. or, cosmic velocity., But, if the velocity rises to about 11-2 km. /sec.,, called the second cosmic velocity or the velocity of escape, the satellite, field and flies away into, passes right out of the earth's gravitational, the cosmos, within the range of the solar system., This formidable problem, can, as mentioned earlier, be easily, solved by carrying the satellite on a multi-stage rocket, for no single, rocket can possibly (at any rate, not yet) achieve the requisite veloWe have already discussed the essential, city all by itself alone., 175. Let us now see how exactly to, in, rocket, a, features of such, launch the rocket, carrying the satellite, into the required orbit., Apparently, the shortest route, 180. Launching of the Satellite., the, from, launching base to its assigned orbit, for the satellite to take, would be the vertical one. This, however, is not feasible in actual, reason that the gravitational pull of the earth, practice, for the simple, to its motion and counthe, in, will then be, directly 'opposite direction, it can gather theteract the pull of the engines. So that, before, fuel -stock may get exhausted, resulting in, limited, its, necessary speed,, down. Vertical, its first coming to a stop and then starting falling, a, not, practicable propositian., launching of the satellite is, therefore,, ,, , ,
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PROPERTIES OF MATTEB, , 388, , To ensure that the satellite does not fall back to the earth, it is, Its upward flight, essential to give it a sufficient horizontal velocity., is, therefore, so arranged that it is brought into its orbit in the, shortest possible time, acquiring meanwhile the requisite horizontal, velocity. It is thus clear that the particular trajectory that will take, the satellite to its assigned orbit has first to be most carefully, calculated., It is usual to arrange the first portion of the flight of the rocket, to be vertical, so that it may pass through the first 20 kms. of the denser portion of the atmosphere the earliest. Thereafter, as it enters the, , rarefied portions of the atmosphere, it is given a gradual tilt by means, of a mechanical pilot, so that it emerges into its orbit with a horizontal, velocity large enough for the centrifugal force coming into play on it,, (on account of its circular motion), to just balance the force due to the, gravitational pull. And, the trajectory of its path is so chosen that the, loss of velocity entailed, due to air-resistance and the earth's pull, is, a small percentage of its required or characteristic velocity. In fact,, to make up for this loss, the actual velocity given to it is a little, higher than the computed value of its characteristic velocity. When, launched laterally to the earth's rotation, however, an increase in its, velocity is automatically obtained at the expense of the velocity of, the earth's rotation, depending upon the latitude of the launching, site. Thus, for example, this increase is the maximum at the equator,, being as much as 400 met res /sec., which is higher than that of the, , fastest fighter planes of the day., , If it be desired to give the satellite an elliptical orbit, instead, of a circular one, the rocket carrying it must either be given a higher, velocity than the perepheral on$ or its velocity, immediately alter, completion of the motor's performance, must not be directed along the, tangent to the circular orbit. In the elliptical orbit, the point nearest to the earth is called 'per the' and the farthest from it, the 'acme'., And it is quite possible that the satellite at the former point may be, nearer to, and at the latter, farther from, the earth than at any, , point in, , its circular, , path., , In any case, the accuracy demanded in the firing of the rocket, into its correct orbital path is really exacting. For, even an error of, 1% in the direction of velocity may produce a height variation of the, perihe and the acme which may be as much as 120 kms. or more. This, firing accuracy is secured by means of proper steering devices, directing the course of the rocket at every stage of its flight. And, clearly,, rudders of the type used in the ordinary jet air-craft, are hardly, suitable for the purpose, since they cannot possibly function equally, effectively both in the denser and the rarefied regions of the atmosphere., , The necessary steering control can, however, be effected in a, number of ways but the one usually resorted to is to so design the, rocket as to enable it to change the direction of the escaping jet by, a mere tilt of the longitudinal axis of its motor with respect to its, This is actually the device adopted in most of the present-day, long-range rockets., The manner in which the angle of inclination of the longitudi-, , .own.
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SATELLITES, axis of the rocket with the horizon is varied, will be clear from, As will be readily seen, the trajectory of the rocket from, Fig. 240., , BURN OUT OF, , IGNITION, , 2nd STAGE, , Jrtf, , STAGf, (10, , MINUTES, , Af TR L A UNCH/H6), OffBH OF SATELUT6, , SEPARATION oft SEPARATISTS, 2nd STAGE, \OF 3rd STA6f, , BURN OUT, AND, , -2SOOOft /Sec), , SEPARATION OF, STAGE ROCKET, , (2OO TO 400 MILES), , i, , Fig. 240., , very start until its longitudinal axis takes up the horizontal position, (/.., until its outward motion towards its orbit) is split up into, , its, , a number of stages, indicated by A 1, , A 2 h 3 etc., depending upon the, height of the orbit. The angles that its horizontal axis makes, with the horizon at each stage is carefully, calculated before hand and, the control instruments set accordingly, to ensure that the rocket, takes its assigned trajectory. And, this very setting of the instruments also regulates the fuel- sup ply in keeping with the predetermined requirement at the lime., ,, , ,, , Now, it will be easily understood that, while going round in it, allotted orbit, the Scitellite passes over different parts of the globe in, its successive rounds., For, by the time it has completed one round,, ihe earth has also rotated about its axis and hence, in its next, naturally passes over other parts that now fall below its, This will always be so except when the satellite goes round, an orbit coinciding with the equatorial plane, in which case, obviously,, it will always pass over the same parts or countries situated at the, equator. It docs not mean that we can launch the satellite in, Any orbit we choose. For, the orbit must be one such that its plane, , round,, , it, , orbit., , passes through the centre of the earth and, , depend upon the, , site, , it, , will,, , therefore,, , clearly, , of launching., , Not only that, but even the time of the day and the season at, time of launching matter a great deal. For, a satellite receives, energy direct from the sun through special type of solar batteries fitted into it, a particular side of which must all along be illuminated, by the sun. The satellite must, therefore, be launched in an orbit, the, plane of which is perpendicular to the rays of the sun, and this, is possible only at a, when, v/z.,, particular hour of the day,, the radius of the earth connecting the starting point of the satellite, with its centre is perpendicular to the sun's rays. And, the season is, important because, with the satellite launched in its orbit, as, explained, the earth which also moves round the sun, comes in-between, it and the sun at a particular time, thus, preventing the rays of the sun, from reaching it. Account has, therefore, to be taken of this occurrence and the season of launching chosen such that the satellite, 'the
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390, , PBOPBBTI1S OF MATTER, , can get the maximum time to store up enough energy from the, sun to suffice for the period when the sun will remain hidden from it, later during its flight., And, finally, it must also be clearly understood that in view, of the uneven distribution of the mass of the earth and, thereits geometric centre,, fore, with its e.g. some 500 km. away from, the satellite in its orbit is subjected to varying forces of attraction at, different intervals, with the result that its real course is neither, circular nor elliptical. It does not even lie in either of the two planes, and is, in fact, a curve of a complicated pattern. For the same, at different, reason, there are variations in the velocity of the satellite, points along its path., It is imperative that, 181. Stability of the rocket during flight., the rocket should, allotted, its, its, trajectory,, along, flight,, throughout, This is achieved by means of an auto-pilot (see 46), not get tilted., and a suitable gyroscopic arrangement., 182. Form of the Satellite. In designing a satellite, attention is, to ensure, naturally paid to the geometrical shape it should be given, this form, is, that, view, The, orbit., its, in, motion, its smooth, present, should be spherical, for, then, it will always have the same area, of resistance and thus help calculation of the air resistance to its, motion at higher altitudes and hence in the assessment of the density, of air at those altitudes. Further, with a spherical shape, there, , be less chances of its getting overturned than if it were cylindriany other shape. At the samo time, a spherical shape, is also a drawback, since it doss not make for an easy setting of, For, as will, the various instruments and other equipment inside it, be easily realised, the instruments must be sot, not haphazardly, but in a definite order so as to ensure both an equitable distribution, of the total weight inside the satellite and a specific position of, This 'balancing* of the satellite, as it is called, is obviously, its e.g., important and must be done with great precision., 183. Weight and size of the Satellite. The weight of a satelliteof the rocket carrier,, clearly depends essentially on the potentialities, and its dimensions, upon those of the last stage rocket, which is, usually the third stage one., The satellite which gets detached from the last stage rocket, not, necessarily be included as part of the rocket itself and may, may, of it. In, simply be arranged to lie inside a cavity in the nose-part, such a case, it is possible to give the satellite a bigger diameter, than the mean diameter of the rocket, as a whole, but only slightly, of the, so, or else it will mean a change in the ballistic characteristics, rocket as also an increase in the air-resistance encountered. The, satellite in the cavity is sometimes covered by a protective streamline, cone, during the course of the flight of the rocket, which is later, discarded and the satellite pushed out by means of a spring or a, compressed gas, when the rocket has actually reached the orbit irk, which the satellite is intended to move. This was exactly the case, with Russian Sputnik /, whereas Sputnik /f formed part of the third, stage rocket itself and did not get detached from it., 184. Material of the frame of the satellite. Obviously, the material, will, , cal or of
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SATELLITES, , of the, , 391, , frame must be both light and strong, the former from, its weight and the latter, to make sure that the, instruments etc. inside it are securely attached and that it 'can, withstand the onslaught of micrometeorites to which it is subjected, during its orbital motion in space. Then, the material must also, be less sensitive to changes of temperature and must be able to, properly reflect radio-waves. It must, therefore, be either aluminium,, magnesium or one of their alloys, with, in some cases, a suitable outer, satellite, , considerations of, , covering,, If, however, it is desired to study the electric currents in the, ionosphere, the frame of the satellite should neither be a conductor, of electricity nor should it possess any magnetic, So that,, properties., in this case, a metallic frame is clearly ruled out in favour of one of, a plastic material, some of the modern varieties of which are just as, tough and durable as steel., , 185. Duration of satellite's existence. It is only natural to, enquire, as to how long can a satellite be, expected to stay in its orbit. Well, if, the space in which it moves along its orbital, path were completely, devoid of air, there would be nothing to stop it and it could go on, But there being air even, perpetually, like the moon, for instance., at a height of 1000 kms. and above, it has to encounter resistance, due to it, however small, this resistance being greater for orbital, paths nearer the earth than further away from it. So that, when, its velocity is thus sufficiently retarded,, it cannot possibly remain, in its orbit and starts falling down, along a spiral path. In doing, so, it either gets burnt up due to the heat produced by friction in, the denser atmosphere or drops down to the earth with the, help of, parachutes., The actual calculation for its 'life' is rather a complicated one,, but it basically depends upon the density of the, upper regions of the, atmosphere, i.e., on the height of its orbit from the earth., , 186. Other Essentials., In case a man is to be placed in* the artior the sputnik, there are, quite a few other problems, to be tackled, as, for example,, provision of an hermetically sealed, ca,bin, with requisite conditions for the sustenance of life, and with, windows fitted with the type of glass that absorbs ultraviolet and, X-rays, a prolonged exposure to which is harmful in its effects. It, is, however, almost impossible to afford, any protection to the cosmonaut inside the cabin against cosmic, which, as we know, can, ficial satellite, , rays, penetrate even through a block of lead, one metre thick., , Luckily,, , although their effect on human or animal life is yet not quite clear,, they do not appear to produce any baneful effects. Then, there are, other problems, like those of, weightlessness etc. All four have now, been more or less overcome, as is evidenced, by four Astronauts,, two Russian and two American, having made orbital flights and, come safely back to the earth., , Another very essential item is the special, type of dress that, an astronaut must wear during his voyage in the cosmos. This is, fittingly called the Astrosuit and must at once be air-tight and, loose-fit to, , allow free respiration. In fact, the astronaut needs one, tifce-off of the rocket from the earth, which, , type of dress during the
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PROPERTIES OF MATTES, , 392, , must be so designed as to free him from the feeling of overload, experienced during a vertical ascent. Then, he should have another, A small cylinder,, lighter dress for free locomotion inside the cabin., provided in the girdle of this dress, creates an artificial pressure, on him to increase his blood pressure (which falls appreciably at, greater heights) as also to counteract weightlessness. The dress should, have provision to ensure normal respiration and the requisite body, temperature and must not restrict movement., 187 Return of Artificial Satellite. For the return of the satellite, back to the earth, the main problem is of sufficiently slowing down, its motion or braking it., There are two devices for it, v/z., (i), utilising air as the resisting, , medium and, , (ii), , using rockets., , is to be used for slowing down the motion, journey back, it must be given the form of a, rocket., For, then, as it enters the denser layers of the air, its, velocity falls but it rebounds back into the cosmos it then re-enters, the air with a reduced velocity and goes a little deeper than before, and there is a further reduction in its velocity. This process is, repeated a few times and the velocity of the satellite is thus sufficiently reduced to enable it to continue falling on specially provided, , If air-resistance, , of the, , satellite, , on, , its, , ;, , slide- wings, , and, , sliding planes., , On, , the other hand, if a rocket is to be used for its downward, journey, an automatically-controlled rocket-motor is necessary, the, reaction of which is in the opposite direction to that of the motion, of the satellite, i.e., it produces an effect opposite to that of the, rocket carrier during upward flight. It is, therefore, called a retroa rocket, taking the satellite back). The velocity, rocket,, (i.e.,, of the satellite is thus reduced and can be controlled by regulating, the fuel-supply to the rocket motor, the distance it thus has to cover, up to the landing strip being carefully estimated with the help of, a radar or other similar appliances. And, an automatic guidance, system is provided to control and manoeuvre the downward descent of, the satellite., , Now, the first method is certainly the simpler of the two, from, the technical point of view, but its great handicap is that it is, extremely difficult to design a landing strip to receive the landing, The second method, although more complicated technically,, satellite., ensures a smooth and an accurate landing on a properly constructed, landing strip., 188. Uses of an Artificial Satellite., , Ignoring the military uses to, be put, we shall concern ourselves here only with, its uses for strictly scientific purposes, among the more important, ones of which may be mentioned the following, , which a, , satellite, , may, , :, , Despite the fact, (/) Proper study of the upper regions of the atmosphere., that the atmosphere is being studied for a long enough time, our present, , knowledge of it is still much too meagre and superficial, particularly about, the region, called the Ionosphere, as also about cosmic rays. The artificial satellites, will, it is hoped, help to improve this., (//) Weather, forecasting. This can be made much more accurate and, dependable with a number of satellites around the earth in various orbits., Meteorological observations over various countries could then be made simultaneously, thereby greatly improving the reliabilr of weather forecasts.
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393, , SATELLITES, , This, (iii) Determination of the exact shape and dimensions of the earth., 4s the task that scientists all over the earth have set for themselves during the, third International Gco-physical year., Detailed study of the solar radiation., , (iv), , Study of meteorites., , (v), , Experimental verification of the theory of relativity., Use of a system of three artificial satellites for universal telecasting., (viii) Study ofpropagation-characteristics of radio waves in the upper regions, *f the atmosphere., (ix) Astronomical observations, without atmospheric and other disturbances, (vi), , (vii), , etc., etc.,, , EXERCISE X, 1., , remain, , in, , Explain clearly the principle underlying an airplane., equilibrium in air and how does it rise up ?, , How, , does, , 2., Differentiate between climbing, diving and gliding of an airplane, explain the co- relation of forces in each case., 3., Name the principal parts of an airplane, but clearly their respective functions., 4., , What do you understand by, , the, , term, , it, , and, , and mention concisely, , 'jet-propulsion' ? Give, in brief,, , in account of jet-propelled planes., 5., What is a rocket 1 How do rocket-planes differ from, Explain the principle underlying a multi-stage rocket., , What is an artificial satellite 1 Explain as, may be placed in its orbit around the earth., , 6., latellite, , clearly as, , et-planes 1, , you can how a, , 7., Mention the essential pre-requisites and conditions for a satellite to, be placed in its orbit and its return back to the earth., Also mention some scientific uses of an artificial satellite.
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CHAPTER, , XI, , AND LUBRICATION- PRINCIPLE OF VIRTUAL, WORK AND ITS SIMPLE APPLICATIONS, , FRICTION, , Static Friction Laws of Friction., In all cases of motion, 189., of material bodies, counter forces come into play in-between their, surfaces which tend to nullify or neutralise the effect of the driving, These counter forces are called resistances, the most, force applied., important among which is friction. As we have already learnt in our, junior classes, when one solid body is sought to be moved over the, surface of another on which it rests, an opposing force, called the, 'force offriction', comes into play in-between the two surfaces, tending to destroy the relative motion between them,* and which is, , usually measured by the force required to produce uniform relative, motion between the two surfaces. It is this force which always acts, in a direction opposite to that in which the motion is desired and, which is called the force of friction or rather static friction^, Experiment shows that friction roughly obeys the following laws, called, the 'laws offriction', discovered by Amontons (1699) and Coulomb, (1779) and hence sometimes referred to as Coulomb's laws, ., , :, , The ffictional force, , a self-adjusting force and increases, with the applied force, so as to be equal and opposite to it,, until motion is just about to ensue, this maximum, , (/), , is, , ;, , fnctional force, value, , is, , different, , called, , is, , the, , friction', , 'limiting, , Before this limiting value of friction is reached,, is just enough to preserve equilibrium. J, (ii), , and, , its, , for different pairs of surfaces., its, , magnitude, , The, , limiting friction between the surfaces of two bodies is, directly proportional to the normal reaction of the support-, , ing surface,, , Thus, if R be the normal reaction of the supporting surface and, F, the limiting friction set up between the two surfaces, we have, , F oc, where p, , R,, , Or,, , FIR, , .=, , M, , ,, , a constant, called the 'Static Coefficient of Friction' or, simply the coefficient offriction for the given pair of surfaces., is, , ^, , ..., , i.e.,, , ., , ,, , static coefficient, ", , rr, , ., , ., , of, JJfriction, , =, , limiting friction, ,--, , ., , normal reaction, , *Strictly speaking, this is not the only force that opposes the relative, l, surfaces., There is also another force, called the force of, to stick or to cling, which is moleadhesion', (from the Latin word 'adhaerere', cular in origin and which tends to make the bodies cling together., , motion between the two, , t'Sffl/Jc',, , because the two surfaces are initially at rest with respect to each, , other., tit will, , be readily ssen that this really follows from Newton's third law oj, , motion., , 394
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FRICTION, , =, , 395-, , =, , =, , nR so that, if R, 1, we have F, Thus, obviously, F, p,, the coefficient offriction for a given pair of surfaces may be denned, as the limiting friction coming into play in-between them, for unit, normal force applied to them, or, as the fraction of the normal force that, is required to keep the two surfaces in uniform relative motion., ;, , i.e.,, , (Hi), , (iv), , The frictional force, , is independent of the surface areas, in, contact with each other* and of their relative velocities., , The frictional force, of the two, , It, , may, , is, , independent of the, , relative* velocities, , surfaces., , be pointed out here that these laws apply only in the, , and smooth or well-polished surfaces., 190., We have already seen how the force of, Sliding Friction., friction between the surfaces of two bodies, one resting over the, other, continues to increase with the applied force and is always equal, and opposite to it until its maximum or limiting value is reached for, case of clean, , that particular pair of surfaces., If at this stage, v/z., when the friction is about to attain its, limiting value, we apply a force in the form of a gentle push to the, body resting over the other, such that it is maintained in uniform, motion over the latter, then, this force measures what is called the, sliding friction between their surfaces, i.e., the frictional force in-between them when motion ensues, It is also spoken of as kinetic or, , dynamic, , friction, to distinguish it, , from, , static friction (that, , comes, , into-, , play before motion actually takes place) and is found to be somewhat less than the limiting friction for the same pair of surfaces. That, is why we find it easier to maintain a body iii uniform motion over the, surface of another than to start it moving., , The, , between, , ratio, , this sliding friction, , and the normal reaction, , then gives the, , coefficient of sliding friction for the given pair of, is also obviously less than the coefficient of static fric-, , surfaces and, tion for them., , small and, , we, , The, , difference, , usually assume, , between the two is however quit, to be the same for all practical, , them, , purposes., 191., , Angle of Friction, , Cone of, , Friction., , If, , we place a body, , Fig. 241., , *This is no longer so, if what are called lubricants, like grease, graphite,, are introduced in-between the two surfaces. For, the normal force, applied is more likely to squeeze out the lubricant from in-between the two, surfaces, when applied to a small area than when applied to a larger area., talc etc.,
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PROPERTIES OF MATTER, , 396, , on an, , inclined plane, [Fig. 241 (a)], it is clear that, simultaneous action of three forces, viz.,, (/), (//), , its, , weight, , W,, , the plane, , downwards, , acting vertically, , the normal reaction, , R, , it is, , under the, , at its e.g.,, , G, , ;, , of the plane, acting perpendicularly to, , and, , ;, , (in) the tlimiting friction F, acting, , upwards along the plane., , If the angle of inclination of the plane be so adjusted that the, body is just on the point of sliding down, it is clear that the three, forces are in equilibrium and, therefore, concurrent, so as to be represented by the three sides of a triangle, taken in order. This angle, of inclination of the plane, ft, at which the body is just on the verge, of sliding down, is called the angle offriction., , W into two rectangular components, we have, W sin \, along the plane, tending to move the body down, bhe plane and component W cos A, at right angles to the plane., its, , Resolving, , (/), , somponent, , Since the body, , is, , F=, , So, , .,, , that,, , we have, , in equilibrium,, , W sin X, -5, , and, , =, , .R, , *, , = W cos \., , =, , tan >., , the coefficient of friction for a given pair of surfaces is equal to, the tangent of the angle of friction for them., From the above relation for /*, we have, tan ft, from, , Or,, , F= R, , F, , R, , and, must lie along the surfollows that the resultant of, face of a cone, with \ as its s^mi-vertical angle, and the direction of, the normal reaction, as its axis., which, , it, , The same is true in the case of two horizontal surfaces, where, the frictional force F acts along the supporting surface and the normal reaction R, perpendicular to it, [Fig. 241 (b}]. Their resultant, P then makes with the latter the angle of friction ft and, again, therefore, the resultant (P) lies along a cone of semi- vertical angle X, such, that tan A, F/R., , =, , is called the cone of friction, and it is obvious that, whatever its magnitude, with its line of action lying within, the cone, can possibly produce motion in the body, its component, along the surface of contact being less than the limiting friction (F), between them., , This cone, , no, , force,, , We, , have just seen, Acceleration down an Inclined Plane., 192., 191, above, that a body placed on an inclined plane will not start, of the plane is, sliding down along "it until the angle of inclination, equal to the angle of friction ft for the surfaces of the body and the, plane ; for, at a smaller angle of inclination than this, tan 0<F/J? or, in, , and ^, the coefficient of friction for, /*, where F is the limiting friction, the two surfaces in question. And, when 0=\, clearly tan 9= tan A, , =, , F/R and sliding just commences., But when > ft, so that tan, , >, , tan ft and hence greater than, Q, F/R, the body slides down the plane with an accelerated motion. Let, us calculate this acceleration of the body., , As, , before, resolving the weight of the, , body, , W=mg, , (where, , m, , is
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FRICTION, , 39T, , the mass of the body) into two rectangular components, along, perpendicular to the plane, we have, , and, , component along the plane, , and component perpendicular, , to the plane, , ==, , =, , mg sin 0., mg cos 6., , Since there is no motion perpendicular to the plane, the normal', R of the plane is equal to mg cos 6 and the two, being equal, and opposite, neutralise each other. And, thus, the only two forces, effective on the body are (/) mg sin 0, downwards along the plane,, and (//) the sliding fractional force F upwards along it. So that, the, resultant force acting on the body downwards along the plane is equal*, reaction, , to, , mg sin, , 6, , Now,, of surfaces,, , F., if, , n, , we, , be, the coefficient, , clearly, , of sliding friction for the given pair, , have, , F=, , /ijR, , =, , n, , mg, , cos, , Q., , So that, the resultant force on the body downwards along the plane, , = mg sin dnmg cos 6 = mg (sin QLL cos, And since acceleration = force/mass, we have, acceleration of the body, , =, , mg, , (sin, , 6)., , downwards along the plane, , Qn cos 0)jm = g, , If the plane be perfectly smooth, , ,, , sliding down the plane would be g sin 6., ation of the body down the plane is, the frictional force between them., , (sin, , 0v>, , cos, , 0)., , the acceleration of the body, Clearly, therefore, the accelerreduced by n g cos 6 due to, , The frictional forces between two sur193. Rolling Friction., when one roils over the other is called rolling friction and is, found to be much less than when sliding occurs between the same, two surfaces. That is why vehicles are provided with wheels and, faces, , their axles, with ball- bearings*, the latter converting the chief frictional loss of the wheel that occurs at the axle or the journal in the, form of sliding friction, here, called journal friction into rollingfriction., , It was shown by Osborne Reynolds that in rolling an appreciable amount of slipping or sliding of one surface over the other, occurs and that the frictional resistance to this slipping, or sliding,, As extreme cases of this slipping, really constitutes rolling friction., between two rolling surfaces may be mentioned (?) an iron cylinder, rolling over a plane rubber surface or (ii) a rubber cylinder rolling, over a plane iron surface. In the former case, the cylinder covers, a distance equal to only nine- tenths of its circumference in its one, full turn and, in the latter case, a distance equal to eleven- tenths of, its circumference, eo that, in either case, there is a slip of one-tenth, of its circumference, In ordinary cases too, some slip always occurs, between two rolling surfaces, even when the two surfaces are of the, same material*, ;, , It follows as a natural consequence that rolling friction between, two surfaces would be zero, (a) T\hen either tie sliding friction, between them is zero, i.e when for them is zero, (b) or when no, ,, , etc.), , v., , *Here, there is a ring of small balls between the wheel, (pulley or disc, axle, so that when the former rotates, the balls all roll also., , and the
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PROPERTIES OF MATTER, , 398, , slipping occurs between them during rolling, i.e., when v> for them is, and oo) there must be, For all other values of \i (between, infinite., friction and, for a particular value of M, it must have its maximum, value., It may as well be mentioned here that while lubrication of, the surfaces always reduces the value of the coefficient of sliding friction, (M) for them, it may or may not reduce the rolling friction between, them. Thus, as is so well and so generally known, lubricating ballbearings only results in increasing friction*., 194., Friction and Stability. When a body, say, a block of wood,, rests on a plane horizontal surface, it does so because the weight, , A, , D, D, , <>, Fig. 242., , of the block W, acting vertically downwards at its, e.g., is just, neutralised by the equal and opposite normal reaction, acting there., And, when a horizontal force Fis applied to the block to move it for-, , R, , wards on the plane, it does not move or slide along it so, long as, F<uR, where u is the coefficient of friction between the surfaces of, the block and the plane, (see page ,196)., The possibility is, however,, , may topple over for, the moment the horizonapplied, at a point P, say, a frictional force F', equal, opposite to F, comes into play in-between the surfaces of the, , there that the block, tal force, , and, , ;, , F is, , =Fx PB,, , block and the plane, [Fig. 242(0)], thus constituting a couple, tending to rotate the block (in the clockwise direction,, case shown), and thus making it topple over., , in, , the, , Now, as Fis gradually increased, this couple formed by Fand, makes the centre of reaction of the plane shift from //towards J5f,, with the force at C progressively decreasing and that at B, increasing,, , F', , until, in the limiting case, the whole reaction R acts at B (that at C, being zero). We thus have another couple, formed by, and 7?,, equal to WxHB, tending to rotate the block in the opposite direction, to that due to the first couple (in this case, anti-clockwise) which thus, tends to restore the block back to its original position. So lon#,, , W, , therefore, as this restoring couple, , W XHB,, , is, , greater than the couple, , *This might raise a question in the mind of the student as to why then, are they lubricated at all ? The simple answer is that it is done only with a view, to reducing wear., tFor, with no force F acting on the block, its weight is uniformly distributed over its base B,
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FBICTION, , FxPB,, , due to, , F, , and, , 399, , the block remains at rest and upright,, if, it topples over., Let us now investigate the conditions under which the block will, remain upright but slide along the plane. For this, let us first determine the resultant of F and, by the ordinary application of, the parallelogram law of fordes and then take the moments of this resultant R' and the frictional force F about B. Let R' be represented, by the diagonal EL of the parallelogram EMLN, with its adjaand, cent sides, representing Fand, respectively, and let it, cut the plane in K, [Fig. 242(6)]., Now, clearly, moment of F' about B is zero, since its line gf, action passes through B. So that, so long as HK<HB, i.e., so long as, the resultant R' passes through the contour of the ba^e of the block,, it will have a restoring moment about B and the block will remain, made by R' with the vertical, or the, But, if the angle, upright., direction of the normal reaction R, be greater than A, the angle, offriction for the surfaces of the block and the plane, (see, 191), it, If however, HK>HB, i.e., if the, will slide along the surface., resultant R of F and W, passes outside the contour of the base, of the block, the block will topple over., i.e., in, , stable equilibrium,, , F',, , but, , WxHB<FxPB,, , W, , 1, , EM, , EN, , W, , r, , ,, , fast, , Another important case of stability due to friction is that of a, vehicle on a curved track, discussed already in, 18 (4)., , moving, , 195., Friction, a Necessity. Taking most of our daily activities, in life as a matter of course, we seldom care to pause and think as to, bow much they are dependent on the existence of friction. Thus, for, example, in the absence of friction, we would find it impossible, to walk or to drive on a road, and if we just start moving, we shall, not be able to stop again, it would be impossible to climb a tree, tie, a knot or even fix a nail in the wall. Brick would not stand on brick, and buildings would tumble down like a house of cards and so, on. Indeed, we find it so much of a necessity that we deliberately increase it for many of our purposes, as, for instance, when \vo, .apply brakes to our bicycles or cars., ;, , In many other cases, on the other hand, we find friction, so irksome, as, for example, in the various parts of our machines,, their, speed slower and their output lower and bringing about, making, a greater wear and tear in them. And, yet, we know that friction is, ;, , necessary even for thorn. What we do, therefore, is just to adopt, ways and means of minimising it in such cases by means of oils, and other lubricants, and ball-bearings etc. etc., (see page 404)., , Simple Practical Applications of Friction Rope Machines. Apart, friction, some of which have been mentioned above,, there are various types of useful machinci based on it. We shall consider here, a couple of them by way of illustration of the principle underlying them., The Prony Brake. This is a simple appliance to measure the power, 1., of machines, which we owe to Baron G. C. F, Prony, a French Mathematician,, (1750 1839). It is in fact a broke dynamometer and consists of two wooden arms, or, *cheeks\ A and B (Fig. 243) in between which can be clamped the shaft of, the machine whose power (i.e., rate of working) it is desired to measure. The, frictional force between the shaft and the cheeks is regulated by tightening or, loosening the screws S and 5. provided on the uoper cheek A % to which is also, attached a small rod /?, about *5 to 1*0 metre in length, carrying a scale pan at, its other end., 196., , from the ordinary uses of
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PEOPEBTIBS OF MATTEB, , 400, , Suppose the shaft of the machine turns in the anti-clockwise direction, at, shown. Then, if there be no weights placed in the scale pan, i.e., if the brake, , Fig. 243., , be unloaded, it will tend to be carried around with the shaft due to friction, between itself and the shaft. But, if the brake be sufficiently loaded, before the, shaft starts rotating, i.e., before starting the machine, the moment due to, the load may be enough to overcome the nioment due to friction between the, brake and the shaft and the brake may turn in the clockwise direction so that,, in the scale pan, i ?., to so load, it is quite possible to so adjust the weights, the brake, that the rod R remains quite horizontal in-between and equidistant, from the stops s t and J 2 placed on its two sides, a little distance away from it., When this is so, obviously, the frictional resistance between the brake and the, shaft is equal to the force F exerted by the machine on the periphery of the shaft, and is exactly balanced by the \\eights or the load, placed in the scale pan. So, that, we have, ;, , W, , W, , to machine) on the shaft = moment of weight, on it., be the radius of the cross-section of the shaft and /, the length of the;, rod R, we clearly have, moment of F on the shaft = JFx r,, = xl., moment of, and, ,,, ,,, , W, , moment of force F(due, If r, , W, , W, , Fxr = Wxl,, , F - W.lfr., whence,, Thus,, If the shaft makes one full rotation in time r, we have, work done by the shaft, i.e., by the machine, in time T, = Fx circumference of the shaft = Fx2nr., And .'. work done by the machine per unit time, i.e., the power of the machine, , ~, , Fx2nr, ', , T, , Or, substituting the value of F, obtained above,, , power of the machine, , =, , =, , ^x, , Thus, knowing W, I and, machine, which, as we can see,, , -, , we have, , W.I, , = _, , x moment of the, , load., , T we, t, , is, , can easily determine the power of th<, clearly proportional to the moment of the, , load., , W, , N.B. If, be taken in dynes,, the machine in ergs per second., , I, , in cms., , and Tin, , sees.,, , we, , get the, , power, , ol, , The Rope Brake. Before discussing any rope-machines, we, minds as to what exactly is meant by a rope and, what, if any, are the peculiar properties possessed by it that make, 2., , must be, , clear in our, , for its usefulness., , A rope, then, is any flexible body or combination offlexible bodies,, capable of transmitting tension. Thus, the string of a violin, apiece ojf, spring, a strap, a band&nd a chain all come under the definition of a, rope., , When, *>ther, , one. end of a rope is connected to a body and, end pulled, stresses are caused at every cross-section of, , its, it,.
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401, , FRICTION, in exactly the same manner as in a rigid rod., bouring cross-sections neutralise each other, directly exerted on the body at the other end., , The, , stresses in neigh-, , and the, , A, , pull, , is, , thus, , rope can thus trans-, , mit a tension, undlminished in magnitude, from one end to the other., Due to its flexibility, however, it cannot transmit any compressional, forces along it, but, at the same time, its flexibility is of great, ;, , advantage in, , it particularly suitable as a, even by a change of, unaffected, force,, , many ways and makes, , means of transmitting tensional, , direction*, as in the case of a pulley etc., , On the other hand, when coiled round a cylindrical body, a, can, exert a very large couple on the body, due to frictiojn, rope, between itself and the body, so, much so indeed, that a man pulling, at one end may even hold a ship,, fastened to its other end., Let us, see how this comes about., , BCD, , Let A, be the cross, section of the cylindrical surface,, with its centre at, and let a rope, PABCQ, coiled round it, leave its, surface at points A and C, (Fig., -, , O, , 244)., , Consider an infinitesimal porof the rope at B and let, the mean tension over this portion, be T, with the angle subtended by, tion, , it, , at, , EF, , O, , equal to, , Fig. 244., , eld., , Then, representing the tensions T, at E and F, by the tangents, FH respectively, we have their resultant force represented by, BJ (by the simple application of the parallelogram law of forces)",, where BJ is clearly equal to BG.dO, T.dO, [v BG represents T, BGJ, and, d6], in the direction BJ, normal to the section EF., , EG, , and, , =, , =, , at B., If,, , therefore,, , T, c3 lindncai surface,, , \i, , be the coefficient of friction for the ropo and the, , we have, , frictional force between the rope, in the direction of the rope at B., , and the surface, , =, , /i.T.rftff*, , Due to this frictional force, there comes about a change in the, tension at the two ends of the rope, which, in the absence of any, In fact, the differfriction, would have been the same, (see above)., ence in the tensions at the two ends is just equal to this frictional, So that, if dT be the difference in tensions at E and F, wo, force., have, , dT, , etc.), i.e.,, , =, , l, , Or,, , T, , .dT, , "This is possible only so long as the body, (e.g., the pulley or the ring, over which the rope is passed does not interfere with its freedom of motion, is perfectly smooth and round., , tBecause frictional force, , J*x normal reaction R* and here,, , R
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402, , PROPERTIES OF MATTER, , we have, , integrating which,, , <..dd., , where g, , is, , the angle, , Clearly, if Q, , EOA, , =, , =T, , l9, , 2\, , log,, , Substituting this value of, log,, , T=, , T = nO+K,, , log,, , and K, a constant of, , T, , 0,, , Or,, , the tension at, , = nxQ+K =, , K in expression, , ne+logt, , Or,, , T,., , (i), , whence, , it is, , =, , Or,, log, 7) 7\, ^0., 7)^, clear that the tension, increases as, if, , <f>, , we have, , =, , rf., , e^,, , T, , Obviously, therefore,, tension at C, we have, , we have, , K., , above,, , =, , Or,, , integration., A so that,, , T-log, T,, , log,, , (0, , increases., , be the total angle AOC, and, , T2, , ,, , the, , It is thus clear that if ^ bs large, i.e if the, rope be coiled many, times round the cylinder, 7 ,/ 7\ is also very, large and a small tension, applied at P rcuiy be made to exert a large pull at Q., ,, , 1, , (3), , above,, , is, , The Band Brake., , made, , use of in, , The, , discussed in, 196, (2), of brakes, for the measurement, of the power (or the rate of, , principle,, , many forms, , doing work) of machines, one, of the simplest of which is the, , Band, , brake., , is a, simple device,, consisting of a pulley fixed on, to the rotating shaft of the, machine whose power is to be, , It, , A band (or a, passed round the, once, twice or thrice, and, has its two ends, etc.,, 245., attached, to, two, spring, suspended from a rigid support, as shpwn in, determined., , cord), pulley,, , Fig., , balances, , A and B, , is, , Fig. 245., , Then,, respectively,, , where, , 7\ and, we have, , if, , T2, , be the readings, , in the, , couple exerted on the pulley due to friction, R is the radius of the pulley., , Now, work done by or, , against a couple, , two spring balances, , =, , (T^T^.R,, , = couple x angle, , of, , rota-, , tion (in radians)., , And, therefore, work done against a couple per second, -= couple, , Or,, , rate, , of doing work,, , i.e.,, , x angle, , power, , =, , turned through, , couple x angular, , in, , one second., , velocity., , Thus, work done per second by the machine against the couple, , due to friction, ^(T^T^.R x 2-nn,, where n is the number of rotations made by the shaft or the- axle of, the pulley per second and hence 2;r, the angular velocity, of the shaft.
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MECHANISM OF FBIOTION, , 4t)3, , This, then, gives the work done per second by the machine, or, the power of the machine. Thus, we have, ri )./?x27m., power of the machine, (T^, , =, , Thus, knowing Tt and Tz (from the two spring balances),, can easily calculate out the power of the machine., , =, , R and, , n,, , we, , e*, Further, we have the relation TZ IT:, is the coefficieat of friction for the cord or band in question, And the surface of the pulley and 0, equal to TT, STT, or STT, efc. according to the number of times the cord or the band passes round the, ,, , where ^, , So that, knowing, , pulley., , out the value of, , 7\,, , T2, , and, , 0,, , we can also easily calculate, , M., , 197., Mechanism of Friction. Since the laws of solid friction were, enunciated by Amontons and Coulomb, much has come to be known as to the, jfiow and why of friction, thanks to^he work of Hardy, Bowden and others., , imay be, , According to Dr. Bowden' s lucid exposition, the mechanism of friction, summed up as follows, :, , The smoothest or even the most polishsd surfaces are, , really not, , smooth, , -enough, having projections and depressions of larger than molecular dimensions, on them so that, the area of the two surfaces, in actual contact is much smaller, ithan the apparent one we see to be so,, being less than ten-thousandth part of, the apparent area of contact, in the case of plane steel surfaces., ;, , It is possible now to form a fairly correct estimate of the actual area of, contact between two metals by measuring their electrical resistance, and it is, ifound to be practically independent of the size and roughness of the surfaces and, to depend only upon the load, The obvious inference is that the projections on, the two surfaces get shorn off or crushed down under the load, until an area,, enough to support that load, is cleared up and comes into actual contact., , Viewed in this light, Amonton's law, regarding th3 independence of the, foice of friction of the surface area, means no more than that, for a given load,, a change in the apparent area makes little or no difference to the area in actual, contact., , A, , logical consequence of this smallness of the area of true or actual contact is that the pressure at the points of contact must be enormously high., Indeed, it is estimated to be of the order of 100 tons per square inch in the case of, to, the result that during sliding between, mild steel. And this inevitably Isads, the two surfaces, the temperature at these points of actual contact must rise to, , All this now stands amply verified by actual exenormous proportions, periment. For, it has been found possible to measure the temperature of the, points of actual contact iq the case of two dissimilar metals by using them as a, thermocouple and by amplifying and applying the ther mo -electromotive force thus, generated to the deflecting plates of a cathode-ray oscillograph. Temperatures,, as high as 1000C*, have been found to obtain at these actual contact points or, 9, the temperatures thus, 'hot spots although they last only for just a split second,, reached depending upon (/) the magnitude of the load, (//) the speed at -which, sliding takes place, (Hi) the thermal conductivity and (iv) the melting points of the", metals in question., -equally, , In most cases, therefore, there is welding at these points during sliding,, due to melting and consequent flow of metal. The surfaces of both metals, thus get damaged by this sliding occurring between them. The surface of the, softer metal gets torn and 'ploughed* by the projections on the surface of the, harder metal and the latter has the softer metal welded on to it, -the maximum, damage occurring in the case of two similar metals sliding one over the other, the, coefficient of friction being the highest in their case., , *Even higher temperatures are reached in the case of poor conductors of, Thus, for example, glass rubbed against glass, or quartz against quartz,, gives tiny sparks of light, changing from reddish to white as the vigour and speed, of rubbing is increased,, heat.
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PROPERTIES OF MATTER, , 404:, , The frictional force between two metal surfaces may thus be attributed, to the shearing of these temporarily- welded points as well as the 'ploughing"", resulting from the projections of the harder metal, the former being by far, the greater of the two forces, depending upon (a) the actual contact area of the, metals and the shear-strength of the softer metal*. To minimise friction,, In almost all, therefore, both these factors should be made as small as possible., cases, however, if one is less, the other is high, as, for example, if a hard metal, is made to slide on steel, factor, (a) i.e , the actual area of contact, is small, but, factor (b), i e., the 'ploughing" is greater, and if a softer metal is made to slide, on steel, the ploughing, [factor (b)] is small, but the contact area [factor (a)J, becomes greatei, As a consequence, for most pairs of metals, the coefficient of, friction comes to he within the same common limits, '6 to I'O., 4, , ;, , ., , 198., , A, , like lead or iridiumi",, like steel, often reduces friction, because of itsreducing the shear- strength without unditlv increasing the contact area. This explains, why we frequently use copper or silver bearings, smeared with lead or iridium in, our modern aero-engines., substance so used to minimise friction is called a, lubricant., , Lubricants., , thin film of a soft metal,, , smeared on a harder surface,, , A, , Even apparently clean metal surfaces in an have a thin smearing of some, between them, e g., a thin film of air or a thin film of oxide, or some such other contaminant, like moisture etc., which greatly reduces, their shear-strength and hence the inction between them., Consequently, the, friction between two metal surfaces is found to be greater in, yacuo than in, the cause of this enhanced friction in* the former case bsing the force of, air,, adhesion between the two surfaces, which grip or seize each other under the, smallest load. Thus, as Dr. Bowden so aptly puts it, it is really fortunate that, all metal surfaces are more or less contaminated in air, or else the world would, be a very sticky place indeed., sort of a lubricant, , All cases of friction fall into one or the other of the following three cateie the frictior.al force enviz.y (/) dry fnotion between solid surfaces,, countered when no lubricant is present between them, (//) boundary friction,, when only a thin film of a lubricating material is present between the two^, given solid surfaces ; and (///) fluid or floatation friction, when there is plenty, of a liquid lubricant present in-between the two surfaces ; so that, the friction,, in this case, depends almost wholly upon the properties of the lubricant, in, gories,, , ,, , ;, , upon, , its viscosity, (sec Chapter Xll)., have already studied briefly the essential details of the first tuo, categories, ^iz friction between solid surfaces in the absence of any lubricating, substance, and 'boundary friction', with a thin film of lubricant present in-between them. The ordinary cases of friction between solid surfaces, which, as we, have already seen, are almost always contaminated, to some slight extent, with, oxide or moisture etc., strictly belong to this second category of boundary friction., Hardy thoroughly studied this type of friction and, according to him, it, is the chemical natures of the metal and th<* lubricant which determine the, strength of the boundary film, a metal attracting more strongly the polar, group (COOH) at one end of the molecules of an oil than the non-polar ones at, the other. This results in the layer of molecules nearest the metal surface being, orientated so as to stand upright (i.e., normal to the surface), with their polar, ends inwards, with possibly further double layers, having their polar ends adjacent,, being formed on them so that, sliding actually occurs between pairs of these, molecular layers ovsr their non-polar ends, Obviously, therefore, the more, strongly doss the lubricant adhere to the surface, the more effective will it be in, minimising friction., In the case of fluid or floatation friction, the thickness of the film makes, most of the molecules lie outside the range of the adhesive force of the solid, surface, so that the only force to be overcome now is that due to the viscosity of, the lubricant., , particular,, , We, , ,, , ;, , *Coulomb had originally suggested that the frictional force between twosurfaces was purely a consequence of their roughness and consequent interlocking of their projections and depressions, so that there should be practically nofrictional force between perfectly polished surfaces. This is, however, not found**, to be so, as we shall presently see., tlridium, , is, , even softer than lead.
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VIRTUAL WORK, , 405, , Beuchamp Tower, in 1883, experimented on a revolving shaft, resting, in a well lubricated bearing and showed that not only was the friction between, the two greatly reduced but that the laws obeyed by frictional force were entirely, different from those in the cass of dry friction. For, (i) while in the case of dry, surfaces* th2 frictional force is proportional to the normal reaction between them, it, 45 practically q ate independent of the normal reaction in the case of lubricated surfaces, (it) while in the case of dry surfaces, the magnitude of the frictional force, depends upon the nature of the two surfaces, it is quite independent of the nature of, it he, lubricated surfaces, depending only upon the nature of the lubricant, and (in}, whereat in the case of dry surfaces, the frictional force is quite independent of their, relative, , surfaces., , velocity, it varies directly with the relative velocity in the case of lubricated, Three y^ars later, in 1886, Osborne Reynolds put forward his funda-, , mental theory of lubrication. Tower had shown that a film of the lubricant, vas formsd in between the shaft and the bearing and Reynolds showed that the, film, which he thought must be several molecules thick, could be maintained, in spite of the enormous pressure between the two surfaces, provided that the, diameter of the bearing were a wee bit, (about one-thousandth part) greater than, ths diameter of ths shaft. This makes the two surfaces slightly eccentric,, with a wsize-shap'd cle&ranc 2 between them, filled in with the lubricating oil,, its pressure increasing with the narrowing down of the wedge and being equal, to that developed between the sliding surfaces., The oil is carried round by the adhesive forces on layers, nearer the shaft,, and by viscous forces on those away from it, and the two surfaces are thus kept, reasonably apart for the quantity of the labricant in this wedge to be sufficient, to have its normal bulk-properties and thus to be able to resist the shear, pureThe shearing of the temporarily welded points between, ly by virtue of its viscosity., the twj surfaces is thus replaced by this inter-liquid shear., Obviously, therefore, liquids, possessing the two essential properties of, But,, viscosity and of adherence to the solid surfaces will be the best lubricants., from the practical stand-point, perhaps by far the most important property required, of a lubricant is that of chemical stability, and mineral oils undoubtedly claim, a definite superiority over vegetable oils in this respect as well as in supplying, the necessary minimum viscosity* required to maintain a multi-molecular film, , between the two, , surfaces., , On, , the other hand, vegetable oils excel over mineral oils in their property, of forming strongly adhering films on the solid surface, (/*., the boundary, films), in view of the polar groups in their molecule, due to the presence of free, fatty acids in them., , To take full advantage of both these properties, therefore, the modern, lubricants used are a mixture of mineral oils with vegetable oils (like castor oil), ike former supplying film lubrication and the latter, bounin proper proportion,, dary lubrication. In addition, it is also quite usual to add to the above mixture, of lubricating oils, a measure of colloidal graphite, a thin film of which is, formed on the solid surfaces, so that even if contact occurs between the two metal, surfaces, seizure takes place between the graphite layers, offering little or no, resistance to any relative motion between them., In addition to all this, other factors, such as change of viscosity with, increase of temperature and pressurcf between the two surfaces have also to be, taken into consideration., , The principle of virtual, Virtual Work., enunciated by John Bernoulli, in the year 1717, and, later, in 1788, quite independently by Lagrange, and applies to all, cases of equilibrium, thus leading to the deduction of all theorems, relating to mechanical transmission of force, (of course, neglecting, 199., , work was, , Principle of, , first, , *A high viscosity of the lubricant being also not quite desirable, in view, of the resistance it would offer to the sliding motion between the two surfaces*, Further, since the resistance to motion, due to viscosity, increases with speed,, a higher viscosity at lower speeds, and a lower viscosity at higher speeds, is, deemed to be more desirable, fAs we know, viscosity decreases with temperature and increases with, pressure.
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406, , PBOPERTIES OF MATTJJB, Let, , friction)., , before coming to the Principle itself or, , us,, , what, , cations, try to Understand clearly as to, virtual work., , is, , its appli-, , meant by the, , te.rm, ,, , we imagine a system of forces, acting on, slight displacement, consistent with the geometrical conditions,, /.e,, compatible with the constraints to which the particle is subjected,, the displacement so imagined is called a virtual displacement, for it, a particle, .to suffer, , If, , a, , may never really occur in actual practice, and is thus purely arbiThe product of each force of, trary, existing only in our imagination., the system and the virtual displacement along its line of action is called, virtual, , work*., , The, , principle of virtual, , work may now be stated as follows, , :, , A, , system offorces, acting at points, connected by any mechanism,, equilibrium, if the total work, accompanying a virtual displacement, is zero., Let us consider a few simple cases, illustrative of this principle :, will be in, , a body in equilibrium on a smooth inclined plane, (/) Case of, under the action of a force. Let P be a body of mass w, on a smooth, incline, of angle 0, (Fig. 246) such that a, force F, acting on it at an angle ^ w ith the, , The, in, equilibrium., of the plane, obviously,, acts perpendicularly to the plane, as shown., In order to eliminate any reference to it, and thus to simplify our calculations, let, us imagine a virtual displacement dr of the, body, along the plane, so as to be perpendicular to R, so that R,dr is zero., keeps, normal reaction, , plane,, , mff, , Then,, , ,, , it, , R, , clearly,, , Fig. 246., , virtual, , work, , and, where, , mg, , sin, , Q, , = mg sin Q.dr,, = F cos, , by the weight mg of the body, virtual work done against force F, and F cos arc the components of, clone, , <j>.dr,, , mg and F along, , (f>, , the plane, the negative sign, in the latter case, merely indicating that, the work is done against the force., , Hence, according to the principle of virtual work, the condition*., that must be satisfied for the equilibrium of the body is that, 0., mg sin 0.drF cos <f>. dr, Or, that, , (mg, , Since, obviously, dr, , mg, , sin, , is, , QF, , sin, , 6F cos, , fy.dr, , =, =, , 0., , not zero, we have, , cos, , $, , =, , 0., , Or,, , F cos, , *, <f>, , mg, , sin 9,, , which gives the condition for the equilibrium of the body, and, which we know, is the one we obtain by the ordinary conventional, method, in which we also take into consideration the normal reaction, , of the, , plane., (//'), , Case of equilibrium of a body on a rough, , Let a body, of mass m, be, , just in equilibrium, , *Very aptly so, since the displacement, , is, , i.e.,, , inclined Plane*, , just, , on the point, , only virtual or imagined,
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VIBTUAL, , WORK, , of sliding down a rough incline of angle, , F, , acts, , upon, , it,, , 407, , when a horizontal, , 0,, , force, , (Fig. 247)., , Here, obviously, the reaction R of, the plane will make an angle, a, say, (the, angle of repose) with the normal PN to, the plane., , Again, considering, , have a, , the, , virtual displacement dr,, , cular to the direction of R,, , R.dr, virtual, , =, , body to, perpendi-, , we have, , And, therefore,, , 0., , work done by the weight, = mg sin (0a).dr,, virtual work done against, , mg, , of, , the body, , Fig, , 247., , and, force F =, F cos ($ a) dr., So that, by the principle of virtual work, we have, for equilibrium,, 0., mg sin (e~*).dr-Fcos (6-a).dr, Again, since dr is not zero, we have F cos (9 a) = mg sin (fl a)., Or,, , F=mg.tan(6-a),, , which thus gives the condition for the, equilibrium of the body., (Hi) Case of equilibrium of a system of two or more connected, bodies., Here, a slight complication arises in that, if we imagine the, bodies to undergo virtual displacements dr dr etc., under the action, 2, lt, offerees Fv F2 etc., acting on them, respectively, these displacements,, since they must be compatible with the constraints of the system,, cannot possibly be completely, arbitrary, but will rather be inter, related to each other, although, they wilf satisfy the relation, ,, , F .dr + F^.dr^+, l, , Fn .drn, , l, , =, , 0., , ..., , We, , can, however, tide over the difficulty by first writing, the above general relation for the different, impressed forces, , /, , down, , Fv F2, and then deducing, from the geometry of the system, the, inter-relation between the different virtual, displacements. This will, ,, , etc.,, , then enable us to obtain a set of completely arbitrary displacements,, , and hence the necessary conditions that the, , different forces must, satisfy for equilibrium of the given system of bodies., Let us, for example, consider the, of two bodies,, , and, , W, , 2,, , tied to the, , ml, equilibrium, round, a, smooth, peg, string, passing, or pulley, (Fig. 248), and subjected to two, impressed forces Fl and F% respectively, as, , two ends of a, , shown., Taking, for convenience, the virtual displacements dr: and rfr2 of the two bodies in the, downward direction (which is quite compatible with the restraints imposed upon the system), we have, for equilibrium, on the principle, of virtual work,, , f, x m%, I, , F*, , q, , (Fl, , +m g).dr +(F,+m g)dr =, 1, , l, , 2, , Now, an examination of the, , 2, , 0., , situation at, , once reveals the relationship between dr^ and, viz., that the string being inextensible,, , dr^
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PROPERTIES OP MATTER, , 408, , So, , that, the, , above relation reduces to, , =, , ).drl, , =, , 0,, , whence,, Fi+mg F2 2 g,, which thus gives the necessary condition for the equilibrium of the, , two, , +m, , bodies., , must be noted here that if we were to use the old convenmethod to determine the equilibrium condition of the bodies, we should have had to take into account the tension in the string., It, , tional, , Since, , not an impressed force,, , it is, , lations,, , we can, , clean ignore, , it, , in our calcu-, , on the principle of virtual work., , We, (zv) Relation between Equilibrium and Potential Energy., express relation /, of case (Hi) above, in Cartesian form, and write, for the/?th particle, in a system of particles,, idxp +\.dyp +li.dzp>, dr,, , may, , Vp, , and, , =, =, , iFA^+jF^+k.Fz/,, , where dxp dyp and dzp denote the virtual displacements of the particle, in the x, y and z directions respectively, and Fxp Fyp and Fzp the, components of the applied force F^ in these directions., ,, , ,, , ,, , The, , principle of virtual, , work may now be expressed, , in, , the, , form, , 2, , (Fxp .dxp +Fy p .dyp +Fzp .dzp ), , == 0., , P=*I, , If the forces be conservative (i.e., for which the total mechanical, energy remains constant or 'conserved'), so that there exists a potential, energy function Vp (xp .yp .zp ), such that, , Then, the total potential energy of the whole system of particles, given by, 2?, , Vp, , is, , K., , *=i, , The, , principle of virtual, , work then takes the form, , 2 dVp =, , 0,, , /-i, , for the virtual displacements considered, the change in the total, potential energy is zero., In other words, for the equilibrium position, the potential energy, must either be a maximum or a minimum for only then, can the variation due to a small displacement be zerof. In the latter case, the, equilibrium is said to be stable, and in the former, unstable., , (i.e.,, , ;, , *In vector notation,, , i,, , j, , and k denote vectors of, , unit magnitude*, , inx, y, , and z directions respectively., mathematically speaking, it is not quite correct to say so, and, reasonably assert, therefore, is that the potential energy will have a, stationary value: In most mechanical problems, however, it is quite enough to, confine our attention to maxima and minima., fStrictly,, , all, , we can
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VIRTUAL WOBK, , 409, , Thus, taking the simplest case of a single particle, acted upon, and resting on a smooth surface, we have, potential energy given by V, mg.z, taking the z-axis vertical,, , t>y the force of gravity, , =, , z being positive upward., , Now, if the particle be in equilibrium on a concave surface, the, total force acts on it normally to the surface, so that z and, therefore,, V is a minimum, and hence the equilibrium, stable for, any finite, displacement of the particle on the surface tends to increase its potential energy. But if it be in equilibrium on a convex surface, we, liave z and, therefore, F, a maximum, and hence the equilibrium unstable, for, a finite displacement of the particle on the surface now, tends to decrease its potential energy., ;, , ;, , Let us now consider the application, (v) Tension in a Flywheel., of the principle of virtual work in a problem on the mechanical, transmission of force and calculate the tension in a rotating flywheel., If the radius of the ring-shaped flywheel be r, , and, , its, , angular, , velocity, to, it is clear that an element Sm of it will experience a cen2, trifugal force Sw.ro>, outwards, at right angles to its circumference,, .and the ring will thus tend to stretch itself, i.e., will be in a state of, ,, , stress., , If we imagine each element of the ring to suffer a radial virtual, 2, displacement dr, the virtual work done by it would be Sm.ro> dr. And,, ., , therefore,, , total virtual, , Obviously,, its, , two ends, , work done by the, , if, , will,, , ring, as a, , whole, , =, , (8m.rar.dr)., , the ring be cut at any point on its circumference,, this centrifugal force, fly apart through a, , under, , distance, , In order to hold the two ends of the ring together, obviously,, an external force F, equal to the tension in the ring, will have to be, applied to, , inwards., , And, therefore,, work done against the force F, F.27T.rfr,, For equilibrium, therefore, we have, in accordance with the, it, , virtual, , principle of virtual work,, , Or,, , Z(Sm.ra>*.dr)-F.27r.dr, rajt.drZSm, , Now,, , E&m, , So that,, , SB, , M,, , r.rfr., , =, , 0., , Or,, , 2(Sm.ra>*.dr), , =, , F.2ir.dr., , F.Zv.dr., , the mass of the ring or the flywheel., , M _ F.^.dr., , Or,, , F=, , '-- - ---,, , which enables the tangential tension in the flywheel to be easily, , cal-, , culated., , The above examples, , hoped, suffice to bring home to the, importance of this rightly celebrated, , will, it is, , student the great utility and, principle of virtual work., , SOLVED EXAMPLES, a, , A, , gramophone disc is set revoking in a horizontal plane and reaches, steady state of motion of two revolutions per second. It is found that a small, 1.
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PROPERTIES OF MATTER, , 410, , coin placed on the disc will remain there if its centre is not more than 5 cms. froiro, the axis of rotation., Explain this and calculate the coefficient of friction between the coin and the disc., (Oxford and Cambridge Higher School Certificate), , When the small coin is placed on the rotating gramophone disc, it is, subjected to a centrifugal force tending to pull it outwards, away from the, axis of rotation of the disc. Its motion is, however, opposed by the force of, friction coming into play in-between the surfaces of the disc and its own. But,, as the distance of the coin from the axis of rotation increases, the centrifugal, a, force pulling it outwards also increases, (being equal to /wrw , where m is its mass r, and is just, , its angular velocity and r, its distance from the axis of rotation), balanced by the limiting friction between itself and the disc, when it is at a, distance of 5 cms. from the axis of rotation., Beyond this distance, the centrifugal, force on it is greater than the limiting friction between it and the disc and it,, therefore, moves outwards., TV in 1 rotation, the disc, f, ~, r = 5 cms. and co, 4rc., .,, ,, Here,, ,, describes an angle 2*L, , =, , ', , So, , that, if, , m be the mass of, , the coin,, , I, , we have, , = mx 5 x (4*r) 2 -* 80 r.*.m dynes., And, clearly, the normal reaction (R) of the disc on it = its own weight mg., And, therefore, the limiting friction between the coin and the disc = PR = P.mgr, mrco 2, , centrifugal force on the coin, , where, , /* is, , the coefficient of friction for them., , Since the coin, , is just in, , =, , equilibrium here,, , we have p.mg, , =, , 80, , 7i, , 2, , m., , 2, , 807i, Or,, P-g, 2, * 80 *'/980 = 4n*/49, r gTaking, = 980 cms.jsec*., /* = 80* /, whence,, ', I* = '8054,, L, Or,, Thus, the coefficient of friction for the surfaces of the coin and the disc, is equal to '8054., Define the coefficient of sliding friction., 2., A uniform ladder of length 21 and weight, rests against a vertical wall, with its foot on the ground at a distance / from the wall. If the coefficient of, friction between the wall and the ladder and between the ground and the ladder is0*4, find how far up the ladder a man of weight 2W can ascend without disturbing:, (Joint Matriculation Board), equilibrium., For definition of coefficient of sliding friction, see 190, page 395., ,, , W, , that, , it, , Here, let PQ be the ladder resting against the wall AB, (Fig. 249), such,, makes an angle with the ground, the ladder being in the vertical plane,, perpendicular to the wall, with, , W, , acting vertically, point O., , downwards, , its, , at, , weight, , mid-, , its, , Let the man climb up a distance x, on the ladder, up to T before the ladder, starts slipping., Then, clearly, the weight, 9, , 2W, , And, , of the, , man, , acts vertically, , down, , at T., , on the, point of slipping, its upper end P moves, downwards towards B and its foot Q awa>, from B, the frictional forces at P and Q act, away from and towards B respectively. If, R and R' be the normal reactions of the, wall and the ground at P and Q, and P 9 the, since,, , when the ladder, , is, , just, , coefficient of friction in either case,, limiting frictional force at P, , =s, /*/?, along the wall, away from, PR', along the ground, towards B., In the equilibrium position of the ladder, clearly., R t*R\, and jR'-f f<R 3^., ...(/), R in relation (//) we have, of, value, the, that, substituting, , Fig. 249., , and, , limiting frictional force at, , Q, , ..(///), , Or,, , have, , B, , =, , W+IW, , So, , we, , ., , .07]
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VIBTUAL WOBK, , FBICTION AND LUBBICATION, whence,, , Now,, , 3FP7(l-f /**), , where,, , /, , Or,, , Or,, ', , So, , 2*r, , Wl+2Wx-2pRl =, , that,, , Or, substituting the value of, , /+2* =,":!;,, ,,a, , Or., , i, , Or,, (j|, Ur, , And,.*., , ...(v), , ;, , moments about Q, we have, , WxQM+2WxQN RxPB+pRxQB., Wxl cos b + 2Wxx cos = jRx2/ sin 0-f/*Rx2/ cos 0,, QM = cos 6, QN = * cos 0, P = 2/ sm and QB = 21 cos 0,, - 2/*/?/ cos 6 = 2RI sin 0., Jf7 co s, 4- 2fP* cos, cos, (HP7+2Wx-2/*/tf) = 2RI sin 0., , Or,, , Or, , /?', , taking, , 411, , ,, , ,, , =, , from relation, , -, , (+v, , ^7r, ZX, , ', , ,, , R, , 2x, , =, , ., , (vi), , above,, , we have, , 6, , J.t', 16, +, , (-4+^3)., N, , [, L, , v *--4., , 1<16, , 2'4x2'132/, -, , (4-411, , -, , 1)/, , ./2'4x2-132, , ~, , /I, , 3-411, , /,, , ., , /, ', , 1-16, , Or., , 2>/3, , ,\, , V, , -, , \, , 1, , 116, , ^, , whence,, , 1, , =, , ~-, , j, , -, , 1'7055, , /., , Thus, the man can ascend up a distance i'7055/ without disturbing, equilibrium of the ladder., A rough plane is inclined to the horizontal at an angle 6, where 6 is les, 3., is placed on the plane., than the angle of friction A, and a body of weight, Calculate the minimum horizontal force required to make the body move (/) down the, the;, , W, , plane and (u) up the plane., at an angle, (/) Let AB be the rough plane, inclined to the horizontal, and let the body of weight, be placed on it,, as shown, (Fig. 250)., , r, , W, , Then, since Q is less than the angle of, friction A for the body and the plane, it will, not by itself slip down, (see page 398)., Let P be the least horizontal force, required to mate the body just move down, the plane., , Then, clearly, resolving Pinto its two, rectangular components, we have its component P cos 0, acting downwards, along the, plane and its component P sin 0, acting, upwards, perpendicular to the plane-, , *, , B, Fig. 250., , W, , into its two rectangular components, we have its, Similarly, resolving, sin, component, acting downwards, along the plane and its component, cos 9, acting perpendicular to the plane., , W, , Thus, since there, we have, , is, , W, , no motion of, , the plane,, , the, , in a direction perpendicular, , body, , to, , R~ W, W, , normal reaction of the plane, cos - P sin 0,, the downward force along the plane, sin 0-f P cos 0., Since the body is just on the point of moving down the plane, it is clear that, , and, , WsinQ+PcosQ =, , where, , F is the, , limiting friction, along the plane., , F,, , ..., , (/), , between the body and the plane, acting upwards.
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PROPERTIES OF MATTER, , 412, Now, F, , =, , /*/?,, , where, , the coefficient of friction for the surfaces of the, , /* is, , body and the plane., , =, , tan A, where A, Again, P, of surfaces. So that,, , F=, , i*R, , Substituting this value of, , =, , tan X, , F in, , the angle of friction for the given pair, , is, , (W, , 0-Psw, , cos, , 0)., , above, we have, tan X (W cos, P sin, , relation, , ( i), , W sin 0-f-P cos &, = W tan X cosQP tan X sin QW sin, P = W tan X-P tan X tan Q-W tan, W tan \-W tan, P+Ptan X tan 8, = W (tan X-tan, P(l-f tan \ tan, tan A tan, W 1-htaw, X tan, , P cos, , Or., , Or,, Or,, , 0)., , 0., e., dividing both, cos 6., [sides by, , 0., , |~i, , 0., , Or,, , e), , Or,, , o)., , ., , ', , = tan (\-0)., Clearly, (tan X-tan 0)/l + /* A tan, So that,, H'tart(A--o), Thus, the least horizontal force required to make the body move down the, , P=, , plane, , is, , W/an(A-8)., In this case, again, let P be the least force required to make the body, plane. Obviously, it will now have to be applied in the oppo.site, *, to that in the first case, as, A, " showndirection, in Fig 251, and the frictional force F, will now act downwaids along the plane., , (//), , lust, , move up the, , p, , riS, , ix, , ttf, lll, , \\^, , Xx, , ", , Resolving P, as before, into its two, rectangular components, we have, component P cos 0, acting upwards, along, the plane,, an<J component P sin, acting downwards, per', , ~^WCOS6, Ps in ft, , pendicular to the plane., So that, the normal reaction on, cos f Psin 0., is now R, , .WAT, , W, , the, , plane, , Fig, 251., for equilibrium of the body, we have, P cos, sin -I-, , F=, , =W, , P, , Or,, , cos, , =, , Jf, , j/ii, , 04- ta/z A, , (W, , cos, , W sin Q + V-R., , Q+Ps!n, , 0)., , p-', , P, , ~, , tanX and, , _, , = W sin Q+W, , Or,, , F=, , Or,, , W ta, , P-P tan \, , Or,, , P(l-tan, , Or,, , X, , tan X cos 64-P ta, taw, , 0), , A4-Pta A, , =, , H^ (tan A + taw, , =, , fiK, , P, , s//f, , {)., , 0., , 0)., , (taw A -f tart 0)., tart, , yy, , whence,, , X, , ta, , A-f tart, ', , 1-tart A, , Since (///, , the, Or,, plane is, , X+tan Q)/i--tan X, , minimum, , W tan(\ +, , 4., , (a), , tan, , =, , /#/* (X-f-0),, , horizontal force required to, , make, , we have, , the, , body, , just, , move up, , the, , 0)., , Define the angle of friction., , A, , uniform rod rests in limiting equilibrium In contact with a horizontal, vertical wall, the rod being in a vertical plane which is perpendicular to, the wall. If the wall and the floor be equally rough, prove that the angle between, the rod and the wall is twice the angle of friction., (London Higher Secondary Certificate), , Boor and a, , (b) What would be the value of the angle in the question above, (/) if the wall, and the floor be unequally rough, (//) if the wall be smooth, (in) if the floor, tie smooth, and (iv) if both be smooth ?
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WORK, , VIRTUAL, , FRICTION AND LUBRICATION, , 413, , with, (a) Let PQ be the rod, of length 2/, resting in limiting equilibrium,, horizontal, upper end Pin contact with a vertical wall AB and its foot with a, floor, (Fig. 252), such that it lies in a vertical, plane perpendicular to the wall and makes an, angle o with it., {#**, its, , W, , acts vertically, its vi eight, at its mid-point O, with the, and R' of tjie wall and, normal reactions, the floor acting at P and, respectively,, as shown., , Then,, , downwards, , R, , ', , ', , Q, , \+y, , Q, , Since ths wall and the floor are, equally rough, the coefficient of friction for, either of them and the rod must be the same., Let it be /*. Then, the frictional forces at P, and Q are clearly t*R and /*', acting in the, directions shown, as explained in answer to, Ex. 2 above., Fig. 252., , The rod being, , in limiting, equilibrium,, , we, , clearly, , have, , ..., R - i*R', (/>, ..(/) and i*R+R' =, So that, substituting the value of R from expression (i) in expression (//),, 2, we have, R' + V**R' = W., ..(///)*, .K'U-f/* ) = W., Or,, , W, , ., , =, , Now,, , taking, , ^x, , Or,, , R x 21, , / .y/Vf, , Wl, , Or,, Or,, , *, , *=*, , $(W - 2t*R), , sin, , tan, , that, substituting the value of, , -, , ~, , R, , from, , =, , X 2f, , ..., , n, , (, , -, , 0-, , V ),, , (vi>, , 2/?/ ro^ 0., , 2RI cos, , 0., , relation (v) above,, , we have, , *, , tan, , '-, , Or -, , Now,, , cos e-f t*R, , Q--PR.21 tin, , sin, /, , sln, , Or, So, , _, , ^ /^//([.j.^, moments about Q, we have, , whence,, , tan A,, , /*, , where A, , tan, , And, therefore,, , (r, , pp3? r, , (tH>, , the angle of fnet i OIL, , is, , *, , =*, , Thus, the angle 8 (hat, , ....., , -i, , rzy^'44, the rod makes, , tan, , ^, , whence, $, , 2A., , with the vertical wall is twice the angle, , of friction., // the wall and the floor be unequally rough, the coefficient, (i), (b), of friction for the rod and the wall will be different from that between the rod and', the floor. Let these be /* and v-' respectively. Then, clearly,, 9, , R = p'x, , R'+WR', , Or,, , So, , that,, , /*'H'(1+/*/*'), , W., p'W., , an d, , R'+pR, , Or,, , JR'(l+ /*/*'), , Or,, , J?(i, , + /*/*'), , **, , =, , W., W., /*W., , ['.*, , A*'U' == >?., , ^=, , Or,, , And, /;, , /7n ;, , -, , --, , _, ., , [See above,, , ., , .(v///>, , Or,, /, , 1, /fln- [2^ /(l, rod now makes with the wall an angle, of, friction for, the, have, co-efficient, be, we, the, wall, shall, smooth,, If, (//), rod and the wall equal to zero, i.e, /* * 0., , Thus,, , //;tf, , ,
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PROPERTIES OF MATTER, , 414, , And, therefore, expression, , above become!, , (vitf), , =, , tan e, , 2/*', , *, , 2 tan, , \., , the tangent of the angle that the rod makes with the wall is twue the tangent of, the angle of friction., rod and the, (I'M) // the floor be smooth, the coefficient of friction for the, i.e.,, , floor is equal to zero,, , =, =, , i e., /*', , tan, , So, , 0., , that,, , 6, , Or,, , 0., , =0., , Thus, in this case, the rod can rest only in the vertical position., friction for, (iv) If the wall and the floor be both smooth, the coefficient of, the rod and the wall as well as that for the rod and the floor is equal to zero, i.e. I, and also /*' = 0., P, = 0,, 8 = 0Or,, Again, therefore, we have tan B, , a, , Thus, here also the rod cannot possibly rest in any inclined position*., If a ladder rests in the limiting position against a vertical wall and, 5., horizontal floor, how far can a man climb up the ladder before the ladder starts, , slipping ?, , Let the ladder PQ, of length /, rest in the limiting position against a vertiAB and a horizontal floor, as shown in Fig. 252, making an angle 6 with, of the ladder acts vertically downwards at its mid.the wall. Then, the weight, So that, using the same symbols as before, in Ex. 4 (6), above,, point, as shown., cal wall, , W, , we now have, , =, , R', , where, , w, , is, , R' + vR = (W+ w\, and, man, also acting vertically downwards., , P'R', , the weight of the, , R'+w'R, , And, therefore,, , (W+ w)., + w') -, , f, , Or,, , t*, , -, , *(l + w'), , Or,, , i*'(lV+w), , 9, , *'(1+^'), , Or,, , R'(l, , - (W+ w)., , t*'(W+w-)., f, , R-, , whence,, , (, , +^-, , ['-', , W=*, , *[, Now, if the man can climb a maximum distance x up the ladder before, slipping just occurs, we have, using the same symbols as in Ex. 4 (b) above, and, , moments about Q,, , taking, , Wl, Wl, , Or,, , sin, , + wx, , sin, , Or,, , ^, Or, , sin B, , ., , 21 cos, , sin, , Q+pR.21, , sin $, , =, , =, , sin Q., , R.21 cos, , 2RI cos, , Q., , $., , 2RI, tan 8, tan B, 2A*', , ., , ., , =R, , Q-pR.21, 6 (Wl+wx+lpRl), , =, , But,, A, , Q+wx sin, , sin, , =, , 2p'l\-w>',, 2RI, , [See Ex. 4 (b), above., , -, , And, therefore,, 2, , Or,, , WWl + lv'wx = 2Rl2w'Rl+4w'Rl = 2Rl+2w'Rl., , Or,, , Substituting the value of, , Or,, Or,, whence,, Thus, the, , a maximum, , R, , from above, therefore, we have, , 2l, , 2/*VA:, , x, , =, =, , 2p'wl., /., , man, , (irrespective of his weight) can climb up the ladder to only, distance equal to half its length before slipping occurs., , 6. A framework ABCD, consisting of four uniform, freely jointed rods, each, of the same length and weight w, is hung from A, and the corners A, C are connected by a string. Find the tension in the string., Let/4Cbe the string, connecting the opposite corners of the framework, ABCD, formed by the four equal and uniform rods, AB, BC, CD and DA, (Fig., 253), in which the tension is to be determined., , *This, , on, , the rod, , is, , obvious otherwise also, , do not, , all, , meet, , in, , ;, , one point., , for, in this case, the three forces acting
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FBITCION, , VIRTUAL WORK, , AND LUBRICATION, , 415, , downwards, , Clearly, the weight w, of each rod acts vertically, midpoint, (the rod being uniform)., , at, , its e.g.,, , i.e., at its, , Then, if 21 be the length of each rod and, the angle it makes with AC, we have, cos d, 21 cos 0., API, a/2/, whence, a, (P being the mid-point of AC, and AP, d)., ,, , AD =, , =, , So that, .4C = 2AP = 2a, Now, let x be the, 4-g.,, , And,, cods, , A, , 2.2/ cos, , x, , =, , /, , aw 0., , 4/, , depth below A, of the, , of each of the two upper, , clearly,, , And, , =, , cos, , Then, , rods., , 0., , =, , AC, , * 4J cos, 4*., 9, therefore,, since the e.g., of each of the lower, pair of, lies a distance x above C, its, depth below, , 4x-x =, , is, , 3x., , If, , we, therefore, imagine the whole system, to be displaced downwards a little, so that the, <c-g. of the, upper rods is shifted through a, KJistince 8x, we have, virtual work done by the weights, of the two, ~ 2w*x., Mpper rods, Similarly, virtual work done by the weights of the, , two lower rods, 6w.8x., 2w.8(3x), And, therefore, total virtual work done by the, =, 2w 8x+6w.$x = &w.8x., weights, And, if T be the tension in the string, we have, , virtual work of the, T$ (4jc) = -47 Bx, string =, the negative sign indicating that Tacts in the, opposite direction to the displacement of C., Since the system is in equilibrium, we have total virtual work = 0., ', , ', , whence,, r = 2w., Thus, the tension in the string is, 2w, i.e., equal to twice the weight of, each rod., A uniform rod oflength 27 lies in equilibrium over a smooth peg, with, 7., nts lower end resting against a smooth vertical wall. If the, peg be at a distance d, from the wa U, show that the rod is inclined to the wall at an, angle sin- \d\ 0*, Let PQ be the rod, resting in, equilibrium over the smooth peg K and, against the smooth vertical wall AB, such that it makes an angle, with the wall, ', , at P, (Fig., 254)., , Obviously, the weight, , downwards, , at its e.g.,, vertical height, x,, , i.e.,, , w, , of the rod acts vertically, at its mid-point 0, at a, , MN, above the peg K, where, x = MN =* P#~PM., = OP cos, f v PN, = OP cos QKM cot, , [and, , *, , T, , v OP, , Differentiating this with respect to 0,, , we have, , x, , Or,, , dx, , /, , =, , Since the rod, , cos, , /, , (, , is, , And,, , therefore,, , Or,, , djsin*, , =, , B-d cot, , sin, , 9., , $+d cosec2, , in equilibrium,, , d cosec* 0-/, /, , sin, , sin, , whence,, , 0), , ., , PM = KM cot, , 0., , and, , d., , we have dx, , =, , /, , 0,, , =, , 0., , 0., , sin* 6, , =, , d\l., , Or,, , Thus, the rod makes an angle siir\d\\)* with the, its position of equilibrium., , wall, in, , fig. 254.
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416, , PROPERTIES OF MATTER, , W, , An elastic ring of weight, rests on a smooth cone of semi-vertical, Find the tension in the ring., Let the elastic ring be in the position shown in Fig. 255, with its centre, at a depth h below the vertex A of the cone, so that its radius r =? h tan 0., 8., , angle, , O, , 6., , Clearly, the forces operating on the ring, ',, acting vertically downwards, weight, at its centre of gravity or centre O, (ii) the tension, T of the ring acting along the circumference of the, ring and (til) the normal reaction, of the cone,, acting at right angles to the surface of the cone*, its surface being smooth (and *hence there being, no frictional component along its surface, , are, , W, , (/) its, , R, , ., , Now, imagine the ring to be given a:, small downward displacement S/z, so that its., distance from th* 1 apex A of the cone now becomes /j-f Sh and hence its radius becomes, (/H Shi tan, Its, , (h+h), and, Fig, , its, , tan, , 9~h, , tan Q=8h. tan, , circumference, by 2nSh tan, , Clearly, then,, , 255., , virtual, , 0., , radius thus increases by, , work done by the weight, , 0,, , Q., , we have, , W of the ring, , = WM., , T of the, , and, ring -^-T.2n8h. tan 0,, ,,, against tension, the negative sign, indicating that the displacement here takes place in the direction opposite to that of T., And, because the ring moves at right angles to the direction of R, no work, is done by or against R., Since the ring, , We,, , is, , in equilibrium, the total virtual, , work done must be zero., , therefore, have, , 0-0., , This, therefore,, , is, , Or,, , 7', , 2*8A tan, , = WM,, , the tension in the elastic ring., , EXERCISE X, , A man, , weighing 140 Ibs. climbs up a uniform ladder, 20 ft. long and, 70 Ibs. in weight, which rests against a rough vertical wall at an angle of 45. If, the coefficient of friction at each end of the ladder is 0*5, how far will the man, be able to climb up the ladder before it begins to slip., (Northern Universities Higher School Certificate), Ans. 13-0/r., 1., , A, , uniform rod is in limiting equilibrium, one end resting on a rough, 2., lorizontal plane and the other on on equally rough plane inclined at an angle, If A be the angle of friction and the rod be in a vertical plane,, * to the horizon., ihow that the inclination of the rod to the horizon is given by, , 3., , coefficient, , Distinguish between static and sliding (kinetic) friction and define the, , of sliding friction., , How, md, , would you investigate the laws of sliding, , friction, , between wood, , 1, , iron ?, , An iron block, mass 10 Ibs., rests on a wooden plane inclined at 30* to, he horizontal. It is found that the least force parallel to the plane which causes., he block to slide up the plane is 10 Ib wt. Calculate the coefficient of sliding, riction, , between wood and, , iron., , (Northern Universities Higher School Certificate^, Ans, 0-5&
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FRICTION AND LUBRICATION, , VIRTUAL WORK, , 417, , 4.. A uniform ladder, 13ft. long and weighing 60 /&$., rests with its, upper end against a smooth vertical wall and with its lower end 5 ft from the, wall on rough ground., By a graphical method find the magnitude and direction, of the reaction at the foot of the ladder. Check your result by calculation., What is the least coefficient of friction between ths ground and the ladder neccessary to maintain equilibrium ?, (Cambridge Local Higher School Certificate), Ans. 61 '3 lb wt. at 78* 10' to the ground 0'2l, To determine the magnitude and direction of the reaction at the, [Hint., foot of the ladder, we must remember that the resultant reaction of the ground, passes through the point of intersection of the lines of action of R and the 50, lb. wt. since, as we know, three forces in equilibrium must all be concurrent., The angle X (i.e.* the angle between the resultant ground reaction and the vertical) is then measured and tan A = /* determined.], 5., Explain the meaning of the term coefficient of friction and describe an, experimental method of measuring it., A rectangular block with a square base of side 10 cm. rests on a horizontal surface., If a horizontal force is applied near the bottom of one vertical face,, ;, , ;, , ., , t, , the block slides. If the force is applied near the top, the block topples over., When the force is applied at 20 cm. from the bottom, the block sometimes slides, and sometimes topples. Find the coefficient of friction between the block and, the surface., (Cambridge Higher School Certificate), Ans. 0-25., , W, , A uniform rod of weight rests with its me end against a rough in6., clined plane AB y of inclination a, and the other end against a smooth vertical, If e be the inclination of tne rod to the vertiwall ED, B being higher than A, 2 tan (A, cal in the limiting position of equilibrium, show that tan, )> where, A is the angle of friction for the plane., , =, , Define the terms friction, limiting friction, angle of friction (A) and, of friction (/*), and show that the coefficient of friction (/*) = tan \., What is meant by the cone of friction ?, A square framework formed of uniform heavy rods of equal weights, 8., is saspended from each of the three lower, W, is hung by one corner. A weight, corners <tnd the shape of the square is preserved by a light rod along the horizontal diagonal., Find its tension ?, (Allahabad and Delhi), Ans 4 W., 7., , :, , coefficient, , W, , 9., Three equal smooth pencils, each of weight W, are tie^l together by, a single loop of fine inextensible cotton S3 that each touches the other t>vo., Prove that the bundle can rest in stable equilibrium on a smooth table only if the, breaking tension of the cotton is not less than JFV3/6., (Oxford Scholarship and Higher School Certificate}, , A, , 10, cylinder of radius 1 cm. and length 4 cms. is standing on end on an, inclined plane, the angle of which is gradually increased. If the coefficient of friction between the cylinder and the plane is 03, find whether the cylinder will, slide or topple first., Ans. Slides first.
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CHAPTER, , X1F, , FLOW OF LIQUIDS VISCOSITY, Rate of Flow of a Liquid. A liquid, for our present purtaken to be perfectly mobile and practically incompressible, and, therefore, the same amount of it fl^ws across every section of a, tube in a given time. The rate offlow of a liquid is, therefore, defined, as the volume of it that flows across any section in unit time., 200., , pose,, , is, , If the velocity of flow of a liquid be v, in a direction perpendicular to two sections A and B, (Fig. 256), of area a, and distance /, apart, and if t be the time taken by the, jj, j|, to flow from A to B, we have, liquid, f^jp, , Obviously, the volume of liquid, flowing through the section A B, in, time, is equal to the cylindrical column, Fig. 256., AB, vtxa. This, there/Xtf, or, the volume of the liquid flowing across the section in time t, , =, , fore, is, , =, , ., , f n, , .1, , /*,., , rate cf flow of liquid, , /., , =, =, , vtxa, , =, , vxa., , \elocity of liquids area, section of the tube., , of cross -, , Sometimes, the rate of flow of a liquid is also expressed in terms,, of the mass of the liquid flowing across any section in unit time ; so, that, in this case,, rate, , =, , offlow of liquid, , velocity, , ==, , mass of liquid flowing across any section, per unit time., , of Hquidxarea of cross-section x density of liquid., , =, , vxaxp., , Lines and Tubes of Flow. In a simple flow of liquid, i.e.,, when it is not turbulent but steady , the velocity at every point in the, liquid remains constant, (in magnitude, as well as direction), the, energy needed to drive the liquid being used up in overcoming the, In other words, each particle, 'viscous drag" between its layers., follows exactly the same path and has the same velocity as its predecessor and the liquid is said to have an orderly or a stream-line, flow., In such a case, if we consider a line along which a particle of, the liquid moves, the direction of the line at any point is the direction, of the velocity of the liquid at that point., Such a line is called a, stream-line., More correctly, a stream-line may be defined as a curve, the tangent to which at any point gives the direction offlow of the liquid, at that point for, it may be straight or curved, according as the lateral, in the latter case the, pressure on it is the same throughout or different,, pressure being greater on the convex side than on the concave one., 201., , ,, , ;, , 418
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FLOW OF LIQUIDS, , 419, , This holds goad, however, only so long as the velocity of the, liquid does not exceed a particular limiting value, called its critical, velocity, bayond which tha flow of the liquid loses all its steadiness or, orderliness, and becomes zig zag or sinuous, acquiring what is called, a turbulent motion. This may be, easily seen by introducing a small jet, :, -.-*, a tube, r --.-'":', of colouring, matter into, be, a, made, which, ., liqui'd may, through, ,, to flow with a gradually increasing, ^ ', when, as long as the velovelocity, -, , -', , -', , ., , ;, , city remains below its critical value,, we see only a thin streak of the, colouring matter along the axis of the, , ..., ^, , ', , tube., [Fig. 257, (a)], representing a, Fig. 257., stream-line motion, but when the, velocity reaches this value, the colouring matter takes a zig zag path,, [Fig. 257, (b)] 9 and later, when this value is exceeded, the colouring, matter spreads put in all directions, filling the entire tube, showing, that the motion is no longer steady or orderly but has become, 'turbulent'., The energy needed to drive the liquid is here dissipated,, for the nmst part, in setting up eddy currents in the liquid., , Consider two areas, A and B, at right angles to the direction of, flow of the liquid, (Fig. 258), and draw stream lines through their, boundaries, then, a tube, of the, This is known as a, liquid is obtained., , AB, , ;, , ^^j^vi?, , tube of flow., , -i**"^, , As explained above, the volume of, , passing through section A is, equal to that passing through section B., For, the sides of the tuba being everywhere in the direction of flow, of the liquid, no liquid can cross the sides but must enter or leave, through the ends. Since the velocity is constant over a section, i.e.,, the motion is steady, (if the tube be narrow), the volume of the liquid, entering section A is equal to a .v l per sec., and the volume of the, liquid leaving section B is equal to a^v 2 per sec., where a lt a%, and, v lt v 2 are the areas of cross-section and velocities at sections A and B, pjg, , liquid, , 258., , l, , ,, , respectively., , we have, p and, , where, , t, , ^i- v i-Pi, , p2, , =, , #2- v -P2>, , are the densities of the liquid at the, , pectively., , =, , The, , liquid being incompressible, p A, , i.e.,, , the volume of the liquid entering section, , pa,, , and, , two, , sections res-, , so, , we have, , A, , is, , equal to that, , leaving section B., , 202., Energy of the Liquid. Since a liquid has inertia, it possesses kinetic energy, when in motion., It is also subject to pressure,, and may also have potential energy, due to its position., have, thus three types of energy possessed by a liquid in flow, viz.,, , We, , (i), , kinetic energy,, (i), , (ii), , potential energy, and (Hi) pressure energy., know that K.E., mv\ so that the, , Kinetic Energy., , kinetic energy of a, , mass, , We, , m of a, , liquid, flowing, , with a velocity, , v,, , is
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PKOPEETIBS OF MATTER, , 420, , m=, , given by \ mv*. If we consider unit volume of the liquid,, density at the liquid, ani, tharefora, we have, , =, if we consider unit mass of the liquid, m =, kinetic energy per unit mass of the liquid = J, We have P.E. = mgh, Potential Energy., , 2, , kinetic energy per unit volume of the liquid, , And,, , the, , p,, , pv, , ., , and, therefore,, , 1,, , v, , 2, ., , so that, the poat a height h above the earth's, surface (i.e., in its gravitational field) is equal to mgh., Again, if we, consider unit volume of the liquid,, p, the density of the liquid,, and, therefore,, (/i), , tential, , energy of a liquid of mass, , ;, , m, , m =, , =, consider unit mass of the liquid, m =, P.E. per unit mass of the liquid =, P.E. per unit volume of the liquid, , But,, , if, , we, , (Hi), , of density, , p.g.A,, 1, , and we have, , gh., , Consider a tank A, containing a liquid, provided with a narrrow side tube T< of cross-sectional area a, properly fitted, with a piston P that can be, , Pressure Energy., p,, , smoothly moved in and out,, Let the hydrostatic, (Fig. 259)., pressure due to the liquid, at, the level of the axis of the side, tube, be p, so that the force on, ~, If* thereP- a, P iston * 8, "rlV: ":T5r lfr~ "^', fore, more liquid is to be introduced into the tank, this much, Fig. 259., force has to be applied to the, Let the piston be moving slowly inpiston in moving it inwards., wards through a distance x, so that the velocity of th liquid be very, small and there may be no kinetic energy acquired by it. Then, clearly, a volume of the liquid a.x., or a ma^s a.x.p of it, is forced into, the tank, and an amount of work p.a.x is performed to do so., This, work, (or energy), p.a.x, required to make the liquid move agdinst, it, thus becomes the enerpressure /?, without imparting any velocity, gy of the mass a.x.p of the liquid in the tank, for it can do the same, amount of work iti pushing the piston bade, when escaping from the, tank. It is referred to as the pressure energy of the liquid., ', , ~^, , ^, , =, , -, , w, , Thus pressure energy of a mass, , tf.x.p, , of the liquid, , is, , equal to, , p.a.x, and, therefore., , /.,,.., =, , pressure energy per unit mass of the liquid, , Now,, , if, , we, , consider unit volume of the liquid,, , pressure energy of volume a.x of the liquid, , and, , /., , p.a.x, -, , =, , of the, , liquid., , pressure, , =~j, , r~, , we have, , = p.a.x,, , pressure energy per unit volume of the liquid, , the pressure, , p, , %, , =, , ^-'--'-, , ==, , /?,
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FLOW OF LIQUIDS, , 421, , The three types of energy possessed by a liquid under flow are, mutually convertible, one into the other. For, consider a liquid of dena vessel, and let its, sity p contained in, depth be h, (Fig. 260). Then, pressure, due to the liquid column h at the bot~ HM"*-!, ,.h p.g. If we, tom of the vessel is p, ar_rLTT.", _^_, fL., IT" - Z"jnr~, -T"_, take unit mass of the liquid from the, bottom B to the surface A, clearly, h.g., units of work has to be done against, gravity, and, therefore, the potential, pig. 260., energy of the liquid increases by this, much amount or this much work is done by gravity if unit mass of, the liquid comes down through a depth h. Hence, potential energy, of unit mass of the liquid is equal to h.g., And, since pressure at a, A.p.g, and pressure energy per unit mass of, depth h, is given by p, the liquid, pressure/density, we have pressure energy per unit mass, h.g, potential energy lost by the liquid in, of the liquid = A.p.g/p, descending through h., Thus, we see that pressure energy and potential energy are convertible, one into the other, and, therefore, their sum for a liquid at, rest is constant., , =, , -._, , ;, , =, ~, , =, , Again, consider the flow of liquid through a tube, (Fig. 261). If, the liquid has a constant velocity, there is no resultant foice acting, upon it. But, if the flow is accelerated,, there must be a pressure gradient along, ^j>, the tube of flow. Let the change of pressure for a distance dx be dp, i.e., let the, , AB, , pressure gradient be dp/dx, which may be, taken to be constant for a short length of, the tube., , Fig. 261., , A to 5, the pressure decreases, pressure at the cross-section B,, that, be greater by Sx.dpjdx, if the small distance, be 8x,, i.e., the pressure at A will be/?, Sx.dpjdx. The resultant force on, the slice AB of the liquid will, therefore, be a.&x.dp/dx, where a is the, cross-section of the tube, (force being equal topressurexarea)., If the direction of flow be from, , from, , A to B., at A will, , If,, , therefore,, , p be the, , AB, , Let the velocity gradient along the tube of flow be dv/dx, then,, be the velocity at A, the velocity at B will be v+Sx.dvJdx, because the velocity increases in the direction A to B, and, therefore, increase in velocity through the distance S* will be Sx.dv/dx., If the liquid covers this distance in time S/, we have, ;, , if v, , Bt, , Or,, , Now,, , =, , v, whence,, &C/V,, the, v, in, limit,, dxjdt., , =, , =, , $x/8f., , =, , acceleration, rate of change of velocity and, therefore,, acceleration at the section, dv/dt, and mass of liquid in the, section =*= a.Sx.p ; so that, force on it, a.Sx p dv/dt, (because force, ss, , AB =, , =, , mass x acceleration)., But force on this slice of the liquid, - dp, , is, , also equal to a.Sx.dpjdx., , dv
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PROPERTIES OF MATTER, , 422, , ve sign merely indicating that the pressure and velocity gradithe, ent! are opposite in sign, i.e. % whereas the pressure decreases, the, velocity increases along AB., dvdx, dv, dv, ~, dp, v, Or>, ', *, dxldt - v, p', , And, , /., , Q fj, , dp =s, , ,, , __, , 1, , [PI, t, , ?-- =, , =, , =, , ~, , Or,, , p.v.t/v., , dp, , =, , dpjp, , Cv t, , (, , v.dv., , I, , j, C, , J vi, , P JPi, , -, , =, , -, , v.dv., , where/?i, p* anJ, , and velocities, , v,, v t are pressures, at sections 1 and 2, res-, , pectively., , -?*., , Or,, , P, J7j, , i, , ji/2, , -, , piessure energy and kinetic eneigy are ccnvertible, one into the other., Since pressure energy is also convertible into potential energy,, follows that the three types of energy are mutually convertible into, , i.e.,, , it, , each other., Bernoulli's Theorem and its Important Applications. Bertheorem states that the total energy of a small amount of liquid, flowing from one point to another, without any friction, remains cons-, , 203., , noulli's, , tant throughout the displacement^, , We, , have seen that pressure energy and potential energy of a, liquid are convertible, one into the other, and so are its pressure, energy and kinetic energy. It follows, therefore, that in any streamline* flow of liquid, the loss of energy in one form is, equal to the gain, of energy in another, or that the sura total of Its energy, viz.,, potential energy +prbss!ure energy -{-kinetic energy, , Or,, This relation, If, , we, , =, , v2, , hg+plp+%, known as Bernoulli's, , is, , divide relation, , (/), , p, A-f-, , Now, h is what, and \ v z /g, the, , is, , by, , g,, , 1, , v2, , =, , a constant., , C, a constant, , (j), , Equation*, , we have, C", another constant., , -+~rt, , ..., , ..., , (//), , called the gravitational head, plpg, the pressure head, , velocity head*., , Thus,, , head +preswre head+velocity head = a constant., We may, therefore, alsD state Bernoulli's theorem in another, way, viz., that at all points, in the stream-line flow of a liquid, the sum, of the gravitational head, the pressure head and the velocity head, gravitational, , remains constant throughout., It follows at once from relation (//) that if the flow of the liquid, be horizontal, the gravitational head h is a constant so that, here,, ;, , p, , 1, , i, , ^._2, , 2, , =, , a constant., , g, Similarly, from relation (/), we would have/?/p-fv* = a 'constant,, since the potential or gravitational energy hg would be a constant,, , Pf, , -, , Or,, , p + }p, , *For, the liquid must, city, , v., , v2, , =, , fall, , a constant, , through, , this, , ., , much, , ..., , (Hi), , height to attain the velo-
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FLOW OF LIQUIDS, , 423, , referred to as the static pressure of the liqtiid and Jpv 1 as, So that, we may express this result, velocity pressure, by saying that for a horizontal motion of the liquid, the sum of its static, and dynamic pressures remains a constant., , p, , He^re,, , is, , dynamic or, , its, , Thus, if in a liquid, flowing herizontall j the pressure and velocity, at one point be p t and v 1 and at another, /> 2 and v a respectively, we have, ,, , which show$ that pressure and -velocity (and, therefore, kinetic energy), can only increase at the expanse of one another, i.e., points of, maximum pressure correspond to thost of minimum velocity, and vicg, versa*. This principle is made use of in various important practical, applications, (see, , 204)., , Important Applications of Bernoulli's Equation, of Efflux of a Liquid. Let the surface of the liquid, Velocity, (/), be at a height h above the level of the orifice O in a tank, (Fig. 262)., If the tank be sufficiently wide, the velocity, at the liquid surface may be taken to be zero,, the pressure there being, clearly, atmospheric., Since the pressure is also atmospheric at the, orifice, where the liquid emerges, it plays no part, TF v be the velocity at, in the flow of the liquid., the level of the orifice, we have, considering, a tube of flow beginning at A and ending, 262^, at 0,, 204., , _, ', , ", , ^, , ', , total energy at, , A, , pressure energy-}- potential energy -{-kinetic- energy,, , =, , P.E. *=* gh and, because pressure at A, 0,, total energy at O, the level of the orifice, , K.E., , ~, , became pressure, , at, , O =, , 0,, , P.E., , ---=, , Sinca total energy remains the same, v, , 8, , =, , Or,, , hg., v, , whence,, , =, , t, , 0,, , arid, , 0., ', , ', , K.E., , =, , *", , ['*, , ", , ', , And,, , !, , -, , 1, , Jv, , ., , we have, , _v =, , 2gh,, , ^/ 2gH., , the velocity of efflux of the liquid at the orifice O., This result was first obtained by Torricelli (in the year 1644), and hence is known as Torricelli's Theorem, or the Law of Efflux, and, may be stated as follows, Th's, then,, , is, , :, , ,, , ,, , The velocity of efflux of a liquid through an orifice is equal to that, which a body attains in falling freely from the surface of the liquid to, the orifice., if the liquid had fallen freely through this heigh<, 8, 2gh, to be equal, velocity would be given by the relation, v, \/2gh, the same as obtained above., , For, clearly,, , h, , t, , its, , to v, , =, , =, , This ideal velocity is, however, seldom reached, for, perfectly free from friction (or viscosity)., This result is also true for compressible fluids and, as Hawksbee's law., , is, , no liquid, , sometimes referred, , is, , t<
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424, , PBOPBETIES OF MATTBB, , Now, the liquid-jet flows out in the form of a parabola^ and, takes time equal to <\/2hJg to fall through a height h t to a plane, in, level with the bottom of the vessel, striking the, plane at a distance d,, called its range, such that, , ~, , v, , x, , =, , 1, , V"?, , For a given height (h+h,) of the liquid column, this range will, And, obviously, if the jet were directed, upwards, it should theoretically rise to-the level A of the free surface, of the liquid. But, again, due to air-resistance and viscosity, the, height attained is actually less than this ideal one., %, , be a, , maximum when h=h lm, , Vena Contracta., , The whole of the liquid entering the, perpendicularly to it, but comes from all directions, as shown in Fig. 263, the stream-lines, near the edges being curved. The liquid coming, from the sides of the vessel, as it enters the, orifice, has" still a lateral velocity due to inertia, and continues to move inwards towards the centre, of cross- section of the jet, until the increasing, outward pressure is balanced by the atmospheric pressure at the jet. The liquid jet thus, F253, contracts at C y a little outside the orifice, to a, Contracta. It is here that the jet becomes uniform, Vena, called, the, neck,, and the velocity becomes the same throughout ^and it is this velocity, which is given by Torricelli's equation, feee, 204, (/), above,, (//), , orifice, , does not, , move, , page 423)., Obviously, the area of the jet at the Vena Contracta, , and, , is, , smaller, , found to be about *62 times the, The volume of the liquid passing out through the orifice in, latter., unit time is, therefore, equal to '62ay/2gh. This ratio between the, area of the Vena Contracta and the orifice is called the coefficient of, than the area of the, , orifice, , is, , contraction., , N.B. If outflow tubes of suitable shapes be used, the Vena Contracta may, be almost completely avoided, but the velocity of efflux always suffers a diminution in its value due to a loss in the kinetic energy of the liquid, caused by its, this diminution being quite independent of the, internal friction or viscosity,, , Vena Contracta., (Hi) Venturimeter. It is an arrangement to measure the amount, of flow of a liquid in a pipe, usually water, when it is called a, , venturi water-meter., , The principle underlying it is that when a liquid flows through a, tube of a varying bore or cross-section, the velocity and pressure vary, along the tube, the pressure being the least where the velocity is the, greatest,, , and, , vice versa., , For, if we have a tube KLM, with a constriction at L, (Fig., the, velocity of the liquid will be greater at L, the narrowest, 264),, or, Let the velocity at L be v,, part of the tube, than that at, and that at K be v*. Then, v/, v^., , M, , K, , ., , >, , Applying Bernoulli's theorem, we have
Page 433 : FLOW OF LIQUIDS, , 425, , (potential energy -{-pressure energy -{-kinetic energy) at, , =, , (potential energy, , Or, h l .g+, , Pl, , +4^, , P, , v, , L, , + pressure energy + kinetic energy) at, , K., , M, , K, -- = n-^i _L, , ?, , -, , =*,.*+ -*where /?,, p l and v/ are the height,, pressure and velocity of the liquid at L, , and k k p k and, , v^, their corresponding, values at K, p being the density of the, liquid, supposed constant, because the, ,, , taken to be incompressible., If the tube be horizontal, h l, hk, so that the above relation becomes, , liquid, , is, , =, , -, , p, , ", , ', , ', , ", , ^, , r', , -, , p, , >, , Fig. 264., , ,, , * V **, , '"^, , *, , p, , p, , Since v/, v^, it is clear that/?* >p /( /.e., the pressure at L is, less than at K., This can be shown by attaching a vertical tube,, connected to, at Zr and dipping it into a liquid, not miscible with, the one in KLM, when the liquid rises up in the vertical tube, as, shown at AB, and it will be seen that the narrower the bore at L, the, , KLM, , greater the rise of the liquid in the vertical tube., Let us now consider a pipe through which water is flowing, such, that it has a cross-section a l at, and 2 at L, (Fig, 265). Then, if, v 1 and Vj be the velocities of water at, , K, , K, , UL.*Jr 2L?!^^>:*--"!!i ^-1 T^yeT i^vf: -, , L, , and, , respectively,, fl, , =, , tf, , 8, , v 2 , [see, , 201, (page 419)., , whence,, , vt, , =, , And, since p is, above becomes,, , 1, , for water, relation, , a l v 1 ja 2, , ., , (i), , 1211, , 1, , Fi 8- 265, , lVl, , we have, , where p and p2 are the pressures at, , -, , K and, , L, respectively., , Or,, , substituting the value of V2, , ,, , I ff -gV - <A-*>Or, Ur>, , -, , we have, Or, , ', , T, , 1, , "i, , (, , ST, , - *- A), , -, , (, , v *, Vl, , 2, , whence v^fl,, ,, , \/ a^^', , S, , Thus, if we know t a, and (Pi~-p^> we can determine, the volume of the liquid flowing across the section, per second., and L is read directly on the, The difference of pressure (p l p^) at, vertical tubes AB and CD joined together to form a manometer, as, ,, , K, , shown., (iv), , Pitot Tube., , amount of, , This arrangement, , flow of water through a pipe, principle as the venturimeter*, , is, , measure the, based on the same, , also used to, , and, , is, , '
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426, , PROPERTIES OF MATTER, It consists of, , lower ends,, , two, , vertical tubes, with small apertures at their, the plane of the aperture of one tube, (Fig., 266), being parallel to the direction of flow of, water and the aperture of the other tube RS,, The rise of the liquid, facing the flow., column in the tube PQ, therefore, measures, the pressure at Q. And since the water is, stopped in the plane of the aperture S of the, , PQ, , tube, , RS, its velocity there becomes 2ero., Therefore, its kinetic energy is reduced from, 2, to zero, where v is the velocity of flow of, v, Fig. 266., increases by an amount iv 2 and the, therefore,, water. Its pressure,, water consequently rises to a higher level in the tube RS than in, If h be the difference of level in the two tubes, we have, *z?, , ^^i^Mim^m^^^m, , ,, , PQ., , jv, , 2 =t=, , Or,, , hg., , v1, , =, , 2g/?,, , =, , v, \/ 2gh., whence,, This multiplied by 0, the cross-section, tubes are placed, gives the Volume of water, section and the, flowing per second past that, amount of flow of water is thus easily, measured., of, Applications, (v) Other Common, Bernoulli's Theorem., It is a, 1., simple, The Steam Injector., , where the, , device to accelerate the ejection of the exhaust, steam from the cylinder of a steam engine, and, consists of a tube A, (Fig. 267), narrowing down, into a nozzle, at its lower end, inside another, tube B, having a side-tube C, which is- connected, , N, , ,, , to the cylinder of the engine., , A, , into A, and as, jet of steam is introduced, it issues out of the nozzle N, its velocity is considerably increased, resulting in a corresponding, fall in pressure, there, and the steam from the, engine-cylinder thus rushes into this region of, , reduced pressure, whence,, the lower, end of B., , WATER FROM TAP, , it is, , ejected out, 2., , The, , through, Filter, , Fig. 267., is also based on, used to reduce the pressure, , Pump., , It, , the same principle and is, in a vessel., Here, a stream of, , water from a tap,, flowing through a tube A, (Fig. 268), issues out in the, form of a jet from its narrow orifice O, which results, in a great rise in its velocity and proportionate fall, in its pressure, which is thus soon reduced to a value,, below that of the atmosphere The air from the, vessel, connected through a fide-tube B to this region, of reduced pressure, then rushes into it, and is, carried away by the stream of water as it flows down, through C., , 268, , In this way, the pressure in the vessel is, ultimately reduced to just a little above the vapour, pressure of water, in a comparatively very short time., If the inlet water tube be a twisted, instead of, a straight, one, the exhaustion proceeds more rapidly,, due to the rotating water-jet in the tube breaking up, more readily and mixing up easily with the incoming, TO SINK air from the vessel., ># The Atomizer. The atomizer or sprayer,, used for spraying scents etc. is yet another example, of a fall in pressure due to an increase in velocity.
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FLOW OF LIQUIDS, , 427, , Here, (Fig. 269), air is blown through a tube T, (usually by compressing a, rubber bulb) fitted on the tube at one end, which, when it rushes out of the, aperture O, where the tube narrows down, acquires high, AiR, velocity. The pressure in ths vicinity of O is thus greatly, reduced, and since O lies directly above the vertical tube,, in, vessel K, the liquid rises up, dipping in the liquid, through it, when, on issuing out of the aperture at the, top, it is blovn into a fine spray by the air stream, 1, , from, , T., , 4., The Attracted-disc Paradox. The following, a simple and interesting experiment, which the, student may well try for amusement at a small gathering, is, , at, , home., , is, , DEis a flat card-board disc (Fig. 270), over which, placed another flat disc BC, fitted with a tube A. the, ,/./A, opening of which is, in flush, , A, , B, , Fig. 269., , with BC., , On blowing, , air, , down through A, on, , to DE, the latter, instead of being blown, away frzm BC, as one might ordinarily, expect, sticks on to -it mofe and more, closely, and might even be lifted up a little., , This seeming paradox is, however,, For, as the air from A, rushes, narrow space in, through the, , easily .explained., , D, , between BC and DE, its velocity increases, and consequently the, there, pressure, so, that it soon falls below the atmospheric, decreases,, pressure on DE, which, thus pushes it up towards BC., This too is a familiar example of a fall of, \5fr The Bunsen-Burner., pressure due to increased velocity. For, as the gas issues out with a great, velocity from the fine nozzle, down bslow, ths pressure, fails in its immediate neighbourhood, and tht air is thus, sucked in through the hole O, (Fig. 271), and gets mixed, >, up with the gas., Fig. 270., , \r, , The Magnus Effect. If a ball, or a sphere be, rotated about an axis through it, perpendicular to the, ?, gftmrii, plane of the paper, the air surrounding it is also set into, 271., mot ton, -rth e streamlines taking the form of concentric, Figcircles in planes, parallel to the plane of the paper, their direction being the, same as that of the rotation of the ball, shown in Fig. 272 (a). And, obviously,, the rougher the surface of ths hall, the thicker the layer of air thus set into, L, , ., , TTTfftt, , ^, , motion., If, however, the ball be given only, aside the air in front of it, to make rrom foi, , ', , liraar, Jf,, , forward motion, it pushes, and this displaced air then, , flows along its sides on to its, back or the rear end, the, form of the streamlines being, as shown in Fig. 272 (b)., , And, finally, if the ball, be given both, a rotatory and, circular motion simultaneously,, , (a), , Fig. 272., , it, , is, , clear, , from, , Figs., , 272 (a) and (b) that the, streamlines due to the two, motions run in opposite directions on the underside of, the ball, but in the same, direction on iti upper side., Thus, there is a decrease of, velocity or an increase of
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*ROPBRTIES OF MATTER, , 428, , or & decrease of pressure on its, sure, on the lower, and an increase of velocity, in the lateral pressure on it,, upper, side. The ball, as a result of this difference, takes a curved p^h which is convex towards the greater pressure side. This is, what is called The Magnus Effect and is easily observed when a tennis or a golf, ball is given a spin., , We have already seen, Bullet., directional, for, that, stability, it is, 46, (//), page 98],, giving, {, desirable to give the shot or bullet a rapid *spin* about an axis,, is achieved by, along its direction of motion, and how this object, or, a, of, barrel, rifle,, the, grooves, cutting, spiral, by, (i.e.,, gun, 'rifling, ~*1., , The Cylindrical Shape of a, , ?, , 9, , inside, , it)., , Now, if the bullet were spherical in shape, there will come, about, as explained above in case 6, a difference in the lateral pressure on it during its passage through air, on account of its simuland the bullet, taneously possessing a rotatory and a linear motion, To, avoid, its, from, this, the, deflected, be, thus, will, straight path, bullet is made cylindrical in shape, so that the lateral pressure on, it remains uniform and it flies undeflected along its path., v^. Streamline Bodies. The student has no doubt heard of, streamline bodies of automobiles, particularly of racing cars etc., We shall discuss in brief here as to what this streamlining of a body, really connotes in the language of Science., air, or through a fluid, in general,, along with itself, pushing the rest on to either, The steamlines of the fluid, directed towards the body, open, side., out to either side to make way for it, as it were, and meet some, This fluid at the rear of the body, enclosed by, distance behind it., the streamlines meeting there is thus carried by the body as a sort, of a 'tail*. Some extra work has thus to be done by the body, in, decrease in, carrying this extra burden, resulting in an appreciable, In fact, the body has to encounter, its kinetic energy and velocity., a double opposition to its forward motion, v/z., (/) an increased, a decreased, pressure in front, called the head pressure, and (it), a backward pull, pressure or the tail suction behind, (which exerts, , As a body moves through, , it carries, , on, , a part of, , it, , it)., , rear region of, Naturally, the surrounding fluid flows into this, decreased pressure or tail suction, and is thus thrown up into vortices (i.e., whirls and eddies) there, which results in a further fall in, for, pressure in this region. These vortices are thus responsible, the body, thus, of, the, of, a, energy, good, fairly, part, away, dissipating, decreasing its velocity or offering resistance to its motion., therefore, the resistance to the forward motion of the body, it should be given a shape similar to that of the, 9, so that there is no tailsuction region firmed, 'tail, its, fluid forming, at all at its rear, and no energy is thus dissipated in the formation of, If,, , is, , to, , be minimised,, , ,, , is thus made with a gradually decreastowards the rear, and having no sharp, corners or edges anywhere. The body is then said to have a streamline, shape and the resistance to its forward motion is considerably, , whirls, , and, , eddies., , The body, , ing cross-section, tapering, , This explains the shape of the bodies of big airliners, decreased., modern cars., most, of, the, of, , and
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429, , VISCOSITY, , 205. Viscosity. When a liquid flows slowly and steadily over a, fixed horizontal surface, i.e , when its flow is streamline, its layer in, contact with the fixed surface is stationary and the velocity of the, the fixed surface, i.e. ,, layers increases with the distance from, the distance of a layer from the fixed surface, the greater its, the, , greater, , velocity., , Considering any particular layer of the liquid, we have the, and the one immelayer immediately below it moving slower than it,, it moving faster than it, so that the former tends to, above, diately, retard its motion and the latter tends to accelerate it. The two layers, thus tend to destroy their relative motion, as though there were a, backward dragging force, acting tangentially on the layers. If, therefore the relative velocity between the two layers is to be maintained, an external force must be applied to overcome this backward, In the absence of any such outside force, the relative motion, drag., between the layers is destroyed ajid the flow of the liquid ceases., This property of a liquid by virtue of which it opposes relative motion, , between, , its, , layers, , different, , known as, , is, , viscosity or internal friction, , of, , the liquid., , 206., , Coefficient of Viscosity (y). Newton showed that the backviscou*, force, acting tangentially on any liquid layer,, , ward dragging, or, , and, directly proportional to its surface area A, and velocity v,, the stationary layer., inversely proportional to its distance x from, Denoting this fofrce by F, therefore, we have, , is, , Foe A, , ;, , Foe -v, , F, , ;, , oc, , ;, , ve sign of v merely indicates that the direction of the, the, that of velocity., , /, , ^, Or,, , where ^, is, , r, Foe, , /, 8, is, , A.v, ~-, , is, , opposite to, , ^A.v, F =-*.-->, , ., , i.f.,, , ,, , force, , a constant, depending upon the nature of the liquid, and, , called its coefficient of viscosity.*, , v/x may be put as dvldx, which gives the rate of change of, so that, we, with, distance, and is called the velocity gradient, velocity, , Now,, , ;, , have, ~~ "~^', , This, If., , A, , is, , know, ~, , 1 sq., , as Newton's law, , cm.,, , and, , ', , dx, , of viscous flow in streamline motion., = 1, we have F = 7., , dv/dx, , Thus, the coefficient of viscosity of a liquid may be defined as the, tangential force required per unit area to maintain a unit velocity, gradient, i.e., to maintain unit relative velocity between two layers, unit distance apart., Arid, clearly, if this tangential force be unity,, the coefficient of viscosity of the liquid is unity, and is called Poise,, after Poiseuille, whose work on viscosity is important., , This, , coefficient, , n, , is, , someumes, , referred to as the dynamic viscosity of the, On the other hand, the ratio /p, , C.G.S. unit (see below)., (where p is the density of the liquid) is called, corresponding C.G.S. unit tor it is the stokes., liquid,, , with Poise as, , its, , its, , kinematic viscosity {), and the
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IpPERTJES OF MATTER, , 430, Dimensions of *n, , It is clear, , from relation / above that, , ~~, , A.dv/dx', ,.So that, the dimensions of y are those of, area x velocity gradient, PL/rn, (..dv, _, - MLT-* _MLT~*1, L, **, ., , ,, , ', , ', , [LI], , Or,, , -, , *7, , T, , rL/ri-L'T-, , L, , J, , UL-^T-\, , Viscosity in liquids corresponds to solid friction in so far as,, like the latter, it also opposes relative motion between two layers., It, however, differs from solid friction in that, unlike solid friction, it, depends upon (i) the surface area of the liquid layer, (ii) its distance, , the stationary layer end (Hi), stationary layer., , from, , -JL07., , with respect, , velocity, , The expression, , Fugitive Elasticity., , re-arranged and put as, , its, , for F, above,, , =,, , to the, , may be, ., , dv/dx, , ., , /, , ., , coefficient, JJ, , i.e..', , ., , ,, , of, J viscosity, J, , =, , tangential stress*, ., , ., , ,., , ., , velocity gradient, , This, ., , ._, , is, ., , an expression similar to the one, F/A, =>, i. =, , for the coefficient of, , _, ~~, , r, , ,,, , -, , rigidity, viz.., , --, , ., , dy, , tangential stress*, , displacement gradient', Maxwell, therefore, considered a liquid to possess a certain, amount of rigidity, breaking down continually under a shearing, stress., Very fittingly, he imagined viscosity of a liquid to be the, limiting ease of the rigidity of a solid, when the latter breaks down, under the shear applied. A liquid is thus regarded as capable of, exerting and sustaining an amount of shearing stress for a short time,, after which it breaks down and the shear is formed over again., In, other words, a liquid offers a fugitive resistance to shearing stress,, which is continually breaking down, and it may thus be said to, possess a fugitive rigidity., , _, , Now, if the rate at which the shear (0) breaks, to be proportional to shear, we have, rate of the breakdown of shear oc 8., Or,, , And,, , ,,, , ,,, , clearly,, , ,,, , =, , \.0 where, t, , *", , is, , 7\, , a constant., , the rate of formation of the shear, , =s, ~, , dj), , ~dt, , _, , ,,, , down be taken, , ~, , _. __^, , ( dy N, , dr( dx, , _, , ~~ _d_ (, , dy \, , _, ~, , dv, ', , ~dx{, , ), [, , v, , dt J, dx, being the velocity in the same plane., , *It will be noted from expressions (a) and (b) that whereas in a fluid, the, viicous drag is proportional to the velocity gradient, perpendicular to the direction of motion, the shearing stress, in a solid, is proporlional to the displacement gradient, perpendicular to the direction of shear.
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431, , VISCOSITY, , Thus, when the motion of the fluid becomes quite steady,, the rate of formation of the shear must be the same as that of its, foieakdown. So that,, , Now, dividing, , */-!;, , relation (a), ', , i?, , ,\, , -, , substituting, , for, , --, , tf, , by, , (b),, , -n\n, , =, , -,, , -.-*, , ., , we have, , in relation (c) above,, , uX, , /, , =, , ->?, , above, we have, , dv, ., , n, , *>, , I/A., , This quantity I/A is called the '//me o/ relaxation of the*, medium' and gives the time taken by the shear to disappear, provided no fresh shear is applied., , ^, , 208., Critical Velocity., It was Osborne Reynolds who first, showed by direct experiment that the critical velocity v c of a liquid, = fc.^/pr, called Osborne Renyold's, is given by the relation, v, c, formula, where ^ is its coefficient of viscosity, p, its density and r,, the radius of the tube, the constant k being called Reynold's number,, its, , value being about 1000 for narrow tubes., , The expression for v c may, however, be, method of dimensions, as explained below, , deduced by the, , easily, , :, , S'nce v c, , is, , fouud to depend upon, , =, , vc, , fc.?7, , pV,, , (/), , ??,, , (ii) p,, , and, , (Hi), , say., , [, , we have, , r,, , k being a constant., , So that, putting the dimensions of the quantities involved, we, , have, , [LT~, , ^, , l, , ], , [Mlr l T-*]*[ML-*\, , b, , [L\*, , r k, , having, , no, , dimen-, , Since the dimensions on the two sides of the equation must be, the same, (by the principle of homogeneity of dimensions), we have, , a+ b =, , ...(/), , So that, adding, , (/), , -a~3b+c =, , ;, , and, , (///),, , we have b, , Substituting this value of, , 1, , =, , (6) in (/),, , -fl=-l, , ...(//);, , we have a, [, , And, substituting the values of a and b in, v c ==, Hence, /:.^7/pr,, , ...(/), , 1., , (ii),, , =, , 1,, , or, directly, , from, , =, , 1., , we have, , c, , (in)., , ^^, , where fc (Reynold's number) is, as mentioned above, near about 1000, for narrow tubes., 100 J.^/pr., Thus, for narrow tubes, v c, , =, , It must be emphasized again that this relation applies only to, narrow tubes. For tubete of wide bores, the value of v c is very, much greater, and may be even a thousand times greater than that, given "by the above relation., , Now, a mere glance, show that, (i), , v, oc, , -7, , ;, , at the expression for v ct deduced above, will, , (), , v, oc 1/p, , ;, , and, , (in) v c oc 1/r,
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PROPERTIES OF MATTER, , 432, , of a liquid is (i) directly proportional to its, inverv^ty proportional to its density and (in) inversely, proportional to the radius of the tube through which it flows., i.e.,, , the critic4&B!Qcity, , viscosity,, , (ii), , It follows, therefore, that narrow tubes, and liquids of high visand low density tend to promote orderly motion, whereas tubes, , cosity,, , of wide bores, and liquids of low viscosity and high density lead to, turbulence., , Again, if wte have a perfectly mobile or inviscid liquid, i.e., a, so that, its flow, which ?7, 0, then, obviously, v c, would be turbulent and not orderly, even for the smallest velocity, and in the narrowest of tubes., Thus, we see that // is the viscosity of a liquid alone, due to, which its flow may possibly be orderly and thus approximate to that of, a perfect fluid., , =, , =, , liquid, for, , ;, , Poiseuille's Equation for flow of liquid through a tube. Imaa cylindrical layer, or shell of liquid, of radius x, flowing, through a capillary tube of radius r. Then, the velocity of flow at, all points on this cylindrical shell will be the same., Let it be v. As, the velocity of the layers in contact with the walls of the tube is, zero and goes on increasing towards the axis, it is obvious that the, liquid inside the imaginary cylinder is moving faster than that, outs'de it, and the backward tangential force due to the outer slower,, moving liquid on the inner faster moving liquid is, in accordance with, relation / above, given by i7.2irx.I.dv/dx, where *n is the coefficient of, viscosity of the liquid, [because, hers, surface area (A) of the cylindrical shell of radius x is equal to 2nx.l, where / is the length of the, capillary tube, and dvjdx is the velocity gradient there]., , 209., , gine, , Let the difference of pressure at the two ends of the capillary, tube be P, Then, he forward force on the cylindrical liquid shell, in, the 'direction of flow, is. clearly equal to Pxirx*, and tends to accelerate the motion of the liquid. If, therefore, the motion of he liquid, be steady, we have, , ~dx, the, A, , ve sign, , And, , showing that the two forces act, ,, , ,, , dv, , .-., , =, , =, , -, , P.ib^dx, --7,, r, , =, , ,, , ,, , Ivl, , 4-*7.2^*./., , Integrating this expression for d\,, , in opposite directions., , P.x.dx, ----', , we have, , v, , =, , ", , C, , PC, , -~ Ix.rfx., ti^l J, , a constant of integration., 0, when x, r, because the layers in contact with, the sides of the tube are stationary., , where, , l, , is, , Now,, , v, , =, , =, , =, , _, , -^-j-f C,,, Pr*, , Px*, , whence, C,, , P, , =
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VISCOSITY, This, therefore,, , is, , 433, , the velocity offlow of ftp liquid &t a distance, , x from the^ axis of the tube, and a glance at the expression, show that the profile or the velocity dis-, , for v will, , tribution curve of the advancing liquid, in the tube is a parabola, (Fig. 273),, , the velocity increasing from, at the, walls of the tube to a maximum at its, centre., Fig- 273., Now, imagine another co-avial, The cross-sectional, cylindrical shell of the liquid, of radius (x+dx)., area between the two shells is clearly 2nx.d* and, since v is the velocity of the flow of liquid in-between the two shells, the volume of, , liquid flowing per second through the cross-sectional area is given by**, If we imagine the whole of the tube to be made up, , dV = 2nx.dx v., , of such like concentric cylindrical shells, the volume V of the liquid, flowing through all of them, i.e.. through the capillary tube, in unit, time, will be obtained by integrating the expression for dV between, the limits, x, r., and x, , =, , =, , Or,, , =, , 4 Jo, , 2-nl, , -r., r*, , 7r, , /C*___, , 2il V 2, , "\, , =, , itP.r*, , ), 7T/V*, , whence,, , ..//, , Thus,, , if, , we know, , P,, , r,, , V and, , /,, , the coefficient of viscosity of, , the liquid (n) can be easily determined., The above relation holds good only, (/), , than, , no radial flow, (in), , %, , when, , steady and streamline, i.e., when its average veloVelocity ;, the pressure is constant over every cross-section, i.e., there is, the flow, , city is less, (//), , "^, , ;, , is, , its critical, , and, , the liquid in contact with the sides, , When, , of the tube, , is, , stationary., , the velocity of flow is small, and the tube is a narrow, It is clear, therefore,, one, these assumptions are mire or less valid., that for tubes of wide bores, the relation, breaks down for, in their case, tlfe value, of the critical velocity is much smaller, B, (v v c oc 1/r) and the flow of the liquid, becomes turbulent. Thus, if we were to, plot a graph between the pressure difference P between the two ends of the outflow, tube, and the rate of flow V of the liquid,, (i.e., the volume of the liquid flowing ouib, of it per second), we get a curve, as shown, (Fig. ?74), where the portion OA of the, Fig. 274., curve corresponds to the velocities lesthan, the critical velocity and the portion AB, to those above its., ;
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r&OFHSTlJttt Of MATTJfiB, , 434, It, , is, , found that when the velocity of the liquid, , is, , below, , the, , the rate of flow V is proportional to P9 the pressure, difference, (as indicated by the straight part OA of the curve). Thus,, within this range of velocity, the rate of flow of a liquid depends, chiefly on its viscosity (^), quite in accordance with Poiseuille'i, formula., critical value,, , critical velocity, however, the pressure different, almost wholly utilized in combating the turbulence set up ir, the liquid, and in imparting kinetic energy to it so that, its rate, of flow is now no longer proportional to jP, and hence no longei, depends upon its viscosity. In fact, it now depends mainly on the, density of the liquid (p) and is approximately proportional to *\/P., The following interesting consequences follow from the above, , Beyond the, , (F), , is, , ;, , :, , Since in turbulent motion, the rate of flow of a liquid ie, quite independent of its viscosity, it obviously follows that all liquids,, (/), , irrespective of their different viscosities, would require the same pressun, difference to be driven through a tube at velocities higher than theii, critical velocities., Thus, for example, a viscous liquid, like treacle, would require the same pressure difference to be driven through a, , tube, at a velocity greater than its critical velocity, as would be needed to drive water through it at the same velocity., (//) Since the criticcil velocity of a liquid is inversely proportional to the radius of the tube through which it flows, it is clear that, liquids of all viscosities would flow equally readily through tubes oi, Thus, in a wide tube, treacle will flow just, sufficiently wide bores., as freely as water., typical natural example of this is the free HOAK, of the highly viscous lava down the sides of an erupting volcano, ita, , A, , rate of flow being about the, , wateivx, , "V210., method*., , same as we would expect, , in the case oi, ^-tfBL, , -", , Experimental determination of y for a liquid Poiseuille's, A capillary tube J, of known length / and radius r,, is fixed horizontally near to the botLIQUID, tom of a vessel A, (Fig. 275), the, , liquid level in which can be kepi, constant at any desired height by, means of an over-flow arrangement O,, A clean and dry beaker, of known, weight, is placed below the outer end, of tube T to collect the liquid flowing, out through it. The liquid is allowed, to flow out in a slow trickle and, collected in the beaker for a known, time, and the beaker is then weighed, The difference of the two, again., 275., weights gives the mass of the liquid, flowing out in that time. Then, knowing the density of the liquid, its volume can be determined and,, dividing it by the time for which, the liquid was allowed to flow, its volume V flowing out per second ia, known. Substituting the value of V, so obtained, in relation II,, , The method, water., , is, , suitable only for comparatively leu viscous liquids,, , liki
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VISCOSITY, , 435, , of the liquid can be easily, , above, the coefficient of viscosity, , cal-, , culated., , There are two important sources of error in the above experi*, (/) part of the thrust, due to the difference of pressure, between the two ends of the flow- tube, imparts kinetic energy to the, liquid and the whole of it, therefore, is not used simply in overcomThis may be corrected, ing the viscous res 'stance of the liquid., for by taking the effective ralue of the pressure difference to be, raent, viz.,, , P, , F2, , p, , 2, , 4, , ,, , instead of, , P, , ;, , (//), , the motion of the liquid, where, , it, , enters, , the flow-tube, is accelerated, with the result that the velocity of flow, not uniform for the first short length of the tube. This is eliminated by taking the effective length of the flow- tube to be (/-f, instead of /., Thus, the corrected relation for *n becomes, is, , ~~8V.(/+1'04, , r), , 8ir(/+l-64, , r), , A much, , better apparatus, however, is the following, in which, the flow-tube F H a long one, and of a uniform circular crosssection, and the difference, of pressure for a length, AB of it is given directly, , by means of a manoA/, whose limbs are, arranged over two fine, holes at A and B, as, shown, (Fig. 276), where, A and B lie at a distance, of at least 10 cm?, from, the two ends of the flowmeter, , J3, , !|f 2, , OVERFLOW TUBE, , Fig. 276., , tube respectively, so that, the velocity of the out-flowing liquid becomes uniform near about, them. This very much minimises the two sources of error referred, to above,, the second one, almost completely. So that, with a slow, rate of flow of the liquid and a fairly sm^ll size of the holes at A and, By no further corrections are necessary., Note. In either of these apparatus, it is essential that the outflow tutje, should have a perfectly uniform bore. The uniformity of the bore may be tested, in a manner similar to that employed in the construction of a mercury thermometer, i.e., by introducing a small thread of mercury into the tube and measuring, its length in the different parts of the tube., In no part should the length vary, by more than 5%., , And, since the 4th power of the radius occurs in the formula for i\ t it, should be determined most accurately. The tube is, therefore, properly dried, and filled with mercury, and the length of the mercury thread measured most, carefully by means of a vernier microscope, making the necessary correction, for the curvature of the ends of the thread., The mercury is then taken out in, clean, dry and weighed watch glass and its mass determined as accurately a$, possible. Then, if m be its mass, p, its density at the then-temperature /', the, length of its thread in the tube, and r, the radius of the tube, we clearly have, ., , tr.r'./'.p, , So that, knowing, calculated., , m,, , /', , and, , p,, , m,, , whence,, , r, , the accurate value of the radius, , r, , of the tube can be
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MATTBB, , PBOFJBBTIBS OF, , 436, , 211. Motion in a Viscous Medium. When a body falls through, a viscous medium, its motion is opposed by a force, frictional in nature,, due to the fact that whereas the layer of the liquid medium in immediate contact with it is carried along with it, that at an infinite, distance from it is at rest., Energy is being continually absorbed by, the medium and is converted into heat. Possibly also, eddy currents, the body, and waves are set, in the medium, particularly when, , up, moving is a fast one, like high speed cars and airplanes or projectilesand these absorb still more energy. That is why cars etc., are, , streamlined these days to minimise the absorption of energy in this, currents, way. Even if the body be moving so slowly that no eddy, or waves* are set up, energy is still wasted due to the viscous drag it, has to overcome., , with the velocity of the, becomes, just equal to the, body,, a constant veloattains, then, the, motive or the driving force, and, body, this opposing force, increases, until, in the case of small bodies, it, , Now,, , city, called its terminal velocity., , Stokes showed that the retardation F, due to to the viscous drag,, moving with velocity v, in a medium, , for a spherical body of radius r,, whose coefficient of viscosity is, , F=, , %, , is, , given by, , 67rvr>7., , This relation, known as Stokes' law,, by the method of dimensions, For slow moving bodies,, , may, , be deduced as follows,, , :, , F, , oc velocity v, , F oc, F oc, , coefficient, , density a, , F oc, , ;, , of viscosity y, of the medium., , F=, , Or,, , where, r,, , K is a constant, , and, , /Cvr a, , b and, , a,, , of the body, , r, , of the, , medium, , [MLT-*], whence, , =, , (/), , b+c, , =, , 1,, , =, , [iT-, , ], , the dimensional coefficients of, , c,, , F=, , .-., , to be 67T, , ;, , K.v.r.y, , b3c =*, b = 1, , () 1+0, , (///),, , =, , ;, , different terms,, , 1, and (iff), and hence from, , 1,, ;, , F=, , K, , of, , downward force on, , the, , =, =, , body, , Equating this against the value of F,, 67rvr*7, , ~, , 8, 7rr .g(p, , Or, , a),, , 2, ., , 9V, , -, , (/), , 2., , we have, , was found" by Stokes, , =, , =, , resultant, , b=, , 6irvr^ 9 as stated above., , If the density of the spherical body be p,, 3, volume x p Xg, its weight, -y.Trr, and the upthrust on it due to the displaced medium, .*., , we, , 1., , and the value, , so that,, , ;, , b, b, [L*][M L~ T-*] [A/'Zr*], , 1, , Therefore, from relation, from (//), a, ; so that,, , ;, , W,, , V and a respectively., Now, putting the proper dimensions of the, , have, , c, , radius, , whence,, , /'gfc^*).., riiiir-.riiim.uin, , -..-__., , -, , -, , .-, , < ., , v, , X p Xg>, , =, , 3, 7rr .a.g., , 7tr*.p.g.iirr*a.g, 3, , 47rr ..(p-<i)., , we have, , =, , 8, , Jirr .(-p, , a)/67rr*7., * ,, , n, , ffJ, ^
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437, , VISCOSITY, , a small size),, Thus, the terminal velocity of a body, (of course, of, is (/) directly proportional to the, a, viscous, medium,, falling through, , 2, to \he difference in, (r ), (ii) directly porporti.nal, and, and, the, (Hi) inversely prothe, medium,, (p-cr),, body, of, the medium (*?)., to, the, of, viscosity, of, coefficient, portional, , square of its radius, the densities, , In arriving at the above result, Stokes made the following assumptions, which the body falls is infinite in, (a) That the medium through, :, , f, , (b), (c), , (d), , (e), , extent., , That the spherical body is perfectly rigid and smooth., That there is no slip between the spherical bodv and the, *, medium., the, as, far, so, is, That the medium, spherical, homogeneous,, body is concerned, i.e., the diameter of the spherical body, of, is large compared with the spaces between the molecules, the medium., That there are no eddy currents or waves set up in the mediin other words,, um due to the motion of the body through it, that the body is moving very slowly through it, or that the, motion of the medium is smooth and nob turbulent. Stokes, found that the relation holds good only when v is smaller, ;, , than, , ^7/crr*,, , called the critical velocity., , A striking example of a body falling through a viscous medium, that of the tiny rain drops that form what we call clouds. These, tertiny drops of water have a radius as small as -001 cm. and their, '00018, comes, minal velocity, as they fall through air, for which ^, That is why they, to about 1-2 cms. I sec., [fiom relation (/) above]., remain suspended in the air and appear to us to be floating about as, is, , =, , clouds., , about 10, Bigger rain drops, on the other hand, have a radius, terminal, their, and, therefore,, -01, times as great (i.e.,, velocity,, cm.), so that, they fall through the air,, comes to about 120 cms. Isec., instead of floating in it, (v being proportional to r 2 )., , =, , ;, , be, Also, if the density of the medium in which the body falls, the, greater than that of the body itself, i.e., if a> p, it is clear that, terminal velocity v will have a negative value. In such a case, thereThat is why, fore, the body will have an upward terminal velocity., bubbles of air or gas can rise up through water or any other liquid,, the smaller the bubble, the smaller its velocity., , v<1U2. Determination of coefficient of viscosity of a liquid, The relation for v obtainad above, ( 211), has been, Stokes* method., used to daterinine the viscosity of a liquid. The method consists in, in, finding the tim3 of fall of small sphores, such as ball-bearings etc.,, the liquid, and then to apply Stokes' relation,, , 4., *It, , ., , **r_>, , ,, , .hence, ,, , was shown by Arnold, however, that, , -, , ., , <*&)., , J-, , in actual practice v should, , ...(), be, , less
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PROPERTIES OF MATTER, , 438, , Care must, however, fee taken to see that the velocity of fall, does not exceed *67/crr, or else tho above relation cannot bs used., I, c.g.s. unit nearly,, Thus, for thin machine oils, for which -n, For, spheres of 1 mm. radius are the maximum size that can be used., thick oils, or liquids like glycerine, whose coefficient of viscosity is, greater than 10 e.g. 9. units, steel ball-bearings of about J" diameter,, are the largest that can be employed. For less viscous liquids,, 'much smaller spheres are required, and these may be obtained, in, any desired size, from radii of 0002 to 1 cm. by flowing melted, Wood's metal into cold water through capillary tubes of different bores., The diameters of these spheres, and hence their radii, are measured, accurately by means of a microscope., The liquid is taken in a tall jar, of a large diameter, and the, time taken by a sphere of suitable size, dropped centrally into it, to, pass three marks at different levels is noted and the velocity calcuIf the, lated for each of the two distances between the three marks., it has acquired, that, the, means, is, it, over, each, same,, velocity, path, its constant Velocity, or terminal velocity, v., If, however, the velocities be different, bein^ greater over the lower track than over the, upper one, a smaller ball-bearing must be tried, until the velocities are, the same, within experimental error., f, , =, , 4, , r, , In actual practice, we have two marks A, and B, (Fig. 277),, some distance (say 10 to 12 cms.) below the top and above the, bottom of the jar respectively, and allow small, spheres of different known radii to fall through, the liquid centrally, noting the time taken by, each to cross the distance 5 between the two, marks A and B, it being assumed that due to, the small size of the *>phere, it has already, acquired its terminal velocity before crossing, the mark A. Thus, since it moves with a constant velocity over this distance S, it* terminal, velocity, v, , =, , S/t., , Then, putting v = Sjt, pression for ??, we have, 2, , *, Fig. 277., , since r 2 /, , ~~, , y, , r2, , #(p, , in, , the above ex-, , o).f, , S", , constant in the case of a given sphere and a given, 2, temperature remains the same, we plot r for a, number of spheres against their corresponding values of l/t. The, slope of the straight- line curve, thus obtained, gives the mean, value, of r 2 ./, and this value is then substituted in the above expression, , Now,, , liquid,, , providsd, , is, , its, , for ^., , Further, since viscosity of a liquid depends upon its temperature, (see, 214), and the rate of change is fairly rapid in the case of, liquids like oils, a sensitive thermometer must also be put into the, jar of the liquid to indicate its temperature, and the experiments, .with the different spheres performed in quick succession., , For greater accuracy, a proper correction must be applied for, the finite size of the containing vessel, as the relation for 7) above, was deduced by Stokes on the assumption that the medium is infinite.
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naoosmr, Ladenburg has shown that if the liquid column in the tall jar, be supposed to be divided up into three equal compartments, lengththe, wise, and the sphere be dropped into the central compartment,, the liquid medium,, velocity v^ of the sphere, as it would be in, if it were unbounded by the walls of the containing jar, is given by, of, the relation v^, v(l+2'4r/jR), where v is the observed velocity, the sphere and R, the radius of the jar. This is called the correction, for the 'wall-effect.'*, were unhinSimilarly, he has shown that if the liquid medium, the bottom or the end of the containing jar, the velocity of the, dered, , =, , by, , sphere would be given, , by, , v, , =, , v(l, , + 3'3r/f*),, , 4, , the full depth of the liquid column in the, called the correction for the 'end-effect'., , where, , u is, , jar., , This, , is, , Combining the two corrections, therefore, we have the following, relation for, , TQ,, , v/z.,, , =, , ., , __, , _., , !Lf^>, , |., , ...(Hi), , N.B. Obviously, this method may also be used to determine the radius, of a small drop falling through air, if we know the coefficient of viscosity of air,, the method being applicable only to drops, bigger in size than the distance, between the air molecules, for otherwise Stokes" law no loger remains valid., , Rotation Viscometer. If we have two coaxial cylinders,, 213., with the space in-between them filled up with a fluid, and then rotate, the outer one with a constant velocity SI about their common axis,, a torque will naturally be communicated to the inner cylinder also, , through this intervening, the viscosity of the fluid., , fluid,, , and, , its, , magnitude, , will, , depend upon, , 278 represent a transverse secradii, a, and b, with their common axi*, of, cylinders,, and, perpendicular to the plane of the paper,, 0., passing through, , Let the, , tion of the, , full line circles in Fig., , two, , Then, as the outer cylinder rotates with, small angular velocity ft, (within the limits, of a streamline flow), about the inner cylinder,, the layers of the fluid in contact with the former, also rotate with it, with its own velocity ft, the, develocity of the other layers progressively, for the, move, we, as, inwards,, until,, creasing, it is, layers in contact with the inner cylinder,, Fig. 278., reduced to zero. There is thus brought about a, relative motion between the different layers of the fluid., its, , Let us consider a co-axial cylindrical layer C of the fluid, at a, distance r from 0, and of thickness dr. If it be rotating with an, to ra>. And,, angular velocity o>, its linear velocity is clearly equal, , A still more, form, verified, , v-voo[, by Bacon, , accurate correction of this effect, , 1-2-104, , is, , given by Faxcn in the, , (x) + 2 "K 0'~'K ")']', , ****, , ba8, , bceD
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440, , PBOPBBTIBS Of MATTER, , F, , therefore, if its length or height be /, the viscous force, acting on, it, in accordance with Newton's Law, (see page 429), is given by the, relation,, , F=, , where y, , its, , areaxyxdvjdr, , =, , Znrl.y dvjdr,, , the coefficient of viscosity of the fluid and dvjdr, the velocity, gradient at distance r from O., is, , =, , Now,, , =, , a^+r.dw/dr., dv/dr, d(rw)ldr, a constant quantity and would represent the velocity, of the layer in the absence of any viscous slip, i.e., if it were to, rotate like a rigid body, it does not contribute at all towards the, velocity gradient and we may, therefore, take the velocity gradient,, responsible for the viscous drag on the imaginary cylindrical layer, C, to be simply equal to r dco/dr., Hence, this viscous drag or force, , And, since, , w, , is, , ;, , F=, , 2-jir.l.f).r'da)ldr., , And,, , moment of this force, or the, clearly equal to r.ZTtr.l.y.r.dw/dr., dr\r*., may be put as 2?r l.y du\T, , therefore, the, , torque T, acting on the layer (7,, T, Or,, 2Tir*.l.'>?.daj/dr, which, , =, , is, , =, , effect of the whole fluid in-betweeri A and B is, and, by integrating this expression for the limits w =, , So that, the, obtained, , &, , =, , ft, , an d, , T, , =, , The torque, C and A., , a and, , T, , r, , =, , b., , Thus, we have, , tends to accelerate the motion of the fluid, , in-, , between, , But this fluid being in a state of steady motion,, the inner cylinder A must also be exerting an equal retarding torque, on the fluid in contact with it. And, since action and reaction are, equal and opposite, it follows that an equal and opposite torque T is, also exerted on the inner cylinder A, tending to rotate it through an, angle 0, say, until it is just balanced by a resfor ing torsional torque, equal to T, set up in the suspension wire, carrying it., Now, if C be the torsional couple, set up in the suspension wire, CO. So that,, per unit twist of it, we have T, , =, , In the above discussion, we have not taken into account the, on the base of the inner cylinder*, so that if this torque be, , wque, 7\,, , we have, total torque on the fluid between, , A and B, , given by, , In actual practice, we eliminate T b altogether, by repeating the, experiment with a different length or height /' of the fluid in-between, A and B. So that, if now the total torque be Tt and the angle of, rotation of the inner cylinder be 0', we have, , radii a, bases., , _, , *This torque on the base of the inner cylinder depends not only upon the, and 6 of the two coaxial cylinders, but also upon the distance between their
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441, , VISCOSITY, subtracting relation, , (iv), , from, , we have, , (///),, , .-(V), , The, , essential parts of the apparatus used are shown in Fig., is clamped on to a table T, so as to, 279, where the outer cylinder, , B, , be coaxial with its spindle, which can be, rotated by a small electric motor at a, speed, of 20 to 60 rotations per minute., , The, inside, , B, , inner, , A, , cylinder, , is, , suspended, , by means of a long and, , thin sus-, , pension wire, which carries a small mirror, m, to enable its angle of rotation, (caused, by the constant rotation of ), to be noted, accurately by the usual scale and telescope, arrangement., The value of C, the torsional, couple, per unit twist of the suspension wire, is, determined by first setting the inner, cylinder alone into torsional vibrations about, it, and then with a hollow, metallic, disc,, of a known moment of inertia, /', placed, centrally, periods t, , upon, and /', , it,, , and noting, , their time-, , respectively., , Then, as we know,, , = 27r\/y/C~an(U :==27r, where / is the M. L of the inner", cylinder, alone about the suspension wire., /, , =, C =, , So that,, whence,, This value of, , (/'*_/), , C, , 4, -, , *Wl'zV, ^, C, , 4:7i*r/(t'*-t, , Fig 279., , C, , ', , z, )., , then substituted in relation, (v) for % above., It must be, emphasized again, that the speed of rotation of the, outer cylinder must be, kept low, or else the fluid-flow becomes, turbulent and T is then no longer proThis may be clearly -seen, portional to ft., from the, accompanying curve between, ft and T/ft, (Pig. 280)., It will be noted, that T/ft remains constant, up to Q, beyond which it varies, as shown by the, dotted curve, in a somewhat uncertain and, 1, irregular manner, and at higher values of ft,, 2, 1, the relation takes the form (a.ft, p.ft ),, J\, where a and p are constants., P, Note The rotation ^/iscometer may be, is, , +, , used for, , o, , X, Fig. 280., , the determination of TJ for both liquids, Only, whereas for liquids, we take two, observations with different heights / and /' of the, liquid in the outer cylinder, in the case of gases,, we use two different inner cylinders, of lengths /, , and gases.
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442, , PROPERTIES Of MATTKB, , and /', the distance between their bases and that of the outer, cylinder being, Kept thp same in either case., , V214. Variation of viscosity of a liquid with Temperature. As we have, seen above, to determine the viscosity of a liquid we have to measure its rate of, flow through a capillary tube, (Figs. 275 276)., If, therefore, we wish to see how the, viscosity of a liquid is affected by a change in, its, temperature, the capillary tube must, obviously be immersed in a bath of known, temperature. This is not possible in the case, , of the horizontal tubes of the apparatus shown, in Figs. 275 and 276,, A simple modification, of it, as shown in Fig 281, is, therefore, used, for the purpose., , Here, the liquid, whose coefficient of, is to be determined, is taken in, a beaker B and maintained at any desired, temperature, which can be noted on the, thermometer T. A capillary tube C is then, arranged vertically, completely immersed in, the liquid, and is connected to a wider tube, DEF, bent as shown, to siphon the liquid, over through it into a weighed vessel /., viscosity, , The method of procedure, , is, , the, , same, , as in the case of the horizontal capillary tube,, with the difference that here the flow of the, is caused, by the liquid, liquid through, , C, , DF, , or the head of liquid h ; for,, clearly, no liquid will flow if the end D of the, 281, capillary tube were open to the atmosphere., The pressure due to the column h is, therefore, the effective pressure difference P, responsible for the flow of the liquid through C., , column, , -, , Now, as the liquid flows through the tube, the liquid level falls in B, thus, To maiptain it constant throughout the experialtering the head of liquid h, ment, B is gradually raised as the liquid flows through C and DEF, so that the, liquid level in B is always maintained at D. This is easily done by attaching a, bent pin in the tube, as show**, and making the liquid level in, always touch its, tip at, , D., , is thus allowed to flo^v into / for a known time t and its mass, then, dividing it by its density and the time / (in seconds), the, volume flowin? out per second, i.e., its rate of flow V is determined. Substituting, the values of V, P, r (radius of the capillary tube C) and / (length of the tube C), in Poiseuillc's formula ?), Prcr*/8W, we can easily calculate the value of rj for, the liquid, at the temperature of the bath., , The liquid, , determined, , ;, , =, , The experiment, tures, , is, , repeated with the liquid at different constant tempera-, , and a graph plotted between, , t, , and, , It will, , vj., , be found that, , vj, , usually falls, , with a rise in temperature of the liquid, though there is no definite or universal, the variation being more pronounced in some, relationship between the two,, caseVthan in others., , ', , 215. Comparison of Viscosities, Ostwald Viscometer. The vistwo liquids, or the same liquid at different temperatures,, , cosities of, , be conveniently compared with the help of the simple apparatus,, in Fig. 282, and known as the Ostwald Viscometer, a well, known form of the so called commercial viscometers, which avoid, the exact measurement of the dimension of the viscorneter, so tedious,, yet so essential, for the absolute determination of viscosities., , may, , shown, , \s will be readily seen, this viscometer consists of a U-shaped, two bulbs A and B, a capillary portion CDE,, and a side-tube T, fitted with a tap S three marks being engraved OB, , tube, , OGDEQT with, , t
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nsoosiT*, it, , N and Q, , at Af,, , immersed, , can be kept, , The, apparatus, respectively., in a suitable bath, to keep its temperature constant., whole, , introduce 1 into the bulb A through the open end, O of the tube, until, with the tap 5 closed, it occupies the space, of its bent portion, thus ensuring, that the same volume of the liquid is, taken in each case., Now, first with one liquid thus, , Each, , liquid, , is, , QDM, , taken into the tube, some air is, abstracted from it by connecting, T to a suitable pump through the, tap S, until the liquid column rises, above the level of Q, and the tap S, is then duly closed., The tap is now, opened again, when, due to the, increased pressure on it, the liquid, flows from the portion QE of the, tube into the portion CG. The time, when the liquid meniscus just passes, , mark, , the, , Q, , downwards, , is, , carefully, , noted, and again when it just passes, the mark N. The difference gives, the time t, say, taken -by the liquid, into, to flow from the position, its rate of flow, the position, , NG, , QM, , Fig. 282., , y, , being determined by the capillary, portion EC of the tub3, tha wid^r pirbs of the tube hardly affecting, it, because, as we know, the retarding force due to viscosity varies, Thus, if V, inversely as the fourth power of the radius of the tube., be its rate of flow, we have, , V, where, , is, , /, , =, , TT/V 4 /817/,, , the length of the capillary portion EC, P, the pressure, it and 77, the coefficient of viscosity of the liquid., ;, , difference across, , then repeated with the second liquid, of, be, Then, if, (at that very temperature)., rate of flow, determined as above, we have, , The experiment, , is, , coefficient of viscosity, , ?)', , its, , V, , ,, , Now, although the pressure difference keeps on changing during, the flow of the liquid it is proportional to the density of the liquid, for every position of it, And, therefore, if in the corresponding, and p' respectively,, positions of the two liquids, of densities p, the pressure differences be P and P', we have, , P, And, , oc p, , /. for the first liquid,, , for the second, , and, ", , Q, ., So, that,, ., , V, -, , f//, , liquid,, , and, , P' oc, , p'., , V, , oc TrpH/8^7/,, , V, , oc trp'r, , = -TrprW, , p, , 4, , /Sy /., , V, , of the variation, Again, the rates of flow, V and V ', also vary (because, in the pressure difference P and P') during the flow of the liquid, but, 1
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444, , PROPERTIES Ot MATTEE, , the times for the flow of unit volume of the two, liquids in corresponding positions, vary inversely as K and K', and it follows, therefore,, that the total times t and t', for the flow of the two liquids respecIn other words,, tively must likewise vary inversely as V and, , V, , ///', , From, , relations, , (i), /, , fn, , -~, , and, , (//),, , (ii), , we have, *n, , whence,, , ,, , ., , V'lV, , therefore,, , t, , =, , p^7, , =, , *, , ty, , =,, p, , t, , ..., , ...(ill), , t, , Thus, knowing p and p', the densities of the two liquids, (or of, the same liquid at different, temperatures), and t and t' the times for, their corresponding flows, we can easily compare their coefficients of, t, , viscosities,, , ^ and, , ??'., , N.B. It will be seen that we may not be merely able to compare the viscosities of the twj> liquids with the help of this viscometer, but may also, determine the viscosity of a given liquid. For, as have seen above, f\ oc p.r, where, p is the density of the liquid and /, the tim; for its fljw through the distance QN., , = a ?.r,, And, therefore,, 75, where a is a constant oF th: viscometer and may be determined once for all, for a, given viscom:ter, by noting t for a liquid of known density (p) and viscosity (73)., Or, from relation (///) above, we have, k __ t, Y)/P _ /, r/, , where k and, , Or, , i, , ff', , >, , ^7, , ~j~/*, , two liquids (see foot note on, page 429), which can thus be compired easily In fact, this was the relation actually used by OxtwM, In practice, it is found t j be more satisfactory, however,, to plot a curve between k and / for a number of liquids, so that, from the noted, values of /, the corresponding values of k can be obtained straightway., Among other commercial viscometers, based on a similar capillary, principle, may be mentioned the Redwood viscometer, used in England, the Engler, viscometer, used all over the continent and the Saybolt viscometer, designed by the, Standard Oil Company, and used in America., k' are the kinematic viscosities of the, , 216. Determination of Viscosity of Gases,, Since the definition, of viscosity of a liquid deduced in 205, applies equally well to a gas,, it, might at first sight appear that Posieui lie's formula for the rate of, flow of a liquid should also apply in the case of a gas., But the* snag, is that whereas the, density of a liquid is practically independent of, the pressure on it, (liquids being almost incompressible), that of a gas, varies directly with it and hence whereas in the case of a liquid, the, volume (as well as the mass) of it flowing through any section of the, tube in a given time can be taken to be constant, in the case of a gas,, it is the mass of it, (and not the volume) flowing across a section of the, tube in a given time that alone can be taken to be constant., ;, , Thus, if V be the volume of a gas flowing across a section, per second, at a distance x from the inlet-end of the tube, p,, its density at the uniform, pressure P over that section, we have, Since, , pK = a constant., we have, PV a constant., Now, if we consider a section dx of the tube, p oc, , P,, , at distance x from, the inlet-end, with a pressure difference dP across it, we have, in, accordance with Poiseuille's formula, the volume of the gas flowing, per second through the section given by, , Y, , ^^
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445, , VISCOSITY, the negative sign merely indicating that the, as distance x increases., , Or, multiplying both sides of the expression, , m,, PV, , P, , pressure, , by, , P,, , decreases, , we have, , P.dP, constant., T- =, dx, So that, if Vl be the volume of the gas entering tha tub, fthe cross section of the, P Vl = PK., at pressure P 19 we have, , =, , 7rr, , 8^7, , l, , Or, \ji, , PV, K x, jj, , ,, , ~-, , ube being constant., , P.dP, "~dx, , P, , Or,, , l, , Vl, , =, , ,, , -^, =, , and, , P2, , ,, , ./>.<//>., , =, , /, and x, Integrating this expression for the limits x, Pl and P, P2 whore / is the length of the tube and Pl and, the pressures at its inlet arid outlet ends, we have, , =, , P, , =, , -dx-, , ,, , Or, , ', , Or,, , whence, , ?,, , the coefficient of viscosity of the gas,, , may be, , easily deter-, , mined., This was the method actually used by Grindley and Gibson, who, noted the difference of pressure bstwecn the two ends of a flow tube, through which the gas was made to flow from one container to, The volume of the, another by forcing water into the former., gas passing through pc?r second could thus be easily determined and, *7 for the, gas evaluated from relation / above., 217., , Rankine's method for the determination of the viscosity of a, by Rank in e for determining the, , gas. The simple apparatus used, coefficient of viscosity of a gas, , is, , shown, , in, , ABCD, , is a glass tube, one metre in, Fig. 283, where, length, having a capillary section AB (of about, 2 mm. bore), and fitted with two stop cocks /and, O, which serve as inlet and outlet for the gas, is introA small mercury pellet, respectively., duced into the part of the tube opposite to the, capillary section AB., , o, , M, , When the tube is held vertically in the position shown, the mercury pellet starts falling down, under its own weight, forcing some of the gas into, the capillary BA, and its rate of flow is observed by, noting the time taken by the pellet to fall through, a measured distance. This is then equated against, its calculated rate of flow, whence the value of, *7 for the gas can be easily obtained., To, , start with, suppose the tube is laid horizontally on the table, so that the gas acquires, Fig. 283., a uniform pressure, all along the tube. Then, if p, be the density of the gas at unit pressure, and if the total volume of, the gas enclosed in the tube be K, we have total mass of the, KAI enclosed, p.P.K., , P
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PROPERTIES OF JBLATTBB, , 446, , Let the tube be now held vertically, so that the mercury pellet, at any given instant,, down. Let it be in the position, and let the volume of the gas above it be v v Then, if pa be the, pressure of the gas above M, it is clear that the pressure of the gas, below, will be pa +mg/a where mg is the weight of the mercury, Thus,, pellet and a, the area of cross-section of the tube DC., , M, , starts falling, , M, , f, , M, , =, mass of the gas above, p-/>a v i=, and, mass of the gas below, pGpa + ^/a)(K-~v ),, where (V vj is the volume of the gas below M., .-., tital mass of the gas in the tube = p./V v i4-p(Pa+wg/a)(F~-- vj., , M, , n, , which solves out to, , And,, , J, , = P.^V! + p(, ^, =P, a, +-, , 9.P.V, , Or,, , if, , -, , />, , ^, , -, , ...(/), , -p~., , p b be the pressure of the gas below M, we have, , A, , Or,, , Let the mercury pellet fall down to the position M', so that the, now becomes v 2 Then, if the pressure of, volume of the gas above, the gas now becomes p' a here, we have, as before, , M, , ._, , ., , __., , ., , ., , And, if p\ be now the pressure of tho gas below the mercury, we have, , pellet,, , Now, with the mercury pellet at M, the mass of the gas below, up to B is equal to P./? (VvJ and that of the gas below it, when, So that, the, it takes up the position M' is equal to p p' b .(V, v,)., difference of these two gives the masi of the gas forced into the, to M'., capillary tube by the fall of the mercury pellet from, tube, the, the, mass, capillary, Thus,, of, gas forced through, it, , ft, , ,, , M, , =, , P-, , P(V -, , v i), , -, , ?/>'*, , (, , V ~~ V J*, , If the position of the mercury pellet be so arranged that, , (v l, , +v 2) = F,, , we have, mass of the gas forced through the capillary = p.P.(v 2 Vj)., Hence, if t be the time taken by the mercury pellet to traverse the, distance MA/', we have, mass of the gas flowing per unit time through the capillary tube, , Now,, , be the volume of the gas flowing per unit time, , if v, , pellet is at, , when the, , M, we have, Sce relation, , where r, , is, , the radius and, , /,, , the length of the capillary section.
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447, , VISCOSITY, , And, similarly,, , when the, , be the volume of the gas flowing per unit time, , if v', , M, , pellet is at, , we have, , ',, , Hence, the average rate of flow of the gas is proportional to, out to be equal to Zmg.Pja., So that, mass of the gas flowing per unit time through the capillary, , (Pb*Pa2 )+(P'b*p'a 2 )l2y which works, =Px2w.-^.7rr 4 /16W., , tube, , Thus, from relations, , PX, , we have, , (II),, , 2*!*./>7rr*, , _p.P(v,-v, ~~, 16W~~, , a, , ., , ,, , whence,, , and, , (/), , ..(//), , *, , -, , 1, , ), ', , mg.iir*.t, , 8^(7.^)', , from which the coefficient of viscosity of the given gas, determined., , may, , be easily, , As can be readily seen, the apparatus may also be used to study, the effect of pressure on the viscosity of a gas. Rankine determined, the viscosities of many gases with the help of this apparatus and, showed that viscosity is quite independent of pressure, as predicted, by the kinebic theory of gases., For extreme accuracy, however, Rankine 's method is far from, with its inherent defect of a capillary tube experiment., FOP such purposes, therefore, a rotating cylinder apparatus is found, to be more satisfactory and the one form of it almost universally, used is that due to Bearden, who gave the value (1-82462 ^, , suitable,, , 0-00006), , X 10~ 4, , poise for air at, , 23C., , SOLVED EXAMPLES, A, , 1,, , layer of glycerine, , i, , mm., , cm. per sec., what force, per second ?, , We know, Here,, , >j, , * 20, , cms. is separated from a large plate by a, the viscous coefficient of glycerine is 20 gms. per, required to keep the plate moving with a velocity of 1 cm., , of area 10, , flat plate, , thick., is, , sq., , It, , that the viscous force, , gm./cm., , sec-,, , A =, , F=, , 10, , is, , given by, , *=, sq. cms., v, , ~ =, , 1, , F=, , 10x10, , 20 x, , ti.A.v/x., , cm.Jsec.,, , =, , andx=l, , mm.-*'\ cm., , 2000 dynes., , Hence, the force required is equal to 2000 dynes., 2. Water flows along a horizontal pipe, of which the cross-section is not, constant. The pressure is 1 cm. of mercury, where the velocity is 35 cms./scc., Find the pressure at a point where the velocity is 65 cms. sec." 1 ., Here,/?!, , 1, , cm., v,, , Applying, , 1, , Or,, , =, , 65 cms. I sec., , Bernoulli's relation,, , j v,, , Or,, Or., , = 1x136x981, , -^', , ;, , dynes/cm*., , f, , =, , 1, , ;, , vt, , gm./c.c., , =, ;, , 35 cms. I sec., , p*, , =, , ?, , we have, , Or, J(65 -35 ) = (13'6x981)-/> f, -/7 1 ~/? 2, i(65+35)x(65-35) - (13'6x981)-/>,., 1, , f, , ., , ix 100x30, />,, , -, , 13350-/?,., , 13350-1500, , Thus, the pressure at the point, , Or, 1500 - 13350-/?,., 11850 dynes/cm*., , -, , is '8884, , em, of mercury., , .
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FEOFEKTIBS OF MATTEB, , 448, , 3., If the diameters of a pipe are 10 cms. and 6 cms. at the points where a, vtnturimeter is connected and the pressures at the points are shown to differ bj, 5 cms. of water column, find the volume of water flowing through the pipe pei, second., , pipe, , We know, , that the, , the same,, , i.e.,, , is, , two different, sections., at, , Further,, , and, , /> a, , volume of the water flowing across any section of th<, a a v 2 where a l and 2 are the areas of cross-sectior, points, and v x and v,, the respective velocities of flow at lhes<, , a^ =, , we know, , that, , a^ \ I ?(^CT 'P*, , *, , fljVj, , V, , are the pressures at the two points, , 2, , ffi, , Hence, a\, .'., , =, , nr-f, , =, , TT., , 5, , (p\pt), , (^P)*, , == 25rc, sq., , cms., , ;, , =, , at, , = 5x1, , cwj of water column, , 22 5^, , V, , page 425), where p, , Trr 2, , =, , 2, TT., , (.)*, , and a, , 9rc j#. c/?75., , we have, , Or,, , V, , - 225rM / .981, 34rrxl6n, , (see, , x981 dynesjcm*., , substituting these values in the relation above,, , A / 2x5x981, , ,, , of cross-sectional areas a, , respectively., , and, , ', , -tf 2 2, , A, , 9810, , 3002, , V/ 34X16, , Thus, the volume of water flowing through the pipe per second, 3-002 litres., , 3002, , is, , or, , c-cs.,, , 4., A tube having its two limbs bent at right angles to each other is hcl<, with one end dipping in a stream and opposite to the direction of flow. If thi, speed of the stream be 6 miles/hr , find the height to which water rises in th<, vertical limb of the tube., , Clearly, the flow of water will be stopped by the lube dipping in th<, stream and facing the flow, so that the loss of K E. per unit mass of water is Jv a, This much must, therefore, be the gain in the pressure energy, i.e., /?/p., />/P, , Since p for water, , =, , we have, , 1,, , p, , - iv*., = v*,, , lv, v, , Now,, , -, , ^, , it, , .1, , 6 miles jhr., , = 6x1760x3 =, -^Q^^Q, , Therefore, water risss to a height of 1*21, , =, , iv, , h.p.g, , =, , hg, , or,, , p, 88, 10, , />. in, , =, , =, , s, ,, , Hg, as p ==, , 1, , ^, 8 $ ft. /sec., ., , rt, , the tube., , 5, , Calculate the velocity of efflux of kerosene oil from a tank in which th, The density o, pressure is 50 Ibs. wt. per sq. inch above the atmospheric pressure., kerosene is 48 Ibs. per cubic foot., (Bombay, , Let h be the height of the level of kerosene oil in the tank above the axi, , of the, , orifice., , Then, pressure due to, poundalstft*.,, , But,, , pressure, h.p, , Now,, , i.e.,, , -, , =, =, , it, , at the level, , A.p. Ibs. w/.///., , 50, , 50x144, , velocity of efflux, , Ibs. wt., , per, , Ibs. W/.//V, , i, , s, , of the axis of the, , orifice, , =, , a, , sq., , 2, .,, , inch, , ~, , whence, h, , 50 x 144, , =, , Ibs. w/.///, , 50xl44/p, , ___, given by v== >/ ~2gh, , h, , p, , 2, ., , 50xl44/48/f., , [v, , p, , 48, , Ibs.lc. ft, , -v 2X32X^4, , - 97-97 ft. Isec., The, 6,, , through, , it., , velocity of efflux of the oil, , A, , vertical tube of, If the pressure be, , is,, , therefore, 91 '97 ft. / sec., , 4 mm. diameter at the bottom has water passin, atmospheric at the bottom, where the water emerge
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FLOW OF, , 449, , VISCOSITY, , Lt$tTlDS, , t the rate, f 800 gins, per minute, what is the pressure at a point in the tube 25, cms. above the bottom, where the diameter is 3 mms. ?, ,= *. gms.lsec., Here, rate of emergence of water = 800 gms.fmt., , = 40/3 c.cs. per sec., 1 gm.fc.c., water, of, ttecause, density, This will be the same across any section of the tube., , =, , Now, diameter, , of the tube at the bottom, , radius, , and, , =, , 4 mm., , =, , '4/2, , =, , =, '2, , *4, , cm., , cms.,, , area of cross-section of the tube, at the bottom, 2, -04TT sq. cm., =Trx('2), , .*., , =, , volume of water passing through any cross-section, = cross-sectional area x velocity, we have, velocity of water passing through the bottom, 40, volume ----- _per sec., __ ----_ 'flowing, Since,, , --, , ', , cm. sec., , =, , p, , =, , ., , ..., , cross- sectional area, , 3x*04rc, , ", , v,., , say., J, , K.E. per unit mass of water at the bottom, , it, and, , 1, , /, , 40, , bottom, =/?i== 76x13*6x981 ews., , pressure energy per unit mass at the, , =, , /> ( /p, , [., , 1, , gm., , c c, , K.E. -\-pressure energy., , total energy, , Again, diameter of the tube, 25 cms. above the bottom = 3 mms. = '3 cm., radius of the tube, 25 cms. above the bottom = *3/2 = *15 cm., And .'. area of cross section of the tube = ?r x('15) 2 sq. cms., 40, = So that,, cms. I sec., velocity of water, here,, -, , -, , ., , ^, , K.E. of water per unit mass, here,, , Hence,, , 40, Let the pressure here be p 2, Then, pressure energy per unit mass, in terms ot mercury column, ,, , Also P.E. per unit mass of water, here,, , And,, , .'., , total energy, here,, , =, , =, , =, , kg, , 25x981, , ergs., , K.E. i-pressure energy -\-P.E., , 2, 9, , in accordance with Bernoulli s theorem, the total energy, at the two places, so that, we have, , Now,, , same, , must be the, , ;, , ), , +'*"'> +, , (/>,x, , 13-6x981)+ (25X981V, , ROft, , Or,, , -^.j^, -, , Or,, , Or,, Or,, , ., , 5627, , f 1014000, 133507> a, , -, , 177804-13350^,4-24520., 5627 f 1014000-17780-24520., , -, , 977327, , The figure is rounded, logarithmic table., , +(76x13-6x981)., , -, , 977300, sa>*., , off thus, to, , be able to use the ordinary four-figure
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PROPERTIES OF MATTER, , 450, , is, , =, , Pt, , Or,, , 73-24 cms., , Thus, the pressure at the point where the diameter of the tube, 73*24 cms. in terms of mercury column., , is, , 3, , mm*., , A, , 7., pitot tube is fixed in a main of diameter 15 cms., and the difference, of pressure indicated by the gauge is 4 cms. of water column ; find the volume of, water passing through the main in a minute., , =, , radius of the main, , Here,, , 7*5, , 15/2, , cms, , of cross-section of the main ** n x(7 5) a sq. cms., Loss of K.E per unit mass of water due to the stoppage of flow by the, area,, , tube, , =, , 2, , iv ergs., , \nd, gain of pressure energy = p/p = p, 2, a, Or, v, Therefore,, iv - 4x981., , = 4x981 ergs., 8x981 * 7848,, , <*, , h.g, , -, , p, , ['.', , *, , 1, , gjn.jc.c., , v = v/7848 = 88 '51 cms.! sec., whence,, volume of water flowing per second across the section, = area of section x velocity ~ TC x (7'5)* x 88*57 c.cs., \nd .*. volume flowing through the section per minute, - re x (7*5) 2 x 88*57x60 = 9'396x 10 5 c.cs., .'., the volume of water passing through the main is 9*396 x 10 5 c.cs. per minute., '., , A water main of internal diameter 8" is fed by a pipe of internal dia8., meter 2", which delivers water at the rate of 1 c. ft per second. If the pressure of, water in the pipe be 50 Ibs. wt. per sq inch, calculate the pressure in the main., radius of the main, , Here,, , =, , 8/2, , =, , 4", , and radius of pipe = |, cross-sectional area of the main, and, , =, , rate, , nx, , = _J_, ft., , =, , 2, (, , J), , 8, , nx, (-, , 1, , =, , Rate of flow of water, , Now,, , ft., , 1", , pipe, , ,,, , , ,, , ** J, , of flow, , velocity offlo*, , =, =, , 1, , n/9 sq. ft., , =, , ), , cross-sectional area, , (v a ) in the, , =, , the, , density, , -, , pipe, , =, , 1, , /, , it, , =, V, /, , A ad ^T.E per unit volume in, , -, , X 62, , 5, , x, , -!*, , 2, , ^2, , =, , ^TTX*, , .'., , ", , Or,, , whence,, , P!, , ----, , =, , i pv,, , 1, ,, , where, , = 256 4 /' Poundals., *, the pipe, Jpv, . -A2 5 144)8 . 65660 //., 2, , 2, , ., , a, , /?! /A5., , w/ /^. inch., , /? A, , x 32 x 144, , poundals/ft*', , = 50, we know,, /? a, wt.jsq. inch., - 50x32x144 = 230400 pottndalslft*., 8, 2, = /> 2, equation, we have /?i + Jpvj, 4-ipv2, //>5-., , ., , -, , x 32 x 144, , =, , 230400+65660 * 296060, 296060 - 256-4 - 295803 6,, , P!, , =, , 295803-6/32 x 144, , x 32 x 144+256*4, P!, , ,, , 144, , 8l, , in the pipe, as, , applying Bernoulli's, , Or,, , 9, , .-^, 2n*, , Let pressure in the main be, , And, pressure, , of flow., , ., , y, , volume (/ e., per 1 c. ft. of water) in the main, of water, (equal to 62-5 Ibs. per c.ft.), , x 62-5 X (, ^, 2, \, , velocity, , =, , ], , -, , /. K.E. per unit, is, , ft., , cross-sectional area, , TT, , and velocity of flow of water, , X, , - -, , ---- rate, , thit, velocity of flow of water (v x ) in the main, , So, , */144 sq, , c. ft. I sec., , 64-19, , Therefore, the pressure in the main, *, , 64-19, , Ibs. wt., , per, , sq. inch., , Ibs.
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FLOW OF LIQUIDS, , 461, , VISCOSITY, , 9., Water is escaping from a cistern by way of a horizontal capillary tube,, 10 cms. long and 0*4 mm. in diuneter, at a distance of 50 cms, below the free surface of water in the cistern. Calculate the rate at which the water is escaping., (Bombay), , We know that, , the rate at which a liquid escapes out from a capillary tub4, r.Pr /8TQ./,, liquid flowing out per second, is given by K, where Pis the difference of pressure between the two ends of the capillary tube,, r its radius, /, its length and *], the coefficient of viscosity of the liquid flowing, , ing, i.e., the, , through, , volume of the, , it., , Here,, , P, , and, , r, , -, , 50 x, , =, , x981 dynes/cm 9 .,, *2, , '4/2, , mm., , =* '02, , I, , =- 10, cms.., , cm., , 4, nx50x981x(02), v_ _/___, , .,, K, , Hence, , 1, , 4, 3'082xlO~, _., , ^, , 8r)XlO, , where, , >), , c, , *), , the coefficient of viscosity of water., , ii, , Thus, the rate at which the water, 3*082 x 10~ 4 /?) c.cs. per second., , escaping from the capillary tube, , is, , is, , A gas bubble of diameter 2 cms., rises steadily through a solution of, 10., Calculate the coefficient of visdensity 1*75 gms./c.c. at the rate of '35 cms /sec., cosity of the solution. (Neglect the density of the gas)., , We, lecting, , Here,, , r, , Hence,, , p,, , 2, '', , -=, , have the relation, 1, , ., , 2/2 =, , 1, , cm., g, , =, , 7), , 5, , -, , ,, , we have, , 981 cms. /sec*.,, , 2x1x981x1-75 =, , f, , v, , (the density of the gas bubble),, , =, , ^ P ~ a), , 9, , a, , =, , 2x981x1-75, , =, , ., , is, , ^ =*, , page 436), so that, neg-, , -g, , 1*75 gm.lc, , 3 ]5, , Thus, coefficient of viscosity of the solution, , (see, , c., , and, ., , v =, , "35 cms. / sec, , , Aft, 1A s, l'09x, 10 poise., , equal to, , T09x, , ., , 10 a poise., , EXERCISE XII, , Why, , Derive the formula for the flow of a liquid through a capillary tube., 1., does the formula fail in the case of a wide bore ?, (Agra], , 2., What is meant by the term 'coefficient of viscosity' ? Obtain an, expression for the rate of flow of a liquid through a capillary tube of circular, cross-section. Note the precautions to be adopted in the experimental determination of this coefficient, using this expression., , Define, , 3., , cient of viscosity, , of viscosity' for a liquid., determined for water., , 'coefficient, is, , Describe how the, , coeffi-, , Water is conveyed through a horizontal tube 8 cm. in diameter and 4, kilometres in length, at the rate at 20 litres per sec. Assuming only viscous resistance, calculate the pressure requued to maintain the flow., (Coefficient of, (Bancras), viscosity of water is 0-01 COS. units)., Ans. l'274x!0 7 dynes/ cm*., Define coefficient of viscosity of a liquid and find its dimensions., Describe the wiy in which the different parts of a viscous liquid move, \vhen flowing through a fine tube. What changes take place if the motion is increased ? In an experiment with PoUeuille's apparatus, the following figures were, obtained, 7'08 r.cv. ; Head of water = 34-] cms., Volume of water issuing per minute, 56 45 cms. Radius of the tube = *0514 cm., Length of the tube, 4., , :, , =, , Find the coefficient of viscosity., , (Calcutta), , Ans, , 0-01377 poise., 5., Calculate the mass of witer flowing in 10 minutes through a tube, <0 1 cm. in diameter, 40 cm. long, if there is a constant pressure head of 20 cms. of, water. The cofficient of viscosity of water is 0*0089 c g.s. units. (A-M.IE., I960), Ans. 81*19 gmt.
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PROPERTIES OF MATTER, , 452, , Define coefficient of viscosity., , 6., , Give examples of some viscous sub-, , How would you determine the coefficient of viscosity of a, Why do you find cloud particles floating in the sky ?, , stances., , liquid ?, , (Agra, 1944}, , Define coefficient of viscosity., , 7., , Describe, , fully, , what you would do to compare the, , viscosities of, , two, , (Madras, 1949), , liquids., , 8., Explain what is meant by the 'viscosity of a liquid ? How will you, study experimentally the variation of viscosity with temperature ?, What are the 'dimensions' of viscosity in terms of length, mass and, time ?, (Allahabad, 1946), 9., Define the coefficient of viscosity of a liquid. What is the effect of, temperature upon it ? How would you determine the viscosity of water at, Derive the formula you use., different temperatures ?, (Punjab, 1941), A square plate of 10 cms. side moves parallel to another plate with a, 10., If the viscous, f velocity of 10 cms. per sec., both plates being immersed in water., force between them is 200 dynes, and the viscosity of water is 0*01 gm./cm. sec.,, Ans. 0*44 cm., what is their distance apart ?, Enunciate and prove Bernoulli's theorem, and mention some of its im11., 1, , portant applications., 12. Two tubes, with small apertures at their lower ends, are held verticalwith their lower ends dipping in a pips carrying water, such that the aperture, of one faces the flow and that of ths othsr has its plane parallel to the direction, of flow of water, which rises in the former to a height 10 cms. above that in the, latter., Determine the velpcitv of flow of water in the pipe. If the pipe has a, diameter 20 CTI*'., what is the volume of water flowing aloig the pipe per, Ans. (/) 140'1 cms. /sec. (//) 26-4 x 10 s c.cs., minute ?, ., ly, , Calculate the velocity of efflux of alcohol (sp. gr. -80) from a cylinder, is 2 atmosphere*., Here, pressure due to the alcohol is one atmosphere, equal to 76, [Hint., Ans. 15-92 metres./sec, cms. of mercury column., 13., , in, , which the total pressure, , 14., , A, , two limbs, , venturimeter, , and 15 cms., , are 20 cms., , differ, , is, , connected to two points in a main where its radii, and the levels oF the water columns in the, H:>w much water flows through the pipe per hour ?, Ans. 43'H x 10* litres., , respectively,, , by 10 cms., , Water flowing in a horizontal main, of a non-uniform bore, has a, 15., velocity 100 cmv./w. at a point where the pressure is l/l9th of the atmospheric, pressure. What will be the velocity at a point where the pressure is one half of, that at the first point ?, Ans. 251 '1 cms /sec., 16., , Deduce ths eipre^sion, , for ths rate of steady flow of a liquid, , a capillary tube of circular section., , through, , A, , vessel of cross section 20 sq. cm. has at the bottom a horizontal, capillary tube of length 10 cms. and internal radius 0*5 mm. It is initially filled with, water to a height of 20 cm*, above the capillary tube. Find the time taken bv, the vessel to empty one-half of its contents, given that the viscosity of water is, O'Ol C.G.S. unit., (Madras, 1947), , Ans. 9, , A capillary tube,, , mm., , mm., , 36 sees., , diameter and 20 cms. in length, is fitted, horizontally to a vessel kept full of alcohol, of density '8 gm.lc.c. The depth of, the centre of the capillary tube below the surface of alcohol is 30 cms., If the, viscosity of alcohol is 0'012 c.g.s. unit, find the amount that will flow out in 5, Prove the formula you use., minutes., (Bombay. 1933, Ans. 57'74gms., 010 gm. cm." 1 .vc.~ .) is escaping from a tank, Water at 20C(7j, 18., by a horizontal capillary tube, 20 cms. long and 1'2 mm. diameter. The water, stands 10D cms. above the tubs. At what rate is the water escaping ?, Ans. 2-5 c.cs./sec., 17., , 1, , in, , -, , !, , ., , it, , 19., If in question 18, the area ff the tank bs IQ* sq. cms,, how long will, take for the water level to fall to 50 cms. above the tube ?, Ans. 91 hours.
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FLOW OF LIQUIDS, , 453, , VISCOSITY, , A funnel is attached to a vertical capillary tube of length 49*6 cm*., poured into the funnel and kept at a height of 68*3 cms. above the, lower end of the tube, whilst 78 c a. are collected in 10 minutes. The tube wai, then removed and entirely filled with mercury 3-695 gms. were required. If, the density of mercury is \3'56gms. per c.c., find the viscosity of water at the, t emperature of the exp eriment., Ans. O'0 1 247 gm. I cm. sec,, The radius of the tube can be determined from tht amount of mercury, [Hint., mass, volume x density., filling the tube., For,, mass ofmercuryjits density., .%, volume of mercury filling the tube, volume of tube = 3*695/13 56 c.cs., Or,, Now, volume = wr 1 ./, where r and I are the radius and length of the tube, 20., , Water, , is, , ;, , respectively., , wr1, , Hence, , r, , -, , volumell, , ?? AQ, 71x13*56x49*6', >, , ,, , ,, , ~, , 3*695/13*56x49*6, or,, , r~, , cms., , JT *_, , cms., \/, V rrx 13*56x49*6, , This value ofris then substituted in the relation, 21., , sq., , *, */], , P.^r*/8, , 7., , v/.], , Give Poiscuille's method of, , measuring the viscosity of liquids., Indicate a method which could be employed to, , Derive the formula used., measure the viscosity of liquids at different temperatures., (Banaras), 22., What kind or kinds of energy result from the work done by a fluid, against viscosity ? How can the viscosity of a liquid be determined ? (Bombay), Define "coefficient of viscosity"., Derive Stokers formula for the velocity of a small sphere falling through, a viscous liquid. Explain how this is utilised to determine the viscosity of a, Mention one more application of Stokers formula., liquid like castor oil., (Madras), 24., Determine the radius of the drop of water falling through air, if the, terminal velocity of the drop is 1*2 cms.lscc. Assume the coefficient of viscosity, for air =* l*8xlO~ 4 and the density of air = 1*21 x IQ-* gm.,'c.c. (A.M.I.E., 1961}, Ans. O'OOl cm. (approximately), 25., What is a rotation viscometer ? Explain clearly its construction and, working., 26., How may the viscosities of two liquids, or the viscosities of the same, liquid at two different temperatures, be compared with the help of Ostwald, 23., , mcometer, , ?, , Explain the limitations of Poiseuille's formula for the rate of flow of, i liquid, through a capillary tube., Why does it fail in the case of a gas ?, Explain, with necessary theory, Rankings method for the determination of, ihe viscosity of a gas., 27., , ), , \
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CHAPTER, , XIII, , DIFFUSION AND OSMOSIS, Diffusion., If we place a solution of, say, copper sulsolid, phate, (or, crystals of potassium permanganate or potassium dichromat e), in a beaker, and cover it with pure water, disturbing it the least, so that the line of demarcation between the two is, clearly seen, we shall find that on standing for some time,, may be,, , 216., , even, , even days or months, the characteristic colour of the salt ascends, upwards and fills the entire space occupied by the liquid, the concentration of the entire solution becoming the same all over., The same is true for gases also. For, if we divide a tall glasn, jar into two compartments by a movable horizontal glass plate and, put a heavier, or a denser gas, like carbon dioxide, on the lower, compartment and a lighter or a rarer gas like hydrogen, in the upper, one, we find that, on removing the glass plate, the molecules of the, two gases intermix with each other, until we get a mixture of the, two, of a uniform density throughout., This process, by virtue of which the molecules of a solute, (in this, case, copper sulphate), move upward from the lower portions of greater, concentration to the upper ones of lower concentration, or by virtue of, t, , which the molecules of one gas mix with those of another, even against, the force of gravity, is called diffusion., , The solute is pushed up as though under some pressure, until, equilibrium is attained, and the concentration and pressure of the, solution become uniform throughout., Solids too, although they possess a definite crystalline structure,, have been known to exhibit the phenomenon of diffusion, if placed, in good contact with each other over a long enough time, the, diffusion taking place more readily between two different solids than, between two portions of the same solid. Thus, for example, the, diffusion of gold into lead has been clearly shown by Robert Austen,, by fusing a small lead cylinder (about 7 cms. long and 1*4 cms. in, diameter) into a thin gold plate and subjecting it to pressure for well, over a month, ia a constant temperature enclosure, the temperature, being kept below their lowest melting point. And, more recently, it, has been clearly established by Groh and Hevesy that radioactive lead, can diffuse into ordinary lead, if the two be kept iri contact for some, reasonably good time, / e., well over a year. This then, is a case of, what is aptly called self-diffusion, for we have, here, the case of a, , substance (lead) diffusing into its own self, (i.e., into lead), for, although the atomic weights of the two types of lead are different,, they are otherwise identical in their chemical properties., The phenomenon of diffusion of liquids was first investigated by, Graham, in the year 1851. It would, therefore, be of interest to, give a brief account of his experiments here., , 454
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DIFFUSION AND OSMOSIS, , He filled a large- mouthed bottle with the salt solution under, investigation, to a level very near the top, floated a cork-disc centrally on the surface of the solution and carefully filled the rest of, the bottle with pure water, (the solvent), by lightly squeezing it on, the discs from the sponge., Then, covering the mouth of the bottle, with a glass disc, lowered it lightly into a larger jar, containing, water, so that the mouth of the bottle was some distance below the, water surface in the jar. Then, allowing the motion of the two, liquids to subside, he carefully uncovered the mouth of the bottle,, causing the least possible disturbance to the liquids, and finally covered the jar with a glass plate to minimise evaporation, maintaining, the jar at a constant temperature to avoid convection currents., Due to diffusion of the salt into the pure water above, the concentration of the solution began to change, and this was determined, from time to time by withdrawing portions of the solution by means, of a pipette. From the change in concentration of the solution,, Graham estimated the amount of the solute diffusing into the solvent, or the water above., Graham repeated his experiment with different salt solutions,, and, although he was unable to formulate any exact law for the rate, of diffusion, he arrived at the following conclusions, :, , (/), , that solutions of different, , centration, diffuse at different rates,, pends upon the nature oj the solute, , salts,, , of the same strength or con-, , and thus,, , the rate, , of diffusion, , de-, , ;, , con(//) that solutions, of the same salt, of different strengths or, centrations, diffuse at rates proportional to their concentrations ;, (Hi) that, in general, diffusion alters the proportion of, in a mixture, , two, , salts, , ;, , that the rate of diffusion inceases with temperature, may be divided into two distinct categories, viz., (a) crystalloids, consisting of mineral acids, salt solutions, sugar,, etc., which diffuse comparatively quickly, and (b) colloids, consisting, of albumen, gum, caramel and gelatine, etc., which diffuse more, slowly than the crystalloids., (iv), , (v), , ;, , that solutions, , Diffusion, in the case of liquids, may easily be explained on the, basis of the kinetic theory of liquids., For, according to this theory,, the molecules of the solvent possess kinetic energy of translation,, which is directly proportional to their absolute temperature. This,, they share with the molecules of the solute present and the latter, thus roam about in the solvent, so much so that they even rise up, against the force of gravity., , Four years later, in 1855, Fick formulated, 217. Fick's Law., a law of diffusion, based on Fourrier's law of conduction of heat, due, to a close analogy between the process of diffusion of a solute and, He established the law that the, the flow of heat through a solid., rate of diffusion in any direction is proportional to the concentration, gradient of the solute, as Pick's law., , in that direction., , The, , law, after him,, , is, , known, , Thus, if we imagine a rectangular slab of a solution, of thickness x, such that the concentrations of the solute all over its two
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456, , PROPEBTIES OF MATTER, , ,, , C, , Cl being greater than t, opposite faces are C\ and C% respectively,, then, the amount of solute passing into one face and out at the, other, i.e., passing any cross-section of it in a given time, is proportional to the concentration gradient, (C l, 2 )jx f, , C, , If we denote the change of concentration for a small distance, dc, the concentration gradient is dc/dx, and the quantity (Q) of, the solute diffusing in unit time through an area A is given by the, , dx by, , =, , Q, K.A.dcfdx,, a constant, called coefficient of diffusion, or the diffusivity, of the Solute^ and depends not only on the nature of the solute in a, given solvent but also on its concentration, i>. % mass (in grams) of, the solute per c.c. And, since the dimensions of Q are [AIT- 1 ], of A,, relation,, , where, , [L, , 2, , ], , K, , is, , and of dcldx, [ML-*], we, [Mr-*], , whence,, , [K], , heave, , =, =, , [K] [L*][ML~*l, , [L*T~, , 1, , ]., , the concentration at the first layer be e, the concentration at a layer opposite, distance Sx apart, will be c-&x.dcldx, because the concentration gradient is dc/dx and, therefore, a change of, concentration in a distance Sx will be Bx.dcjdx. Thus, the quantity, of the solute entering the first layer in unit time is K.A.dc/dx and, that leaving the second layer in the same time, (i.e., unit time), is, , Now,, , if, , t, , given by K.A., , ~, , c, , =, , - -*.*, , (, , -, , K.A., , K.A., , *.8x,, , ), , i.e.,, , the, , rate at which the solute is leaving the second layer is lower than that, at which it enters the first layer by K.A.Bx.d 2 c/dx 2, This amount, of the solute is, therefore, added in unit time to the volume of the, ., , solution between the, A.Sx., , =, , first, , and the second, , layers,, , i.e.,, , to a, , volume, , ~, , K.A.8x.d 2 c/dx 2 IA.Sx, Thus, the rate of change of concentration, this, rate, of, concentration by, of, Denoting, change, , K.d2cldx 2, , ., , dcjdt, we, therefore,, , have, dc/dt, , =, , K.d 2 c/d\*,, , a relation of fundamental importance in diffusion of liquids., , The analogy between Pick's law of diffusion and Fourrier's law, of conduction of heat on the one hand, and Ohm's law of conduction, of electricity on the other, will at once be apparent. For, just at, heat always flows from a point at a higher temperature to that at a, lower temperature, and just as electricity always flows from a point, at a higher potential to that at a lower potential, so also does a, solute flow from regions of higher concentration to those of lower, Not only that, but like conduction of heat and, concentration., electricity, diffusion too is, to all intents and purposes, an irreversible, process, because, if the solvent, or a portion of it, gets impregnated, with the solute, it can never divest itself of it without external aid., 218. Relation between Time of Diffusion and Length of Column., Suppose we have two columns of solution, of lengths /r and /a respectively, having, identical concentrations at different points along them at a given instant. Let the, two columns again acquire similar concentrations at corresponding points after, intervals ti and r s respectively.
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DIFFUSION AND OSMOSIS, , 457, , Quite obviously, these time- intervals must be proportional (/) directly to, the distances covered by the solute and (//) inversely to the absolute velocity of, the solute at each layer. And, therefore, since the distances covered are in proportion to these lengths of the columns, we have, , where, , and, , vt, , v,, , are the solute- velocities at corresponding layers of the, , two, , columns., , Now, by, But, , Q, , tion, , =>, , Pick's law,, A.v.c, if v, , Q = K.A.dcfdx., be the velocity of the solute at a given layer, of concentra-, , c., A, , A.v.c., , ^ A&, = K.A-, , ^, Or,, , ', , ,, , d*, , v, , K dc= --c, , dx, , v oc dcldx,, , [Kfc being a constant for the given layers, But dc/dx, will be inversely proportional to the length oC the column, if the twe>, columns have similar distribution of concentrations., And, therefore,, v oc I//,, [where / is the length of the column., i.e.,, , Hence,, , So, , v 2 /v,, , -, , /,//,., , that, substituting the value of v 2 /v A in relation (ij above,, fi/'i, , - /iW, , /,, , we have, , t <=< /"-, , Or, the time of diffusion from one distribution of concentration to, another, in a given solution, is directly proportional to the square of the length, of its column., *221., Experimental measurement of Diffusivity. The diffusivity of a, solution may be determined by measuring its concentration at a chosen point,, from time to time, and applying Fourrier's theorem to the relation,, , dcldx, , =, , K.d c\dx\, , Different methods have been used for the purpose. Thus, for example,, Kelvin determined the densities of the different layers of the solution by introducing into it, a series of beads of different but known densities and noting their, equilibrium positions. Tms method, however, is open to two very valid objections, viz., (i> some salt from the solution crystallizes on the beads and (11) air, bubbles are formed on their surfaces. Both these factors tend to alter the buoyancy of the beads and the results obtained are thus far from reliable., , Other methods used to measure concentration are based on the measurement of (a) the refractive irfdex of the solution, (b) the contact potential diflerence, and (c) in tbe case of optically active solutions, like that of sugar, the rotation of the plane of polarisation etc., etc., Littlewood has succeeded in measuring the concentration of the solution at a given la>er within *05 gm. per litre by an, optical method, based on the bending of light rays., , For aqueous solutions of sails like NaCl KCl, KNO Z etc., however, the, method devised by Clack, (1942), is perhaps the best. We shall, therefore,, study this in some detail., Clack's Method. He took a 'diffusion cell\ which was just a rectangular, tube, about 5 cms. long, 1 cm. wide and with a horizontal thkknei>s 4 cms., made, up of glass plates, and fitted it vertically into the bottom of a glass box containing, %, , an, , air-free, saturated solution of the salt in question, to maintain which, at its saturation point, crystals of the salt, or the solute, were also placed in, the solution. Outside the tube, and above the glass box, were arranged compartments carrying pure distilled water, at a level about 5 cms. higher than, , An inlet-tube for the water to enter and an outlet-tube for its, also provided, so that there was a constant flow stream of the distilled, water flowing across the upper free end of the cell* resulting in the incoming water, being carried away as a feeble solution of constant concentration, due to the upward diffusing solute from the cell getting mixed up with it. The whole s>stem, was allowed to come to a steady state, which took as many as 12 days, or more and the flow of water was maiatained at about 50 c.cs. per day, the tem,perature being kept constant by means of a suitable thermostat. The theory, underlying the method will be clear from the following :, the tube., exit, , were, , Consider a layer P, say, distant / from the top of the cell, and let Q be the, mass of the solute crossing upwards here per second. Then, it c be the concentra-
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PROPERTIES OF MATTER, , 458, , tion of the solution at this layer, have, by Fick's law,, , and A, the area of cross-section of the cube, we, , K=, where, , K is, p, , is, , [See page 455., , the diffmivity of the solution., , ~, , dl, , Now, W>, where, , .~,, , dc, , dl, , d, , <ti*, , ac, , ', , the optical refractive index of the solution at P., , d, , Q, K =,...,., A, A dp dc, dl, , ., , *',, , Now, apart from diffusion, the motion of the solute is also affected by, the mass motion or bulk-motion of the solvent, (i.e., water) downwards into, the tube., If v be the uniform, downward velocity of this mass-motion of the solvent, it isob\kus that mass A.v.c. cf the solute and a mass A vXp- c) of, the solvent flow downwards at P, where p is the density of the solution at Pand,, therefore, (p c), the concentration of the solvent there., be the mass of the solvent entering the cell per second, at the, Hence, if, , M, , and m, that of the solute, , top,, , M, , But, since, we have, from relation, , that,, , putting, , A.v., , =, , Or,, , Q =, , we have, , ....., , m+A.v.c., , ., , [from relation, , Af/(?-c),, , ..., , (/), , ..(//), , ., , (i), , above., , (//),, , M\m =, , &,, , a constant,, , ^ =, ?, , =, , Or,, , tnc csll per second there,, , =- /l.v.(p-c)., , m^Q-A.v.c., , and,, , So, , b wing, , ,, , we have, , / Sm ^, , >+(-->, , =, , /w(p-c)-f-8.mc, , -j^r-, , ........., , '^8.c)., , (/v), , Now, the net loss in the ma>s of the system per second is clearly equal to, mass of the solute per second minus gain in mass of the solvent per second., Denoting this net loss by i, therefore, we have, i = m, m S.ra m(\ S), whence, m ==//(!)., Substituting this value m, in relation (/v) above, we have, loss in, , M, , ,, , putting this value of 2, in relation / above,, __, A ~, , /(P-C+S.C), , we have, rf/, , ^, , ^a-S)( P -c)' ^'^c', , ", , The various quantities involved in this relation for K are obtained, below, (i) / and S are determined either by chemical analysis or by drying and, , as indicated, , :, , weighing the solution., , () dl\dv> is measured by making a narrow horizontal beam of monochromatic light incident on layers of increasing densities, when the beam gets refracted in the vertical plane and emerges out downwards at an angle a with the horidp - sin a, being the horizontal, t, ,, --., zoDtal, such that, -^, f, [thickness of the cell., where a is measured by noting the vertical displacement of the central fringe in, the interference pattern produced by two narrow horizontal slits, close to each, other, and illuminated by the green bght from a mercury arc, when this light is, allowed to pass through the cell., (///) dp/dc is determined by means of Rayleigh's interferometer, for solutions of different concentrations c, and, (iv) the concentration c at P is computed from the curves (/,, and (c,, ,, , ,, , R
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DIFFUSION AND OSMOSIS, Thus, all quantities in relation // above for, calculate K, the diffusivity for the given solution., , 459, , K being known,, , we can, , easily, , 219. Graham's Law for Diffusion of Gases. Gases diffuse much, more rapidly than liquids, due to their high molecular speed, their, ^diffusion being governed by the following law, known as Graham's, Jaw of diffusion, "The rates of diffusion of two gases are inversely proportional to, :, , the square roots, , of their densities.", Thus, ifr,, r 2 and p 1? p 2 be the rates of diffusion and densities, respectively of two gases, we have, Pa,, Pi, , The difference in the rates of diffusion of gases was usecf, by Graham to separate gaseous mixtures, to which process he gave the, , name, , 'atmolysis'., , Transpiration and Transfusion., a gas escapes from a vessel into a vacuum, through a, small hole in a thin plate, such that the width of the hole is greater, in comparison with its length, the gas is said to effuse, and the, Graham showed that the rate of effusion, process is called effusion., varies directly as the square root of the difference ofpressure on the two, 220., , (/'), , sides, , Effusion, , When, , of the hole, and inversely as the square root of its density, and, of the passage of any other gas at the same time., , is, , <quite independent, , m,, , Thus,, , ///*, , velocity, *, , of, J effusion, , oc, , difference, , A /pressure, A/, density, -, , V, , It must be noted that in this process, the gas flows, as a whole,, through the plate, there being no separation of a mixture of gases into, its, , constituents., , be not too fine,, (//) If, on the other hand, the hole in the plate, -and the thickness of the plate be greater than the diameter of the, hole, the process of escape of the gas through it is called 'transpiration'. Here, the flow of the gas is controlled by viscosity alone, which, is subject to the same laws as are applicable to the flow of a, gas through along tube., , Here, too, the gaseous mixture, as a whole, passes through the, plate, there being no separation of it into its various constituents., (Hi) If, however, the holes be so fine that their diameters, are comparable with those of the molecules of the gas, the passage of, the gas is known as diffusion or 'inter-diffusion., Here, partial separation of a mixture of gases, i.e., atmolysis,, takes place and its constituents may thus be easily separated., 1, , Osmosis and Osmotic Pressure. If two liquids, which can, one another, be initiallv separated by a membrane, they, This process of inter -diffusion of two, inter-diffuse, one into the other., 221., , diffuse into, , a membrane is called osmosis., To demonstrate the phenomenon of osmosis we use what, , liquids through, , are, , membranes, which have the property of selective, transmission, allowing some liquids to pass through them and not, There are various membranes of animal or vegetable origin,, others., called semipermeable
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PROPERTIES OF MATTER, , 460, , which have this property of selective transmission and are, therefore,, Examples of such membranes are ordinary animal, , semipermeable., , :, , bladder, cell-walls of plants, and certain inorganic precipitates etc., , For instance, if we fill an animal bladder with alcohol, tie its, tightly and place it in a vessel of water, the bladder begins to, If, on the other hand, we fill, swell, and may ultimately burst, it with water and place it in alcohol, it shrinks., This clearly shows, that the bladder is more permeable to water than to alcohol. For, water, can pass through it easily whereas alcohol cannot. Graham made use, of the semipermeable properties of pig's bladder to separate crvstalloids from colloids,, a process he termed 'dialysis'. The mixture of, crystalloids and colloids is placed in a tray wi'h a bottom of parchment, paper and the tray is then floated in water, when the crystalloids pass, through the parchment into the water, leaving the colloids behind in, the tray. Poisons in the viscera of poisoned animals can be detected?, in this manner., , mouth, , The systematic study of the laws of osmosis was, , first, , made by, , who used a semipermeable membrane, , of precipitated copper, ferrocyanide, supported by the pores of an ordinary battery-pot or, porous biscuit- ware, as ordinary parchment membranes are too feeble, to withstand high pressures. The pot is first soaked in water to drive, out all the air from its pores. It is then filled with a solution, of copper sulphate and placed inside a solution of potassium ferrocyanide. B Jth these salts diffuse into the wails of the pot, where they, meet to form a brown semipermeable precipitate of cupric cyanide., The pot is then washed with distilled water., Pfeffer,, , If such a pot be now filled with a salt or a sugar solution, closed, tightly with a rubber bung, carrying a mercury manometer, as, shown, (Fig. 27:i), and then placed in a vessel, containing pure water, (the solvent), it is, found that water passes into the pot and an, increase of pressure is recorded by the mano-, , meter, showing thereby that more water is, entering the pot from the outside than is escapMore and moreing out from the inside., water ihus gets into the pot and the pressure goes on steadily rising until a certain^, This will, definite, pressure is reached., , MERCURY, soLuno,, , happen when equilibrium, when \iater passes in at just, , SOLVENT,, , Fig. 273., , is, , attained, i.e*f, the same rate at, which it passes out. This increase of pressure,, or the excess pressure inside over the at-, , mospheric pressure on the water outside, now prevents any further, passage of water into the pot and measures what is called the, osmotic pressure of the solution. Osmotic pressure is thus the pressure, which must be applied on the solution-side of the membrane to prevent, Its value, any flowing in of the solvent through the membrane., depends upon the nature , concentration and temperature of the, -solution., , If the, in, , it, , is, , manometer be open, , inconveniently large., , at the top (A), the travel of, , To prevent, , this,, , mercury, , the upper end, , is
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DIFFUSION AND OSMOSIS, , 461, , and if the volume of air in A B be known, when the pressure, atmospheric, any subsequent reading of the volume of air in it, enables the pressure to be calculated., This method for the measurement of osmotic pressure, however,, is not quite a suitabk one, for, closed,, is, , ;, , due to the solvent entering the pot, the concentration is, lowered, and hence the osmotic pressure measured is for this final, lowered concentration of the solution, and not for the original one, (i), , ;, , of the solution is not uniform throughout, and the pressure measured is thus for the concentration of the, solution in the immediate neighbourhood of the membrane, which it, (/i), , is, , the final concentration, , not quite so easy to determine., , A, , better, , method, , is, , due to Berkeley and Hartley, who, in 1900,, , measured the, , osniotic pressure by determining the external pressure, necessary to be applied to the solution to just prevent any solvent, , [passing into, , it., , Their apparatus, (Fig. 274), consists of a porous tube MN,, (with a membrane of cupric cyanide formed inside its walls), which is, IL*, filled with the solvent and placed, n;, ,^\, ii, inside a metallic vessel B, which, as filled with the solution, through, The porous tube, tthe tube E., v!is connected to two tubes at, two ends, as shown, one, dts, fitted with a stop-cock *S, and the, other being a graduated capillary, F, 2 74., tube of glass, D, |, , j, , MN, , -, , Pressure, , is, , applied to the solution through the tube, , ,, , so that, , no solvent can enter into it from the tube MN, i.e., the meniscus, -of the liquid column in D remains stationary, or the condition of, The external pressure applied through E, -equilibrium is attained., thus gives the osmotic pressure of the solution in B (plus the small and, .almost negligible hydrostatic pressure of the solvent in the tube MN)., Osmotic pressures, as high as 130 atmospheres, may thus be, -easily measured with the help of this apparatus., , 222., Laws of Osniotic Pressure. Pfeffer's results led to the, -establishment of the following laws of osmotic pressure for dilute, solutions, , of non-electrolytes., , The osmotic pressure (P) of the solution of a given solute, 1., is proportional to its concentration (c), provided the temperature, einains constant or,, P oc c., ;, , Since concentration is inversely proportional to volume (K), this, jorresponds to Boyle's law for gases, viz., that P oc 1/F., , The osmotic pressure of the, , solution of a given concentratemperature or P oc T. This,, obviously, corresponds to Charles law for gases., Combining the two laws, therefore, we have P oc c.T., 2., , tion is proportional to its absolute, , ;, , 9, , P, , Dr,, , Or, putting, , c, , = K.c.T,, 1/F,, , where, , K, , c, , is, , is a constant., [where, the concentration of tho
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PROPERTIES OF MATTER, , 462, , solution, in gram- molecules of the solute pere.c.,, , containing, , 1, , gram-moelcule of the solute,, , and, , we have P, , =, , V* its, , volume, , K.(\jV).T., , PV = KT., , Or,, , This constant K for very dilute solutions, is found to be identiso that, for dilute solutions, we have, cal with the gas constant R, !"/>., the same as the standard, PV, RT., ;, , =, , [.equation for a gas., , Solutions of non-electrolytes exerting equal osmotic pressures contain the same number of gram-molecules per c c., i.e., are, of concentrations proportional t> the molecular weights of the solute., Solutions, This corresponds to Avogadro's hypothesis for gases., exerting the same osmotic pressure are called isotonic or isomotic, 3., , solutions,, , Solutions of electrolytes exert a greater osmotic pressure, 4., than those of non-electrolytes, due to their splitting up into ions., [n the case of electrolytes, the relation between P, K, R, and T, IB given by PV = iRT, where / is a factor depending upon its degree, uf dissociation, and is equal to the ratio of the observed osmotic presso that, knowing /, the percentage, flure to that calculated as above, Association of an electrolyte can be calculated *, ;, , 223., Kinetic Theory of Solutions. Van't Hoff propounded the, kinetic theory of solutions, similar to the one for gases, and deduced,, for infinitely dilute solutions, the relation, RT, from purely, close, This, considerations., similarity with the, thermodynaraical, behaviour of gases led him to suggest that the osmotic pressure oj, a dilute solution is the same as would be exerted by the solute, if it, could exist as a gas, and if it occupied a volume equal to that occupied, by the solution, at the same temperature. This is known as Van't, Hoff law., , PV =, , This parallelism between dilute solutions and gases has been, amply verified by the accurate results obtained by Berkeley and', Hartley., , Osmosis and Vapour Pressure of a Solution. The vapour, 224,, pressure of a solution is always less than that of the pure solvent, as, -vcill be clear from the following, :, , If we take a solvent enclosed in an evacuated cbambrr, <Fig. 275), and place vertically in it a long glass tube, containing i, solution, with its lower end closed by means of a gemi permeable, membrane which allows the solvent to pass through it, but not th<, solute, we find that the liquid risen in the tube up to //, say, unti, , H, , is in contad, equilibrium is attained. Now, clearly, the vapour at;, with the solution, and at /, with the holvont. Let the maximunr, be p and p respectively. Taen, obviously, p i*, pressures at J and, greater that) p' by an amount h.v.g. where or is the density of the, vapour of the solvent and is practically the same as that of the vapoui, of the solution at //, the column h being small., , H, , f, , Pp' =, , Thus,, *See solved example, , 9,, , h.v.g., , ..., , (/
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DIFFUSION AND OSMOSIS, "f, , P, , 46:, , be the osmotic pressure of the solution, we have, , =, , P, excess pressure inside the tube over, hat outside at Jon the solvent., ivhere p is the density of the solution., />, , 3r,, , whence, h.g, , tion, , =, =, , A.g.(p-<r),, , /, f, , P \, -:-}, o, , Substituing this value of h.g. in rela, above, we have, , ^, , (/), , ', , P-P', p, , Jr,, , p', , SLMI-PERMtABLE MEMBRANE,, , <*, , =, , Fig. 275., , [neglecting a in comparison with, , Po/p,, , (pp, , the lowering of the vapour pressure, tional to the osmotic pressure (P)., i.e.,, , Now, a, , =, , 1/y,, , where, , v is, , So that,, , the volume of, , p, , p', , =, , I, , is, , ),, , directly propor-, , gm. of vapour at pressure p, , P/t'.p., , ...(m)>, , If P be the atmospheric pressure, v the, its density, we have, this pressure and o, ,, , fit, , p., , f, , volume of the vapour, , ,, , Thus,, , And,, , P.OQ, clearly,, , = P /v = P, v = P, , O, , Or, a =p.<J /P, , .CT., , .v, , 1, , ..., , ., , (fv), , //?., , So that, substituting this value of, , v, , in relation, , (///), , above, we, , have, , P-P, , *=, , P, , X p, , pp', , ==, , J/?,, , ft, ', , [Putting a, , for, , l/v, , we have, , P, , P, where P/P, , v, , =, , /, Or, putting, , = P.?9.a, p, , /7, , P, , the osmotic pressure in atmospheres, and a /p, tho, ratio of the densities of the solvent in the gaseous and the liqidl, states at N.T.P*, is, , 1, , This relation thus gives the ratio between lowering of the vapour, and the vapour pressure of the solvent. It ia true only,, however, if h be small, which will be the case, when the concentraIf, tion, and, therefore, the osmotic pressure of the solution is small., these be large, and, therefore, h also fairly large, the relation between, , pressure, , /*, , and, , p' is, , deduced as follows, , :, , ., *N.T.(*. stands for normal temperature and pressure,, perature and 76 cms. pressure, or a pressure of one atmosphere., , i.e.,, , for, , 0C tem-
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PHQPEBTIES OF MATTBB, , 464, , Considering a very small length, dh, of the column,, , =, , dp, , we have, , dh.a.g,, , (the negative sign merely indicating that pressure decreases as h increases), , = p.a /P, , a, , Now,, dp, , --*'; JH., , AA, , ., , _*-, , Or,, , +, , 6' CT, , [See (/v) above., , r, , Integrating this expression for dh between the limits,, , p and, , />',, , we have, , h, , Or,, , h, , Or,, , -, , ., , =, , .logp, .(i, v, , T = -^l., g-Q, , J^, , ^, p'), p-logB F, , g, *, , ', , =, , Or., But, , e, , 1, ., , P, , h.f.g,, , log, 6, , ., , .a, , .<J, , -, , p', , -;,-^, , whence, h, , -, , ', , p', , /, , P.g. />, , P, , This gives the relation between p and p' for large values of, concentration and osmotic pressure of a solution., Since a dilute solution hehaves as a gas, the volume occupied, at N.T.P. by one gram molecule of a substance in solution would be, the same as it would occupy under the same conditions, (i.e., at, N. T P.) in the gaseous state, viz., 22-4 litres for, we know that one, gram-molecule of a gas at N.T.P. occupies 22-4 litres., ;, , It follows, therefore, that the osmotic pressure for a solution of, one gram-molecule of a substance in a litre would be 22-4 atmospheres., Now, the volume occupied by 1 gram-molecule of water vapour,, 18, gms. of it, is 22-4 litres or 22,400 c.cs. at N.T.P., i.e.,, .-. density of the vapour CJ Q at N.T.P., 18/22400 gms./c.c., t, , =, , And, density of water, p = 1 gm./c.c. so that, for a molar solua solution containing 1 gram molecule of the solute, \\e, have, applying the above relation,, P cr, dp, ;, , tion, ie.,, , ~n, , P/P, , Here,, cr, , =, , =, , =, , P, , ', , rP~'~n, P, Q, , [, , 22-4 atmospheres,, , 18/22400 gms.jc.c., and, , p, , =, , See relation, , (v), , above., , 1 gm./c.c., , p, a, , -'22400x1, , Thus, the lowering of the vapour pressure, non-electrolyte in water is 1'8%., , of the molar solution of, , As already, , indicated, the lowering will be greater in the case of, due to its molecules dissociating into ions,, Osmosis and Boiling Point of a Solution. We know that, , ,an electrolyte,, , 225., , &, , liquid begins to boil, , when, , its, , maximum, , or saturation vapour prcs-
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DlIJTUSIOfl, , 465, , AHD OSMOSIS, , its surface, i.e.,, is equal to the external atmospheric pressure on, the boiling point of a liquid is the temperature at which its maximum, This value of, vapour pressure is equal to the atmospheric pressure., for a liquid increases with increase of, the maximum, , sure, , vapour pressure, , temperature., of a solution, at a, seen, the vapour pressure, the pure solvent so that, at, of, that, than, lower, given temperature,, the temperature at which the solvent begins to boil, its vapour pressure is equal to the atmospheric pressure, but that of the solution, heatwill be lower than it, and the solution will, therefore, have to<be, ed further in order that its, may become equal to the, , Now,, , as, , we have, , ;, , is, , vapour pressure, , begin to boil., In other words, the boiling point of a solution fr higher than that, of the pure solvent., Let us see how this rise or elevation of the boiling point of a solution is related to its osmotic pressure., The elevation of the boiling point of a solution may be obtaini.e., it, , atmospheric pressure,, , may, , ed with the help of Claussius-Clapeyron's latent heat equation,, , IT, , -, , v/z.,, , ^^'^-fa, , are the volumes of the vapour and the liquid resheat of the liquid, T, the boiling point of the, the, latent, pectively, L,, of the solution, and, solvent, dp, the lowering of the vapour pressure, of, the, elevation, the, boiJing point produced., dT,, , Vvap and V lig, , where, , to be used in the above equation, >, / .<r /'o-P227, above, v/z., dp\p, , The value of dp, from relation, Or,, , it, , (v),, , may, , is, , obtained, , =, , be determined directly as follows, , :, , and B be the solution and the solvent, separated by a, a bell jar, (Fig. 276),, aemipermeable membrane and enclosed inside, such that both are at their boiling points (T+tTT)' and T Absolute,, are equal, and their, respectively, so that their vapour pressures, each, with, vapours are in free communication, other above the membrane, the system being in, Let, , A, , equilibrium., , Treating the arrangement as a reversible heat, engine, working between the solution and the solvent as source and sink respectively, consider the, following cycle of operations, Let a small quantity v c.cs. of the, (/), solvent pass through the membrane into the solu:, , dilute it., Then, work done in this exis equal to the osmotic presof, solute, the, pansion, sure of the solution x change in volume, i.e.,, , tion, , and, , P XV, , P is the osmotic, , ergs,, , A, SOLUTION, , T+dT, , B, SOLVENT, , T, , Fig. 276., , pressure for the solution in dynes per sq. cm., or v,p gms., of the solvent be evaporated, these, v, Let, c.cs.,, (ii), from the solution in A, where p is the density of the solvent., If L be the latent heat of vaporisation of the solvent, in fr^s, , where
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FEOPERTIES OF MATTEB, per gm., the total heat absorbed by the mass v. p gms. of the solvent, will be equal to (v.p L+P.v) ergs., Let this vapour pass on from the space above A to that, (Hi), above B. Obviously, no work is dvne during this passage of the vapour,, as the pressure is the same above A and B., , And, finally, let this vapour condense back into liquid in, up v.p.L ergs of heat, thus completing the cycle., Then, we have, from the theory of the reversible heat engine,, , (/v), , B, giving, , heat absorbed from the source (A), heat given to the sink (B), , ^, , v.p., r, -, , Or,, , _, , temperature of source, , ~~~, , temperature of sink, , +vP.v = T+dT, =--., , V.Q.JU, , J, , y.p.L-ffv-~v.p., , ~~, , "**, , T+dT-T Q, , P.v, , T~~, , v.p.L, , ~'~, , v.p.Z/, , dT, , whence,, , =, , 1, , _, ~", , ', , dT, , T, , ', , p, p L, , 2 .--_-, , Thus, the elevation of the boiling point of a solution is directly, proportional to its osmotic pressure. It follows, therefore, that the boiling points ofisotonic solutions, in the same solvent, will be the same., , Let us now calculate the elevation of the boiling point for a, solution of one gram-molecule in one litre of water at 100C, under the, , normal atmospheric pressure., , We know that the osmotic pressure of a solution of 1 grammolecule in one litre is 22-4 atmospheres, at N. T. P., (sec page 464)., Therefore, osmotic pressure at 100C and normal atmospheric [ressure, (76 cms.), will, , be, , = 22*4x374/273 atmospheres., / p<xT., 22*4 v 37*3, = 275X 76x13-59x981 dynes/cm*., L = 540 calories., = 540 x 4-19 x 10 ergs/gm.,, / /, 4-19x10' ergs/cat, ~, 0-958 gms.fc.c., at the boiling point of water,, p, T = 100C = 100+273 = 373 Abs., (, , Now,, , 7, , [, , and, , Substituting these values in the relation for, 373, X 22 4x 373x 76 x 13-59, ", , dT, , The, 100, , ', , =, ~~, , dT above, we have, x 981, , 273x0 -958 x 540 x4-T9~xT6 7, , elevation, , of the boiling point of water per, , 1, , gram-molecule, , in, , known as, , the molecular elevation of the boiling point,, and will clearly be 0-534 x 10 or 5-34C, (because dT oc c and c, here,, will be 10 times as great)., c.cs., , of it, , is, , Osmosis and Freezing Point of a Solution. A liquid will, its temperature is reduced until its, vapour, pressure equals that of its solid. Therefore, since a solution has a, lower vapour pressure than the solvent, it will freeze at a lower temperature than the pure solvent. In other words, the freezing point of, e solvent ti lowered or depressed, when a solute form* a solution, 229., , freeze, as, , we know, when
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467, , DIFFUSION AtfD OSMOSIS, with, , ft., , Let us calculate this depression of the freezing point of a, , solution., , ,, , Consider, again, a solvent B and a solution A, separated by a, semipermeable membrane, and- enclosed under a bell jar. Let both, be at their freezing points, Tand T~ irrespectively, the system being in equilibrium. Treating the arrangement, again, as a reversible, heat engine, we take the solvent here to be the source, and the solution to be the sink, because the former is at a higher temperature than the latter. Now, let the following cycle of operations be, , performed, Let, (/), :, , membrane, , v, , or v p gms., of the solvent pass through the, Then, if P be the osmotic pressure for, done is equal to P.v ergs., , c.cs.,, , into the solution., , the solution, work, Let the solution be frozen, so that v c.cs. of pure solvent, (il), separates out, giving up heat, equal to (v.p.L P.v) ergs, whefce p. is, the density of the solvent at temperature T Q and L, its latent heat of, liquefaction, or latent heat of fusion of its solid, (in ergs per gram), ., , Let this frozen solid be now transferred to the solvent B., And, finally let this solid be melted back to liquid in 5,, (iv), taking up heat equal to v.p.Z/ ergs, during the process., Then, as before, we have, (in), , heat absorbed from source (B), temperature of source, ==, Heat rejected 10 sink (A), temperature oflrink, V'p, , Or, , -, , L, , -, , vp.L-P.v, UT, , (T-dT)-T, ~, _, ~, T, ~, , ', , -, , dT, , ~, , P.v, _ IF/S/, , ---, , Or, , "~"/7, , 1, , T-, , T-ff, (v. P, , .-P.v)-v, , P .L, , v.p.L, , _, , V.p.jL, , P^dT, __, W _, , _, , ^, , -, , T, , Or, , 1, , p./>, , -, , P, , "/, p.L, , Or, the depression of the freezing point t dT, T.F/p.Z/,, a result similar to that for the elevation of the boiling point of a, solution., Since the value of L is smaller in this case than in the first case,, (because the latent heat of solidification of a liquid is always smaller, , than, , its latent, , heat of vaporisation), the value of, , dT, , will be greater, , Let us calculate the depression of the freezing point of water,, (i.e.* dT), when 1 gram-molecule of a solute is dissolved in a litre of, and under normal atmospheric pressure, (76 cms.). We have, it at, *= 273 Abs., P = 22-4 x 76 x 13-59x981, T, dynesjcm*, here., , 0C, L, , = 0C, = 80 calorieslgm., , p, , =, , =, , 80x4-19 xlO 7, 1, , ergs/gm.,, , and, , gm./c.c., , Substituting these values in the relation for dT, above,, 273 x 22-4 x 76 xi 3-59x98 1, , we have, , The depression of the freezing pqinf <?f water produced by 1 grammolecule of a solute dissolved in 100 c.cs. of it is called the molecular, depression of the freezing point? and will clearly be equal to 1 '85 x 10, , *, , 18'5C,
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PROPERTIES OF MATTER, , 468, , 230., Determination of Percentage of Dissociation of an ElecIt must be emphasized again that the above relations are, , trolyte., , true only for non-electrolytes, i.e., for substances like sugar, where, the solute exists in the form of single molecules., greater effect,, than, (i.e., elevation of boiling point or depression of freezing point), that obtained by the above relations is observed in the case of an, The deviation of, electrolyte, because of its dissociation into ions, the obberved effect enables the percentage dissociation of the solute, to be determined. For example, if we take a substance, like potassium, chloride, sodium chloride, or sodium nitrate, each molecule of which, breaks up into two ions, we can calculate its percentage ionisation as, follows, , A, , :, , .Suppose there are a molecules of the solute, and p of them distwo ions each, so that the total number of particles is, Then,, (oc+p)., (oc-p+2p), , sociate into, , =, , observed, , effect, , calculated effect, , dT', dlr, , ^ a+p, , ', , cx~, , djf'and JJare the observed and calculated elevations of boilpoint, or depressions of freezing point, respectively., , tvhere, , ng, , dT =, , Or,, , Or, , dTA/oL, , =, , d., , percentage of ionisation, , Or,, , ^ = dT'dT, ft, , dT'-dT, whence,, , =, , --, , ul, , a, , ^, , xlOO., , Similar treatment will also apply to substances whose molecules, more than two ions each., , dissociate into, , A perfectly, its, , so, , dilute solution is, , supposed to dissociate completely into, , decreases with concentration,, ions, and the degree of dissociation, that a concentrated solution of an electrolyte behaves, more or, , a solution of a non-electrolyte., The reverse of the above happens in the case of colloids. The, substance in a solution, here, forms clusters or aggregations of molecules, having different sizes in different cases. In such cases, the osmotic pressure, and, therefore, the elevation of the boiling point, or the, is very much smaller than in the, depression of the freezing point,, normal case of a non-electrolyte, where the whole solute exists in the, form of single molecules., less like, , 231. Determination of molecular weight of a substance from elevation of boiling-point, or depression of freezing-point of a solution ol, the substance., the boiling point. Suppose we have x gms,, (a) From elevation of, of the given substance dissolved in 1 00 c.cs. of water, and the elevaThen, since an elevation'oJ, tion of the boiling point observed is dT., 5*34*C is due to 1 gram-molecule in 100 c.cs. the number of gramThis number of gram, mol'ecules in the aolutipn must be dr/5-34., molecules is obviously present in x grams of the substance" taketa., t, , So that,, , <T/5-34, , gram-molecules, , of, , the, , substance, , weigl
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466, , DIFFUSION AND OSMOSIS, , x grams., , And, therefore,, , 1, , gram-molecule of the substance would, , weigh 5-34:Xx/dT grams., the molecular weight of the substance, (M),, , Or,, , =, , x xjdT grams., , 5-34, , x grams of, Again,, (b) From depression of the freezing point., the substance dissolved in 100 c.cs. of water lower, or depress, its, if, , by dT, we'have, number of gram-molecules in % grams of the substance = dr/18-5,, because a lowering of the freezing point by 18'5C indicates the prefreezing point, , sence of, Or,, , And,, , .-., , gram-molecule in 100, , 1, , c.cs., , dT/lS-5 gram-molecules of the substance weigh x grams., 1, gram-molecule of it would weigh 18-5 Y^x/dT grams., molecule weight of the substance, IfrfrxjdT grams., , =, , Or,, , SOLVED EXAMPLES, One gram, , of sugar is dissolved in water, making a solution occupying, a, 1000 c.cs. Find ihe osmotic pressure for the solution at U C and 1000 C. Molecular weight of sugar = 342 gms ; R for 1 gm molecule = 8*4 X 10 7, 1., , ., , We know, , that the relation, , temperature of a dilute solution,, (0 At, , 0O., , 1, , RT., , And, therefore,, , RT/V., , 7?, , V, , because a mass of, occupy 342 x 1000, , connecting osmotic pressure, volume and, , PV =, P=, , 8'4x 10 7 ergsldeg.C.* T - 04-273 = 273 Abs.,, 342 x 1000 = 342 x 10'c.w.,, gm. of sugar occupies 1000 c.cs., a mass of 342 gms. would, , Here,, , and, , is, , =, , =, , 342 x JO 8, , ~, =, , P===, , ,\cs., , 4xl07x273, l, , ~, , 8, , 8'4xlOx273, , 342 xlO 3, 6'707, , 342, , 10 4 dynes[cm 2 ., the solution, at, , x, , Or, the osmotic pressure for, , 0C,, , is, , equal, , to, , 6*707, , XlO, , 1, , dynes/cm*., , At, , (//'), , 100C, R, V =, , Again,, , and, So, , T, , 8'4x 10 7 crgsJdez.C.,, 342 x 1000, , =, , 342 x 10*, , 100 + 273, , =, , 373' Abs., , c.cs., , that, substituting these values in the relation,, , p = RTIV we, 9, , have, , _..__, , -, , _, , ., , ^^^, , =, , 9*162, , xW* dynes /cm*., , 0C =, , the osmotic pressure for the solution, at, and, at 100C, , ."., , =, , 6'707x 10 dynes/cm*., 9-162xl0 4 ^5/cm a, ., , One gram, , of salt is dissolved in water to make a solution of 100 c.cs, Assuming the salt to be entirely dissociated, find, in atmospheres, the osmotic pressure of the solution at 20 C., 2., , Atomic weights. -Na, gms/c.c. and, , R, , =, , 8 4 x 10 7 for, , 23,, , =, , Cl, , 35*5., , Density of mercury, , =, , 13'(, , 1 gm.-molecule., , Each molecule of the salt (NaCl) dissociates into two ions, i.e., Na an<3, And, since the salt is completely dissociated, there will be twice as many, particles present in 1 c.c. of the solution as the number of whole molecules., It, follows", therefore, that the osmotic pressure will be double of that obtained, b}, the relation, P = RTIV, i.e., this osmotic pressure, in this case, is given by, Cl., , Now,, , R=, , and, , y, , x 10 7 ergs/deg. C, mol wt. of sa.lt x 100, , 8'4, , ;, , T, , =, , <r.w,,, , 20, , -f-273, , =, , 293
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470, , PROPERTIES OF MATTER, , because, , gm,, , 1, , in mol. wt., , ontained, , is, , x 100, , 100, , in, , c.cs.,, , its, , and, therefore,, , mol. wt., , contained, , is, , c.cs., , V-, , Or,, , (23+35-5)xlOOc.C5., , -, , ~, , 58'5xlO*c.w., , --__, 2x8'4x, , 7, 2x8'4xl0, x293, .., _, --jor, 5g>5, 6, 2x8'4xl0 x293, , 10 5, , x293, , 58^X76 xl 3-6x981, , (v 1 atniosphere = 76x13*6x981 dynes/cm .), 8*299 atmospheres., Or,, The osmotic pressure for the solution is, therefore, 8*299 atmospheres., 3., When 1 gm. of iodine is dissolved in ether making 40 c.cs. of solution,, the osmotic pressure at 20C is 24 atmospheres. Show that the molecule of dissolved iodine consists of two atoms. (At. wt. of Iodine = 127)., 9, , =, , />, , 2'4 atmospheres., HeYe, observed osmotic pressure, let us calculate the osmotic pressure from the given data,, , Now,, , P = RT/V., 8'4x 10 7 ergs/deg., T =- 20-1-273 = 293 Abs., and, V = 127x40c.c5.,, V iodine being moncatomic, its mol wt. is equal to its atomic weight., , We have, , R *, , Since,, , C, , ;, , ., , =, , -, , *m., , 4-778, , 4*8 atmospheres (nearly),, , equal to 2x2'4 atmospheres., Thus, the observed osmotic pressure is half of the calculated osmotic pressure, and, therefore, it follows that the two atoms of iodine in solution combine to, form one molecule, or that the molecule of dissolved iodine consists of two atoms., , which, , is, , Calculate the strength of a cane-sugar solution whose osmotic pressure, one atmosphere. Mol. wt. of sugar = 342 gms., and R = 84xl0 7, ergs./deg. C., We have the relation PV RT., V = RT/P,, Or,, 4., , 270C, , is, , where V, , is, , at, , substance,, , the the volume of the solution, containing, in this case, 342 gms. of sugar., , Thus,, , V c.cs., , 1000, , 1, , gram-molecule of the, , of the solution contain 342 gms. of sugar,, ., , c.cs.,,, , gnu. of sugar., , =, , 342 x 1000/K., Or, strength of the sugar solution, 342 X 1000 X PI RT grams/ litre., , -, , P, R, , Here,, , =, , ., , ,.., , .., , .*., , atmosphere. == 76xl3'6x981 dynes /cm*., 8*4 x 10 7 ergs/deg. C., and, T 274-273, , [ '.', , V, , 1, , ,., , strength of the sugar solution, , -, , 300 Abs., , 342x1000x76x13-6x981, ~, -----, , =, , 8-4, , x, , 10', , *, , x 300, , 13*76 gms. /litre., Thus, the strength of the given cane sugar solution, 5., The osmotic pressure of a solution containing 6 gms. of cane sugar in, 100 c.cs. of water is 307 cms. of Hg. at 13C. What is the osmotic pressure of a, solution of one-sixth this concentration at 50 C ?, ., (London Inter-Science), Let the molecular weight of sugar be = m gms., , Then, the volume containing, i/, y, , and, because, , l, , *, , 1, , gm. mol. in the, , c.cs. t, , 2, , gm., , is, , contained in 100, , c.cs., , case, say,, , Vlt, , where, , 6 gms. are contained in 100 c.cs., 100 c.cs., Land /. m.gms., o, , the volume in the second case, say, K,, 1, , first, , TV, , mxlOO, , xm, , mx 100 c.cy.,
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DIFFUSION AND OSMOSIS, 13-h273 = 286 Abs., temperature 7\ in the 1st case, Tt, 2nd case * 50-1-273 - 323, be the osmotic pressures in the two cases, we have, , And, If, , Pa, , P! and, , P = RTJV19, , R, , the constant for the solution,, , is, , ~, ~~y~~~, , *~LT", , l, , l^f~, , f~, , the, , ^n^, , ==, , ', , p~, , P, =*, , and, , x, , where, , RT^V t, , ., , same as the gas constant., , ", , ', , P!, , y, , ~rpT, , "Pl, , TV 7a Klt F2 and P lt we have, 323 xmx 100x307x1 3-6x981, ', 1, 286x6 xmx 100, 307 cms. of mercury, 307 x 1 3*6 x 98 dyneslcm*., ^ - 323x307x13-6x981^, , Substituting the values of, , ,, , ^, , Pl, , because, , =>, , Or, , ., , I-, , 286 x"6, , 323x307, , 323x307x1 ^'6x981, , Thus, the osmotic pressure in the second case = 57-77 cms. of mercury., Find the percentage lowering of the vapour pressure of water produced, by dissolving 1 gm. of sugar in 100 c.cs. Mol. wt. of sugar = 342 gms., We have the relation for lowering of vapour pressure,, 6., , .*., P, , = p, , CT, , ._, , "o, , o, , [Sec p a g e 463> relation, , ., , (v)., , P, , percentage lowering of vapour pressure, , .*., , P, , Po, , P, , the osmotic pressure of the solution in atmospheres, and ff 'P tn e, ratio of the densities of the solvent in the gaseous and the liquid states, at N.T.P., , where P/P, , is, , P=, , Now,, , T~, , and, here,, , 273 Abs.,, , RTIV,, , V =, , R, D, , P, , And, hence,, because, , P =, , and, , p, , or water at, , dp, , V, , .'., , ,, , =, , By how much, , above, be depressed ?, Latent heat of, , dynes./cm, , ., , ., , =, , "", , 76 x 13'6x981 dynes (cm*., 1 ^w. mo/, of water vapour, or, of water vapour, at N.T.P. occupies 22'4, ^->>,nn, 22400, L ///rc5, or 22400 c.cs., , rv, , 18, , gm.jc-c., , 84xl0 5 x273, , 342x76xT3T6"x98r, is, , =, , 80, , 18, , 22400, , X, , equa to 0-05316%., , will the freezing point, , ice, , ', , =*, , 0-053 16/,., lowering of vapour pressure of <vater, 7., , 0C., , 8'4 x 10*x 273, atmosP neres, 342 X 76xl3-6 981, , N T.P. =1, X, , [taking the solution at, , dyneslcm, , 76 cm^. of Hg., <J, , 342 x 100 c.cs., x 10 7 ergsldeg.C., 8'4xl0 7 x273, 8-4, , cals./gm.,, , 1, , of the sugar solution in question, , and, , atmospheric pressure, , =, , 6,, , 10*, , 2, ., , We have, here, 1 gm. of sugar in 100 c.cs. of water, or 1/342 gm. molecules, ilOOc.c*. of water, and, therefore, 10/342 gm. molecules in 1000 c.cj., or one, itre of water., Now, 1 gm. molecule in, atmospheres at N.T.P,, .'., , 10/342 gms. -molecules in, 22*4 x 10 224, p, , _, , 1, , litre, , 1 litre, ., , of water exerts an osmotic pressure, , of water will exert an osmotic pressure, 224 w iA. ., ._., f 1, , 22*4
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PROPERTIES OF MATTER, , 472, , dT, , Now,, , ** T., , J= 0C -, , Here,, , =, , p, , i, , 273 Abs.., , =, , and L, , gm.lc.c., , P-, , JT, ai, , 342, , - dynes[cnt. 9, , 80 ca/*./#w.=80 x4'2 x 10 7 ergs per gm., , the depression of the freezing point,, , .*., , [See page 467, , ~~., p.L, , i.e.,, , 273x224x10*, _ 273x224, 342xlx8t)x4-2xl0 7 ~342x80x4, , =as, , The freezing point of the sugar solution, by, , ft, ;, , ., , nwrr, Z, , 2, , will, therefore,, , be depressed, , 0'053C, , 8., Determine the elevation of the tailing point of ether produced by dissolving 10 gms. of carbon hexachloride (C 2 CI 6 ) in 100 c.c. of ether., , =, , 0'695 gm. per c.c., boiling point of ether is 35*C, its density at 35C, heat of vaporisation is 81*5 cals./gm. (Take atmospheric pressure, , The, and, , its latent, , = 10, , 9, , 2, , dynes/cm, Here, molecular weight of carbon hexachloride (C2 C7 6), ., , =, , 24+213, , =, , 237, , gms., , Hence, 100 c.cs of ether contain 10 gms. of carbon hexachloride, or, 10/237 of 237, or 10/237 gram-molecules of carbon hexachloride., '., , 1000 ccs. or, , 1, , litre, , of ether will contain, , yr, , x, , 7oO~ f, , or, , ^ ram " mo ^", , 237, , of the hexachloride., Now, osmotic pressure for a solution of 1 gram-molecule in, atmospheres at N.T.P., osmotic pressure for a solution of 100/237 gram- mole cules, cules, , 1, , ', , litre is, , in, , 22*4, , 1, , litre at, , 1, , litre, at, , N.T.P., , -, , -, , j, , And, , .*., , x22*4 atmospheres., , i, , osmotic pressure for a solution of 100/237 gram-molecules in, , 35C, or (273+35) or 308 Abs. is given by, ., 2240x308x10"., 100x22-4^308 atmos, ,, ", dynes cm, P hreSx, "237x273, ~~237~ 273~, ., , ,, , Now,, , elevation of the boiling point, , T=, , Here,, P, , =, , 35+273, , ., , 0*695 gmsfc.c.,, , (taking /, , =, , 4-2, , x, , L, , dT, , - ft"" cut., I, , *= 81 '5 cals.lgm., , =, , 81'5x4'2x 10 7, , 10 7 ergsjcal.)., , 308>2240x308xlO, , -, , ', , [See page 467., , T. Pfr.L., , P-, , 308 Abs.,, , and, , ', , 237"x 273 x 0~'695, , x 81, , _^, , 5, , x 4'2 x, , 7, , ', , 10, , 237 x 273 X -695x81 5x4'2, Or,, , the elevation of the boiling point of ether, , =, , 1-38C., , A, , one per cent solution of potassium chloride (KC1) has an osmotic, C. Determine its degree of dissociation., pressure 6*604 atmospheres at 27, K - 39, Cl - 35 5)., (At. wts., 9., , =, , 74'5 gms., Here, molecular weight of potassium chloride is equal to 39 +35 5, .'., 74*5 gms. of potassium chloride dissolved in 1000 c.cs. of water would exert, a pressure of 22 4 atmospheres at N.T.P., if undissociated., , The solution given, however, contains 1 gm. of KCl in 100 c.cs. or 1/74-5, gm.-molecules in 100 c.cs., or 10/74'5 gm.-molecules in 1000 c.cs., 22'4 x 10/74-5 atmospheres at N.T.P., .'., osmotic pressure of the solution, , =, , Hence,, , osmotic pressure of the solution, at 27C.
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473, , DIFFUSION AND OSMOSIS, , 3*303, , Or, osmotic pressure of the solution, if no dissociation takes place, atmospheres., 6*604., But the observed osmotic pressure, observed effect, 6 604, rt, ,_, 1, , -, , -, , ', , 1, , ., , 999, , 3-303, , bccause, , >, , ', , -, , [535553 tfT, , This means that instead of there being 1000 particles, 1999 particles are, actually exerting pressure., , Thus,, , 1000 particles dissociate into 1999, , And, , 100, , .*., , :^, 1UUU, , dissociation, , Or,, , =, , particles., , x 100, , -, , 199'9 particles., , 199'9-100 -99 9%., , Thus, degree of dissociation of the KCl solution = 99'9%., 10., 31*88 gms. of a substance is dissolved in 2 litres of water. The boiling, 266C higher than that of pure water, under normal, point of the solution is, atmospheric pressure. Calculate the molecular weight of the substance., Here,* 2000 c.cr ol water contain 31 '88 gm*. of the substance., -11, , /., , 1, , 00, , c.cs., , of water contain, , Now, we know, , QQ, , 2 ooo, , x, , O<^, , 20 sms, , ^ tlie, , dT, , su ^ stance, , -, , i.e.,, , - gms., , =, , and, Here, x, 31*88/20 gms., .'., molecular weight of the substance,, , Or,, , ^1, , that the molecular weight of a substance,, , M=, .,, , =, , *, , [See page 469., , =, , 266'C., , i.e.,, , 5-34x3188, , the molecular weight of the substance, , is, , EXERCISE, , equal to 32'00 gms., , XIII, , Enunciate Pick's law of diffusion and find the dimensions of the coof diffusion., , 1., , efficient, , Starting with the same concentration of a given solute at the bottom of, columns of the same liquid, show that the times required to set up a given concentration at the top are proportional to the squares of the, , columns., 2., , State Pick's law of diffusion in liquids., , Compare, , heights of the, (Madras, 1947), , diffusion in liquids, , with heat conduction., Indicate a method of determining experimentally the coefficient of diffusion of a salt in a solution., (Madras, 1949), 3., What do you understand by osmosis, dialysis and diffusion ? State, the laws of osmosis and describe an airangement for measuring osmotic pressure., (Calcutta, 1944), 4., State the laws of osmotic pressure., , Give an account of the relation between osmotic pressure and other properties of a solution., , (Madras, 1947), , Calculate the osmotic pressure of a sugar solution, (moL wt. = 342, Ans. 8*478 cms. of Hg., gms.), containing 1'5 gms. of sugar per litre, at 37-0 'C, 5., , 6, , What, , the strength of a sugar solution whose osmotic pressure at, 10 dynes/cm 2 ., and R, (1 atmosphere pressure, 8*4 x, 10 7 ergsjgm. mol. per degree Centigrade)., Ans. 21 -06 gms. /litre., , 17Cis, , is, , 1*5 atmospheres., , =, , =, , fl, , How, , 1., Define osmotic pressure and state its laws., will you demonstrate this pressure ? Deduce an expression for the elevation of the boiling point, of a liquid by a non- volatile substance dissolved in it., (Madras, 1949), -, , Calculate the osmotic pressure of a one per cent solution of sodium, 8., chloride (NaCl) at 27C, assuming the dissociation of the molecule to be 99 5%., ** 23'0 Cl ** 35, Ans. 8"013 atmosphere*., (At- wts.-Na, 5), f
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474, , PBOJPEKTIJBS OK MATTJKH, , 9, Determine the percentage lowering of the vapour pressure of an, aqueous solution of a non-electrolyte, of mol. wt. 100 gms., when 1 gm. of it is, dissolved in 100 c.cs. of water at N.T.P. (R = 8'4x W/gm. mol. per degree C)., Ans. 0'1818%., 10., Deduce the relationship between osmotic pressure and relative lower(Madras, 1949), ing of vapour pressure. How is osmotic pressure measured ?, 11., A solution of 5 gms. of an electrolyte in water, total volume 1000, 0'279C. What is the molecular weight of the substance in, c.cs., freezes at, solution ? (Molecular lowering of the freezing point = 18'6C)., Ans. 33*3 gmsWhat will be the boiling point of a solution of cane sugar, (mol. wt ~, 12., 342), containing 1 gram of sugar per 100 c cs. of it, under the normal atmospheric, pressure ? Density of water at 100C may be taken to be 0'9580 ^m./c-c., Ans., , 1000156C., , =, , 12-5 gms. of sugar (mol wt., 342) dissolved in 1000 c.cs. of water, lowers the freezing point by, 068*C. Calculate the molecular lowering of the, Ans. 1 8-6C., freezing point for water., 13., , 14., In terms of the molecular theory of matter, discuss the phenomena, associated with (a) diffusion, (b) osmosis., (Bombay, 1944), 15., Calculate the freezing point of a salt solution containing 1 gm. of salt, in 1000 c cs. of water, assuming that the salt is completely dissociated into, sodium and chlorine ions. (At. wts.Na, Ans. -0'064C., 23, Cl =- 35 5), , =, , 16- The osmotic pressure of a solution of 2 pms. of acetone dissolved in, water to make 100 c.cs. is 7*75 atmospheres at, Calculate the molecular, 7, Ans. 60., weight of acetone. (R = 8 4x 10 ergsjgm. mol per deg. C)., , 10C, , 17., 1*2 gms. of a substance dissolved in 24'5 gms. of water caused a, freezing point depression of 1*05C. VVhat is the molecular weight of the substance ? Molecular lowering of freezing point for water may be taken to be, 18-6C., Ans] 8
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CHAPTER XIV, , SURFACE TENSIONCAPILLARITY, Since surface tension is essentially a, 232. Molecular Forces., molecular phenomenon, we had better have first a clear idea as to, what forces operate between molecules., adheThere are two types of molecular forces, (/) forces of, of cohesion or cohesive, sion, or adhesive forces, and (Hi) forces, :, , forces., , Adhesion, , the force of attraction between molecules of different, different for different pairs of substances, e.g.,, has a greater adhesive force than water or alcohol., (/), , substances, , gum, , ,, , and, , is, , is, , (//) Cohesion, on the other hand, is the force of attraction between, This force is different from the, molecules of the sum substance., ordinary gravitational force arid does not obey the ordinary inverse, square law, the force varying inversely probably as the eighth power, of the distance between two molecules and thus decreases rapidly, with distance, -in fact it is 'appreciable when the distance between two, molecules is inappreciable and becomes inappreciable when the distance, It is the greatest, in the case of solids, lets in the, is appreciable.'*, case of liquids and the least in the case of gases, almost negligible at, ordinary temperature and pressure, when the molecules lie very much, further apart for it to, be appreciable., This explains at once why, a solid has a d< finite shape, a liquid has a definite free surface and a, gas has neither., , 0^233. Molecular Range Sphere of Influence. The maximum, distance up to which the force of cohesion -between two molecules, can act is called their molecular range, and is generally of the order, of 10~ 7 cms. in the case of solids and liquids, being different for, different substances. A sphere drawn around a molecule as centre, with, a radius equal to its molecular range is called the sphere of influence^, of the molecule. Obviously, the molecule i affected only by the, molecules inside this sphere, i.e., it attracts and is, in turn, attracted, by them, remaining unaffected by the molecules outside it, as they He, beyond its ran^e of attraction. Laplace (18CH) and Gauss (1830), were the first to have evolved this theory of cohesive force between, molecules in order to satisfactorily explain the various effects of, surface tension, like capillarity etc., /, , 234., , Surface Tension., , in small quantity, free, , It is a general experience that, force, like that, , from any external, , a liquid, , due to, , N, , K. Adam remarked in his "Physics and Chemistry of Surfaces,' if the, *As, force were gravitational in character and obeyed the inverse squaie law, 'the, surface tensions of the ocean would be far greater than that of a cup fail of water,, ', because the distant parts would act with sensible effect, tit is also sometimes referred to as the sphere of molecular attraction, or,, molecular activity., , 475
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476, , PROPEBTIBS OF MATTER, , gravity, will always assume the form of a spherical drop, e.g., rain, drops, small quantities of mercury placed on a clean glass plate etc., Now, for a given volume, a sphere has the least surface area. Thus,, a liquid always tends to have the least surface area. The following, experiments beautifully illustrate this tendency of a liquid to, decrease its surface area., tf we take a wire-ring and dip it in a soap solution, we find that, on, If we place a moistened, out, a thin film is loimed across the ring., cotton loop lightly on the film, it will remain in any form or position ,in which it is, placed, as shown in Fig. 277 (a) ; for, the, soap-film lies both inside and outside the, loop and at every point on the loop, therefore, there are equal and opposite forces,, tending to pull it outward (due to the outside film) and inwards (due to the inside, film\ thus cancelling each other out. But, if the film inside the loop be pricked, so, that the film there disappears, we find that, the loop at once gets stretched into the form, of a circular ring, as shown in Fig. 277,(),, V4/;, because the inward forces having all vanish277,, ed, only the outward forces are left acting perpendicularly to it at every point. Now, for a given perimeter, a circle, In, encloses the greatest area, so that the loop now encloses the maximum area., other words, the aiea of the film left between the loop and the- wire ring is now, reduced to a minimum, clearly showing that the film has a tendency to contract, or shrink, or that there is tension in it., 1., , taking, , it, , ^, , 2., If we place a greased needle on a piece of blotting paper and put the, paper lightly on the surface of water, the blotting paper will soon sink to the, bottom but the needle will remain floating on the surface Careful observation, will show that there is a small depression formed below and around the needle,, and that the free surface of water is slightly extended, The weight of the needle, is here supported by the tension in the depression., If one end of the needle be, made to pierce the surface of water, it rapidly goes slantingly down to the, , bottom, , ^, , If we immerse an ordinary camel hair paint brush in water, its hair all, 3., spread out, presenting a sort of a bushy appearance, but the moment it is withdrawn, they all come closer together in a more or less compact mass, as though, bound down by some sort of a contracting membrane., v/ 4. Yet another beautiful, experiment, often performed for fun by junior, students, is the rapid movement of a camphor scorpion on water. What they, do is simply to arrange pieces of camphor together, in the shape of a scorpion,, and put it on water, when, due to the reduction in the surface tension of water,, on account of the camphor gradually dissolving into it, the camphor is drawn- or, pulled a-ide by th a surrounding uncontaminated water of a higher surface tension. And, since we have camphor dissolving more rapidly at some points than, at others, this force due to surface tension is not uniform all round, with the, result that the 'scorpion' scampers about haphazardly in different directions., If, however, the witer be already contaminated with some grease etc., its, surface tension may be reduced to an extent that the camphor has no further, In such a case, therefore, the movement of the campossibility of reducing it., phor may altogether stop., , ^, , \sThe above experiments clearly show that the surface of a liquid, behaves as though it were covered with an elastic skin or membrane,, having a natural tendency to contract, "with the important difference,, however, that whereas in the case of the membrane or fkin, the tension, increases as the skin is stretched, or its surface area is increased, in, accordance with Hooke's Law, it is quite independent of the area of the, surface in, the case of a liquid, unless the liquid film is reduced in thick"
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SURFACE TENSIOK, ness to less than 70~ 7 cms., when the tension in it decreases rapidly., This tension or pull in the surface of a liquid is called its surface, tension, and may be defined a* the force per unit length of a line drawn, in the liquid surface, acting perpendicularly to it at every point, and, t, , tending to pull the surface apart along the, , line., , L, , An extension of the, 235. Explanation of Surface Tension., kinetic theory of gases to the case of liquids easily explains the, phenomenon of surface tension and the credit for it must go to, Laplace,, , who, , first, , attempted, , it., , Consider four molecules, A, B, C and D of a liquid, with their, spheres of influence drawn around them, as shown in Fig. 278,, sphere A being well inside the, liquid B, near to the free surface, of the liquid, C, just on the free, surface and D, above the free, surface., , Since the sphere of influence, lies wholly inside the liquid,, it is attracted equally in all directions by the other molecules lying, within its sphere of influence,* so, Fig. 278., that there is no resultant cohesive, force on it one way or the oTSsr^and it, therefore, merely- possesses, its thermal velocity., The sphere of influence of molecule B, on the other hand, lies, partly outside the liquid, and this part contains only a comparatively, few molecules of the gas or vapour above the liquid, so that the, upper half of the sphere contains fewer molecules attracting il, upwards, than the lower half, attracting it downwards, and so there, , of, , A, , a resultant downward force acting on B., The molecule C lies on the surface of the liquid, so that, one-half of its sphere of influence lies above the BUT face of, , is, , ful, th(, , liquid, containing only a few molecules of the gas or vapour, whereas, there are liquid molecules in its entire lower half, and thus, th<, , resultant downward force in this case is the maximum, This down, ward or inward force exerted per unit area of a liquid surface is callet, its internal, intrinsic or cohesion pressure, and is the cause of cohesion, It is this pressure whi^h is represented by the term a/v 2 in van de, , Waal's modified gas- equation. /, In the case of the molecule D, which has passed out of, liquid surface, only a part of the sphere of influence lies inside, , th<, th<, , so that the downward force on it decreases, and when th, sphere of influence passes entirely outside the liquid suiface, there i, no downward force on the molecule at all, and it is free to wande, about as a molecule of the vapour or gas., thus see that all ove, the surface of the liquid there is a downward pull due to the attractio, , liquid,, , We, , between the molecules., , V, , If a plane be draw:, 236. Surface Film and Surface Energy., to, the, a, free, surface, and, distance, at, parallel, layer, equal to the molt, cular range from it, the layer of the liquid, lying in-between the fre
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4?8, , PROmiflJES, , Ott, , J&ATTJSH, , this plane is called the surface-film., Clearly, all the, the film are being acted upon by a resultant cohesive, force in the downward direction, the magnitude of which increases as, the free surface of the liquid is approached., , surface, , and, , molecul, , s in, , When a molecule is brought from the interior of the liquid to, the surface-film, work has to be done against the downward cohesive, It, force acting on it, and its potential energy is thus increased., follows, therefore, that the potential energy of the molecules in the surface-film is greater than that of those inside the liquid., Since, however, the potential energy of a system tends towards a, (i.e , a system tends to have the lowest possible potential, the, film tends to have the least surface area, (its thickness, energy),, being fixed) in order that the number of molecules in it may be a, minimum. The potential energy per unit area of the surface -film is, , minimum,, , called its surface energy., , Now, before a molecule can be brought to the surface of the, room has to be made for it, and this is done by separating the, Work is, therefore, done against, adjacent molecules on the surface., liquid,, , the cohesive forces between the molecules in the surface, and not against, the internal or downward cohesive force, for the molecules are moved, It thus appears that, in a direction perpendicular to this latter force., in increasing the surface, , area of a liquid, work, , is, , done as though the, , surface under a state of tension were being extended, very much like a, rubber sheet being extended. The analogy, however, does not go far, for, as already indicated, whereas in the case of a rubber sheet, the, tension increases with extension or increases in its surface area, it is, quite independent of the surface area in the case of a surface-film,, and is the same at all points in it., ;, , l/^yi. Free Energy of a Surface and Surface Tension. Take a, rectangular framework of wire PQRS, (Fig. 279), with a horizontal, wire AB placed across it, free to move up and, down, and form a soap- film across AQRB, by, lj, dipping it in a soap solution. The wire AB, is pulled upwards by the surface tension of, the film, acting perpendicularly to the wire, and in the plane of the film. To keep the, wire in position, therefore, a force has to be, applied downwards, equal and opposite to the, upward force due to surface tension. Let, this downward force be equal to F including, the weight of the wire AB, which is also actvtf, ing downwards. Then, if T be the surface, ^, tension of the film, i.e., if T be the force per, Fig. 279., unit length of the film and /, the length of the, 2 I.T., bewire AB, we have upward force acting on the wire, cause the film has two surfaces and each has a surface tension T., Since the film is in equilibrium, it is clear that, , AB =, , Z.LT, , Now,, , if, , the wire, , AB, , F., , be pulled downward* through a, , m*U
Page 487 :
479, , AtTBFACB TBHSlQH, distance x into the position A'B',, area 1.x on each side, we have, , work done, , The, , film gets cooled, , i.e.,, , if, , the film be extended, f*X>, , =, , x, , ^F, , x, , by an, , T^'^*>, , 2.1.T x., , on being stretched, because the drawing out, , of the molecules from the interior against the attractive force results in, a retardation of their thermal agitation, with a consequent lowering, of temperature*. It, therefore, takes up heat from the atmo*|ih're, , come to its original temperature. This heat absorbed together, with the mechanical work done, forms the energy of the new surface area, 2lx of the film formed., to, , If, therefore, E be the surface energy of the film and Q ergs of, heat be absorbed per unit area of the new surface formedf we have, ,, , Ex 2l.x =, , T.x+Q.2.l.x., , T + Q., [Dividing throughout by 2lx., = (surface energy heat energy per unit area), = potential energy per unit area., T = work done in Beating unit area of the film., , E, T = (EQ), , Or,, Or,, i, , 2.1, , e.,, , Thus, the surface tension of a liquid may be defined as the amount, of work done in increasing the surface area of the liquid-film by unity,, or as the mechanical part of the surface energy of the liquid film., This mechanical part of the surface energy of a liquid-film is free, energy so that, the surface tension of a liquid is equal to the free, energy of the liquid film or surface.^, ,, , u, , 238. Pressure Difference Across a Liquid Surface Drops and, Bubbles. (/) Suppose the free surface of a liquid is plane, as shown, in Fig. 280 (/), then, the resultant force due to surface tension on a, molecule on its surface is zero, and the cohesion- pressure is, therefore,, just nominal. J, , 280 (//)],, (//) If the free surface of the liquid be concave, [Fig., the resultant force on a molecule on the surface would be upwards,, and the cohesion pressure is, therefore, decreased. \/, (in), , Fig. 280, , finally, if the liquid surface be convex, as shown in, the resultant force due to surface tension on a molecule, , And,, , (Hi),, , (0, , O'/), , (///), , Fig. 280., , *This is clearly an example of Le Chatelier's principle, viz., that 'if one of, the factors of any system in equilibrium is changed, thus disturbing the equilibrium,, the effect produced tends to restore that factor to its original value** Thus, a, lowering of the temperature of the surface results in a rise in its tension, which, increases the force opposing enlargement of its surface., is equal to Q.dTldQ, where 9 is the, "fit can be shown that the value of, absolute temperature and dTldQ, the rate of change of surface tension with, , Q, , temperature., tSurface teniion of a liquid ii generally, but erroneously, defined as the, surface energy of the liquid surface. But, obviously* the surface energy can bt, ,tquaJ to the surface tension only <f the beat absorbed by tbt film bt
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480, , t*KOPBETilfiS, , on the surface, pressure, , will, , be, , OF MATTEB, , directed downwards, so that the cohesion*, , in this case, increased., , is,, , Excess Pressure inside a Liquid Drop., , \^ 239., , It, , must be, , clear, , from the above that the molecules near the surface of a drop, (which, a convex surface), experience a resultant, The pressure inside it must,, pull inwards., therefore, be greater than the pressure outside it., Let this excess pressure inside over, is, , the pressure outside the drpp be p. Then, if, r be the radius of the drop, and T, its surface, tension, we have considering the equilibrium, of one-half of the drop say, the upper half,, the upward thrust, or the upper hemisphere,, on the plane face ABCD, (Fig, 281), due to, the excess pressure p is equal to p.irr z, ,, , ., , And, force due to surface tension, acting downwards on it and, round its edge, is equal to T. 27ir. Since the hemisphere is in equilibrium, we have, p.itr- = T.Znr,, , 2T, whence,, Excess Pressure inside a Soap Bubble. If, instead of a, of, drop, liquid, we consider a bubble, there are two surfaces to be, considered, and not one, because it is like a spherical shell or a hollow, so that, the force, cylinder, having an inner and an outer surface, due to surface tension in this case is 2 x ?7tr.T, (i.e., 2vr.T due to, each surface). Therefore, for equilibrium of the hemisphere, we have,, ;, , 2, , in this case,, , /?.7rr, , ., , p, , whence,, , =, , =, , 2, , x, , 27JT.7, , Inr.T, ,, , Trr*, , =, , 7, , =, , 47rrT,, , 4T, -, , r, , It will thus be seen that the excess pressure inside a drop or a, bubble is inversely proportional to its radius (i.e., p * 1/r) ; so that, the, smaller the bubble, the greater the excess pressure inside it., , This can be beautifully shown by blowing two soap bubbles, of, unequal sizes, at the two ends of a tube of the form of the letter T,, as shown in Fig. 282, and then putting, them into communication with each, The air passes from the smaller, other., one into the bigger one, (because the, pressure inside the smaller one is greater, than that inside the bigger one), so that, the smaller bubble goes on shrinking,, and the bigger one goes on swelling,, until the smaller one is reduced to a, hemisphere for, then, any further shrinkage would mean an increase, in its radius and, therefore, a decrease in the /pressure inside it., Equilibriuto between the two bubbles is attained when the curvatures, of the two become the same., ;
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SURFACE TENSION, , 481, , 241. Determination of the Surface Tension of a Bubble., expression for the excess pressure inside a bubble, deduced in, affords us a simple method of determining ihe, surface tension of the bubble. For, if wo have, , a vertical glass tube AB, (Fi#. 283),, fine orifice or aperture at its lower, , The, 240,, , with a, , end B,, , connected to a manometer, , M, and a stop-cock, then, on dipping AB into the experimental, liquid, a liquid film is formed at the orifice., This film is then blown into a bubble at B, by, opening the stop-cock for a while and allowing, some air to come into AB. The difference h, in the heights of the liquid columns in M_j$i^, noted., Then, if P be the density of the, manometer -liquid, the excess pressure inside, the bubble is clearly equal to A.P.g. But, as, Fig. 283., we know, the excess pressure inside the, bubble is also equal to 4:T/r, where T is its surface tension. Hence,, equating the two, we have, 5*,, , =, , =, , T, whence,, r.h.p.gj*,, where r is the radius of curvature of the bubble which is obtained, from its diameter, measured accurately by means of a travelling, 4T/r, , //.p.,, , microscope., It will easily be seen that for accuracy of the result, the value of, the excess pressure inside the bubble should be, i.e, , h should be large,, , ,, , This would be, aperture at B be small., large., , the bubble be small or tho size of the, , Work, , done in Blowing a Bubble. If, for the sake of, n3glect the cooling produced when a film is stretched,, done in blowing a bubble is easily calculated out as follows, , 242., , simplicity,, , the work, , so, if, , we, , :, , We know,, , from, , 237 above, that, , =, , in creating a film, surface tension x area of the film formed., therefore, the radius of tho bubble blown be r, the area of the, 2 x4irr% for it has two surfaces, an inner, film forming the bubble, , work done, If,, , =, , and an outer one, each of surface area 4vr 2, Therefore, work done in blowing the bubble, ., , 243., , = Tx 87rr =, 2, , Sirr*T., , If we have a, Curvature, Pressure and Surface Tension., rest, the inward pressure on it due to surface, tension must be balanced by an equal, pressure acting outwards., , curved liquid surface at, , Consider a, portion A, of a, liquid surface, (Fig 284), cylindrical in, form, i.e.. curved only in one direction., P, Then, the force of surface tension T, acts at right angles to every unit, Fig. 284., The forces, length of its boundary., and, are equal and opposite and hence cancel out, but, over, and BC, though equal in magnitude are inclined to, those over, each other, and have thus a resultant p normal to the surface, as, , BCD, , AB, , CD, AD, , t, , ehown.
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482, , PROPERTIES Of MATTER, , To determine the vajue of /?, consider unit length of the surface, ng the direction of DA. Forces T and 7, , *t 0, (Fig. 285), taken alo, , 1 ntT55rr^r, , T-^-TT3CtTj, , \P, I, , 1, , \f, , ff, \j, , C, , OP =, , their, , Thus, a force T.dQ acts inwards on, , !, , J, , LJ, , act at the point 0, as shown,, resultant being, T.dO., , an area equal to, f, , 1, , x r.d6, where, , r, , i*, , the radius of curvature of the surface,, (because we are considering unit length, of the surface of breadth r.dB). If, , p acts outward on, total force directed out-, , resultant pressure, this area, the, , ward on the area is equal to p.r.dQ., Fig. 285., For this element of the liquid surface to be at rest, therefore, we, , = T.dO,, p = T/r dynes/cm*., , have, , p.r.dQ, , whence,, This resultant pressure is, therefore, the difference of pressure, on the two sides of the surface, which is required to balance the effect, due to surface tension. It follows, therefore, that the pressure must be, to balance the effect, greater on the concave side than on the convex side,, on, the, of surface tension, surfaces., If the surface be curved in two directions, as shown in Fig. 286,, and the radii of the two curvatures be r l and ra respectively, tht, pressure due to the curvature of AB and, CD will be J/r t and that due to the curvature of AI) and BO will be T/r 9, and, therefore, the total difference of pressure on the, two iides of the surface will be given by, f, , ;, , And, if one of the surfaces be convex, and the other, concave, the radii r l and ra, of the two surfaces will have opposite signs., So that, in such a case, we shall have, , Combining the, , cases, therefore,, , Fig. 286., , we may put the general, , relation aa, , us consider a few special cases :, Case of a spherical surface. la the case of a spherical surface, like that, af a liquid drop, or an air bubble inside a liquid, we haver ], r t *- r, say, *o that, excess pressure inside it is given by, , Now,, , let, , (/), , 1, , ;, , Inner, , In the case of an air or a soap bubble, because there are two surfaces, ao, and an outer one, we have, **T, , P-2X*-, , 4T, -., , (Sec, , In this case, one of the radii, (//) Caw of a cylindrical surface., whereas the other is the sa ne as the radius (r) of the cylinder., So that,, for one single surface ,, p ** T/r,, , tod, , for two surfaces,, , p, , 2F/r., , is, , 240, , infinite.
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StJBFAOB TENSIOH, with, (Hi) Case of a catenoid. Here, the surface being one of revolution,, 0., difference of pressure, we have p, And, therefore,, , no, , =0., , An example of such a surface is that of a soap-film, supported in-between, t vo parallel rings with its two ends burst, to maks me pressure difference zero., It is common ex244. Layer of liquid between two plates., one, over the other,, of, be, two, glass, placed, plates, perience that if, there is no difficulty in separating them,, P|, but if a drop of liquid, say, water, be, pig. 287,, placed in between them and squeezed into, ft thin, layer, it may, require considerable, The reason will be clear from the, force to pull them apart., ,, , following, , ., , :, , The thin, circular area,, , layer or film of water wets the plates over an almost, is concave outwards, as shown in Fig. 287., , which, , If d be the thickness of the water layer in-between the plates,, the radius of curvature of the two concave edges of the liquid layer, 4s nearly d/2., Thus, if r be the radius of the circular area or film of, water, the excess pressure p inside the liquid-film over the outside, ,, , atmospheric pressure, , where, , T is, , is, , given by, , the surface tension of the water-film., , is very large compared with d,, comparison with 2/d so that, we have, , Since r, , iu, , Ijr, , is, , almost negligible, , ;, , Thus, the pressure inside the film is less than the outside atmosve sign of/?), by 2T/d&nd, thereclear from the, pheric pressure, (as is, on the two plates pushes, of, the, excess, the, atmosphere, pressure, fore,, them closer together, making d still smaller and r larger, thus*, , further increasing p., Now, if A be the circular area over which the water wets the*, them together is equal to, plates, the total force which squeezes, , This much force, perpendicular to their surface, will, therefore,, be needed to pull them apart. Obviously, the smaller the value of, thinner the layer of water, the greater the force required to, </, i.e., the, them., separate, , Shape of Liquid Meniscus in a Capillary Tube. We, liquid meniscus in a capillary tube is concave for a, and convex for a liquid, like mercury. Let us see, water, like, liquid,, its, determines, what, shape., Let a capillary tube of glass be dipped vertically in a liquid,, meeting its surface at P, (Fig. 288). Then, a liquid molecule at P,, In contact with the tube there, will* be attracted outward by thfr, solid molecules of the tube near to it, due to the force of adhesion*, 245., , know that a
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PBOPBBTIES OF MATTER, , 484, , and "attracted inwards by the near molecules of the liquid, due to th<, force o^ cohesion the resultant force on it will, therefore, be th<, resultant of these two forces of adhesion and cohesion., ;, , 288., , Fig., , Now, the, , resultant, , force of adhesion acts, , at right angles t^, the tube at the point P., Let it bo represented by the straight line, PQ. And, the resultant force of cohesion acts at an angle of 45 to the, vertical, and is represented by the straight line PS, so that, the twc, forces acting on the molecule are inclined to each other at an angle, of 135. T&eir resultant, obtained by the ordinary law of parallelogram of forces, is represented bv the diagonal PR of the parallelogram PQRS, with PQ and PS as adjacent sides. Its direction, will natunlly depend upon the relative magnitudes of the two forces, PQ and PS, and the following different cases arise, :, , (i), , tant,, , =, , \/1.PQ, the resul, equal to l/v/2, i.e., if PS, along the vertical, as shown in Fig. 288 (i)., , JfPQIPSbe, , (PR) wilt, , lie, , () If PS be smaller than V%-PQ, the resultant (PR), side the liquid, as &hzv\n in Fig. 288 (//)., , mil, , lie out-, , PS be greater than \/2.PQ, then, PQ will lie inside the, in Fig. 238 (///)., shown, as, liquid,, What happens to molecule P happens to all other molecules in, contact with the glass of the tube. And since a liquid cannot permanently withstand a shearing stress, its surface at every point will be, at right angles to the resultant force there, when the liquid attains, the position ol equilibrium., (HI) If, , y, , Thus, in the first cave, when the resultant force PR acts along, the vertical, i.e., when PS, \/2.PQ, or, the cohesive force is ^/2, times the adhesive frce, the molecules of the liquid are neither raised, nor lowered and the liquid surface remains flat or plane., , =, , In the second case, when, , PR, , outside the Jiquid, i.e., when, than ^/2 times the adhesiv*, molecules, the, near, the walls of the tube are, of, the, liquid, force,, raised up against the tube, those in the middle remaining practically, unaffected, thus making the liquid surface concave upwards, as in the, case of water and other liquids which \*et the walls of the tube., , PS<\/ 2.PQ,, , or the cohesive force, , PR, , lies, , is, , less, , the liquid, i.e., when, greater than -\/2 times the adhesive force, the, liquid molecule* near the walls of the tube are depressed there,, making the surface convex upwards, as in the case of mercury and, other liquids, which do uot wet the walls of the tube,, , And, when, , or,, , lies inside, , the cohesive force, , is
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485, , SURFACE TENSION, Angle of Contact., Case, of a liquid in contact with a, (0, 246., , When, , solid., , a liquid, , plane of contact with the solid is,, In general, curved., The angle between the tangent to the liquid surface, at the, pnint ^f^nntnrt qftd th e solid surface, irisidetheUquid, is cdTSA, the angle of, coqtactybr that pair of solid and liqttict., , meets a, , solid, its surface, , near, , its, , ~, , and 180. For most, than 90, for mercury and glass it is, and the, // wally depends upon, thena^, solid, andjs not altered by a change in the inclination oftheTjSlStL, In the figures shown, [Fig. 289 (/) and (//)] /. PQR is the angle, of contact. It is acute in (/) and obtuse in (//), for, in the former, , The angle may have any value between, , and, about 140., liquids, , glass,, , is, , it, , less, , ;, , ', , ;, , ;, , case, the liquid rises up a little, alongside the glass plate, dipped, in, the Iquid, and the angle, , between QR, the tangent to the, of, liquid surface and the part, the plate, inside the liquid, is, acute, whereas, in the latter case,, , QP, , the liquid, , where, , is, , depressed, , a, , little, , comes into contact with, Fig. 289., the glass plate, and the angle, between the tangent PR to the liquid surface on the part, it, , QP, , of the, , plate, inside the liquid, is obtuse., , For pure water and clean glass, the angle of contact is 0. For, and if the surface of the, ordinary water and glass it is about 18, glass be contaminated with grease, its value may be as much as 35., ;, , s, , If, , Case of two liquids in contact with each other and with air., liquids, not miscible with each other, be brought into contact, as at 0, (Fig;. 290); both being in contact, , (ii), , two, , with air, three surface tensions are to be, taken into consideration, (a) that of the, surface between air% and liquid* /, v/z.,, (b) that ot the surface between air, r,, ;, , and, and liquid // viz., T, between liquid / and liquid, tl, , r3, , ;, , (c), , that, , J7,, , viz.,, , ., , -p^ r, equilibrium 7\, jT2 and T8, should be represented by the three sides, of a triangle, taken in order. This, , ,, , Fig. 290., , triangle of forces is known ae Neumann's triangle. In actual practice,, across no two pure liquids for which the Neumann's triangle, may be constructed, one of the surfafce tensions being always, greater than the other two so that, the equilibrium condition shown, in the figure is never attained., Thus, for example, in the cage of, water, mercury and air, the water drop, when placed over mercury,, spreads all over its surface, prov ided both water and mercury are, This is so, because the surface tension of mercury is about, pure., , we come, , ;, , 550 dyneslcm., and that of water, only 75 dynes/cm. But, if the, mercury surface be contaminated with grease, its surface tension.
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PBOPEETIE3 OF MATTEB, , 486, , decreases and some water drops may stay on it, so that, in that ease,, the construction of Neumanns triangle can be possible., tx(iii) Case of a solid, liquid and air in contact. This is, more important case than the previous one, for we have to consider, three surface tensions, v/z., 7\ for air-liquid, Ta for air-solid and T~, for liquid-solid surfaces respectively, (Fig. 291)., Let $ be the angle of contact of the liquid with the solid,, In case (a) and obtuse in case (b). For equilibrium, therefore,, , the component 7\ cos, , by, , TI,, , of, , T, , in the direction of, , 19, , i.e. t, , cos e, , TS+T!, , Or,, i, , e.,, , = T -T, , T! cos, , 2, , Clearly, therefore, if, t will be less than, , 3,, , ro, , is, , 90, , ;, , =, , ,, , plus, be balanced, , TV, , whence,, greater than, , and if T2, , $>, (OJ, , T3 must, , acute,, , T9, , cos e, , T3, , ,, , is less, , =-(T^T^T v, , cos B will be positive,, , than, , Ty, , &^SN^*?^ y, , cos, , will I*, , I, , Fig. 291., , and 180., If,, negative, and 9 will lie between 90, howevet>, ^a > ^1+^3* there will be no equilibrium, arid the liquid will spread, over the solid, as happens when a water drop is placed over a perfectly clean plate of glass, or a grease- free mercury surface., 247. Measurement of the Angle of Contact. For mercury and, glass, the angle of contact may be determined by the following, simple method due to Gay Lussac., , A small round-bottomed glass flask is nearly filled with mercury, (more than Iths of it), and its mouth closed tightly by a rubber, bung, through which passes a glass rod R to adjust the level of, mercury in it. The flask is then clamped in the inverted position, a*, shown in Fig. 292 (a), and the rod (R) is moved in or out, until the, , (a), , surface of mercury is, or curved portion (or, appears there. .This, printed sheet of paper, , Fig. 292., , (), meets the glass, i.e., no meniscus, capillary curve > as it is sometimes called),, can be tested by observing the image of a, held against the flask, by the light reflected, plane where, , it
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SURFACE TENSION, , 1, , 48 ?, , *t grazing incidence from the mercury surface. An undistorted image, indicates that the surface of mercury is plane where it meets the glass., , The mercury surface in the flask forms a circular sheet of diameter AB, which is measured by means of a pair of calipers, whence, the radius AC of the sheet is known. Let it be denoted by a. Then,, if f be the radius of the bulb of the flask, and /_BAO, $, we have, , =, , a, Or,, , r, , <f>, , =, , cos~, , l, ., , AD, , be drawn to the spherical bulb at X, the, Now, if a tangent, angle of contact 0, for mercury and glass, is clearly equal to the, angle BAD., 0*, Or,, Or,, So that, knowing <, from relation (/) above, the value of 8 can, be easily determined., , A, , better arrangement for, , the one shown in Fig. 292, , making the surface of mercury plane, , Here, the level of mercury in the, spherical bulb, which is open at both ends, is adjusted by raising or, lowering the reservoir of mercury (/?), connected to it by an Indiarubber tubing. The proceure otherwise is the same as in the first, Is, , (b)., , experiment., , Another simple method to determine the angle of contact, mercury and glass is to insert a small slanting glass plate, Into mercury, as shown in Fig. 293, and to, adjust its inclination until the mercury, meets the glass at P without curvature, i.e.,, the surface of mercury is horizontal there., Then, the angle of contact for mercnry and, <., 6) where /_APC, glass is equal to (180, , for, , AB, , =, , Is, , To measure the angle, dropped from A., , Then, clearly,, , tan, , <f>, , =, , <f> t, , p^-., , a plumb line, Fig, 293., , Or,, , ^, , =, , tan- 1, , p(^<, , For measuring the angle of contact between water and glass,, coated with wax, a similar method was used by Adam. A glasa, trough is coated on the, inside with wax, so that, , may be filled with, water above the level, of the sides, and, 10, it, , supported by two screws,, S and 5, (Fig. 294), auch, that, by working them up, or down, the top of the, , trough, , is, , made, , filled, , is, then, with water and any, , impurities on, , skimmed, , off, , lightly across, Fig. 294., , perfectly, , It, , horizontal., , its, , surface, , by moving, it, , coated glass plate*, , a wax-
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PROPERTIES OF MATTER, , 488, , A glass plate A, also coated with wax, is clamped, as shown, so, that it can. be moved up or down in the vertical plane, about, a horizontal axis at D, by m^ans of a screw S'. It is then dipped, into the water, and its angle of inclination adjusted until the water, surface touching it is perfectly horizontal. Then, the angle 6 between, the lower part of the plate and the horizontal water surface is the, angle of contact for water and the wax-coated glass plate, and is, measured as in the above case for marcury and glass., ', , The angle 0, in this case, is found to depend on whether the, plate is lowered into tho wator or raised up and hence two readings, are taken one, whsn tho plate is lowered an 1 the other, when it is, raised, and their mean is taken as the correct value of 0, or thd, required angle of contact., , \^-^4R. Rise of Liquid in a Capillary Tube. One of the most, striking effects of surface tension is to rais3 a liquid in a capillary, tube dipped into it, a capillary tube being just a tube of a very, fine bore (from the Litin word, c^pillus, It is for thi*, a hair)., reason that surface tsnsion is also sometimes called capillarity., , When, wets, , it, , a capillary tubo is dippad in a liquid like water, which, for which tho angb of contact may be taken to be zero,, the liquid immediately ris^s up into it, and if the, tube be &fine one, tin shape of the liquid meniscus, , and, , is, , spherical, , and, , co-icave upwards, as, , shown at B,, , (Fig. 295).*, , B, , A,, , Let r bo the radius of the tube at B, the point, t^, which the liquid rises into it. Then, it will be, up, practically the same as the radius of the concave, meniscus, so that the excess pressure above the, maniscus over that immediately below it is 2T/r, i.e., the pres3ur3 in tli3 liquid, just below the meniscus, is less than the atmospheric pressure above it, by 2T/r. And, since the pressure on the liquid, t, , surface, outside the tube, is atmospheric, the liquid, will bo forced up into the tube, until the hydrostatic, the, of, liquid column in the tube equtls this excess pressure, pressure, 2T/r. If the liquid rises to a height h, the hydrostatic pressure due, Fig. 295., , to the liquid column in tho tube on the surface of the liquid will, clearly be A.P.g, where p is the density of tho liquid. f, , 2r/rA.'.g., whence,, , Or,, , 2T, , =, , rA.V.g,, , T, , *The rise of a Ii4uid in a capillary, a book of his, published in ths year, tbe glass for the liquid. Observing that a, walied tube than in a thia-WAied one,, , tube was, , first, , explained by Hawksbet, , 1709, as being due to the attraction of, liquid does not rise higher in a thickhe aiturallv concluded that only the, molecule close co the surface of the glass must be concerned in this attraction., As meationed already, ue now know that the forces between molecules causing, sirfac? tension are precisely the same as those operating in the cases of cohesion,, solution or chemi;al reaction., tit will thus be seen that ths capillary tubs acts like a manometer, giving, the difference of pressure above and immediately below the liquid meniscus,, in*
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489, , SURFACE TENSION, Thus, knowing r, A, p and, can be easily determined., , g,, , the surface tension, , T, , of the liquid, , of contact for the liquid be not zero, surface tension T of the liquid acts, inwards along the tangent to the liquid meniscus at every point of, its contact with the inner surface of the tube, making an angle 8, is the angle of contact, with the wall of the tube, ^Fig. 298). where, for the liquid and the glass of the tube., , In case, however, the, , ansrle, , and the tube be not narrow, the, , There, , is, , thus exerted an inward pull on the glass in this direc-, , tion at all these points., Since, in accordance with Newton's third law of motion, action, and reaction are equal and opposite, there is an equal and opposite, reaction R exerted by the glass on the, RcosO RcosO, , This reaction R (equal to T), be, resolved into two rectangular, may, components (/) R cos B/cm. = T sin, 0/cw., along the vertical, in the upward direction and (ii)R sin (t/cm.= T sin Ofcm.^, at right angles to it, in the outward direction, as shown., Taking the whole menis-, , =Tco$0, , liquid., , cus into consideration, the horizontal or, components all cancel each, other out, and only the vertical components are effective, which are thus, , outward, , added up., , Now, if r be the radius of curvature, of the tube at the height of the meniscus,, 296., then, ohviously, t ie moniscus touches, it along a, length 2nrr, tho circumference of the circle of radius r, so that, the total upward force on the liquid in the tube is Znr.T cos 0., It is this force which supports the weight of the column A of the, liquid in the tube, (where h is the length of the column: from the, l, , ;, , horizontal surface A of the liquid, outside the tube, to the bottom of, the meniscus at B) plus tho weight of a volume v of the liquid,, in the meniscus itself,, i.e*, the weight of a total volume of the liquid,, rrr, , 2, , ./i+ v,, , (where, , 2, 7rr .h is, , 2-Trr.T, , where, , p is, , tho volume of the liquid column, 2, cos, (?rr, /t+v). p.g,, , =, , h)., , ., , the density of the liquid., 358, , \27jr.atfW', , p '^", , If the volume of the liquid in the meniscus be negligible, in, comparison with that in'the column A, i.e., if'th* tube of a very fine, bore, we have, , 2-rrr, , cost), , 2 cos V, , ', , '", , the relation being known as Jurins Equation., N.B. In case the capillary tubs is not vertical, but inclined at an angle, to, , it., , we, , take into consideration only the vertical height (h) of the liquic, , *For experimental details of the method, see page 504.
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PBOPERTIES OF MATTES, , 400, , column in it, which will obviously be equal to / cos, the tube occupied by the liquid. So that, here,, , T=, , cos, , r.p.g.l cos a/2, , a,, , where, , /, , is, , the length, , f, , i, , 0., , be not negligible, its value may be determined bj, If, however,, taking the meniscus to be hemispherical in shape, (since the tube is, narrow), of radius nearly equal to r, i.e., the same as that of the tube, at that place, so that the volume of the liquid in the meniscus is, equal to the difference between the volumes of a cylinder of radius r, and length r and a hemisphere of radius r., f volume of a, wrV, cylinder, v, , v, , Of,, , SB 7rr 2 .r, , 3, , f.irr, , =, , Tir 3, , i.Trr, , <ofa, , 3, ., , 8, sphere, 4nr /3,*and of, , L hemisphere, , Or,, In this case, therefore,, 27rr., , Or, , =, , Tcos, , r, , >, , v, , =, , 7rr, , 2, , ?!^'!dLi, , ., , 3, , in, , case, , 7rr, , )p^, , )'!y? *, , %nr. cos B, , And,, , 2, , .(/i+r).p.g., , //ie, , $'*, , (column, , = 0, coy & = so that,, r = l^m^ dyneslcm., 1, , ;, , Affective height, is, , of the, , now h+^ r, and not, , liquid, , h., , we then have, , But, column, , a, , 2^r 8 /3, , 3, , (7rr .A-)-|irr, , r, , =, , if g be greater than 90, cos 6 is negative and the, in the tube is depressed below the liquid level outside, , liquid, it, i.e.,, , his negative. Hence it is that w_e_fmd it so difficult to introduce, mercury, (forj^UliJi-sJ40^jnearh )7lnt6 a fine capillary tube., Again, if we introduce into water, a capillary tube, with its inside coated with paraffin wax to make #>90, the surface of the watei, column also, inside the tube, lies below that in the outer vessel., r, , And, further, we may come across two liquids for which the, T are the same but whose angles of contact (0) are different., They will naturally rise to different heights when the same capillary, values of, , tube, , is, , introduced into them., , 249., , Rise of Liquid in a Tube of Insufficient Length. We, rises up into the capillary tube, dipped, into it, until the weight of the liquid in the, tube is just ^balanced by the force due to its, surface tension., If, be the angle of contact, between the liquid and the tube, and R, the, radius of the liquid meniscus in the tube, we, have r, R cos 0, (Fig. 297), where r is the, radius of the tube ; so that, relation (ft"),, , have seen above, how a liquid, , J, , =, , above,, Fig. 297., , now becomes, -T=^ 2c cost*, e, , g, , = ~ h 'Z, 2, , ', , h is the height of the liquid column in the tube., R.h, Hence, clearly,, 2iyp.g a constant., Now, with the tube sufficiently longer than h, it is the value of, h alone that changes to satisfy the above relation for T., But, if the, tube be smaller than the calculated value of A, the only variable in, the above relation is R, because now h, I, the, length of the tube (s, i, , =, , f
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8TJEFAOB TENSION, , 491, , is, a constant for the given liquid and the tube., liquid thus just spreads over the walls of the tube at the top and, , constant) and so, , The, , meniscus acquires a new radius of curvature R' 9 such that, or, that R'.l=zR.h^& constant., vWSSo. Rise of Liquid in a Conical Capillary Tube. Suppose we, take a concial capillary tube, (i.e., a capillary tube with a fine conical, bore) ABC, with a hole at its upper end, (Fig., ^, ~f, A, 298), and dip its in a liquid, like water, which, wets it walls. We shall find that the liquid, rises up into it to a height A, above the liquid, surface," in accordance with the relation,, its, , K.7=277p.g,, , where a, , and, , /?,, , the semi-vertical angle of the tube,, the radius of the tube at the liquid, , is, , meniscus FG., , AD, , Let the vertical height, of the apex of, the cone from the liquid surface be denoted by, /, and the radius of the tube, where it meets the, liquid surface, (i.e., of the portion EC of it), by, f, Then, from the nimilar right-angled triangles, ., , AEF and ADB we, t, , Fig. 298, , have, , BD, R lh, K, -= l-h, -{-, , >, , whence,, , d-h, fln\, /?=^_-J., O, , Putting this value of, , R, , in relation, , (/),, , above,, , r, , 7, , ,, , TH, , we have A = /f J-jA, , *, , fi, , So that,, Or,, , which, , is, , a quadratic equation in, , h., 21. T., , So that,, , A=, , cos*, , 2, ,, , whence the height h,, be easily determined., , up to which the liquid, , rises into the tube,, , CAB, , if the tube be only slightly conical, we have cos a=l, very, In that case, therefore,, , And,, nearly., , N.B., , The, , ve sign is not usually indicated., , Further, in such a tube, R is practically equal to r, /.., it is practi., cally a uniform capillary tube, as shown in Fig. 295, and, therefore,, for the liquid to rise up to the top into the tube, the minimum, value of r should be greater than r./.p.g/2.
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PROPERTIES OF MATTER, , 492, , We, 251. Energy required to raise a liquid in a capillary tube., have seen above how when a capillary tube is dipped vertically into, a liquid which wets the walk of the tube, there is a rise of the liquid, inside the tube., The rise, obviously, takes place against the action of, The, gravity and the liquid, therefore, mw>t gain in potential energy., question, therefore, arises as to where does it get this increase in its, For, according to the Jaw of conservation of, potential energy from., energy, energy can only be converted from one form into another,, but cannot bo created. The explanation is, however, simple., We have three surfaces of separation to consider when a, capillary tube is immersed in a liquid, viz., (/) an air-liquid surface, a glass- liquid surface, each having its, (ii) an air-glass surface and (//'/), own surface tension, different from the others, and equal to its free, surface energy per unit area., , Now,, ture,, , (i.e.,, , as the plane liquid surface in the tube acquires a curvabecomes concave), the air-liquid surface increases and, as, , the liquid rises in the tube, the glass-liquid surface increases, the airThus, the surface, glass surface decreasing by an equal amount., energy of the air-liquid and the glass-liquid surfaces increases w-hile, that of the air- glass surface decreases by the same amount. In, other words, the energy required to raise the liquid in the capillary, tube is obtained from the surface emrgv of the air-glass surface., , On the other hand, a liquid, which does not wet the walls of, the tube, gets djpressed inside it, below its level outside the tube., In this case, obviously, the glass- liquid surface decreases, whereas, the air-glass surface increases by an equal amount, resulting in a net, increase in the surface energy of the whole system., This energy it, derived from the depression of the liquid inside the tube, whose, gravitational potential energy is thus decreased by an equal amount., Rise of a Liquid Between Two Parallel Plates. An almost, 252., similar case to the above is that of two vertical plates kept parallel, and close to each other in a liquid, when the liquid rises in between, them, (if it wets the plates). Let us calculate the height to which it, rises., , If, , d be the distance between the two, , of the meniscus, (which, , is, , plates and,, , cylindrical, in this case),, , whence, 2 r cos Q=d, or rdj2 cos, tact for the liquid and the plates., , Now, we know that the pressure, than the pressure just above, , it, , where, , Q,, , Q, , is, , r,, , we have, , the radius, d!2, ', , ~=cos, , 0,, , the angle of con-, , just below the meniscus, , by an amount equal to T(, , -f, , is less, , r\, , where r and r' are the radii of the two curvatures, at right angles to, each other. Since the meniscus is cylindrical, one of the curvatures, has a radius equal to that of the cylinder and the other, a radius, equal ts oo, (the surface being plane), so that r' = oo and/ therefore,, l/r'=0, and hence the excess pressure just above the meniscus over, that just below it is equal to T/r. Or, substituting the value of r, we, ,, , have excess r, pressure above the meniscus, , = -^, a/2 cos $, , ', , =-, , c s, *., , d
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493, , SUEFACB TENSION, Therefore, the liquid column will rise, , up in-between the two, , due to it becomes equal to, 2T. cos 6jd. Thus, if h be the height up to which the liquid rises, and, p, its density, we have 2T cos 6/d^h.p.g*, plates, until the hydrostatic pressure, , Or, , 2T.cos, h, , ', , Or, h, , is, , 9, , -~d^T', , inversely proportional to d,, , i.e.,, , the rise* of the liquid is, two plates., , inversely proportional to the distance between the, , In case, however, the two plates be not parallel but inclined at, a small angle to each other, meeting along a common vertical edge, the, liquid does not rise uniformly in-between, them., But, at any point on the liquid, column, the distance (d) between the plates, is, , proportional to the distance, , x of the point, , from their common edge. And, since, as wo, have seen, h oc l/d, it follows at once that, h oc, , 1/jt., , Now, from the, above, we have, , h.d~2Tcos, It is,, , relation for A, deduced, , 6 /p., , g= a constant., , therefore, clear that, , h.x=^a constant,, , Fig. 299., , also., , [, , dx&x., , In other words, the liquid surface in-between the two plates will,, be a pan of a rectangular hyperbola, (Fig. 299)., Force Between Bodies Partly Immersed in a liquid. It is a, 253., common observation that pieces ot cork and such other li^ht bodies,, when floating in water, cling and collect, together into clusters. This, is due to capillary action, i.e., the rise of water into the small spaces, in-between the pieces., in this case,, , /There are three cases to be considered, viz., (7) when the liquid, it does not, and (Hi) when it wets one and, (//) when, , wets the iwo bodies,, not the other., , In the first case, say, for example, two glass plates, partly immersed in a liquid, like water, [Fig. 300 (/)] the liquid rises in between them to a level higher than that outside them. And, since the, below the meniscus in-between them is, pressuie in the liquid just, smaller than the pressure due to the atmosphere at the same level, outsida them, they get pushed towards each other, i.e., they seem to, attract each other., , In the second case, as for example, when the two plates are, a liquid, like mercury, (which does net wet them),, partly immersed in, the liquid is depressed between them below the level outside them,, above the liquid menis[Fig. 300 (//), and the atmospheric pressure, in the liquid at, the, than, smaller, is, in-between, pressure, them,, cus,, will be Htgctive, and the liquid will get deprewed, *ln case 0>90, cos, in-between the two plates iubkaa oi ri&itg up.
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PROPERTIES OF MATTES, , 494, the, , same, , level outside, , them, and no this excess pressure on their out, , side again pushes them towards each other,, seem to be attracting each other., , and again,, , therefore, they, , Fig. 300., , In the third case, as for example, when one of the glass plates,, be clean so that it can get wet with water, and the other B, be, y, coated with wax, (so as not to get wot with water), and both be partthe liquid meniscus is concave near to the, ly immersed in water,, and convex, near to the other plate B,, plate A wetted by the liquid,, not wetted by it. Due to the pull of one meniscus on the other, therefore, the concave meniscus on the inside of A is a little lower than, the concave meniscus on its outside, and the convex meniscus, on the, Inside of B is a little above the convex meniscus on its outside, [Fig., 300 (Hi)]. Since the pressure just below the concave meniscus outside of A is lower than the atmospheric pressure at the same level and, above the concave meniscus on its inside, it is pushed outward, away, from By as shown by the arrow-head. Again, since the atmospheric, pressure above the convex meniscus outside B is lower than the, pressure in the liquid below the convex meniscus at the same level on, its inside, it is also pushed outwards, away from A, as indicated by, the arrow head, and thus both A and B move away from each other,, i.e., they seem to repel each other., say,, , A, , t, , This explains why a small piece of wood, (wetted by water) and, a needle (not so wetted), when floated close together in water, steadithe former drifting towards the edge and the latter, ly get apart,, the centre., towards, moving, , above is true only when the two, near, if the distance between them, aro, bodies, For,, together., floating, be large, the portion of the meniscus where it changes from convex, to concave, or vice versa, will become quite straight or horizontal, i.e.,, In a level with the rest of the liquid outside the plates, and there, will, therefore, be no resultant force acting on them, one way or the, On the other hand, if they be very close to each other, the, other., point of inflexion of the meniscus disappears altogether, resulting in, It should bo noted that the, , the rise of liquid in-between them, and they get pushed towards each,, other., , 254., , Shape of Liquid Drop on a Horizontal Plate., , There are, , forces acting on the drop of a liquid, placed on a horizontal, v/z., (i) the force due to gravity and (ii) the force due to surface, plate,, , two, , tension.
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SURFACE TENSION, , 496, , Whereas the former tends to flatten it and to spread it out, becomes horizontal, i.e., tends to 'squash* the drop,, &s it were, the latter, by trying to decrease its surface area, tends to, gather it up into the form of a sphere. It is, therefore, a case of graNow, the gravitational force depends, vity versus surface tension., upon the mass of the drop, which is proportional to its volume, i.e., to, the cube of its linear dimensions, and the surface tension depends, upon the surface area, i.e., upon the square of its dimensions. Therefore the effect of gravitation is more pronounced on a large drop and it, until its surface, , ,, , gets flattened out, whereas the surface tension has the upper hand, in, the case of a small drop, and gives it a spherical shape. That is why the, , small dew and rain drops, or those of mercury or oil etc., are "all, This can also be readily seen by placing small, spherical in shape*., and large drops on a plate of glass or water drops on paraffin wax,, when it will be found that small drops assume a spherical shape, but, the large ones get flattened out, until their upper surface becomes, horizontal. In the case of mercury on a glass plate, a large drop will, assume a long elliptical sort of shape, its upper surface, in the, , middle, being plane, with the edges protruding on either side, the, angle of contact being about 140., It is possible to calculate the surface tension of mercury or the, angle of contact, by considering the various forces that keep it, The method was developed by Quincke, and latet, in equilibrium., improved by Edser, [see 255 (4)]., 255., Experimental Determination of Surface Tension. We, shall now consider some of the usual methods employed to determine, the surface tension of liquids., Searle's Torsion Balance Method., This is perhaps the, direct, and, the, most, method, for, the determination, quickest, simplest,, 1., , of, , surface, , tension of, , of, the, , liquids, particularly, , those, , for, , which, , angle, , of, , contact, , is, , zero., , The essentials of, Che apparatus used are, *s indicated in Fig. 301, where, is a rigid rod,, fixed to a fine torsion, wire w, (about 1-25 mm., in, radius), stretched, horizontally across a, frame work, as shown., The rod terminates in, , R, , a pointer, moving over, a vertical scale, , and, , S, , at, , one, , end,, , Ing, , a sliding weight, *A striking proof of the, , carryFig. 301., , rain drops being perfect spheres, , is, , the natural, , phenomena of the rainbow and the halos, whose arrangement of colours and shape, Dan only be explained on this basis. The slightest deviation from the spherical, ihape of the dropt would materially affect both these.
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PROPERTIES OF MATTER, , W, , at the other, with the help of which it is counterpoised, so as to, remain horizontal., From a point, near the front end of the rod, is suspended a, scale pan, which, in its turn, carries a rectangular wire-frame or a, and '02 to *()5 cm. in thickness),, glass plate, (about 10 cms. Jong, fitted in a metal socket or clip, with its lower edge perfectly horizontal., A vessel D, containing the experimental liquid, is placed on an, as desired., adjustable table, which can be raised up or lowered down,, , To start with, the vessel D is raised up until the wire-frame, of the glass plate just dips a little, dips irto it or until the lower ed<re, into the liquid. It is then lowered down until this lower edge of the, wire-frame or the plate lies exactly in a level with the liquid surface,, A film of the liquid is thus formed in the wire-frame, or in-between, the plate and ths liquid surfaco, exerting a downward pull on the, frame or the plate due to the surface tension. This results in the, downwards and its position is, pointer-end of the rod also deflecting, read on the scale. The vessel is removed from under the frame or the, The film is now punctured or the plate allowed to, glass plate., dry up. The liquid film thus disappears and with it also the, downward pull on the frame or the plate, and hence on the rod, which,, therefore, returns back to its initial position., Weights are now placed, , in the scale pin until the pointer-end, again deflected downward to the same extent as before., to this weight, mg., Clearly, then, the downward fbr^e on the rod due, due, to surface tension,, on, it, the, downward, as, same, the, is, pull, say,, in the first case., Now, if / be the length of the wireframe or the plate and /, its, thickness, it is in contact with ths liquid along a total length equal to, cos 8, where 6 is the angle of contact for the liquid. Hence,, 2(1 +t), if T be the surface tension of the l-quid, the downward pull on the, frame or the plate due to surface tension is equal to 2(1+1). cos 0.T., , of the rod, , is, , ., , Hence, , 2(/+/)., , 6.T=mg., , T=*, , mg!2(l+t). cos 0., In the case of the wire frame, / is negligible, and, therefore,, , =, , And,, BO that,, the case, , if, , T =, , T mg/21 cos 6., be zero for the given liquid, we have cos 9, m.g/'2(l+t), in the case, , of, , the plate, , and, , T=, , =, , 1, , ;, , m.g/2l, , t, , in, , of wire frame., , And thus, knowing mg, / and /, the value of surface tension (T) for, the liquid can be easily calculated., As will be readily seen, the method may also be used to comfor which is zero., surface tensions of two given liquids,, the, pare, N.B. It happens sometimes that, despite all care the lower edge of the, in a level wiih the liquid surface, dips a little, say,, plate, instead of being just, If this be so, it is obvious that downward pull on, to a depth h inside it, the plate, due to the surface tension, is reduced >y an amount equal to the upward force on it, due to the buoyarcy of the displaced liquid, which as we, know, is equal to the weight of the liauiddispliced by the plate, ie.. equal to, the density of the liquid and g, the acceleration due to, (/ x t x h x p x#), where p is, , gravity at the place.
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SURFACE TENSION, Thus,, , weha^, , Or,,\ '>', , 2(/+0- cos 9.T.-(lxtxhXpxg), , 2(/+0- cosB.T= mg+U.h.p.g,, , whence,, , T=, , =, , mg., , ^^r~\, , We know that excess pressure inside an, 2.^ Jaeger's Method., bubble in a liquid is equal to 2F/r, [239J, where T is the, surface tension of the liquid and r, the radius of the bubble., Jaeger, by a simple and ingenious method measured this exc3ss pressure, so that, knowing p and r,, p, necessary to produce such a bubble, the surface tension (T) of the liquid can be easily determined., has,, , ;, , The apparatus simply, , consists of a long thin glass tube AB,, ending in a fine jet of about -2 to, , (Fig. 302), with its lower portion, 5 mm. in diameter, and with its, , tip cut quite smooth and square*,, so as to be perpendicular to its, axis., , This dips in the experi-, , mental liquid, contained in a, beaker, with about 4 to 5 cms., length inside the liquid., then connected to a manometer, and a WoulfT s bottle,, fitted with a dropping funnel F,, containing water, as shown., The liquid used in the manois Xylol (a liquid hymeter, of, It, , its, , is, , M, , M, , in, drocarbon), preference to, because of its lower density,, water,, in the two limbs may be large., , Due, , Fig., , 302., , so that the difference of level, , some liquid rises up into the tube AB,, meniscus being nearly hemispherical. Some air is, now forced into the tube by dropping water into the Woulff 's bottle,, which displaces its own volume of air from it. The liquid column, in AB thus slowly moves down until it reaches B, when a bubble is, formed there, The process must be regulated to be so slow that, about 10 sees, are needed for the bubble to form. The radius of, curvature of the bubble gradually decreases with increasing pressure, inside it, until it reaches the minimum value, and the bubble acquires, a more or less hemispherical shape, with a radius r, equal to that of, to capillary action,, , the shape of, , its, , the aperture at B< the pressure inside being now the maximum, as, indicated by the difference of levels (H) in the two limbs of the, , manometer., , The bubble now becomes unstable, for, any further growth of, tends to increase its radius, which results in a, crease in the pressure inside it due to surface tension, thus, ing the equilibrium between its internal and the constant external, It, therefore, now gets detached from the tube, and the, pressure., whole process starts all over again., ;, , it, , clearly, just before the bubble breaks away from B, the, it is, inside, pressure, equal to that at C, i.e., equal to P+H.p.g, where, , Now,, , * This, must be so, not only to the naked eye but cvengyoder a microscope, with no trace of any roughness or iraggedness at its infofcf tfr wfer edges.
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PBOPEBTIBS OF MATTER, , 498, , P, , the atmospheric pressure and H.o.g> the pressure due to the, in the manometer, (p being its density)., liquid column, And, when the bubble just breaks away from B the pressure on, it is equal to that at the level of B in the beaker, i.e., equal to, P+h.d g where h is the depth of the tip B from the surface of the, liquid in the beaker and </, its density., la, , H, , t, , t, , And, therefore, the excess pressure inside the bubble, , =, , (P+H+4)-(P+h.d.g), , = g(H.?-h.d)., , But the excess pressure inside the bubble, we know, must be, equal to 23P/r, where T is the surface tension of the liquid in the, beaker., , Hence,, , 2T/r, , T, , Or,, , Thus, noting H>, , h, p, , = g(H.p-h.d)., , =* r.g, (H.?-h.d)l2., , and, , d,, , and determining, , of a microscope, fitted with a micrometer eye-piece,, culate the value of T for the given liquid., , with the help, , r, , we can, , easily cal-, , Despite all care, however, there is no absolute certainty as to, the radius of the bubble, when it gets detached from the tube, and it, may not be hemispherical and of quite the same radius as the, aperture at B. In fact, quite the contrary. For, it can be hemispherical only in the case of extremely narrow tubes, and its radius is, found to be always a function of the radius of the aperture. For a, greater accuracy in the result, therefore, the following relation, is often used, :, , This method, (i), , (11), , (iii}, , is, , of importance in that, , it, , can be used, , for determination of the surface tension of molten, mqtals ;, for determining the variation of surface tension of a solution,, different concentrations of the solute ;, for, , comparison of surtace tensions of different liquids, , with, , ;, , determining the variation of surface tension of a liquid with, temperature, (as the temperature of the containing vessel can be, easily controlled and the bubble is formed inside the liquid itself) ;, , (iv) for, , and, (v), , for studying the molecular aggregation of the liquid, (i.e., the number, of atoms in its molecule^, from the slope of the curve between temperature and surface tension., , '", , 3., The Drop-Weight Method? This is a simple, though perhaps, not buite so accurate a method for determining ths surface tension of, a liquid by considering the vertical forces that keep a email, drop of liquid in equilibrium, just before it gets detached from, the end of a vertical glass tube of circular aperture., , At the instant the drop gets detached, it assumes a, cylindrical shape at the orifice of the tube, (Fig. 303), so, be the surface tension of the liquid and r, the, that, if, , T, , radius of the, , o, Fig. 303., , orifice,, , we have, , excess pressure (p), mospheric pressure, Tjr., , =, , inside, , the drop over the outside at[see, , 243, page 481., to this, , Hence the downward force on the drop due, , =, , vrMT/r.
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SURFACE TEKSIOK, , 499, , And, since the weight mg of the drop also acts vertically downwards on it, we have, total downward force on the drop = irr*.Tjr + mg., , Now, the liquid touches the tube all along its circumference 27rr,, and hence the upward force acting on the drop due to surface tension is, equal to 2itr.T., It is clear, therefore, that while the drop is yet in equilibrium,, immediately before its detachment from the tube, the two sets of, opposing vertical forces acting on it must just be balancing each, other., Hence,, 2, 7rr .r/r, 2nr.T, irrT, mg, mg., , Zirr.T, , Or,, , =, - vr. T =, T, , whence,, , =, , +, , mg., , Or,, , Trr.2, , 7, , =, , +, , mg,, , mg/irr., , Thus, knowing the mass of tha falling drop and the radius of, the lower end of the tube, we can easily calculate thQ value of jP, the, surface tension of the liquid in question., , The actual experiment, , A, , is, , carried out as follows, , :, , clamp and a thin,, clean and dry tube of glass (of about 4 mm. bore) is attached to its, nozzle by a piece of India rubber tubing, carrying a pinch-cock on it., The burette is filled with the experimental liquid and its flow, through the glass tube regulated by the pinch-cock, so that small, drops* form slowly at its lower end, their rate of detachment from, burette, , is, , fitted, , vertically in a suitable, , being about one per minute, when an accuracy of about '2% may, possibly be attained., The drops, as they fall, are received in a clean, dry and an accurately weighed beaker and the average mass (m) of a drop determined, by weighing the beaker again with its contents. The diameter of, the orifice of the tube is also determined carefully by means of a, , it, , travelling microscope., , As indicated above, the rnathod is far from accurate, for the, simple reason that the liquid drop seldom gets detached from the end, of the tube, under the ideal statical conditions assumed, the radius, of the 'neck of the ^rop as well as the amount of *t that actually gets, detached being more or less uncertain quantities, even if the end of, the glass tube be smeared with wax, thus making the whole problem, a complicated dynamical one., 9, , =, , N.B. Lord Rayleigh has suggested the relation T, m^/3-8r, which yields, better results, and Harkins and Brown have, experimentally shown by using, liquids of known values of T, that the relation T, mg.FIr holds true, where, a, the symbols v and r standing for the volume of the, is a factor related to v/r f, , =, , F, , drop and the radius of the tube respectively., 4., Quincke's Method. This method is applicable in the case, of liquids which do not wet the surface in contact with them, as for, example, mercury. Not only can it be used to determine the surface, tension of such a liquid but also its angle of contact with the solid, , for, as, , *rhe drops formed at the narrow orifice of the tube need* must be small,, they grow in size, the liquid skin thet e is unable to support their weight., fThis relation too is found to be true only up to the limit where the, , maximum, , value of v/r 8 does not exceed 10.
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PROPERTIES OF MATTER, , 500, surface on which, for, , it rests., , which the method, , is, , We, , shall consider hore the cas3 of, , mercury,, , so frequently used., , It has been indicated already thit in the mittsr of the shape, of a liquid drop resting on a solid surface, which it do3s not wet, it is, a tussle between surface tension and gravity, the, X, former having the upper hand in the case of, CTr^.*.^X&f, small drops, which thus assume a spherical shape, and the latter holding sv\ay in the case of the, F>, 3, larger ones, tending to spread them out until, their surface beoornos horizontal., Tais explains the gradual flattening up of a dr>p as it grows in size, until its top becomes quite, horizontal, beyond which no more flattening occurs due to the limit, imposed by the angle of contact of the liquid and the solid in, question. So that, the shape ultimately acquired by a large drop of, the liquid, (in our case, mercury), is as shown, (Fig. 304), with the, central part of the top surface horizontal and the two ends protruding outwards, such that the tangents to them make an angle a each,, with the horizontal solid surface, wh?re a is ths supplement of the, its value being, angle of contact (6) for the liquid and the solid,, about 40 in the case of mercury and glass, (the angle of contact, for them bsing nearly 140)., The actual shape of these protruding, edges of tin drop is of littb or no concern to us for our present, purpoas, Except that, at their mi^t protruding part, such as at D,, (Fjg. 305), they are more or less vertical., , ^i^^S^S^, , Let us consider the equilibrium of the drop in two ways (/), without involving the angle of contact, or, when it is not known, and, (//) involving the angle of contact, or, when it is known., Without involving the angle of contact,, , (i) Imagine a large drop, central part horizontal, to be cut into two halves by a vertical plan3 perpandicular to its length and passing centrally, through, it, and consider the equilibrium of a thin slice ABDEFG, of one-half of, , with, , its, , the drop, thus obtained, with the vertical faces cut parallel, (by two, other planes, parpandicular to the first) and a horizontal width, GA, FJ -=b, , =, , As, , is, , evident from tha very symmetry of the drop, (a) the forces, GCEF and the one opposite to it at the back,, , acting on the two sides,, , must be, equal, and opposite, and, (b) so also must, be, , the, , TA, , pressure, , immediately above, and below, the, , 'YDROSTATIC, , 'PRESSURE, , horizontal, part, of it at the top., , Let, , us con-, , sider the portion, , or the slice, lying above the, horizontal plane, DLKM, a distance h bjlow the top and passing through D, where the
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501, , STJBFAOK TENSION, , drop protrudes most and which, therefore, represents the greatest, horizontal surface, i.e., the surface having the maximum area in the, , of the drop., , section, , (i), , (/i), , forces acting, , slice are, , :, , the surface tension at, , towards the, (i/i), , on this, , the following, acts vertically upwards and, which, D,, has, therefore, no component along the horizontal ;, the lateral hydrostatic pressure (P) on the plane face GLKA, due to the neighbouring part of the liquid on its right, acting, , The, , left, , ;, , the pull due to surface tension over, to, , it,, , towards the, , GA,, , acting perpendicularly, , right., , Thus, the oaly horizontal forces acting on the slice axe the latter, two,, opposite directions, and since the slice is in equilibrium, they, must be equal in magnitude., Now, lateral hydrostatic pressure due to a liquid being equal lo, in, , the pressure half-wuy, , down, we have, , =, , =, , \h p g,, \GL.$.g, hydrostatic pressure P over GLKA, h p, the density of the liquid and g, the acceleration, where GL, , =, , ;, , due to gravity., .-., , hydrostatic thrust over, , =, , h.p.gxhxb, , \, , And, the pull on, , GA, , =, , =, , area, , h\p.g.b!2., , ;, , GLKA., , h*.p.g.bI2., , due to surface tension, , th& surface tension of the liquid, , T.b, , P\, , GLKA, , ~, , T.b.,, , where, , T, , is, , so that,, , T = h^.g/2, , Or,, , ...(A), , whence, T, the surface tension of the liquid may be easily determined,, the angle of contact, /f h bo measured (see below), without involving, for the liquid and the solid in question., of contact. In this case, we cpnsider, (ii) Involving also the angle, So that, the surface-tension pull, the equilibrium of the whole slice., of, T.b., due to mercury on glass, at E, i.e., on the line of contact EF, the, slice in the direction, to, acts, and, ially, tangent, mercury,, glass, ES, with an equal reaction of the glass on tie slice, in the opposite, The horizontal component of this react ional force due, direction EQ., to glass, along EF, is clearly equal to T.b. cos a, where a is the, and solid, supplement of the angle of contact for the given liquil, And, so does the pull due to, surfaces, and acts towards the right., so, surface tension over GA, viz., T b. 9 acting perpendicularly to GA, =. T.b. cos a, the, towards, force, horizontal, total, the, right, that,, ;, , +, , T.b., , This, face, , over the, opposed by the horizontal hydrostatic thrust, , is, , GFHA,, , acting towards the, |, , where,, , GF =, , left,, , H.p.gxHxb -, , H, the, , -this, , thrust heing, , now, , equal to, , \H*.?.b.g,, , total height of the slice., , So that, there being no other horizontal forces to be considered, we, have, for the equilibrium of the slice,, T.b cos a+T.b, Or,, | H*.p.b.g., Or,, , =, 2T+2T cos a =, , H*.p.g., , Or,, , 2T (1 +cos a) =, T cos a+T, , \ H*.p.g., , H*p.g.
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PBOPBRT1ES OF MATTER, , 502, , T, , Or,, , ...(B), , T may be calculated, if H and a be known., The quantities A,, and <x, involved in relations (A) and (B), above may be measured as follows, (0 Measurement ofh. For the measurement of A, the distance, between the top of the drop and the horizontal surface of maximum, area, we must determine the point Z>, where the free surface of the, whence,, , H, , :, , drop, , is, , perfectly vertical., , This has been beautifully done by Edser,, , by means of the arrangement, shown, , in Fig. 306., , A, , special vernier microscope, having a plane glass plate P,, cemented on to its objective at an angle of 45, is arranged horizon-, , with, , axis horifront of the, most protruding part of the, drop., Light from an incandescent lamp S is focused by, a lens L and the plate glass, P, acting as a mirror, on to, the edge of the drop, when a, bright, thin, horizontal line is, seen at D, where the drop, tally, (i.e.,, , its, , zontal), right in, , protrudes out, surface, , moat and, , is vertical., , Ue, , The, , aperture of the microscope is adjusted until this bright horizontal line, coincides with the horizontal cross wire of the eye-piece, and its read-, , ing is noted. The microscope, which is capable of both a vertical, and a horizontal movement, is then raised up and moved towards the, , drop until the image of the top flat surface of the drop coincides with, the horizontal crosswire of the instrument. This latter adjustment, is greatly facilitated by sprinkling a little lycopodium powder over, the top of the drop and mikin* the imige of a speck or two of the, The, povyder C3incide with the horizontal crosswire of the eye-piece., distance through which the instrument has had to be raised up, h., directly gives the required distance GL, This may be easily done by means of a, the latter being focused on the top,, or, a, microscope,, spherometer,, (i.e., on a speck of the lycopoiium powder on its surface, as before),, and then on the paint of the edge at the bottom, which is in contact, with the surface of the glass plate., , Measurement of H., , (//), , (in), 6., , We, , Measurement of oc, and, therefore, also the angle of contact, A and B above,, , have, fron relations, , ZJTt, , Or,, , 2, +cos a = # 2 /A whence, cos a = (H*lh*)-l., knowing H and A, we can easily ca^ulate the value of cos, , 1, , So that,, and hence that of, , ,, , a,, , a., , Clearly, this also gives us a method of measuring 6, the angle of, contact for the given liquid and solid ; /or,, (180- <x), the two, , =, , angles, , and PL being supplementary, , angles.
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SURFACE TENSION, , 603, , (/) Relation (A) above tells us that h> the distance, surfaca of the drop from the horizontal surface of, maximum area, is quite independent of the size of the drop, but, depends only upon the nature of the liquid and the solid in contact,, for them)., (i.e., upon the angle of contact, , Corollaries., , of the top, , flat, , We, , (11), , Hm =, *, , have, from relation (B) above,, , ~, , 4T cos*, , 2T(l+cosa), ~, g, , (a, , 1, , 2), , *, , CV, , +cos a, , 1, , -, , 2 a'(/2)., , ff=2c<*-, , Or,, , p', , whence,, , it is, , clear that, , (a) all flat drops of a liquid, resting on a horizontal solid plate, which they do not wet, must have the same height., (b) all liquids that wet the solid surface spread out indefinitely on, TT ; so that, it for, in their case,, and, therefore, a, , y, , 0=0, , cos a/2, , =, , and consequently,, , 0,, , =, H = 0., , A liquid jet, issuing horizontally, 5., Rayleigh's Jet-Method., out of an orifice, shows a strange recurrence of forms in its surface, a phenomenon, in which surface tension plays its own part. This, has been utilized by Rayleigh, Pedersen, Bohr and Stocker for the, determination of the surface tension of a liquid. We shall consider, here only the simpler Rayleigh's method, by way of an illustrations, to how this may be done., We know that,, , under ordinary conditions, a liquid jet is quite, the hydro -dynamical sense of the term, for, although, made up of drops in motion (i), its, surface is fixed in space and (ii) the, velocity at any point remains constant., But, on making a closer and careful, examination of it, we find that, in its, initial formative stage, its various parts, are in motion with respect to each other,, its 'form' oscillating this way and that, 1, , steady',, , in, , about mean, lateral, , ;, , spherical, , dimensions,, , with, , one,, , its, , measured along a, , particular direction, exhibiting a cyclic, Thus, for example, points, change., and Q in the jet, (Fig. 307), include one, such cyclic change in its cross- section., , P, , pjg. 307., , The time-period, t, of the oscillation of such a drop, may be, For, obviously, it, easily determined by the method of dimensions., will depend upon (i) r, the radius of the orifice (O) for the horizontal, tube, whence the jet emerges, (ii) T, the surface tension and, the density of the liquid. Thus, let, , = K.i*.?*.T*,, K is a dimensionless constant., Now, the dimensions of (time) = [T] of r, of density, or mass/volume = [AfL~8, , p,, , (ft'i), , t, , where, , t,, , ;, , ], , ;, , (radius), , =, , [L], , ;
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PROPERTIES OF MATTER, , 504, , T or force/ length = [M LT~*].[L- = MT~*., = K[L]" [ML-*]* [MT~*] C = K.L^.M^'T.-", 0-36 = 0, 6+c=0 and -2c = 1., 1, , and, of surface tension, , Hence, , [T], , So that,, , = |, b = \ and a = |., - .rJ'p.*2 .~~*, - K yV /P^, constant K was experimentally, c, , whence,, , And, , ], , ?, , /., , r, , 3, , f, , Or,, , The value of, Rayleigh to be, , found out by, , 77/, r, , So that,, , Or,, , *p, , 2T, , takes a parabolic path, its horizontal velocity, v being given by \/2gh, where h is the vertical height of the liquid, surface above the orifice, i.e., the height of the liquid head above the, , Now, the, , jet, , horizontal., , And,, , if, , =, , x, , have, , x be the distance betwean P and Q,, , But, , /, , *, , So that,, , =, , Or, x*, , v.t., , 2, , 2, 2, , /v, , =, =, , 7T, , 7r, , 2, 2, , .r, , 3, , v, , 2, , 2, ./, , ., , Or,, , =, , t*, , (Fig. 307),, , x 2 /v 2, , [From, , .p/2r., , 3, , r .p/2r. Or,, , 2, , * /2gA, , =, , TrV, , 3, , Or,, , x*[gh, , 77, , 2, , .r, , 3, , .p/r,, , r=, , whence,, , I,, , above., , p/2r., 2#/i for v, , [Substituting, , =, , we, , ., , irV.p, , 2, ,, , 2, , g/r/.x, , ., , 80 that, measuring x with the help of a travelling microscope,, (by properly illuminating the jet) we can easily calculate the value, of, , T, , 9, , the surface tension of the given liquid., , The Capillary Rise Method., , 1x6., , =, , T, , A, , direct application of the, , 248 page 488), connecting the rise of, a liquid in capillary tube with its surface tension, it is a simple, laboratory method for the determination of surface tension of liquids, which do not wet the walls of the tube, i.e., for which the angle of, relation,, , contact, , A, , is, , r (h, , -f-, , r/3)p.g/2,, , (, , zero., is taken and the uniformity of its bore careby introducing a thread of mercury inside it and measuring its length, by jerking it into diffe-, , capillary tube, , fully tested, , rent positions along the tube. If the, thread measures the same everywhere,, When, the tube has a uniform bore., a proper tube Las thus been selected,, it is thoroughly cleaned by rinsing it, first with caustic soda, then, with nitric, acid and finally with distilled water., If the liquid, whose surface tension is, to be determined, be water, the tube, , be used straightaway, but if it, be some other liquid than water, the, tube must be properly dried by passing, a current of warm air through it, and then fixed vertically, alongside, a plumb line, with its lower end immersed in the experimental liquid,, , may, , Fig. 308.
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SURFACE TENSION, And,, , if, , 507, , the liquid be one which wets the tube, i.e., for which 6, 1, the expression for T reduces to, , so that, cos, , =, , =, , ;, , r=r./i.p/2,, from which the surface tension T of the given liquid can be easily, calculated, , ., , There is only one slight error possible in this msthod, viz., the, shape of the liquid meniscus at Q may not be really perfectly spherical (as assumed), due to the distortion effects on account of gravitational forces. But Ferguson and Kennedy have shown that they can be, safely neglected, if the bore of the tube be of a really small diameter,, about I cm. or less., Before proceeding with the methbd, Ripples Method., first try to understand tlie difference between waves, and ripples. We are all familiar with the waves travelling over the, surface of liquids., Their velocity of propagation depends on both, the force of gravity as well as surface tension, For an amplitude, smaller than the wave length, the wave-curve is given by the successive, positions of a point fixed to a circle rolling along a straight line, the, amplitude (a) and the wave-length (A) being equal to the distance of, the point from the centre of the circle, and the circumference of the, circle respectively., In other words, each particle of the liquid in the, wave describes a circle in the vertical plane, the wave itself advancing, forwards through a distance \, (its wave length), during the time, that a particle takes to complete its one full round along the circle,, the direction of motion being ant i- clockwise, for a wave travelling, 8., , proper,, , from, lel, , we must, , left, , to right., , Let us imagine a vertical section of the liquid by a plane, paralto the direction of propagation of the wave. (Fig. 311), and con-, , sider first ihe effect, of gravity alone., , Then,, , if, , V, , be the, , velocity of the, , wave, , along the horizontal, r, the radius, of the circular path, of the particle and, f, the time it takes, ;, , ;, , to, , describe, , it,, , we, , have, the crest A, at, velocity (v x ) of the particle at, , any given, , instant,, , =, , v1, (i), V-^r\t., given by the relation,, relathe, at, the, trough 5, given by, And, velocity (v s ) of the particle, , tion,, , v2, , -, , (), , F+27rr//, , to be solely due to, , Taking this increase in the velocity of the particle, h = 2r,, its having fallen from A to B, through a vertical distance, under the action of gravity, we have, f, From the kinematic lelation v 11*, (, f =, =, ind, a, a, u, u, v, v, where, g, lt, va, t, =2<,, 4g.r...(iii), 2g.2r, Vj, ,, , C, , j~2r.
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PROPERTIES OP MATTER, , 508, , And, squaring and subtracting relation, , So that, from relation, 877, , Now,, , (Hi), , and, , (/v),, , (/), , from, , (//'),, , we have, , we have, , Kr//=4g.r, whence, F-f 4g.r.f/87rr=g//27r., , obviously,, , t, , = \jV., , And,, , .-., , V^g^TrV., , Or,, , V^^, , ............, This, then, is the expression for the velocity of the, the action of gravity alone., , Let us now see, velocity of the wave., , how, , (v), , wave due, , to, , the surface tension of the liquid affects this, , 312 represent a vertical section of the liquid, with, as the harmonic wave travelling over it and At, its undisturbed surface., Let Fig., , A BODE, , ;, , Let the displacement, , PQy^, , of a, , particle, , of, , liquid,, , at, any, distance x, , the, , a, , instant. at, , Q, X~~~, , from an arbitrary origin, be, A,, by the, given, relation,, , Fig. 312., , y==a, , where a, , sin-- +C, , ........., , 9, , (v/j, , tho amplitude of the p \rticle and C, the phase constant., , is, , Now, due to the action of gravity alone, the vertical pressure at, would increase by jy.p g, where p is the density of the liquid and g,, the acceleration due to gravity., , Q, , to the surface tension (T) of the liquid, however, there, to 7\( I //?+!//') acting normally, at P, from the concave towards the convex side, where R and R' are, the respective radii of curvature of the liquid surface in the plane,, and perpendicular to the plane of the paper. But since the wave, is infinite, system is a cylindrical one, one surface of which is plane,, and hence 1/R'=0. The pressure at P due to surface tension thus, Since the amplitude of the wave is small,, reduces to T(l/R) or T/R., with, its, wave-length, the normal at P almost coincides with, compared, the vertical through P. Tho net increase in the vertical pressure at, , Owing, , will also be, , an excess pressure equal, , R, , P is,, , therefore, given, , ', , J__ *y, R, , ~"Jx*, , Hence, , To determine, twice, when we get, , f, , by, , ', , if, , a<, , <:iand, , ', , ~y, , ', , dx, , p=y.p.gT.d*y/dx, , 2, , d 2yldx2 =, , we, , iir^/A, , =, , f, , differentiate relation (v/), , 1, ., , l, , Putting this value of d*y\dx in relation, , l', , (vii), , ., , the value of d*y/dx*,, , (Seepage 332,, <<, Ex. 4 (solved), ........, , (vii), , above, we have, , 47i r, y+(g+ -j^-, , /, , ,
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509, , SURFACE TENSION, , It is thus clear that the effect of surface tension is tantamount, to increasing the value of g by 47r*77A 2 .p., Clearly, therefo e, if V be the velocity of the wave under the, action of both gravity and surfa3e tension, we obtain its value by, 2, 2, substituting (^+47T r/A .p) for g in relation (v) above i.e.,, ;, , r, , A.p, , mere glance at this relation tells us that V = oo, both when A, and when \ = oo. In-between these two extreme values of A, there, must be a certain value of it for which V has the minimum value., Clearly, the product of the two terms Ag/2ir rtnd 27r77?\.p, viz., gj/p, a miniis a constant, and it follows, therefore, that their sum will be, mum when they are equal i.e., when, , A, , =, , *g/2;r, , Or,, , when A 2, , =, , ;, , 27r77A.p. Or,, , when, , when, , =, , 2, , 47T r/g.p., , Or,, , 7\, , 2, A*.g.p -_ 47i T., , This value of A, for which the velocity of the wave is the minimum,, is oil led the critical wave-length, and miy ba denDbol by the symbol, ?v Thus,, , __, , =, , .........., , A,, (ix), 2irV77g.p, Substituting this value of A, therefore, in expression (viii) for V,, above, we get the minimum velocity (Vm ) of the wave given by the, relation,, , Vm, , =, , Now, examining relation (viii) again, we find that, (/) If A>?^, the first term Ag/27r becomes more important, increases and, therefore, neglecting the second t3rm ia, , with, , as A, comparison, , we have, , it,, , Disturbances of this type, whose wave-length is greater than the, known as waves. Their propagation is mainly due, to the force, of gravity and, as can be readily seen, their velocity, , critical value, are, , increases as A increases., (ii) If < A r the second term becomes more predominant, and the, term may, therefore, be neglected. So that, in this case., ,, , first, , Waves, , of this type for which the wave-length is less than the, length, are called ripples or capillarity waves. Their pro, pagation is, in main, due to surface tension, and, as can be easily, seen, their velocity decreases as A increases., , critical, , wave, , *For, substituting the value of, , above,, , v^, , 2'/p, , =, , r, , A*5 /4T*, , from, , (/JT), , in, , relation, , we have, , \l, V, , *? r, 4., 2*, , Now,, , 2rt, , x$*, ', , A, , 4r, , substituting 2w, , ~~, , A, \/, V 2n, , VX/^ P", , i.-^L, ^, , for A,, , 2n, , _, ~, , we have, , \/, \, , ^, 2n, , _ \/A?, ~~, , \, , T:, , ', , (v///),
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PROPERTIES OF MATTER, , 510, N.B., are, , get A*, , Taking the case of water, for which, - 1'7 cmj. and Km - 23 cms. (sec., , T75 dyneslcm- and, , />*!, , grn.fc.c.,, , The Method Proper, iorrf Rayleigh (1890) was the first to have, used ripples, excited on the surface of a liquid, to measure its surface, tension, by the direct application of relation (>'///) above, although, the method has subsequently been greatly improved by a host of, other workers, including Dorsey, Ghosh, Banerji and Datta. And,, recent improvements have been effected by Tyler and, still more, Brown, enabling the ripples to be photographed and their images, thrown on a screen., We shall deal only in brief outline, here, with the essentials of, the method employed. The experimental liquid is taken in a shallow, rectangular, , porcelain trough,, , about 10" long, 6" wide and, 1-5", , above, , deep,, it is, , (Fig., , 313),, , arranged an, , and, , electri-, , cally maintained tuning fork, F, of a large frequency, with, its prongs horizontal and one, above the other in their posi-, , tion of rest, so as to vibrate in, the vertical plane., light, of polished, style or 'dipper P,, silver or aluminium, about 3*, lower, long, is attached to the, , A, , 9, , prong of the fork, with its, plane also vertical but perpendicular to the plane of vibration of the fork, such that it, , Fi g 313., just touches the liquid surface in the trough., Now, as the fork is set vibrating, the style or the dipper alternately moves in and oat of the liquid, thus exciting trains of ripples, on its surface. These, on reflection from the walls of the trough,, give rise to stationary ripples., ., , To enable these ripples to be observed and their wdve-length, measured, the liquid surface, must be properly illuminated. This is, done by completing the tuning fork circuit through the primary P' of, a small induction coil, to the secondary S of which is connected a, so that, evory time the tuning fo.k circuit is, neon -discharge-tube, made, an electric discharge passes through the neon-tube, the light, from which then brilliantly illuminates the liquid surface in the, trough, thus enabling the irave-form of the ripples to be easily seen, ;, , and photographed., Now, during one vibration of the fork, there is one alternate make, and break of the tuning fork circuit, giving us one view of the ripples, on the liquid surface. The same recurs, when the fork has made one, vibration, and when, therefore, the ripples have advanced through, a distance equal to their one full wave-length so that, as the liquid, surface is lit up again, as before, we get a second view of it, identical, with the first, the ripples appearing to be stationary in their earlier, This is so, because we do not see the liquid surface conposition., tiauouily but only at intervals equal to the time-period of the, ;
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SURFACE TENSION, , 511, , ripples, tho frequency with which the liquid surface is lit up .being, the same as that of the fork and, therefore, of the ripples themselves., The succsesive views we have of this liquid surface are thus, only those in which one ripple has just replaced another similar, ripple and hence, due to persistence of vision, (the frequency of, the fork being large), the surface of the liquid appears to us to, be exactly the same as before, with the ripples in their original, , position., , The distance between as large a number of ripples as possible, measured with the help of a travelling microscope and their mean, wave-length A thus determined. Then, if n be the frequency of the, fork (which is known to us), we have V = n\, where V is the velo*, is, , Substituting this value of, , city of the ripples., , V, , in relation (v/w), , above, we have, , n\, , =, , -, , f, , \, >-v, , > ., , A#, , >, , n-\*, , Or,, , ^, , =, , ^TT, , 4-, , ), , STT!/, , *< YI, , ., , -,, , Or, , ., , AD/, , +*TT, , .7., 7, , whence,, , i/, , =, , T^.p, r, , 9^ ., , .fl*V, , ^7T, , A*P, , .j*, , ^, A.p, 77", , A, , Tj^TT, , ., , >, , r=.^:P-*>*, , Or,, , from which,, , being known, the value of T, the surface tension of, , ft, , the given liquid,, , may be, , easily calculated., , The Ring Method. This method derives, , 9., , importance from, , its, , enables us to study the changes that come about in, the surface tensions of different liquids with the passage of time. It, has been used with great accuracy by Harkins, Young and Cheng,, , the fact that, , and, , it, , being increasingly employed in Applied Physics., A metal ring of a wire of circular cross-section is suspended in, the exparimental liquid, with its plane horizontal. It is then raised, gradually out of the liquid, when, in addition to its own weight, the, extra downward pull on it due to surface tension passes through a, maximum value. If the wire, constituting the ring, be a thin one, this, maximum pull is approximately given by P, ^rrRT where R is the, radius of the ring and T9 the surface tension of the liquid. More, correctly, however, P is only proportional to 4-n-JRr, so that, <xP, To avoid calculat47r/?r, where a is a non-dimensional factor., ing the value of a, Harkins and his co-workers assumed that, is, , =, , y, , =, , DT, , AtTT/xJ, , /, ., , P, , T), J\, , \., , /, , \, , T, , j, , \, , ,, , ~*, , the radius of the wire, F, the volume of the liquid held up, pull of the ring, equal to P/p.g, (p being the, of, the, density, liquid)., They used three liquids of known surface tensions, v/z., water,, benzene and bromobenzene and measured the maximum pull on three, different rings, with different values of R and r, but the same consiant ratio R/r, and then plotted a graph between J 3 /F, (along the, jc axis), and 4?rRTIP, (along the j-axis), when the points for all the, three liquids* were found to lie on the same smooth curve. From this,, they came to the legitimate conclusion that, it the same three rings, -where r, by the, , is, , maximum, , t
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PROPERTIES OF MATTER, , 512, , were to be used with other liquids, the points corresponding to the, values of jR 8 /K and ^nRT/P for them too would lie on the same curve., Thus, the procedure to determine the surface tension of a given, by this method becomes the following, , liquid, , :, , We take, , a ring, identical with one of the rings used by Harkins,, Young and Cheng, (i c. of the same material and the same value of, R/r), measure, with its help, the maximum pull as the ring is raised, out of the given liquid and then calculate the value of R*jV for it,, which is equal to R*.g.plP. Next, we locate the point corresponding, to this value of R*jV as abscissa, on the graph, plotted by them for, the ring of the same value of Rjr. The ordinate y, corresponding, to it, then gives the value of ^rrRT/P for the liquid and is carefully, noted., Thus,, y, ItrRT/P, and, therefore, T =yP/4:7rR,, t, , whence the value of T for the liquid can be easily calculated., The method has the merit of quickness, ease and accuracy, but, the following precautions are necessary for its success, This is ensured, (/) The liquid surface must be perfectly clean., by sweeping the surface of the liquid by means of what are called, 'barriers', before performing the actual experiment., :, , inver(//) The liquid surface must be kept properly covered by an, ted glass funnel, to avoid evaporation., (Hi) The dish containing the liquid must be wide enough, so that, errors due to curvature of the meniscus etc. are almost altogether, eliminated., (iv) The whole apparatus must be arranged inside a thermostat,, to ensure that the temperature of the liquid remains constant throughout, and the thermostat must be supported independently of the, ring and other apparatus, to avoid duturbanee or agitation of the, , experimental liquid., 256., Surface Tension of Liquid Interfaces. If we consider a, of, immiscible, system, liquids in contact, we naturally expect a new, phase to develop at their interface, with a definite energy of its own,, depending upon the nature of the two liquids., , Antonow gave a, , tension between two, , rule that the interfacial, , liquids, in equilibrium, is equal to the difference between their individual surface tensions. Thus, if 7\ and z be the surface tensions, of two liquids separately, their interfacial tension 12 when they, , T, , =, , T, , ,, , are in equilibrium, is given by, T x T2, jT12, This rule would, however, apply only to mutually saturated, solutions, for then alone would they be in equilibrium, or else the, addition of one to the other would reduce its surface tension. Being, the difference of ths individual surface tensions of the two liquids,, the surface tension of t*ieir interface is obviously a small quantity,, in general, increasing with the decrease in the solubility of one liquid, into the other, and decreasing with a rise in temperature., ., , Since a knowledge of interfacial tension is of importance while, considering problems like spreading of one liquid over- another or, those relating to chemical constitution, we shall now proceed to see, how it may be determined for a given liquid-liquid interface.
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51$, , SUEPACB TENSION, , Most of the methods discussed above, for the measurement of, surface tension of pure liquids would, in principle, be equally applicable in this case also., Thus, for example, the drop-weight method?, can be easily used for the purpose and probably gives the best results^, shall discuss this method in proper detail., , We, , Here, we proceed, (/) The Drop-weight Method., manner as in determining the surface tension of a pure, , in the, , same, , liquid, (see, fine orifice, of, , page 498), with the difference that the tube, with a, about 3 to 5 mms. radius, containing one liquid, dips inside the other,, so that the drop of the first liquid is formed inside the second, instead of, in air the weight of the drop being obtained by collecting and weighing a known number of them., Then, if p A and p a be the densities of the two liquids respecThis is obviously also tbe, tively, we have volume of the drop, m/p t, volume of the second liquid displaced by it. So that,, weight of the second liquid displaced or, upthrust on the drop, 9, , =, , = m, , ., , r~m being the mass, of the drop., , ?, , L, , PI, , And, therefore, apparent or, , effective weight, , of the drop, , Now, assuming, as before, that, just at the time of being detached from the orifice, the drop is cylindrical in shape and has the, same cross-sectional area as that of the orifice, the excess pressure, inside, , it is, , where, , fl2, , =, , ri2 /r*,, p, given by, the surface tension of the interface., , is, , The downward, , force, , on the drop due to, , r=pX7rr, And, therefore,, , 2, , =, , it is, , T, , -, , Trr, , 2, , =, , thus, , *.Tn .r., , downward force on the drop, effective weight of the drop+7T.ri2 .r,, , total, , =, , And, the upward force on it due to surface tension = Tn .2-Trt, 2irTn r where 2itr is the circumference of the cross-section of the, ^, , =, , t, , .drop., , In the equilibrium position of the drop, therefore,, ., , Pi, , Or,, , we have, , &=*) -, , ,.ru .r., , Pi, , T, 1 10, in the above treatment, we have assumed that the drop attain*, when, having attained its cylindrical form, it ii, to, be, This is, however, not so, the detachmem, detached., about, just, of *the drop being essentially a dynamical process, for which w<, obtAin th6 following relation, by dimensional analysis, , static equilibrium, , :, , *, , *The radius of te other face of the cylinder beog, , infinite.
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PROPERTIES OF MATTER, , 514, where, , <f>, , is, , an arbitrary function of the non-dimensional variable, , Lord Rayleigh, , has, however,, , drops, F in air that the value of, , shown from, , ~-^, <f>, , large variation in the value of, , \, , L^*(Pi, , his, , work on water, , comes to, , 3-8, , for, , a, , P)J, , r., 2, , So that, we have, , =, , ri2 .r.3'8., , Pi, , Tn, , Or,, , whence, the surface tension, , ~, , T12, , for the given liquid interface can be, , easily calculated., , Wilhelmy had suggested a straightforward method, (11) Wilhelmy Method., of determining the surface tension of a liquid by measuring, with the help of a, balance, the additional force necessary to counteract the vertical pull of surface, tension on a verticle metal plate, suspended from the balance arm, vihen it is, dragged away from the surface. Thus, if the lower edge of the metal plate be, in level with the undisturbed liquid surface and if / be the length of its, line of contact with it, the vertical pull on it due to surface tension is equal, contact of the liquid- plate surface., is the angle of, to T.I cos 0, where, And, therefore, if mg bs the additional weight required in the scale-pan to, balance this additional force, we have, mg -= T.I cos 0., It we use a torsion balance, in place of the ordinary balance, the method can be, easily used for interfacial tension also, as will be clear from the following, The two liquids are taken in a beaker, one above the other, a vertical, plate suspended from the arm of the torsion balance and adjustment made for, equilibrium with the plate wholly immersed but well above the interface of the, two liquids. The beaker is now gradually raised and the equilibrium continues, :, , to be maintained, until the plate just approaches the interface, when the additional downward pull on it, due to the interfacial tension 7\ 2 of the two liquids,, disturbs the bilance. The torsion head has thus to be turned through an additional angle a, say, to restore the plate back to its equilibrium position. This, angle of twist (a) measures the force / r, a on the plate, where / is the horizontal, perimeter, the angle of contact (0) being assumed to be zero (or cos B very nearly, value of Tn for the given interface can thus be easily, equal to 1). The, '", determinedMack and Bartell used the following, (///) Mack and Bartell Method., simple method for the measurement of interfacial tension of water and organic, liquids. Besides its simplicity, it has the great, , merit, of precision and of requiring only a very small, of the liquid,, quantity, just 2c-cs. of it or so., The apparatus used, , by them, Fig., , 314,, , is, , as, , shown, , where, , in, , A and B, , are two wide glass cups,, with, a, communicating, central, wide, tube, C,, through two capillary tubes, , Pand Q, , of slightly differing radii R t and /?, sealed, on to them, where /?f> the, larger of the two, is less, , than I mm., poured into the cup A, connected to the narrow capillary, P, to ensure that no air bubbles get entrapped, and a larger quantity of it is, then poured into cup B, a little over and above that required to fill the two, Fig. 314., , Water, , capillaries., , is, , first
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SUE FACE TENSION, 259., Surface Tension and Vapour Pressure over a Liquid, Let a capillary tube be dipped vertically in a liquid, (which, Surface., wets the tube), of surface tension J, and density p,, and let the whole arrangement be enclosed in an, exhausted bell jar or chamber, (Fig. 316), so that, the effect of the atmosphere may be neglected., Obviously, then, there is nothing but its own va-, , when, , pour over the liquid, and, , is, , equilibrium, , attained, the space above the liquid becomes saturated with its vapour., , Suppose the liquid rises into the tube up to a, height h above the horizontal liquid surfac^ A in, the vessel., , Then, if P be the vapour pressure at the horizontal surface A,, and a, the density of the vapour at this pressure, the vapour pressure, above the concave surface B of the liquid, in the tube, will clearly, be less than P by an amount equal to that of a column h of the, vapour, i.e., by an amount h.a.g = p, say so that vapour pressure, = Ph.a.g. And, clearly, the pressure, just above the meniscus at B, in the liquid just below the meniscus is equal to (P--/?.p.g.) where p, ;, , the density of the liquid., If the tube be narrow, the meniscus may be regarded as spherical of radius r, nearly the same as that of the tube, and so the excess, will be, pressure, just above the meniscus over that just below it,, is, , equal to 27>., Clearly, this pressure just above the meniscus, pressure just below it by, , =, Now,, A, , h.Q.g, , =, , ;, , /'.*.(P-<0, so that, h.g, , vr, , p, (p-o)-__,, V, i, , i, , And,, , p, , =, , h.p.gh.a.g, , .*., , x, , greater than, , =, , h.g. (p, , is, , greater than the, , a)., , 277r., , ~ p/a., , [See above-, , i., , whence, />, , = 27V, , a, , (_, , -, , f, , the expression for p is posit ive,.and,, therefore, the vapour pressure above the concave surface of a liquid is, ame, less than the vapour pressure at the horizontal surface of the, we, an, or, as, a, amount, 2T, near, may, (p, by, a),, a/r, liquid, approximation,, 2T.<r/r.p, because a is very small compared with p., say that/?, Since p, , is, , a,, , =, , Now,, , py, , =*, , if, , R.Q, , t, , we, is, , instead of by, , treat the vapour as a gas, for which the relation, the absolute temperature by r, T, which we have used for surface tension here), we, applicable, (denoting, , have, , V, , Now,, , o, , = R.6/P., = 1/K., , K, , [Where, , And, , is, , the volume of the vapour., , 1/K or a, , .-., , =, , P/l?0,, , Hence, substituting this value of o in the expression for excess, pressure p abave, we have, excess vapour pressure on the horizontal surface of a liquid over, that on its concave surface given by, , P, , ^, , 2T, ^, , 2T, , a, r", , ^^>, , "**, , ~, , m, i, , P, n, , *, , -
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520, , PROPERTIES OF MATTER, Thus, the relative lowering of vapour pressure, , surface, , is, , given by the expression, , --=, P, , -., , above a concave, , ., , rR.0.p, , Similarly, if we have a capillary tube, dipping in a liquid which, does not wet the tube, and enclose it, as before, in an exhausted, chamber, the liquid column in it will be depressed, below the horizontal surface of the liquid in the, vessel, as shown, (Fig. 317), through a distance h,, say., , Then, if P be the vapour pressure at the, horizontal liquid surface, outside the tube, the, vapour pressure just above the convex meniscus in, the tube will be equal to (P -\-h.a.g), where o is the, density of the vapour at pressure P or, putting, h. a g., p, as before, we have h.g, pfe., , =, , =, , ;, , Now, pressure in the liquid just below the, Fig. 317,, convex meniscus is clearly equal to P+the hydrostatic pressure due to, the liquid column h, or equal to (P+h.p.g), where p is the density of, the liquid., Since p is greater than a, clearly, (P+h,G.g)>(P+h.v.g), and,, therefore, excess pressure just below the meniscus over that just above, it is, , equal to (P-\-h.p.g)(P+h.o.g), , i.e.,, , o, , But we know that excess pressure just below, , the meniscus over that, is the surface tension, just above it is also equal to 2Tfr, where, of the liquid and r, the radius of the meniscus, (supposed to be, spherical)., t\ rn, v, O1, T1 / G, jv, 1, P, x, f, \, whence, p, (, (p, o), "_~ ), , T, , ,, , ., , =, , =, , ,, , ,, , Thus, the vapour pressure above the convex surface of a liquid is, greater than that on a plane or horizontal surface of the same liquid by, an amount equal to 2jT.a/r.(p a) or 2jT.a/r.p, as a near approximation,, , =, , ZTIr.R.Q. p., p = (2r/r). P/-R0.p, whence, p/P, be noted that the excess pressure p is inversely proporso that, the smaller, tional to r, the radius of curvature of the surface, the value of r, or the greater the curvature, the higher the value of, the saturation vapour pressure at the curved surface. It thus follows, that the saturation vapour pressure over a small drop of a liquid will, be greater than over a large drop of it., Let us take a large drop of water of dicmetcr 1 rr.m., or r, -5, '05 cm., mm., Then, putting 0=273 Abs., p l gm./c.c., d-=-6l x 10~ 8 grn./c.c.,, 75 dynes I cm., we have, and T, a, 2x75x'61xlO~ 3 ,, , Or, as before,, It should, , ;, , =, , =, , =, , =, , p, , __, , ----, , ^, , 1-, , For a drop of water, of diameter 1/1000 mm.,, 1*83 x 10 dynes/cm*, p = 1-83 x 1000, , =
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21, , SURFACE TENSION, , And, , for a, , as, , the case, , is, , ckop of water, of diameter equal to one-millionth of a, when it just starts forming, we have, , p, , =, , 1-83, , x, , 1C 3, , x 10 3, , =, , 1-83, , x, , mm., , 10* dynesjcm*,, , greater than a pressure of one atmosphere., Thus, we see how the saturation vapour pressure rises with the, diminution in the size of the drop., , which, , is, , on Evaporation and Condensation. We have seen, the maximum vapour pressure is less for a concave, liquid surface than that for a plane or horizontal surface and also, how the vapour pressure for a convex liquid surface is greater than, that for a plane surface., If, therefore, we place a drop of water in, a space in which the vapour is at the >aaturation value for, a, for the vapour, plane surface, the drop will begin to evaporate, pressure in the space will be less than the saturation vapour pressure, for the drop, and it will, therefore, be converted into vapour, in, order to increase the vapour pressure to its own saturation value., 260., , Effect, , how, , above, , ;, , This will result in a further decrease in the radius of the drop, or an increase in its curvature (i.e., convexity) and a consequent rise, in the saturation value of its vapour pressure, and it will, therefore,, evaporate more and more rapidly. That is why a saturated vapour, does not condense into drops for, as soon as a tiny drop is formed,, it begin to evaporate., Thus, condensation may not take place even, when the vapour becomes supersaturated., ;, , If, however, dust particles or charged ions be introduced into, the saturated vapour, they offer a flatter surface to it and condensation at once starts on them, for, the radius of curvature of the drop,, so formed, is not very small, even in the beginning, and hence it has, little tendency to evaporate, And, as it 3 radius increases, and,, therefore, its curvature or convexity decreases, its tendency to evaporate becomes smaller still. For, the saturation value of the rapour, pressure for it goes on decreasing, and it continues to grow in size., ;, , Thus, dust particles or charged ions play ai important part in the, condensation of vapours. And it is precisely because of the absence, of these dust particles, (which act as nuclei for the vapour to, condense on), that dust- free vapour does not condense, even if its, temperature be lowered be km its normal temperature of condensation., , Again, because the saturation vapour pressure over a concave, is less than on a plane or horizontal surface, the vapour will, condense more readily on a concave surface than on a plane surface., This might be clearly seen by closing the bottom of the tube in the, last experiment and removing some liquid from it, when condensation would set in on the liquid inside the tube and will go on, until, the liquid column in the tube attains its previous height., surface, , SOLVED EXAMPLES, A, , sphere of water, of radius 1 "mm , is sprayed into a million drops, of equal size. Find the work expended in doing so., Breaking the liquid drop means an increase in surface area and, therefore,, work is required to be done for the purpose, which is equal to the product of, the surface tension and the new surface area formed. This work done becomes, the surface energy of the new surface area. We shall, therefore, first calculate, 1.
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PROPERTIES OF MATTER, , 522, , the initial surface energy of the sphere of water, when it is intact, and then the, Thesurface energy of the million drops into whicft it is broken up and sprayed., difference of the two will give the work expended in the process, Then, this is, Let tne surface tension of water be taken to be 72 dynes\cm, equal to the free surface energy of the sphere of water, (see page 478)., free surface, , Or,, , Radius of the sphere, , ==, , energy, 1, , mm., , surface area of the sphere, , And, , .-., , And,, , = 72 ergs/cm*., = "1 cm., = 4nr = 4" x, , I), , (, , sq. cm., a, , 72x47rxCD, energy of the sphere, volume of the sphere= *WA^=^X n x ( I) 3 c.cs., , initial surface, , Now,, number of drops, , into which, , it is, , broken=10, , volume of 10 drops = YXTrx(-l), 6, , And, , 3, , l, , =, , .'., , 8, , -, , 9 048 e rgs., , fi, , ., , c.c^., , volumeof 0rtedrop=x*x(-l) 8 /10 8 =!x*x('l/10V, , c.f., , 2, , Hence,, , radius of one drop=**l/10 cw.=*001 cm., a, surface area of one drop 4* x ('00 1 ) sq. cm., , And, , surface area of all drops 4 ('00 l)=x 10 a =4*x l =, surface energy of all drops, or, final surface energy, , Hence, increase, , 4rc sq., , cms., , energy =* (904-8- 9'048) ergs., 895*752 ergs., , in surface, , Or, work expended in the process = 895*7'52 ergs., A glass plate, of length 10 cms., breadth 1 54 cms. and thickness, 2., 20 cm , weighs 8'2 gms. in air. If it is held vertically, with its long side, horizontal, and its lower half immersed in water, what will be its, apparent weight ? (Surface tension of water = 73 dynes per cm.), (Cambridge Higher School Certificate}, Here, apart from its weight, there will be two other forces acting on the, plate, viz., (/) the upthrust of water, and (//) downward force due to surface tension along its edges inside the water., Now, upthrust of water=weight of water displaced by it, half the volume of the plate x density of water xg t, , because half the plate, , ~, , Or, , >, , is, , immersed, , ---, , in water-, , 10xi-54x-20, ----- X, 2, , ,, , l, , ft01, X981, , [', , ., , *"", , 1-54x1x981 dynes~l'54gms., , volume of the plate, , UlOx 1-54X -20 c cs., wt., , And, the force due to surface tension acts downwards along a distance, equal to twice the sum of the length and the thickness of the plate, i.e., along, a total length2(10+-2)=20-4 cms., total, , downward, , =20;4x73, , force on the plate due to surface tension, 2, 1-518 gms. wt., wt., dynes., "ol, , = ~~~gm>, , =, , net upward thrust on the plate, 0-022 gm. wt., 1*54 1-518, Hence, apparent weight of the plate in water=*weight in air upthrust on it., =8-2 022*=8-178 gms, wt., 3., The pressure of air in a soap babble of 0*7 cm. diameter i, 8 mms. of water above the atmospheric pressure. Calculate the surface, tension of the soap solution., (Delhi 1944), , We know that ^excess pressure inside a soap bubble over that outside it is, given by p 4Tjr where T is the surface tension of the soap solution and r, the, radius of the bubble, is, , Here, the excess pressure inside a soap bubble over the outside atmosphere, given to be equal to 8 mms. or *8 cm. of water column., cm. of water column., ("Taking density of water, f, 1 gm.lc.c* and value of, -8xlx981dy/iw/cm ., -=784-8 ctyMilctn.*, L^98i, , P'S
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623, , SURFACE TENSION, r='7/2 = *35 cms., p and r in the above relation, we have, 4T= 784-8 X -35., 784-8~47Y 35,, whence,, T= 784-8 X '35/4= 68'66 Vyneslcm., , and,, , substituting the value of, , Or,, Therefore, surface tension of the soap solution is 68'66 dyneslcw., 4., Why is the upper surface of mercury in a glass capillary tube, convex upward, while for water it is concave ? Assuming the surface, tension of rain water to be 72 dynes/cm., find the difference of pressure, inside and outside a rain drop of diameter -02 cm. What would this differ., ence of pressure amount to, if the drop were to be decreased by evaporation to a diameter of 0-0002 cm. ?, (Punjab}, , For answer to pirt one of the question, see, , 245, (page 483)., that the excess pressure inside a liquid drop over that outside, it is given, by/>=277r, where Tis the surface tension of the liquid drop, and r,, its radius., , We know, , r=72 dynes/cm., and r= '02/2= '01 cm., 2r/r=2x72/-OI = 144/'Ol==14400l'44xlO*^/iej/fm., If the diameter of the drop be reduced to '00002 rm., its radius becomes*00002/2=-00001 cm., and we, therefore, have, a, a, p = 2 x 72 /-O 300 1 -14400000 4y/tfs/cw === I -44 x 10 dynesjcm, 44 x 10*, 1, be, Thus, the excess pressure inside the drop in the two cases will, 2, 2, 7, dynes/cm ., and l'44x 10 dynes /cm ., respectively., 5., What will be the pressure in a spherical cavity within a mass of, The cavity is at a depth of 20 cms. below the surface and, paraffin oil ?, has a diameter of 0*0026 cm. The specific gravity of the oil is 0-85 and its, surface tension is 26 c.g.s. units The pressure of the air over the liquid, surface is equal to 76 cms. of mercury., (Bombay)*, Here, pressure on the surface of the oil = 76 cms. of mercury columnHere,, , p, , =, , 7, , ., , =76xl3-6x98l~l-014xl0 6 <fjms/cm 8, , ,, , pressure due to the oil column, 20 cms. long, = 20 x '85 x 981, 1-667 x 10* dynes/cm*., because, , =, , P- A.p #, , on the spherical cavity, *>., on the bubble, = l-014xlO + 1-667 xl0 4 =10*(l-014x 10*46-667)., = 10 4 (1014 f 1667)= 103-067 x 10* 1030670 dyneslcm*., , total pressure, , Now,, outside, , Here,, , it, , the pressure inside the spherical bubble, , is, , in excess of the pressure, , by p~2T/r., , and, , T**26dyneslcm.,, , r= '0026/2 = '0013, , cm., , />=2x26/-00132x2/-0001 = 40000, Hence pressure, , inside the ca vity, , =*, , dynesjcm*., 1030 670 -f- 40000 - 107067., , A minute spherical air bubble is rising slowly through a column6., of mercury contained in a deep jar. If the radius of the bubble at a depth, of 100 cms. is 0-1 mm., calculate its depth when its radius is 0-126- mm. ;, given that the surface tension of mercury is 567. Assume that the atmospheric pressure is 760 mm. of mercury., Here, pressure on mercury surface in the jar * 760 m/j., x, 76 cms. of mercury = 76 x 1 3 *698 1 dyne si cm**, and pressure due to 100 cms. of mercury column 100 x 13 '6 x 981 dyneslcm**, on air bubble*76 x 13-6 x 981 -f 100 x 1 3 6 x 981., -(76+100) x 13'6x981 -176x1 3*6x981 - 2348000 dynes/cm'*, , total pressure, , And, exce*i pressure inside the, , air, , Zr/r - 2 X 567/-01, , because, , T-567, , dyn*slcm* 9, , bubble, , - 1 13400 dynes/cm*., *0!, and r*l mm, , cm., , total pressure inside the bubble at depth 100 cms., , =2348000+ 13400=2461400, 1, , dvneslcm*.
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PBOP1BTI1S OF MATTER, volume of the bubble, , And,, , Let depth at which, , Then,, , its, , total pressure, , *Trr**.7t( '01 ), , mm- be h, , radius becomes 1-26, , on the Ijubble, , c.cs., , cms., , at this depth-, , -76xl3-6x98I+/ixl3-6x981., , =, , -, , (76-f-fyx 13-6x981, , And, excess pressure inside the bubble over, , ", , T, , =, =, , on, , 2T - 2x567 OArtnrk, =89, *""/"',, -0126-, , ", , total pressure inside the, , ., , 13350(76 +- h) ctynes[cm*., , this pressure, , 1, , ., , f, , ,, , I, , its, , oulside, , .-., , =, , here,, , r-126, , /', , -0126 cm., , bubble at depth h cms., , 33 50(76 + /04- 89990 dynes /cm*., , = 03990 + 13350A= ^Tc(0126) c.cs., , 1014000 + 13350/1-1-89990, , And, volume of the bubble, , 1 1, , 3, , at this depth, , Now, in accordance with Boyle's law, the product of pressure and volume, of the bubble must be the same in the two cases so that,, ;, , 2461400 x*.(-01) 3, , -=, , (1103990 -fl3350/0x.7r.(-0126)., 3, 2461400x('01) = (1 103990 + 13350/0 x('0126)*., -1 103990 x('0126)3 -f 13350 x(-0126) 8'xA-, , Or,, , 13350 x(-0126) 8 x/*2461400x(-OI) 3 -1103990x(-0126)*., , Or,, Or,, , -02669A, , ~, , h, , ,, , whence,, , -, , 2-461-2-209, , '^, , the required depth of the, , -252,, , =*9'441 cms., , bubble^ 9 441 cms., , A, , capillary tube of 0*5 mm. bore stands vertically in a wide vessel containing a liquid of surface tension 30 dynes/cm. The liquid wets, the tube and has a specific gravity of 8. Calculate the rise of the liquid, 7., , in the tube., , diameter of the tube, , Here,, , and, therefore,, , 0-5, , radius r, , p, , = 0'8 and, , -05/2, , angle of contact=0 (because the liquid wets the tube)., , 7=, , Now,, , Thus, the liquid will, 8., , meter 5, , -05 cm., , =, , '025 cm., , 30 dynes /cm., , surface tension (T), , Sp. gravity, , mm., , h?g, , 'P*., , 2 COS 6, , 2, , '', , rise to, , (, , [-.-, , e, , - Oand, , .-., , cos, , =, , 1, , 1, , a height =3*061 cms. in the tube.,, , A capillary tube of internal diameter 1 mm. and external diamms. hangs vertically from the arm of a balance, the lower end, , of the tube being in a liquid of surface tension 40 dynes/ cm., Assuming, wets the tube, what is the change in the apparent weight, 8, of the tube due to surface tension ? (g=980 cms. sec.- )., , 'that the liquid, , (London Higher School, , Certificate], , Here, clearly, the force of surface tension will act downward on the capillary tube along the inner as well as the outer circumference of its lower end dipping into the liquid, i.e., along a length 2*x-25-f2ftX-05=27r(-254-' 05)., *60ir cms*30 cms., f v the external, , =, , force acting downward on the tube due to surface, tension, '60n, dynes ~'60nx 40 dynes., , =, , xr, , _____, Or, , ,, , ,, , A -, ft ,, ,, -07696, gm.wt., , !, , ], , ., , increase in apparent weight of the tube, , [, is, , radius5/2, 2-5 mms., -25 cm., and the internal radius, -05 cm., , equal to *07696 #m. wt.
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526, , SURFACE TENSION, , The stem, , 9., , common hydrometer, , of a, , mms., , a circular cylinder oC, , is, , It floats, with its stem wetted, in alcohol, whose specific gravity is -796, and surface tension, 25*5 dynes /cm., Calculate, deeper it floats than if alcohol had had zero surface tension., , diameter 2, , how, , much, , (Cambridge Higher School Certificate), Because of the downward force due to surface tension, the hydrometer, goes deeper down into alcohol than it would otherwise do, such that the weight, of the alcohol displaced by this additional immersion of the stem, or the upthrusf, due to this displaced alcohol, equal to the force of surface tension acting on it., 1 mm- *>* *1 cm., Now, diameter of the stem 2 mms. and radius of the stem (r), = '2* cm*, =, .\, 2/rr, x, 2*, !, the, stem, of, circumference, And,, , So that, the surface tension acts along this length ; and, since its value is, 25-5 dynes/ cm., the force on the stem due to surface tension, 1603 dynes., 25-5 x -2n, , -, , -, , Let the stem go further down through a distance x cms. than it would do, if alcohol had had zero surface tension., Then, additional volume of stem im1, mersed, or the additional volume of alcohol displaced is equal to wr .*. c-cs., , where, , And, , mass of this alcohol displaced * *.r*.x. p gms., the density of alcohol., weight of alcohol displaced, or upward thrust due to alcohol displaced, z, 2, ==Trr XA:xp gm. wt.**Kr, dynes., , p is, ., , xxXpXg, * 24-53 x dynes., , *x(-l)*x*X'796x981 dynes, , Since upward thrust due to displaced alcohol, due to surface tension, we have, , 24-53*, , x, , whence,, , 16'03/24-53, , =, , downward, , equal to, , is, , force, , 16'03,, , '6530 cms., , Thus, the hydrometer floats deeper by 6'53, had had zero surface tension., , =, , 6-53, , mms, , mms., , than, , would,, , it, , if, , alcohol, , Water rises to a height of 5*0 cms in a certain capillary tube. In the, 10., same tube the level of mercury surface is depressed by 1*54 cms. Compare the, surface tensions of water and mercury, (the specific gravity of mercury, the angle of contact for water, , is, , 13*6,, , and for mercury 130)., , is, , (London Inter-Science}, , We know that surface tension of a liquid is related to the rise or depression of a liquid in a capillary tube, density and radius of the tube, by the relation,, , T=, , ;>, 2.., , COS Q, , ,, , where, , r is, , the radius of the tube, h, the rise or depression, , of the liquid, p, the density of the liquid and Q, the angle of contact for the, liquid and the tube., , Let, , Tw, , Tm, , be the surface tension of water and, , that of mercury., , Then,, , we have, , Tmsa 'X(-l-S4)x_13-6xg, 2 cos 130, , v, , /, , ['54 cms. (depression),, , h, , rm, , Ur>, , And,, , v, , h, , p= \y6gms. jcc. and, , =, , 5 cm., p, , ~, , 1, , 9, , ~~, , $, , ~ l'54x!3'6xrxg, , = 130., 1-54 x 13*6, _, xrxg, ~", 2X-6428, , 2x(-'6428), , 7r = 5xrXlX#/2xl = 5xrxg/2, = 0, so that eos = 1., gm.jc.c. and, Tm, l-S4xl'36xrxg y, 2 x -6428, , _, .*., , "6428x5, , _, ~ 6M6%, 6 .si 6, , Or ', Ur, , 11., respectively., , A, , U-tube, , is, , The tube, , 6-516, , 1, , :, , 6*516., , capillaries of bore 1 mm. and 2 mms., held vertically and partially filled with a liquid of, , made up of two, is, , -., , Tm, , the surface tensions of water and mercury are in the ratio,, , 9
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PROPERTIES OF MATTER, , .526, , cm" 1 and, , surface tension 49 dynes, , 4he, , zero contact angle. Calculate the density of, of the menisci is 1-25 cms., , ,, , liquid, if the difference in the levels, , (London Inter -Science), , We, , 7=, , is, , I)., , contact, , have the relation, zero, (cos $ being =, , r.h.p, , gj2 for a liquid for which the angle of, , Let hi and h* be the heights of the liquid columns in, and r l and r a their respective radii. Then, we have, , the, , two, , r, , T=, And,, , T=, , n, , =, =, , -, , h*, , mm., , =, , IT \, 2T x, /)= P* x[\, , *1, , =, , *5 /w/w., , 1, , 1, , \, , fi, , ra, , /), , *05 cm*, , cm-, , 98, , 1'25=-, , Or,, -whence,, , x(20~10)., , y9, , ^, , P, , density of the liquid., , -, , >>?, , =, , =, , p, , r 2 .p-g, , 1'26 cms., r x, 1, , [Where, , r$.g, , -, , I?, --, , V'VP-, , r2, , =, , Or,, , *\l., 2, (, , hi, , HI, , and, , hj., , r, , =f, , L^, , (hi- h t ), , Or,, , -'&-,, , 2, , similarly,, , .-., , ^*, , ', , limbs,, , I, , ,, , 1'25, , Or,, , =, , 980/px981,, , 7991, , Therefore, the density of the given liquid, , is, , '7991 gm./c.c., , Find the difference in the levels of mercury in the two limbs of a, II -tube, if the diameter of the bore of one limb is 1 mm. and of the other 8 mms., The surface tension of mercury is 44!) c.g.s. units, its density 13*6 gms./c.c. and the, angle of contact with the walls of the tube 140., (Joint Matriculation Board Higher School Certificate), ,, 12., , We, , have the relation,, , iLet the, , T=, , p, , depression of mercury in one limb be, , Then, since, , ^=, , *5, , mm., , =, , 2 x 440, , ,, , h, , !^, ^, whence,, 2 cos, and, , *05 cm., , =, , ra, , /* t, , 4, , and that, , =, , mm., , x -7660, ', , t=~, , -, , I, , v, , -4, , in the other,, , cm.,, , /;,,., , we have, , cos, , 2 x 440 X '7660, 7, , 4x, , 13-6x981', , = 2x440x, (h -/n, ''**, Vl, , ', , 7660, ', , 13-6x981, , fJL_, V/05, , !, , 'N, , 2x440x-7660, , -4, , y, , 13-6x981, , 35, , X, , T"', , __440 x '7660x_35 - *.,8843 ', 13~6~X981, , .*., , the difference in the levels of the, , What, , work done, , two columns, , is, , equal to, , 8843 cm., , blowing a soap bubble of radius 10 cms. ?, (T = 30 dynes per cm.). What additional work will be performed in further, blowing it, so that its radius becomes 15 cms ?, 13., , is, , the, , in, , (/) We know that work done in blowing a soap bubble is equal to its, surface area (inner and outer) x its free surface energy, i.e., equal to its surface, areax its surface tension, (v free surface energy, surface tension)., , =, , and, /., , 2, 2, Here, surface area of the bubble = 2x4nr =8*,10 =800w5?. cms,, T 30 dynesjom., , =, , work done, , in, , blowing the bubble of 10 cms. radius, , =8007tx30=24000*=7-541 x 10 4 *r#s., when its radius becomes 15 cms., 2, 8wx225 = 1800rc,y0. cms,, 2x4*X(15), , (//), , Surface area of soap bubble,, , .*., , increase in area of the soap film, , **, , =, , (1800;r-800*) =* 1000*, , sq., , cms.
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627, , SURFACE TENSION, .'., , work done, , new area of soap, , in creating this, , 1000* X 30=30,000*, , =, , film, , 9-426, , x 10 4, , ergs., , Hence, work done in increasing the size of the soap bubble from 10 cms., <-a<Mus to 15 cms. radius is equal to 9*426 x 10* ergs., a pin hole, 0*1 mm. in diameter, at the bottom of a vessel, depth of mercury may be put in the vessel without any, 1, leak occurring ? (Surface tension of mercury =550 dynes /cm.- , and density=13'6, t?ms./c.c. Neglect the angle of contact.), , There, , 14,, , is, , -containing mercury., , What, , We have to calculate, here, the length of mercury column that would be, supported by the surface tension acting along the circumference of the hole., r, where h, , is, , Or,, , the height of mercury, since, , we have, Here,, , T=, , T, , r, , r', , ', , r, , h, , .'., , p><?, ,, , ., , =, , V, , column without acy leak, = 0, and cos 9 1,, , =--=, , 550 dyneslcm.,, , r, , -2, , occurring., , *., , -1/2, , -, , --- - -., , h, , whence,, , =, , 0'5 mm., 2x550, , =, , and, , '005 cm., , p, , =, , 13-6 gms./c.c*, , -005 x 13-6x981, mercury can be poured into the vessel to a depth of 1649 cms., without, , any leak occurring., inside a soap bubble of radius 1 cm. balances a 1*4 mm., of specific gravity 0*80. Calculate the surface tension of the soap, , The pressure, , 15., , column of, , oil,, , solution., , We know thU, , the excess pressure inside a soap bubble is equal to 47/r,, the surface tension of the soap solution and r, the radius of the bubble., "Since this is balanced by the column of oil, we have, , where, , 7 is, , 4T/r, Avhere h is the length of Ihe oil column,, acceleration due to gravity., A.T, H, ;, , =, p,, , h.?.g,, , the density of oil, and g, the value of, r, , =-14x80x981., , f, , 1, , ', , o, Ur, , J, , 80x981, ~_'14x, ~, ~, ", , J, , ', , |, , 4, , =, , '14, , x -2x981, , [, , V, , r, , h, , =, =, , 1, , cm., , 1-4, , mm., , --14 cm., and P- -80 gm./c.c., , =, , 27-47 dyms\cm., Thus, the surface tension of the soap solution is 27'47 dyneslcm., , EXERCISE, 1., , part of, , XIII, , Show, , that the surface tension of a liquid is equal to the mechanical, Calculate the work done on the film in blowing a soap, a diameter of 4 cms. to one of 30 cms. t if its surface tension be 45 in, , its surface energy., , bubble from, , Ans., , <;..$. units., , 2*5, , x 10 s, , ergs., , Show, , that the excess pressure inside a soap bubble of radius r over the, atmospheric pressure outside it is equal to 4r/r, where T is the surface tensio0, of the soap solution., 2-, , How may, , the surface tension of a bubble be determined ?, Define surface tension, and show how it can be regarded as involving, potential energy. From consideration of energy show that (i) oil will spread on, the. surface of water, (//) mercury will collect on a clean glass plate in drops of, -different shapes according to their size, and (///) water will rise in a capillary, tube., (Bombay), 3., , What would be the pressure inside a small air bubble of 1 mm. radius,, 4., situated just below the surface of water ? Surface tension of water may be, taken to be 70 dyneslcm. and the atmospheric pressure to be 1*012 x 10* dyneslcm*., AIM., dweslctij, , \027xW
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PROPERTIES OF MATTEB, , 52B, , Calculate the loss of enerygy if 1000 drops of water, each of diameter, coalesce to form one large drop. The surface tension of water is equal to, 72 dynes/cm., Ans. 8143 ergs*, 6., Calculate the work done in breaking up a drop of petrol, of voluroj, 1 c.c, iato a thousand million drops. The surface tension of, petrol is 26 dynes/ ctr, Ans. 1255 74 erg*, 5., , '2, , mm., , 7., Calculate the amount of energy needed to break a drop of water 2 mm,, in diameter, into 10* droplets of equal size, taking surface tension of, watei^as 73, dyne si cm., (Madras), , Ans., , 9160, , Define surface tension. Show that the excess pressure acting on, curved surface of a curved membrane is given by, 8., , ergs,, ffi, , '[*, where r l and, membrane., , r2, , are the radii of cuivature and S, the surface tension of the, (Punjab), , Calculate the amount of energy evolved when eight droplets of wa, N, 1 12 mm. each, combine into one., (Punjab, Ans. 9*05 ergs., 10., A soap bubble is spherical in shape, and has a diameter of 10 cms. if, the surface tension of the surface separating soap solution and air is 40 c.g-s., units, what is the excess pressure of the air in the bubble over the atmospheric, Ans. 32 dynes /rm 2, pressure ?, 11., Find in the terms of mercury column the excess pressure inside a rain, 74 dynes/cm. Ans. 2'22 mms of mercury., drop 1 mm. in diameter, for which T, Calculate the force required to separate two plates of glass, of area 10, 12., The surface, sq. cms. each with a layer of water -001 mm., thick in-between them., 72 dynes/cm., Ans. 1*44 x 10 7 dynes., tension of water, 13., Descrrbe a method of determining the surface tension of a soap, bubble Deduce the formula used., 9., , ;, , (surface tension 72 dynes per cm.), of radius, , *, , ;, , ., , =, , The equal spherical soap bubbles coalesce if V is the consequent change, volume of the contained air and S, the change in the total surface area, show, that 3FK = 4Sr, where Tis the surface tension of the soap bubble and P, the, ;, , in, , atmospheric pressure., 14., If a number of, , (Allahabad), of water, all of the same radius r cm.,, droplets, coalesce to form a single drop of radius R cm. show that the rise of temperature, of water will be given by, , little, , -., , (T, , "jr )>, , where S, , is, , the surface tension of, , water and J, the mechanical equivalent of heat., (Saugar), 15., Find the relation between the radiu* of (a) a spherical drop, (b) a, spherical bubble of a liquid, the surface tension and pressure., , Two soap bubbles of radii 2 and 3 cms. coalesce into a single bubble of, radius R cms. If the surface tension of the soap solution is 25 dynes f *r cm. and, the atmospheric pressure is 76 cm. of mercury, (whose density, 13-o#m./c.c.),, find the equation to determine R., (Madras), 3, 1-014, Ans., x 10.(/? -35) + (fl f -13) = 0., , =, , A, , 16., soap film. '001 mm. thick, and at 0*C, is stretched adiabati'colly,, What is toe resulting fall of temperature, if we assume, until its area is doubled., that 1he specific heat and density of the film are both unity, that dT/dff at 0%., = '15 dyne per cm. per 1C, and that J 4-2 x 10 7 ergs per calorie ?, , =, , -, , Ans., , 1, , 0195'C, , ,, , Explain the terms surface tension and angle of contact. Show that, the pressure inside a spherical bubble of radius r exceeds that outside it by, If this excess pressure is balanced by that due to a column of oil,, 4T/r., (sp. gr- 0*8), 2 mm. high, when r=rO cm., find the surface tension of the Soap*, 17., , ble., bubble., , Ans. 39*24 dynes per cr$., Calculate the difference of pressure between the inside and outside of a, (Agra), spherical bubble blown inside a liquid., \, , 18.
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529, , SURFACE TENSION, , of length 7 cms.,, soap film is formed on a rectangular frame-work, the arm of a, from, framework, This, solution., hangs, a, into, soap, dipping, to, Balance. An extra weight of 0'38#m. must bs placed on the opposite pan, What is the surface tension of the soap solution I, Balance the null of the film., 19., , A, , Ans. 26-6dynesjcm., Calculate the density of a liquid, a column 2'09 cms. of which balances, for, the excess pressure inside a soap bubble of radius 1 mm. Surface tension, Ans. *78 gm-jc-c., the s< ap solution may be taken to be 40 dynes, cm., 21., Describe and explain how the surface tension of a liquid may be meaDiscuss whether the result obtained, sured by forcing bubbles of air through it., this way should be the same as that given by the capillary tube method., What would be the pressure inside a small air bubble of 0-1 mm* radius,, -= 72 dynes per cm., ituated just below the surface ? Surface tension of water, (Bombay}, 1'013 x 10 6 dynes per sq. cmaid atmospheric pressure, 2, t, Ans. I'0274xl0 d.ym?5/cm ,, diameter, of, 22., Show that the excess pressure inside a drop of water,, i, '100th of a mm. is '0137 cm. of mercury column at 0*C. Density of water at, *, and 769 mm. pressure^'61 x 10, C =1 gm-lc c density of water vapour at, m.jc-c.y and surface tension for water=75 dynes/cm., the, 23., Explain the method of finding the surface tension of a liquid by, method of drops., the, diop of water, 05 cm. radius, is split into 1000 tiny drops. Find, mechanical work expended. Calculate the pressure inside one of these small, (Madras }, drops (surface tension of water = 75 dynes per cm ), Ans. '675 nr ergs (P 3000) dynesjcm*.,, (where P is the atmospheric pressure)., In a drop-weight determination of the surface tension between water, 24., and chloroform, a glass tabe of 4 mm. external diameter was used and 50 drop*, of chloroform, density 1'5 gm. per c-c., were allowed to fall in the water. The, weight of these drops was 3'43 gms. Find the interfacial surface tension., Ans. 29'5 dynes per cm., 20., , =, , r, , ., , 0C, , A, , ;, , Define surface tension., , 25., , Show how, , it, , is, , related, , I, , to surface energy in a, , liquid., , plates are pressed together with a very thin film of water betExplain clearly why the two plates firmly adhere to each other., (Agra, 1929), 26. (a) Define surface energy. Give thi theory and practice of the method of, (Punjab), determining the surface tension of a liquid by weighing drops., 100 equal, (b) If a globe of water of diameter 2 cms. suddenly splits into, which, globules under isothermal conditions, determine the gain in surface energy, I.E. 1950}, occurs, given that the surface tension of water is 75 dyne* per cm., Ans. 3432 ergs., , Two, , glass, , weed them., , (AM, , Explain clearly from where the energy comss whei a liquid rises, Derive an expression for the height h, in a capillary tube., through \v ,ch the liquid of surface tension T will rise in a capillary tube of, radius r. What will happen if the length of the tube is smaller than h ?, (Punjab), 28. Describe Jaeger's maximum bubbb pressure method of determining, 21., , against gravity, , surface tension., , A, , R contains inside it a smaller soap bubble of, bubble now bursts isothermally, with no leakage of air, om the system as a whole, so that a new bubble of radius R' is formed., ^how that the radii of the three bubbles are connected by the relation, P CRf 8 /? a )-f4r(/?' a r 1 .K' a )=0, where P stands for the atmospheric pressure,, and 7", for the surface tension of the soap solution, Give the iheary and experimental details of a method for determining, 29., ^c surface tension of mercury, and the angle of contact for mercury and glass,, Calculate the work done in spraying a spherical drop of mercury, of one, millimetre radius, into a million droplets, of the same size, the surface tension, if mercury being 550 dynes/cm., (Agra), Ans. 6839 ergs., idius, , large soap bubble of radius, , r., , If the smaller
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PROPERTIES OF MATTER, , 530, , 30. Define surface tension and angle of contact., Describe Quincke's drop method of measuring these quantities for mercurj, in contact with glass. Give the theory of the method., (Banaras\, What forces determine the shape of a liquid drop on a horizontal plate 1, 31., Show that all large drops of mercury resting on a clean glass plate will have the, , satne height., 32. Describe the laboratory method of finding the surface tension of a, liquid by the rise of the liquid in a capillary tube., , A, , tube of conical bore is dipped into water with apex upwards. The length, of the tube is 20 cms. and the radii at the upper and lower ends are 0*1 and, Find the height to which the liquid rises in the tube. (Surface tension, 0*3 cm., (Allahabad], of water --^80 dynes /cm.), Ans. 0'55 cm., , Show how the existence of an acute angle of contact and of a pressure, 33., difference due to curvature accounts for the rise of liquid in a capillary tube-, , A capillary tube of internal diameter 1 mm. and external diameter 5 mm,, hangs vertically from the arm of a balance, the lower end of the tube being, Calculate the change in the apparent weight, in water (angle of contact=-0*)., of the tube due to surface tension. (Surface tension of water ^78 dynes/cm, and, #=980, 34., , (Allahabad], Ans. 0'15 gm<, , cm./sec*.)., , Describe an experiment for determining the surface tension of a liquid., , Deduce the formula used., , (Punjab], , Explain the capillary tube method of determining the surface tension, of a liquid. Why is this method not suitable for temperatures other than that, of the surroundings ? Suggest and explain other methods for doing so, in which, the surface tension may be calculated by measuring the pressure necessary to, force the meniscus back in a level with the surrounding liquid., A capillary tube is dipped in water. Water rises to a height of 4 cms., 36, above the surrounding liquid. If the angle of contact is zero and the radius of, the tube is 0-1 mm., what is the surface tension of the liquid ?, (A.M.I ., 1961}, Ans. 19'63 dynes/cm., 37., A verticle U-tube containing mercury has one limb of diameter 5 cms., and the other, of diameter *1 cm. Calculate the difference in level of the, mercXiry columns in the two limbs. (Tfoi m2rcury=550 dynesjcm., density of, mercury =1 3*6 gms. per c.c. and its angle of contact with the walls of the lube, Ans. i'04 cms., 140)., 38. A liquid of density 1 05 gms. Ic.c. and angle of contact 20 has a vertical, If the surface tension of the, capillary tube of 2 mm. diameter dipping into it., liquid be 23*5 dyne&jcm., find the rise of the liquid in the capillary tube., Ans. 4*29 cms, The surface tension of water is 72 dynes/cm. Calculate how far water, 39., Ans. 1*47 cms., will rise up a circular tube, 2 cms in diameter., 40, A U-tube, whose ends are open and whose limbs are vertical, contains, oil of sp. gravity 0'85 and surface tension 28 dynes'cm., If one limb has a, 8 mm., what is the difference, diameter of 2*2 mm. and ths other a diameter of, in level of the oil in the two limbs ?, Assume the angle of contact between the, oil and the glass to be zero., (A.M^l Mech Engineering}, 35., , ,, , Ans. 1*07 cms, Define surface tension. Explain how you can determine the surlace, tension of a solution with the help of a bubble blown out of itThe limbs of a capillary U-tubs have the internal diameters of 1 mm- and, 2 mm. The tube is held vertically and is partially filled with a liquid of, surface tension 50 dvnetfcm, Find the density of the liquid if the difference of, levels in the two limbs is 1-25 cms., Assume that the angle of contact is zero., 41., , (Bombav), Ans., , -XI, , 55 gm. Ice., , 42., Deduce the relation between surface tension and vapour pressure at a, curved surface and discuss its effects oa evaporation and condensation, , (Bombay ,1946 and 1948}
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531, , SURFACE TENSION, , Calculate the difference in vapour pressure of water for a plane surface, that for a drop of radius 0-1 mm. Density of water vapour**6-l x 10~, 75 dynes/cm., gms.jc.c. and surface tension of water, (Bombay), Ans 9*15 dynes / cm*, 44., ShDv that the maximum vapour pressure over a curved liquid surface, .differs from that over a phne surface, DMuce the value of this difference in, terms of the densities of the liquid and the vapour, the radius of curvature of, <tli2 surface and the surface tension, of the liquid, Discuss the bearing of the, above fact on the formation of rain drops., (Madras), 43., , and, , =, , 45., Show that the vapour pressure over the curved surface of a liquid drop, of radius r exceeds that over a flit surface by an amount equil to 2r/or.(p 9, where Tis the surface tension of the liquid, p, its density and ff, the density of, <us vapDur., Discuss the application of this result to the condensation of supersaturated vapour on dust particles and other nuclei., (Bombay), , 0C, , 46., If the aqueous vapour tension at, be 4-6 mm. of mercury, calculate, the radius of a water drop, at, C, which would be in equilibrium with its, vapaur at twice this pre^ure. (Surface tension of water=75 dynes per cm., 18, Ans. 2*42 x iO~ 6 c/w,, g>ms. of water at N.T.P. measure 22'4 litres)., , A, , 47., capillary tube is immersed in water and, by exerting a pressure of, 15 65 cm. of water, the, menisjus in the tube was kept 1'25 cm. below the surface, of the water outside. The radius of the tube was 0-104 cm. Calculate the, surface tension of water., Ans. 73'5 dynes/cm., , 48., , Explain concisely, but clearly, the following, (/) pieces of camphor scurry about on the surface of water but their, motion slows down if we immerse our figure in the water ;, :, , (ii) it is difficult, , to introduce mercury in a fine glass tube, , (in) small pieces of cork, attract each other ;, (iv), , and straw, , etc., floating, , ;, , V^-6, , -, , ^a'O, , on water, appears to, , small drops of mercury on a plane glass sheet are spherical in, shape, but large drops of it are flat at the top, ploughing of fields helps ictain moisture in them., ;, , (v), , How, , 4)., Distinguish between a wave and a ripple., may the surface, tension of a liquid be determined by the method of ripples ?, 53., What is the effect of temperature on thi surface tension of a liquid ?, Describe how it may be studied experimentally.
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CHAPTER XV, , GASESKINETIC THEORY, The Kinetic Theory, , In order to connect!, Introduction., accumulated by experience, some, hypothesis or theory became necessary, and the kinetic theory of matter, The, is the one that has proved most helpful for the purpose., distinction of being the founder of the modern Kinetic Theory goes,, by common consent, to Daniel Bernoulli (1730), as he was the first, to explain Boyle's law on its basis, though the theory may be said, to have had its beginnings in the speculations and ideas of Thales oj, Miletus (640 to 547 B.C.) about the possible structure of matter, amj, a host of other early workers. And, the credit for having established, it on a firm mathematical basis is due to Claussius and Maxwell,., The triumph of the, in whose hands it attained its present form., theory lies in its success in explaining known results and predicting, 261., , and, , new, , co- relate, , the, , facts, , many, , ones., , The theory is based on two assumptions, That matter is not continuous, but consists of small aggrega(1), :, , tions or lumps, called molecules, very much like a handful of sand,, composed of fine granules so that, even when the molecules are in., contact with each other, there are inter-spaces in-between them, ;, , ., , The molecule of a substance is the smallest part of it that, possesses the characteristic properties of that substance, and can, have an independent existence of its own. It can be broken up into, smaller bits, called atoms* by various methods, but then, these no, A, distinctive properties of that substance., longer exhibit the, molecule may consist of one 01 more atoms, e.g., the molecule ol, Helium is monoatomic, i.e., consists of only one atom, that of, Hydrogen or Oxygen is diatomic, i.e., consists of two atoms, and, that of Carbon dioxide is triatomic, i.e., consists of three atoms,, and so on., It should be, , noted with care that although there, , may, , be any, , different kinds of molecules, there can be only a hundred, and odd different kinds of atoms (including isotopes), corresponding, to the different elements. These atoms are the smallest particles that, , number of, , can take part in chemical reactions., , That the molecules are generally not in contact with eaclv, (2), other but are in a continuous state of agitation, moving about with', great speed, haphazardly, in all directions, their freedom of movement, however, being different in the three states of matter, viz., the, solid, the liquid and the gaseous., , Now, there is a huge mass of evidence in favour of both thce^assumptions, e.g., (a) the phenomena of diffusion and solution, which, clearly suggest the molecular structure of matter and agitation of, molecules ; for, we find heavier gases, like Carbon dioxide, diffusing, 532
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OASES, , 53J, , KINETIC THEORY, , into lighter ones, like Hydrogen, in defiance of gravity, as it were,, and even a gas diffusing into lead, (b) the familiar phenomenon of, expansibility of gases, which is clearly due to the tendency of the, molecules to fly away from each other, (c) the phenomena of vapour, , pressure and evaporation, and (d) phenomenon of Brownian movement,, which is perhaps the one single experimental fact, which enables us, to actually see, so vividly, the molecules of a substance moving about, hither and thither before our very eyes., , This last phenomenon derives its name from Robert Brown, an, English Botanist, who first discovered it in 1827 while observing,, under a high-power microscope, the suspension of plant pollen in, water, when he found the pollen grains dancing about in the wildest, manner, and thought that they were perhaps very tiny living creaThe phenomenon has been fully investigated by Parrin (in, tures., (1908) arid may be readily observed by examining a colloidal suspension under an ultra -microscope, when a strong beam of light is passed, 7, through the liquid. The particles, as small in size as 6x 10~~ cms.,, whi(;h just appear as mere points of light, surrounded by diffraction, rings, are visible by the light, they scatter at right angles to the, beam, and are seen in a spontaneous and eternal dance of a most, irregular, , now, , and haphazard manner, now spinning, now, and so on and on., , resting,, , and, , rising again,, , Now, this mad movement of the molecules is the clearest proof, of molecular agitation. For, the movement of the tiny particles is, due to the large number of molecular impacts they receive simultaneously on all sides. Since these impacts are not necessarily uniformly distributed, there is a resultant unbalanced force on the tiny, And, because the force varies, particles which causes them to move., most haphazardly and irregularly, the motion of the particles also, exhibits the same haphazardness and irregularity., Quite obviously,, the smaller the particles, the more raadily are they subject to these, irregular motions for, on a large or a heavy particle, the impacts will, almost balance and no resultant motion will ensue, e.g., when a large, body like a glass bead, or a marble piece, is immersed in water or any, other liquid, it is not tossed about in this manner., ;, , Whereas the kinetic theory of, Kinetic Theory of Gases., is still in a formative stage, the kinetic theory of, gases has made rapid strides and can fully explain the various proThe reason is not far to seek and will be clear if we, perties of gases., try to picture to ourselves the structure of a solid, a liquid and a gas., 262., , -solids, , and, , liquids, , In the case of a solid, due to the great force of cohesion, its, (/), molecules are all compactly or closely packed and every molecule is, more or less fixed in its position, having only the freedom to vibrate, about this position. It is not free to move over the whole volume of, the solid/ much less to escape away from it., , In the case of a liquid, the cohesive 'force is still there, but, (//), not so strong, with the result that although it is sufficient to hold, the liquid together and to give it a, free surface, it cannot prevent a, molecule from roaming over the whole volume of the liquid, with the, result that while a liquid has a definite volume and a well defined free, surface, it has no shape of its own. The average distance between two, is
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PROPERTIES OP MATTER, , 634, , molecules of a liquid is estimated to be more or less of the same order, as the size of its molecule., , In the case of a gas, on the other hand, these cohesive, (Hi), forces are almost negligible under ordinary conditions of temperature, and pressure, so that the molecules lie $c$ aparb from each other, the, average distance between two molecules being about a hundred times, the size of a molecule. A molecule is, therefore, free to wander over, the entire space available to it, with the result that a gas has neither a, volume nor a free surface of its own., It will thus be clear from the above why the problem becomes, simpler in the case of a gas, compared with thac in the case of a solid, or a liquid., It is obviously so, because in the case of a gas, (i) the, lie far apart and we can, therefore, neglect, under ordinary, conditions, the actual volume occupied by the molecules themselves, (if, compactly arranged), compared with that occupied by the gas, as a, , molecules, , whole. Clearly, this cannot be done in the case of a liquid or a solid,, the molecular forces* can be neglected because of the large distance, between the molecules., (//), , Now,, , in order to further simplify our problem,, , we, , shall first, , consider, here, a gas whose molecules have negligible size, i.e., whoso, molecules are mere mass-points, and in which the molecular forces, are also negligible., Such a gas, with zero molecular size and zero, molecular forces, is called an ideal or a perfect gas. No such gas, however, exists in reality, and the properties of an actual or a real gas, only approximate to those of this ideal gas., , Pressure exerted by a perfect Gas. Due to the constant, of a gas and their high speed, the}, 'bombard' the walls of the containing vessel, and thus exert pressure., To calculate this pressure exerted by a gas, wo make the following, further assumptions, to simplify matters., 263., , random motion of the molecules, , That the molecules of gas are all alike, (though different, (1), from those of another), and are perfectly elastic spheres, and that, no force of attraction or repulsion between them, or between, In other words, thai all, vessel., their energy is kinetic and that they do not suffer any loss of momentum, or kinetic energy, on a direct or 'head on' impact with the \\alls, of the vessel, only their direction of motion being reversed., That large numbers of molecules exist in the smallest volume, (2), of a gas with which we can deal. Under ordinary conditions of temand pressure, the number of molecules present in 1 c.c. of a, perature, 19, their size or diameter being, is estimated to be of the order of 10, gas, the actual volume, with, the, inter-molecular, small, space,, compared, very, in, 10 litres of the, about, 3, the, molecules, cxs., only, being, by, occupied, there, , is, , them and the walls of the containing, , ,, , gas., , That due to their large number and ceaseless haphazard, ranging from zero to infinite,, there are frequent collisions against one another and consequent, changes in the direction and magnitude of their velocities at each colli(3), , motion in, , all directions, .with velocities,, , * These molecular forces are, entirely different from New ton's gravita, tional forces. They are electrical in nature and do not obey the ordinary gravitational inverse square law, as has been indicated already.
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GASES, , KINETIC THEORY, , 535, , ; and that this does not affect the molecular density of the gas, i.e.,, the molecules do not, in the steady state, collect more at one place, than at another., , sion, , That the molecules, being material bodies, subject to the, (4), laws of motion, move in straight lines with uniform velocity between any, two, , collisions., , That the time for which a collision lasts is small compared, (5), with the time-interval between two collisions , or, the time taken to cover, the distance traversed between two collisions., This distance is called, the mean free path of the molecule and depends upon the temperature, and pressure of the gas., , We are now, , in a position to calculate the pressure exerted, on the walls of the containing vessel., , by an, , ideal or a perfect gas, , Let there be a gas, enclosed in a cubical vessel of unit edge, with, walls perfectly elastic, (Fig. 318), and let the number of molecules, present in it be n, the mass of each molecule being m., its, , Since the molecules arc constantly moving about with different, they are bombarding the walls of the, cube., Consider one molecule m, having, a velocity C 1 at a given instant. This velocity of the molecule may be resolved into, three rectangular components, w 1} v l and w l, along the axes of x, y, and z parallel to the, three edges of the cubical vessel respectively., , velocities in different directions,, , ,, , Then,, , clearly,, 2, , Ci, , = uf+vS+w*., ^, , Now, consider the motion of this molecule, Fig. 318., along the axis of x, i.e., perpendicular to the, walls A and B of the vessel so that, striking the wall A with velocity u lt it rebounds with the same velocity, in the opposite direction (the molecule as well as the wall being perfectly elastic), strikes wall j5, rebounds back to A and so on., ;, , momentum of the molecule as it approaches A is ww,, obviously directed towards A. When, however, it collides, against A> it rebounds with only its velocity reversed so that, its momentum still remains the same in magnitude but is now directed opposiClearly, the, , and, , is, , ;, , tely, i.e., is, , now, , m^., , Therefore, change in, , its, , momentum, , =, , mi/j, , (mu^ =, , 2 mu^., , This, then, is the change in momentum of the molecule at each, with wall A and, clearly, as it collides against the wall, it imparts this momentum (2 mu^ to the wall., , collision, , Hence, momentum imparted by the molecule, , =, , 2, , mu x number, , of collisions, , it, , to, , makes with, , wall, , A per second, , the wall per second., , Now, for each successive collision or impact with wall A, the, molecule must traverse the distance from A to B and tfeck, i.e., a, distance equal to twice the length of the unit cube, or a distance of 2, cms. Since it covers a distance u lt along this direction, in 1 second, it will cover the distance 2 cms. (from A to B and back) in 2/1^, ,
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PROPERTIES OF MATTER, , 536, , seconds ; and, therefore, the number of collisions, wall in one second is equal to i/j/2., , So, , momentum imparted, , that,, , =-, , to wall, , 2mut x u, , 1, , /2, , A, , will have with the, , it, , by the molecule, per second, 2, muj, , =, , ., , In other words, this is the force /15 say, exerted by the molecule, on wall A for, in accordance with Newton's second law of motion,*, force is just rate of change of momentum. Thus,, ;, , / = mu^., x, , /, , exerted, Similarly,, 2 /3 ,. ....... ,/n, molecules, having velocities c 2 c 3 ............... r w, forces, , ,, , ,, , ponents, , (u l9 v,. Wj),, , ft, , w 2 ), ...... (n n, , v2 ,, , (i/ 8 ,, , respectively, are given, , by, , = ww, , 2, 2, , ,, , ,, , ,, , ,, , ,, , v wt n' n ),, , on wall A by other, and rectangular comalong the three axes, , /3 = ww 3 2 ..., /n =/ww n 2, , ,, , the mass of each molecule being the same m, in each case., Thus, total force on wall A due to all the n molecules is given by, , 4=1x1=1^., , Now, area, , of wall, cm. [/ each edge of the cube=l cm., So that, FA is the force exerted by the molecules on unit area and is, thus equal to pressure Pl exerted by them on A because, as we, know, pressure is equal to force per unit area. Thus, pressure Pl on, t, , wall, , =, , A, , FA., , P -, , m(w 1 +w/+w 3 + ...... w w )1, same, manner, considering the motion of the moleExactly, cules along the axes of y and*z,, i.e., perpendicular to walls C and D, and walls E and F of the cubical vessel respectively, we have pressure exerted by the molecules on wall C given by, Pt = m(v a +va 8 +v 8 2 + ..... + v * 2 ),, and pressure exerted by the molecules on wall E given by, PS = m(w 1 *+Wt*+Wt* + ...... +u n *)., Since the pressure exerted by the gas is the same in all directions, we, have P1 = P 2 = P 3 = P, say., Or,, , 2, , 2, , 2, , in the, , j, , 3?, Or,, , =, , =, , Pi+P*+P*, , And, therefore,, So that,, , 3JD -, , m(u*+u,* + ..M n *)+m(v1 *+v z *+...v n *)+m(w 1 *+^^^, 2, ^[w; +w/+..., , P=, , P =, , Or,, , i^.-^^, , Now,, , c stands for the root mean square velocity of the, molecules*, usually written, for brevity, as r.m.s. velocity., , where the symbol, , So, , that,, , And,, , therefore,, , 2, , (i +c 2, , + ...+c n =, P = %mn+c, , *, , 2, , ), , nc\, , 2, , ., , Thus, the pressure exerted by the gas enclosed in the vessel, , ...(/), , is, , to, */., the square root of the mean square velocity of the molecules., , equal
Page 539 : G1SES, 264., , m.n, Or,, , KINETIC THEORY, , The Value of c. Now,, mass of the molecules, , =, =, , mass per, , Substituting this value of, , P=, , in relation, in unit, , m, 2, , Jp.c, , Or,, , n in relation, , y, , c, , =, , gas., , density of the gas,, , p., , we have, , for P,, , (/), , =, , c2, , whence,, , ,, , we have, , above,, , (/), , volume of the, , volume of the gas, , unit, , 537, , 3P/p., , = V3/Yp., , This enables us to determine the value of c for any, particular, gas, at a given temperature, if P and p for it be known., Thus, for example, taking, , P for, , air, at, , 0C,, , 76x 13'6x98, , to be, , =, and, , w, , p, , for air, at, , =, , 0*C, , have, , c, , 1/100(M of that of water, , ^, -=, , -, , ~, , cms '< sec, , V^IO-'*, 9, \/3~xl0 =, 5-5xl0 4 /100, , 1 mile I sec-, , =, , ^, , dynesjcm*., 2, ,, , (approx.), , %, , 10- gms.jc.c.,, , i/1000, , V / 3xTOT xl6r, , V30X10 8 =, , *, , 5-5x10*, r/ze, , 10 8 dynes /cm, , velocity, , 5-5, , X 10 4, , rms./s<?c., , SSQmetreslsec-, , of a, , rifle bullet., , We thus sec how tremendous is the speed with which the, molecules of the as move about haphazardly in, any given volume of, it, and, obviously, the, the, the, lighter, molecule,, faster it moves., The velocity of the molecule of any other gas may also be calculated in the same manner, at any desired, temperature., 265., Relation between c and T. We have the experimental, gas equation, connecting Boyle's law and Charles' law, viz.,, , = RT,, , PV, where, , P and, Nw,, , R, , is the gas constant, T, the absolute, temperature of the gas, F, its pressure and volume , respectively., , =, , p, , PV =, 'Or,, , *, , putting, , nV, , =, , TV*, the total, , of the gas, we have, Or,, , ^m.TV.c, , 2, , =, , PV =, j?r,, , i m.n.c*., , i m.n.V.c*., , number of molecules, -ZV., , J/w, , whence,, , c 2 oc, , Or,, , and, , c2, , c, , in the, , volume, , V, , 2, ., , =, , SRTjmN., [v, , J/m., , 7?, , and, , AT are coastants., , Thus, absolute temperature of a gas is directly proportional to the, of the r.m.s velocity of its molecules, the greater the velocity, <and, therefore, the greater the kinetic energy) of the molecules, the, -greater the temperature of the gas., , .square, , 266. Deduction of, 1., , Boyle's Law., , Multiplying, , it, , Gas Laws on, , We, , the basis of the Kinetic Theory., , have the relation,, , P, , by F, the volume of the gas,, , PV =, , =---, , %m.n.c, , z, ., , we have, , m., , N, , *For the sake of simplicity,, is taken to be the total number of molecules in a gram-molecule of the gas ; for, at the same, temperature and pressure,, its value will be the same for all gases., _ It is called the Avogadro Number, and, dts value is found to be 6-0 x 10 28 .
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638, , PROPERTIES OF MATTER, , where, , nV, , =, , N, the, , total, , number of molecules, , in the given, , volume V, , of the gas., Here,, , N being constants, it follows that PV oc c, PV = a constant, if c be constant., , m, , and, , Or,, , 2, ., , c is constant, if T, (the temperature of the gas) be consSo that, at cnstant temperature, PV, a constant, which is our, Boyle s Law., , Now,, , =, , tant., , Alternative Deduction of Boyle's Law., A more rigid derivation, of the law from the kinetic theory is the following, :, , N, , Lot, molecules of a gas, each of mass, be enclosed in a spherical vessel of, , m,, , radius, , r, , and volume V, , ;, , so that,, , V, , ~, , -J, , nrr, , 3, ,., , Consider one molecule striking the, wall of the vessel at A with a velocity, (i.e.,, the root mean square velocity), c at an angle, 6 with the normal at A, (Fig. 319). Let the, , molecule, , then follow the path, , ABCDE., , The component of the, cular to the wall, , is, , velocity c perpendiclearly c. cos Q, and gets, , the other component c sin 9, at, Fig- 319., right angles to it, remaining unchanged., The change of momentum at A is, therefore, equal to 2 in c. cos 0,, reversed,, , inwards., , distance travelled by the molecule between collisions2 r cos B., 2AO. cos 6, equal to AB, , Now, the, at, , A and B, , is, , =, , = 2AM =, , And, therefore, the time taken by the molecule to travel this, distance is equal to 2r. cos Bjc sees. so that, the number of collisionsin cne second, cj2r cos B., ;, , =, , .-., , change of, , mcmentum due, , momentum, , is, , one second; or, the rate of change, , =, , 2m., , 6xc, --- - me*, --- per, , c cos, -------., , 2r cos 6, Since, , momentum, , there are, , =, , of, , to one molecule is, , N, , second., , r, , molecules in, , all,, , the total rate of change of, , m.N.c 2 jr, inwards., , Again, since the rate of change of momentum is equal to force,, the force exerted by the molecules on the walls of the vessel, outwards, , Now, the area of the, .*., , force exerted, , vessel, , =, , given by, , Now,, whence,, , [Because, , it is, , spherical-, , by the molecules per unit area of the vessel, m.N.c*, , m.N.c*, , Or, pressure, , 4wr*., , P exerted by the molecules, on the wall of the vessel, P = m. /Vc2 /47rr 3, [v force per unit area = pressure., , J, , volume of the, 47rr, , 8, , vessel,, , =, , 3F,, , V = 4r 8 /3,
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KINETIC THEORY, , GASES, , 539, , 3, Substituting this value of 4?rr in the expression for, , =, PV =, P, , we have, Or,, , And, constants,, law., , P, , above,, , m.N,c 2 j3V., , m.McVS., , ...(//), , m and N, as also c (at a given temperature), are all, PV = a constant, at a given temperature, which is Boyle's, , since, , N.B. Afote how the above treatment may be used to show that, nV> where n is the number of molecules per _unit volume., F0r, 2V, z, this value of, in relation (it) above, we have P, $ m.n c, , =, , =, , N, , P=, , 1, , J/w.w.c, Substituting, -, , -, , Avogadro's Hypothesis., According to this hypothesis, equal, volumes of all gases and vapours, under the same conditions of temperaLet us .see, ture and pressure, contain the same number of molecules., ho\v we can deduce it from the kinetic theory., 2., , Let N! and N2 be the numbers of molecules in two equal volumes of gases respectively, at the same temperature and pressure., , Then, clearly,, where m l and, , PV ^ frn^A* = 1^^*,, , m, , ...(a), , are the masses and the root mean, square velocities of the molecules of the two gases in the two cases, respectively., 2, , ,, , and, , c^, , and, , c2, , =, , 2, the, Now, kinetic energy, \xmassx (velocity) and, therefore,, 2, the, and, kinetic energy of each molecule, in the first case, -J^Vi, 2, kinetic energy of each molecule, in the second case, |w 2 r 2, ,, , =, =, , ', , ., , As shown by Maxwell, the average kinetic energy of any gas, the same at a given temperature so that, \m^c^, \mtf<^., Multiplying both sides by, , ,, , we have %m L c L *, , =, , NZ9, Now, dividing (a) by (b), we have N, the, number, molecules, in, the, of, case, i.e.,, same, which, , is, , is, , =, , ;, , (b], , \nij'^., , of the, , two gases, , is, , the, , Avogadro's hypothesis., , Thus, since a gram-molecule of every gas contains the same number of molecules under the same conditions of temperature and pressure,, it follows that the greater the density of the gas, the greater its mole, cular weight, or, the molecular weights of two gases are proportion, al to their densities at the same, temperature and pressure., ;, , Graham's Law of Diffusion. This law states that the rates, of two gases (wldch depend upon the velocities of the, molecules of the two gases) are inversely proportional to the square, roots oj their densities., Let us see how can we arrive at this result, from the kinetic theory., 3., , f, , diffusion, , M, , Let Mj and, be the molecular weights,, 2, gram* molecules), of the two gases respectively., , (i.e.,, , weights, of, , cm, , Then, since PV = ^tn^N.c^ = $rti 2 .Nc 2 2 at the same temperaand pressure, V being the volume of a gram-molecule of the gas,, N, the Avogadro number, and m l and m z and c x and c 2 the respective, masses and r.m.s. velocities of the molecules in the two cases we, have, m t N Mi and m2 N = 2, ture, , ,, , And, therefore,, , PV = \Mv c^ =, , M, , M, , 2 ,c 2, , 2, ., , .
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PROPERTIES OF MATTER, , 540, Or,, , whence,, , M, , where p x and p 2 are the densities of, But MI oc p! and, 2 oc p 2, the two gases at the same temperature and pressure., ,, , = vWPr, , cjc 2, Since, sion of the, , ^ oc r t and c 2 oc r 2 where rA and, two gases respectively, we have, , are the rates of diffu-, , =, , VP2/Pj>, the mathematical expression of Graham's law., 'ilr*, , which, , r2, , is, , For deduction of Charles* law and Dalton's law, see solved, 3, at the end of the chapter., , example, , We, , Kinetic Energy of a Molecule., 267., pressure of a gas is given by the relation,, , have seen that the, , =, , Multiplying by, *, , we have, , the gas,, , P, \rn.nj?., V, where V is the volume of a gram-molecule, , PV =, nV, , Now,, , =, , PV -PV =, , Also,, , \rn.N.c?, , =, , Im.N.c*., , R.T, where, , |m.c, , But, , I /?/c, , 2, , is,, , 2, , =, , Now, RjN, , is, , we have \rn.N, , the gas constant., , & = |jR.T., , .(RIN).T., , clearly, the kinetic energy of a molecule., , Therefore, kinetic energy of a molecule, is, , R, , R.T., , Multiplying both sides by 3/2,, Or,, , of, , ^m.n.Vc*., N, the Avogadro number., , a constant, , A^,, , =, , ~.(R/N).T., , called Boltzmann's constant., , K.E. of the gram-molecule of a gas, , =, , ~.K.T., , Obviously, therefore, the kinetic energy of the gram- molecule of, a gas is equal to | K.N.T. =- ^(R/N).N.T., r, , A .E., , Or,, , This, , o//Ae molecule of a gas, , =, , ^/?.r., , us at once that the kinetic energy of a molecule (/), depends upon T, the absolute temperature of the gas and (ii) is quite, independent of its mass. Hence a tiny molecule will be more active, than a bulky one., tells, , This fact, , is, , often referred to as the kinetic interpretation of, , temperature., , Again, examining the above expressions for kinetic energy a, carefully, we see that the factor \ appears there because, of its presence in the expression for kinetic energy, and the factor 3, appears there because the molecule has three degrees of freedom of, straight line motion, (i.e., along the three axes)., Therefore, the, kinetic energy per degree of freedom is equal to one- third of the, little, , more, , above., , Thus, K.E, of one molecule of a gas, per degree of freedom^ K.T., And, therefore, K.E. of the gram-molecule of a gas, per degree of, freedom,, , =, , \RT.
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KINETIC THEORY, , OASES, , 541?, , We know that a gram268. Value of the Gas Constant (R)., molecule of every gas occupies the same volume at the same temperature and pressure. This volume is 22*4 litres at the normal temperaand 76 cms. pressure. And,, ture and pressure, (N.T.P.) i.e., at, of R must be the same for, value, the, that, follows, it, since PV, RT,, all gases at the same temperature and pressure., R., Now,, PV\T, , 0^, , =, , =, , So that, putting the value of P, V and Tin the above relation, we, have, p = 76 cms., 76 X 13-6 x 981 X 22400/273, R, = 76x13-6x981, 7, 2, = 8-29 x 10 ergs per degree Cdynes I cm, , =, , ., , Now,, , 1, , calorie, , =, , V, , xlO 7 ergs, (value of/), 829xl0 /4-18xl0, 4-18, , -22-4x1000, , =, , 7, 7, /?==, s=l-98 calories j'degree C., , =, , Or,, , we, , ---22' 4 litres., , 22400, , c.cs., , 2 calories/degree C> as a near approximation., , a gas. If, This, then, is the value of R for one gram-molecule of, consider only one gram of a gas, however, the value of R will,, , be, naturally, be I'Q&jmolecular weight of the gas, and will, therefore,, molecular, their, different, for, different, weights., gases, depending upon, , = RJ, Note. In the Constant Tables, (Kaye andLabys}, the relation PV, used with a different meaning, viz., P is taken in atmospheres, and V, as the, ratio between the volume of the gas at pressure P and temperatute T and its, C, C so that, in this case,, volume at normal atmospheric pressure and, is, , ;, , P=l aim V - 1 and, R=PVIT = x 1/273 =, ,, , And, , 1, , .-., , T=, , 0C -, , 273, , Ahs., , 1/273, for all gases., , Van der Waal's Equation. In our discussion so far, wo, ourselves on! y with ideal gases, i.e., gases in which, concerned, have, the molecular size is zero, and the molecular force, (i.e., force of cohesion, between the molecules), is zero. These assumptions are far from valid, in the case of any real gas, even at ordinary pressures, and they, become absolutely inadmissible at higher pressures., 269., , This fact was recognised as early as 1827, when Desperetz discovered that the resulting volumes of originally equal volumes of air, and carbon dioxide differed from each other, when subjected to a, Surely, then, he argued, they could not, pressure of 15 atmospheres., both be obeying Boyle's law. The problem of obtaining a relationshipbet we0n pressure, volume and temperature of a gas, more in conformity with actual facts, therefore, engaged the attention of many, were proposed. The, investigators arid many a different equation, one, however, most satisfactory from both the theoretical a-nd theone due to Van der Waal, known after, practical stand-points, is the, him, as Van der Waal's equation. He takes into consideration both, the factors mentioned above, viz.,, the force of cohesion betWi en them., (/), , Correction for molecular, , (/), , the size of the gas -molecules,, , and, "**, , size., , A gas, , molecule has a, , finite;, , size and, therefore, occupies some volume; so that, the actual space inwhich it is free to roam about, when enclosed in a vessel, is a little, less than the volume of the containing vessel, (or the gas), and this>, A correction must,., is much more so, when the gas i& compressed, .
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PROPERTIES OF MATTER, , 542, , therefore, be applied for this decrease in the free, the gas., , space available to, , Now, it maybe recalled that we have imagined a molecule to, a perfectly hard or elastic sphere, so that when the total volume, occupied by a gas is decreased, the volume of the free space available, for the movement of its molecules is decreased to a greater extent, due to the incompressibility of the latter, which still continue to, occupy the same volume as before. For precisely the same reason,, when "the total volume occupied by a gas is increased, the free space, Thus, the, available to the molecules increases in a greater ratio., volume of the free space that undergoes change with pressure is not, V but something less than K, and, at first sight, it might appear that, this volume should be K minus the volume actually occupied by the, This would be quite, molecules, when compactly arranged together., a valid assumption, only if the motion of the molecules were orderly,, and if only some of them moved and others remained at rest. Actualmolecules fly about in a chaotic manner, in all possily, however, the, ble directions, and, therefore, they interfere with, and obstruct, each, other to a very much greater extent, thus greatly reducing the space, For this reason, the volume of the free space is, for free movement., h is near about four times the total volume, where, be, taken to, (K b), >f the molecules. We must, therefore, substitute this value of volume, = RT for a perfect gas., {V -b) for K, in the relation PV, be, , 9, , In an actual gas, the, Correction for the force of cohesion., (//'), force of cohesion is not zero and, therefore, every molecule in a gas is, is being attracted by, every other molecule near about, molecule, well inside the containing vessel, is being, attracted equally in all directions, with the result that the resultant, , attracting,, it., , and, , Now, a, , Force, , on, , it is zero,, , and, , its, , speed remains undiminished., , But a mole-, , 3ule close to a wall of the vessel is only being attracted backwards by, nolecules in the body of the gas and hence its speed is somewhat, diminished, with the result that the force with which it strikes the, In other words, the pressure exerted, ^all of the vessel is lessened., ay the gas on the wall is now smaller., it is obvious that if we double the number of molecules, of the gas, this reduction of the observed pressure will become, four times as great, for the simple reason that', , Now,, , per, , c.c., , the, , now be twice as many molecules striking the walls of, containing vessel, and, near to a wall, will now be attracted inwards, (ii) each molecule,, (i), , there will, , or backwards by twice as, , Now, both, of the gas, , ;, , many, , these depend, so that,, , molecules., , upon the number of molecules per, , c.c., , reduction of observed pressure oc (number of molecules of the gas, 2, , per, Or,, , c.c.), , ., , reduction of observed pressure oc, , l' e, , -fafae, , Or, reduction of observed pressure, , tfc 'gas)*', , ", , 2, , tion of observed pressure oc, , =, , <tf, , I/V, 2, a/F where a, ., , ,, , is, , a constant.
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GASES, , KINETIC THEORY, , 543, , It follows, therefore, that if the observed pressure be P for an, actual gas, it would be (P+a/K 2 ) for an ideal gas, and, therefore, we, s= RT., must substitute (/>+.#/ K 2) for P in the relation,, , PV, , Thus, substituting the values, (K b) for F,and (P+a/F ) for P, 4n the perfect gas equation PV, RT, we have, *r., (P+alV*)(V-b), This is known as Van der Waal's equation, and, as stated above,, is more in conformity with experimental results, but is still far from, representing the behaviour of gases accurately, and has its own, 2, , =, , =, , defects., , 273., , Mean, , of a molecule, , is, , sive collisions.*, , Free Path of a Maleculea, , The following, , is, , The mean free path, , (A,)., , the average distance covered by, , a simple, , it, , way, , between two succesof calculating its, , approximate value., Let us assume that only the particular molecule that we are, considering is in motion, all the others being at rest. This moving mole, ctile will, then, naturally collide with all those molecules whose, cen^reg, happen to lie within its sphere of influence, (see p age 439). If the, c, , radius of the sphere of influence of the molecule be r, then, all ttese, molecules lie in a sphere of radius r, described about this molecultFas, centre. If, therefore, c be the r.m.s. velocity of the molecule through, the gas, it will, in one second, collide with all the molecules lying in, the region traversed by its sphere of influence as it covers a distance, c, Now, the region thus traversed in one second is, obviously, a, 2, cylinder of length c, and an area of cross- section Tir i.e., of a volume, 2, 7ir .c., Therefore, if n be the number of molecules per c.c., there will, be 7tr*.cn number of molecules enclosed within this cylinder, and, 2, The, hence, the number of collisions in one second will be Trr .cn., 2, average time between collisions is, thus, equal to 1/7T en, and, therefore, average distance covered by a molecule between successive collisions is equal to velocity x this time, i.e.,, ., , ,, , =, , Or,, , ex, , l/nr, , 2, , .c/f, , =, , 2, , l/7rr, , .rt.,, , 2, the mean free path of a molecule, ft = l/7rr .n., m, of, one, the, mass per unit, be the mass, molecule,, Further, if, , volume, or density, , p,, , of the gas, is equal to mass of one molecule into, unit volume, mxn., , number of molecules per, ^, , =, , =, , mlmn.irr*/f= w/7rr, , 2, , p,, , [v m.n, , =, , p., , inversely proportional to p, the density of the gas; and,, since density varies directly as the pressure of the gas, J\ varies, inversely as the pressure of the gas, and directly as its absolute, i.e., A. is, , temperature., , To have an idea as to the magnitude of X, it might be mentioned that, at ordinary pressures, its value is of the order of 10~ 5 cms.,, but at low pressures, such as 10~4 cms. of Hg column, (as in electric, glow lamps etc.), its value ranges between 5 to 10 cms., In the above discussion, we have made the simplifying assumption that all molecules but the one under consideration, are at rest., *The exact nature of these collisions is not yet known, and it is not quite, whether molecules come into actual contact or whether they recede away, from each other, when at a distance, close to each other., :lear
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PROPERTIES OF MATTER, , Maxwell has taken into consideration the motion of all the molecules, and has shown, by a more rigorous treatment that A ==, which gives a smaller value of A than the one obtained above., 271., The viscosity of a gas is just a, Viscosity of Gases., mechanical property of it, and its viscosity coefficient may be, defined in precisely the same manner as that of a liquid (see Chapter, XII, page 42))., , Suppose we have a gas, flowing from left to right, over a solid, horizontal surface with a velocity which is very small compared with, the velocity of its molecules. This, velocity will be the same everywhere in a plane XOY, (Fig. 320),, parallel to the horizontal surface,, but will increase upward in tho, direction of the z-axis, being the, least, (/., zero) for the layers in, contact with the solid surface,, z, and increasing with the distance, of the layer from it, i.e., there, ._, will be a velocity-gradient dv/dz, Fig. 320., along the z-axis., , __, , Now, we may imagine the molecules of the gas to be divided up, into three distinct parts, moving parallel to the three mutually perpen, so that, the average, dicular axes, x, 7, and z, in either direction, number of molecules moving in one direction along any one axis will, ;, , be one-sixth of the total cumber of molecules in the gas., Consider an area A, parallel to the plane XOY. Let K be the, Let there be two other layers B, velocity of the gas in this plane., and C, parallel to A* above and below it respectively, each at a distance from it equal to the mean free path of the molecules, so that the, molecules from it, moving normally upwards to B or downwards to, (7, do so without any collision., Then, clearly, velocity of the molecules in the layer B is, F+jfrfv/rfz, and, that of the molecules, , in the layer, , C, , is, , equal to, , V-\dvldz., , moving about indiscriminately, a continual interchange of molecules, Since the molecules from B, crossbetween the two layers B and C., have a velocity (F+A^v/flfe), the forward, ing A downwards,, momentum carried by them per unit area of A is equal to mass oj, molecules x (K+ h-dv/dz)., , Now, due, , in, , all, , to molecules of the gas, , directions,, , there, , is, , If n be the number of molecules per unit volume of the gas, and, and w, the velocity and mass of each molecule respectively, the, number of molecules crossing unit area of A downward, in unit limp, is equal to ;ic/6, and their mass = n.c w/6., So that, the forward momentum carried by them downward, j, unit area of A, per unit time is, , c, , -~~, m, because n.m, , =, , p,, , dv, ., , ~-, , 6, , the density of the gas in grams per, , c.c.
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KINETIC THEOBY, , GASES, This, then, is the, U* the direction of flow,, , momentum, by the, , 546, , lost per unit area, per unit time,, , moving layer 5., the, momentum, carried, upward per unit area of A per, Similarly,, unit time, in the direction of flow, from the slower moving layer (7, faster, , y, , n.c, , =, , n-.m, , 6, , /.,, [, , K, , ., , dv \, , A.-v-, , ), , dz ), , V, , =, , dv \, KA. dz, ,-, , P.C /.,, , -I, , ^, , ,, , v, , ],, , 6 \, , and, , ., , this, , J, , ., , A1, , is,, , therefore,, , tho, , momentum, , gained per unit time by the layer B., Thus, the net momentum lost by layer B above A, per second,, , '.'("-), And, cleirly, the same amount of momentum is being gained by, the layer below A., Thus, the layer above A tends to accelerate its, motion, and that below A tends to retard it so that, the backwarddragging force acting per unit area on it is equal to \ p.c..?y dvfdz., This must be equal to the tangential force acting per unit area oj, ;, , the layer A,, the gas., , i.e.,, , =, , y.dv/dz,, , where y, , ^dv, , is, , _, , the coefficient of viscosity, , *oj, , dv, , -J-P-"*"^', v = f p.c.7\., , *'dz, Or,, , Since, at a constant temperature, p increases with pressure and, X decreases in the saim ratio, v is quite independent of the pressure,, This fact is amply borne, provided the temperature remains constant., out by experiment and leiuh powerful support to the kinetic theory, of gases, Farther, ^ is proportional to c, the molecular velocity, and,, therefore, to the square root of the absolute temperature of the gas., This result is not so well borne out by actual experiment and is only, approximately true., 272., Production of Low Pressure Exhaust Pumps., In the, present-day staggering development of Science, the technique ot, producing high vacua is of the utmost importance., Apart from its, well known use in radio and X-ray equipment etc we owe our initiation into the comparatively new realm of atomic physics to the welcome development of exhaust or vacuum pumps and other methods, for the production of high \racui, for it has helped us to study, the behaviour of atoms and molecules under low-pressure conditions., And this in turn, has had the reciprocal effect of enabling; us to make, further improvements in our exhaust pumps and high vacuum, technique, in general, with the result that we can today produce as, low a pressure as 10~ 9 mm., ,, , And, simultaneously with the development of these high vacuum, must go the development of delicate gauges, ttf enable us to, asure the very low pressures proclucdd by them., We shall now, jceed, therefore, to study these twin-devices., ., , t, , '^ps, , 273., ith, , any, , Exhaust Pumps Their Characteristics., Before dealing, types of pumps, it will be worthwhile to understand, , specific
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546, , PKOPBKTIES OF MATTER, , the characteristics of a good vacuum pump., ; (/'/) the degree of attainable vacuum, , pressure, , These are, ;, , and, , (///), , exhaust, (/) the, the speed of the, , Let us take each, in turn,, , pump., , In any vacuum pump, there is an inlet, fine or, (/) The Exhaust Pressure., intake side, from which the gas or vapour from the vessel to be exhausted, is drawn into the, pump, and an outlet or exhaust side, from which it is expelled, out., As its very name indicates, the exhaust pressure is the pressure on the ex, haust side of the pump, and may be just atmospheric or much lower than it,, varying from pump to pump. But, generally speaking, the higher the vacuum, desired to be produced on the fine or the intake side of the pump, (i-e, in the vessel connected to it), the smaller mui,t be the exhaust pressure or, *, the rough vacuum\ as it is alternatively called, on its outlet side., For, the recognised procedure in high- vacuum technique today is to first reduce the, pressure from atmospheric to a small fraction of it, say to *1 mm. or so, i.e.. to, create a 'fore-vacuum', by means of an ordinary pump, here called the auxiliary,, the rough or the 'backing pump*- This fore-vacuum or backing pressure, as it is, also termed, is then reduced further from 10~ 4 to 10~ 7 mm. by means of a suitable fine or high vacuum pump. For this purpose, the backing and the high- vacuum, pumps are arranged in series or tandem, so that the gas or vapour from the vessel, to be exhausted is drawn in at the inlet of the latter and expelled at its outlet, into the fore-vacuum of the former, which, then finally expels it out into the, ,, , atmosphere., (//) The Degree of Attainable Vacuum., By this we understand the lower, limit of the pressure that it is possible to obtain in the vessel, connected to the, pump. This depends to a very large extent on the exhaust pressure- For, if it be, very low, it may result in the passage of the gas or vapour in the reverse direction i.e., in its leakage from the exhaust to the intake side of the pump., , Now, theoretically speaking, there is no lower limit to the attainable, pressure in the case of a diffusion-condensation pump, but, in a molecular pump, a, The, definite limit i* sst by the constant ratio it bears to the exhaust pressure., limit may, however, in general, be considerably extended by using connecting, tubes of wide bores in-between the vessel to be exhausted and the pump,, as it greatly minimises the resistance to the flow of the gas or the vapour from, the former 'to the latter., The speed of a pump may, in a general way,, (///) The Speed of the Pump., be defined as the relative rate of reduction of pressure in a given volume., Thus,, , if PO, , bs the limiting value of the attainable pressure, with the help, , pump, p, the pressure at ai instant t in the vessel of volume V, connected to it, and 5, the speed of the pump at Ms pressure, the rate of reducof a given, , t, , tion of pressure in the vessel,, , we have, , lence,, , i.e.,, , dp/dt,, , is, , dpl(p~pj), , given by the relation,, , ~, , S.dtfV., , ..., , (//], , and p a be the values of the pressure in the vessel at instants, p, aad fj respectively, we have, by integrating expression (u) for the limits p, - Pi, t = ff, t, l and p, , So, , that, if j^, , lt, , ,, , swhich, when p 9, , is, , jo*, , ,, , comparatively negligibly low, reduces to, , 5, , -, , K.log,0>i//>s)/('*-'i)-, , and gives us a definition of 5, the, This is known as Gaede* equation, - e and (tf-tj 1 sec. t we have 5 speed of the pump. For, if pjp,, ;, , irinsic, , ........., , ., , =, , in, , V, , *As, equation enables us to determine the interva, the rush or 'surge' of th<, Cs-fi). taken by the vacuum system to recover from, to /> 4 ,-the working pressure required by the systen, gas, waich raises tbe pressure, will be readily seen, this, , being p t, , .
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KINETIC THEORY, , GASES, , 547, , Thtts, the Intrinsic speed of a pump may be defined as the volume in which it can reduce the pressure to 1/eth (which comes to about 36-79%) of its instantaneous value, in I, , = 2-17828)., = 0,, P, Putting, taking the attainable vacuum to be perfect, we have,, = S-p/V, whence, S *= ----- -. This gives the speed of ex(/) aboveft//?///, , second,, , e, , (*, , ', , from, , haust of the, , So, , pump, or the pumping speed, , E, as defined by, , Langmulr, , -", p dt, , .hat,, ', , Now, , .., , (iv), , dv be the volume of the gas or vapour, measured at pressure p, extracted in the time-interval dt, from the volume K, we have, if, , pv = (pdp)(V+dv),, ['.-pressure decreases., whence, neglecting the product dp.dv, compared with the other terms, we have, dv, j, , and, therefore,, O-i,, , =B, , y.dpjp^, , d\, , ,, , <, , i, , V, , dp, , p, , dt, , dt, , from relations, , (/v), , and, , (v),, , ., , we have, , E=, , ., , (v)', , ., , ,, , dv/<://., , Thus,, of exhaust of a pump may be defined as the rate of change of, volume of the gat or vapour in the vessel at any given instant, the measurement of, volume being effected at the pressure attained by the pump at that \ery instant., Substituting the value of dpldt from relation (/) above in expression (iv), the speed, , for, , ,, , we have, , Or,, This relation tells us that, in the beginning when the pump starts working,, p is very much greater than/? so that p lp is practically zero, and, therefore, E is, almost equal to S i.e., in the beginning, the pumping speed of the pump is pracBut as p is progressively reduced and, tically equal to its intrinsic speed., In, approaches p Q E gradually decreases, and .finally becomes zero when p, /?., other words, a pump loses all its pumping speed at the lowest^ attainable pressure. It, is, therefore, important to design pumps not only with a view to producing high, vacuum, but also with a view to having as high a pumping speed as possible, at, ,, , t, , ,, , all, , pressures., , And, since the pumping speed (E) is found to depend not only upon its, and the lower limit of the attainable pressure (/? ), but also, , intrinsic speed (S), , to its flow, it follows that the wide bores of the connecting, tubes (referred to above) also help achieve this end., , upon the resistance, , Different Types of Pumps., The following shows at a, 274., glance the classification of the different types of exhaust or vacuum, , pumps., Exhaust Pumps, , Oil, , Pumps, , Pumps, , Pumps, (sec, , *~, Piston type, Solid piston, , Rotary type, 276), (see, , pump,, , produces low pressures, 1, only up to 10- mma, backing, pump. Familiar examples Common Air, used, , as, :, , Pump, Pump, , and, , Geryk, , Piston type, , Liquid (mercury), piston pump. Exhausts down, 2xlO" 5 mm-,, , slow, in, , to, , but, and tedious, , action., , Now,, , only in limited useExamples, Toepler, and Springel Putnps, :, , I, Diffusion, , Molecular, , Mercury, , Pumps, , 277), , (see, , 278), , Rotary type, Designed by Gaede,, needs a fore-vacuum, of about 1 mm. for, Slow, in, working., action, but can produce low pressures, , down, , to, , 10~*, , mm-, , extensively used, in the laboratory., Still
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PROPERTIES OF MATTER, , We shall now deal in detail with some of the simple and important types of pumps., 275. The Common Air Pump., It consists of a receiver plate P, connectec, to a cylinder C, through a tube bent twice at, right angles, as shown in Fig. 321, *, The cylinder is fitted with a piston, and botr, the cylinder and the piston carry valves, V, _ x, and F 2 respectively, such that they open onl), upwards. The vessel F to be exhausted h, ,.., .,, /, \, I, placed over the receiver plate in the mannei, |mlr^, I, I, shown., ,, , \, , I, , To start with, the piston is moved up tc, the top of the cylinder, from its initial position, at the bottom, when the valve F 2 remains, closed, due to the atmospheric pressure on it,, and the valve V t is forced open due to the, , 321., , pressure of the air or gis in the vessel F; so that, it comes and collects in, cylinder C. The piston is then moved down- The valve V l now remains closed, due to the increased pressure on it, and the valve V z is thrown open by the gas in, the cylinder, which thus escapes out into the atmosphere-, , The operation, , is, , repeated a, , number of times, , ;, , each time the gas comes, , collects in the cylinder daring the upward stroke of the piston and is forced, out during the downward stroke so thaf, after some time, there is a fairly good, vacuum produced in the vessel F-, , and, , ;, , This pump is unable to give a high vacuum, because of the pressure of the, residual gas or air in the vessel being unable to force the valve V l open and get, into the cylinder., , That complete vacuum cannot be created by, mathemetically as follows, , pump may, , this, , be shown, , :, , Let V c cs. be the volume of the vessel Fand the tube up to the bottom, of C, and v, thac of the cylinder. Then, during the first upward stroke, the, volume V of the gas expands to (V+v) c.cs And, since during the downward, stroke of the piston, a volume v of the gas, (i e., equal to thai of the cylinder),, i <?.,, is swept oat, the volum: of the gas left behind is V c-cs, K/iK-f v) of the, original volume (K + v) c cs., During the next upward stioke, this volume again, expands to (V \-v) c.cs anJ, again, during the downward stroke, vc.cs. is forced, out, leaving behind V\(V -f v) of the volume left after the first stroke, or, VI(V + v) of VI(V + v) of the original volume (V f v) c.ci i.e (K/l/4 v)* of the, original volume., ,, , ,, , ,, , ,, , Similarly, after the third stroke, the volume of air or gas left behind will, b^ (V IV \-vi* of thj original volume, and, therefore, after n strokes, the volume, n, left behind will be (K/K-fi>) of ths original volume (V i-v)., , Now, it is clear that this expression, (V\V, great the value of n., , -f-v), , n can never be, zero,, ,, , however, , In other words, whatever the number of strokes given, there will always be, gas left behind in the vessel, and thus there can be no perfect vacuum, created inside it., , some, , air or, , As a matter of fact, the pressure can hardly be reduced below 1 cm. of, mercury column with the help of this pump, due partly to the inability of the, gas to open the valve V,, and partly to leakage and the presence of moisture in, the vessel or receiver to be exhausted., For obtaining low pressures, therefore,, other types of pumps are uged, the Rotary Oil Pumps being the mo e suitable, for the purpose., , 276. Rotary Oil Pumps., Originally devised by Gaede, these, are of two types, viz., (/) the rotary vane oil pump, and (//') the, stationary-vane oil pump. The principle underlying both is, however,, the same, a massive cylindrical shaft or *rotor\ revolving eccentrically inside a hollow stout steel cylinder, or 'stator* compressing the, a 'non-regas or vapour entering it, and finally ejecting it, , through
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549, , EXHAUST PUMPS, , turn' outlet valve, the whole pump being kept immersed in oil,, which serves a three-fold purpose, (i) providing automatic lubrication,, preventing leakage of gas or vapour into the high vacuum created,, (//), and (iii) making for efficient cooling of the pump. Let us study each, , typo in a, , little, , more, , detail., , Rotary-vane Oil Pump, or the Gaede Rotary Oil Pump., the pump are shown diagrammatically in Fig. 322,, , L^The, , The mam^arjbs of, , C is the hollqw, cylindrical steel chamber, or stator' and S, the stout and massive cylindrical shaft or 'rotor\ rotating eccentrically (by, where, l, , means of an electric motor) such that it is, always in contact with the stator at some peripheral point, such an P., , A, , cut diametrically, right across the, vanes, slid into it, which are, iK)t only kept apart from each other, but also, pressed against the walls of the stator by means, of one or more springs in between them, thus, dividing the space between the stator and the rot, slot,, , rotor, carries, , two, , Fig. 322., >-, , into, , two separate, , compartments., , and the stator remain in, and an outlet port O, the latter being fitted with a spring- operated valve V. The whole, pump is kept immersed in oil for the reasons explained above., As the rotor rotates in the direction shown, the space between, the rotor and the stator, on the inlet side, goes on increasing, while, that between the rotor and the outlet side of the stator goes on, to /, decreasing, so that the gas or vapour from the vessel connected, in the latter, gets, and, the, into, drawn, that,, former,, continually, progressively compressed, until when its pressure becomes sufficiently, The, high, it forces open valve V and escapes out of the outlet O., 3, , On either side of P, where the rotor, contact, the stator is provided with an inlet, , t, , (/), , iiS, , mm. is, process goes on repeating itself until a pressure as low as lQ~, A, the, to, connected, vessel, in, the, self-sealing, special, produced, pump., oil- valve prevents the gas or vapour from being sucked back into the, exhausted vessel, even when the pump stops working. No forevacuum is required for the working of this pump, and it can, therefore, be used directly from the atmospheric pressure., 2., , consists, , The Rotary vane, of, , or, , the, , Oil, C, inside which, , Hyvac Rotary, , a stout outer cylinder, , (III), , (it), , Fig. 32J., , It, Pump., mounted, , is, , (iv)
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550, , rttUrJHiKTlJilS Uif, , MATTJfiit, , eccentrically, a cylinder jR, called the rotor, (Fig. 323). Kept pressed, against the rotor, with the help of a spring $, is a partition, called, the vane V> which keeps the gas or air, already inside the cylinder,, apart from the fresh in coming gas or air. The outer cylinder is, provided with an inlet tube /, \vhich is connected to the vessel to, be exhausted, and an outlet tube (9, which is provided with a valve,, opening outwards. To prevent any leakage, the whole pump is, Immersed in oil, as shown in Pig. 323 (/), but a special type of valve, prevents the oil from getting into the vessel being exhausted, when, the pump is stopped., The rotor is driven at a very high speed by, means of a separate electric motor in the direction shown by the, arrow heads., , Fig. (i) shows the condition to start with, when the inlet tube, connected to the vessel to be exhausted, and when the gas or air, from the vessel has just been admitted into the space in-between, is, , the cylinder and the rotor R., Fig. (ii) shows the condition when the rotor has started, Fresh, rotating eccentrically, and the gas or air is being compressed., gas or air comes into the cylinder through the inlet tube i, behind, the rotor, and m kept apart from that already present by the vane, F, as explained above., Fig. (///) shows the process of compression, taken a step further., Fig. (/v) shows the final stage of compression, when, due to, increased pressure of the gas in (7, the valve at the mouth of the, outlet tube O is forced open, and the gas is expelled out., , The gas or air behind the rotor is similarly compressed arid, forced out and the cycle is repeated, until a high vacuum is produced, in the vessel, connected to it., In practice, the pumping system consists of two such units, in, with each other and mounted side by side, worked by the, same motor. The first unit works directly from the atmosphere, as, explained above, and the second then works from the fore-vacuum, created by it., The maximum speed of working attainable is about, 6 litres per minute and the vacuum obtainable, about 10~ 3 mm., as, series, , mentioned already., It will be readily seen that if a vessel be connected to the outlet, tube 0, the gas or air will be compressed into it, and, therefore, this, pump can also be used as a compression pump., Molecular Pumps. These too are the result of the labours, 277., of Gaede> together with Langmuir, and are based on the principle, that if there be a rapidly rotating surface (called the rotor) very, close or adjacent to a stationary one (ealled the s tat or), the space or, clearance between the two being as srjiall as 03 mm., it exerts, due to, viscosity, a dragging force, in the direction of its own motion, on, the molecules of the gas or vapour in that space. This is so because,, at low pressures, the mean free path (A,) of the molecules of the gas, or vapour is greater than the linear dimensions of this small annular, gap between the rotor and the stator, so that there are too few, collisions between molecules and molecules, compared with those, between molecules and the walls of this gap, and Knudsen has, shown that when such is the case, the molecules acquire the 'drift
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EXFIAUST PUMPS, , 551, , velocity' of the rotating surface they impinge upon and rebound from,, and that this velocity is not altered by any subsequent collisions, , amongst themselves., Unlike the oil pumps, which can start working straightaway, from the atmospheric pressure, down to a pressure of 10" 1 to 10~ s, mm., the molecular pumps (as also the diffusion pumps) operate, only from a reduced pressure or a fore-vacuum. They are, therefore, always used in series with a backing pump, connected to the, exhaust port, which creates the necessary fore-vacuum or 'rough*, , vacuum to receive the gas or vapour driven into it by the rotor,, there being a continuous unbroken communication between the receiver, and the rough vacuum, in contrast with the other types of pumps*,, where there is only an intermittent or interrupted communication, between the two, a solid or a liquid piston first separating out a, part of the gas or vapour in the receiver and then putting it into, communication with the atmosphere or the rough vacuum, into which, it is, , expelled., , Fig. 324 shows a diagrammatic representation of the pump,, where A represents the rotor, revolving roundf about an axis through, its centre, inside and closely adjacent to the walls, of the hermetically sealed shell or stator 0. There, are an inlet and an outlet in the stator at P and Q, with a slot in-between the two,, respectively,, where the annular gap is consequently greater, than at all other points,, , As the rotor revolves in the direction shown,, drags with it the gas or vapour from the inlet, port P to the outlet port Q, thus creating a pressure difference between the two, as indicated by, the manometer M. This pressure difference cannot possibly be duo to the viscosity of the gas, alone for, at ordinary pressure, the viscosity of a, Fig. 324, gas is found to be quite independent of the pressure., Indeed, Gaede, has shown that the pressure difference (p l p 2 ) between P and Q is, given by the relation,, it, , ;, , *>i, , where ^, , P*, , __, -, , 6/w, *', , fa, , ', , the viscosity of the gas in question, z/, the speed of the, //, the length and radial depth of the, slot between P, and Q. Thus, at ordinary pressures, since *? undergoes ro, change,, the pressure difference (p l p 2 ) remains con>tant for a constant, speed, (M) of the rotor., rotor, , is, , and, , /, , and, , At low pressures, on the other hand, it has been shown by, Gaede that it is the pressure ratio pjp 2 (and not the pressure difference), that remains constant for a constant speed of the rotor, and which is, also quite independent of the fore vacuum., This pressure ratio is, cw, where c is a constant, depending, given by the relation Pi!p z = e, ,, , *Bxcept the diffusion pumps., tThe speed of its rotation should never be, minute., , less, , than 5000 revolutions per
Page 554 :
PBOPEBT1BS OF MATTER, , 552, , upon both the nature of the gas or vapour and the dimensions o, the, , slot., , In Gaede's own form of the pump, there are a set of twelve slots, or grooves along the circumference of the rotor 9 their depths decreasing progressively from about *6 cm. in the inner to about "15 cm. in, outer section, the sections being all connected in series. Into these, rotor-grooves fit projections from the stator, the clearance between, the two being '03 mm. and the gas or vapour is swept along thin, small clearance. The arrangement of the slots is such that the, pressure has its lowest value at the centre and goes on gradually, increasing as we procesd outwards to the ends, where we have the, backing pump connected., In the Hoi week type of pump, the working is on similar lines,, but the sl^ts are made in the stator, with no corresponding projections, on the rotor, and the clearance between the two can here be reduced, to a figure even lower than -03 mm., The Jow pressure obtained by, these pumps is conditioned by (/) the speed of the rotor and (//) the, fore-vacuum at which they are worked. Thus, for example, with, a fore-vacuum of about 2 mm. and with a rotor- speed of 10,000 revolutions per minute, the pressure ma;y be reduced by Gaede's pump, down to the figure of 10~ 6 mm. within a matter of minutes. Hoi week, even succeeded in evacuating nitrogen gas down to a pressure of, 10 7 mm. with the help of his pump, with a rough vacuum of only, 15 mm. and a rotor-speed of 4500 rev. per mt., Such low pressures, are, however, welJ-nigh impossible to attain when there are vapours, present., , The one serious drawback of these pumps is the recurrence of, mechanical trouble, due to the small clearance between the rotor, and the stator. And then, while they can easily deal with gases and, vapours, slowly vaporising substances, like mercury, and traces of, grease etc., give a lot of trouble., 278, Diffusion- Condensation Pump., The inter-diffusion of one, gas intc^jMlother has been used to create rapid vacuum in vessels. The, method was first used by Gaede in the year 1815, and depends, upon the principle that, in a mixture of gases, the diffusion of a gas, takes place from a region, where its concentration is great tr to the one,, where i s concentration is smaller, or, in other words, from a region, where its partial pressure is higher, tojhe one, where it is lower, irresThe action of the, pective of the total pressure in the two regions., pump will be clear from the diagrammatic representation of it in, , ^^, , f, , Fig. 325., , F to be exhausted of air, connected by means of a tube, C to a wider vertical tube AB, through, which a regular stream of air-free liquid, vapour is maintained in a direction, shown, the tube C being kept at a, low temperature, by circulating cold, water round it., Now, clearly, the concentration or, the pressure of air, is greater at C than, The bulb, , or gas, , 325, , *, , is
Page 555 :
EXHAUST PUMPS, , 553, , at the othfjr end, because the liquid vapour being air-free,, there is little or no air here and, therefore, diffusion of air, takes place from C into AB, where it is swept out by the stream, of the liquid vapour, so that, again, the concentration and partial, pressure of air are greater at C than at the end opposite and hence, more air diffuses from C into AB. This goes on so long as the, *, pressure of air at C is greater than its partial pressure in A B., , The liquid vapour also tends to diffuse from AB, where its concentration (and, therefore, its pressure) ig greater, towards C, where, itw concentration and pressure are lower, thus driving the air diffusing, from G into AB backwards, bub this is prevented by (/) making the*, aperture of C, opening into AB, narrow, its dimensions being smaller, than the mean free path of the molecules of the gas or air in AB,, so that there are much fewer vapour- gas or vapour-air molecular, collisions and the diffusion velocity of mecury vapour towards 'C, is smaller than the diffusion velocity of air from C into B, and (//'), condensing the mercury vapour by cooling the tube C, thus not only, preventing it from proceeding further and entering the vessel F, but, also reducing the residual vapour pressure there to less than 10~ 3, mm., thereby ensuring an uninterrupted diffusion of the gas from C, into AB, The condensed mercury is then conveyed back to the boiler,, (not shown), to be used over again., This pump can, however, be used successfully only when the, pressure in the receiver or vessel F, is about or less than 1 cm. of, mercury. The pressure in the vessel is, therefore, first reduced to, this value, or this fore-vacuum created, by means of a rotary vacuum, pump, which can be connected to the vessel through the inside lube, T the maximum effect being obtained when the pressure of the mercury, vapour is just above the fore-vacuum, thus produced,, t, , Such pumps may be used in series, each pump, in turn, carrying, the evacuation a step further than the last, and thus a pressure, as low as a millionth of a centimetre of mercury column can be, rapidly attained, although, theoretically speaking, there can be no, limit to the vacuum produced by them., For, under the ideal conditions, mentioned above, there would be a continuous diffusion of, the air or gas from C into AB, until the air or gas pressure there is, reduced to zero., Since diffusion as well as condensation both play their part in, the working of these pumps, they are referred to variously as diffusion-condensation pumps, diffusion pumps or even condesation pumps., , The vacuum obtained with the help of these pumps, 8, , is of a very, or less, with a fore-vacuum of lO" 1 mm.,, and, they are, therefore, being increasingly used for the evacuation of, X-ray tubes and wireless valves and for such other industrial, , high ordcr,--10~, , mm., , ., , purposes., , There arc only two main drawbacks of these pumps,, (i) they ha\e a comparatively slow exhaust speed, and, (ii), , viz.,, , fiey need a rather vigilant regulation of the vapour-temperature.
Page 556 :
554, , PBOPBBTIBS OF MATTER, , A, , simple form of diffusion pump, (designed by Waran), is shown, It consists of a conical glass vessel A, containing, mercury, which is heated, so that the mercury, vapour passes up the tube B, which ends in a, nozzle, into a wider tube C, surrounded with a, cold water jacket., An inlet tube / connects the, tube C to the vessel to be exhausted, through, a Uquid-air-trap, (not shown). The tube B is, lagged with asbestos to prevent any condensation, of mercury vapour in it., , Fig. 326., , As the mercury vapour issues out of the, nozzle, it carries along with it the air, /diffusing, into C through (/) from the vessel to be exhausted, the diffusion of the mercury vapour, into the vessel being prevented by its almost, immediate condensation on entering C. The air, then discharged through the outlet tube and, the mercury vapour collected in the bend D, Fig. 326., Lelow Cand returned back to A. The process goes on, until a, high, vacuum, of the order of about 10~ 5 mm. of mercury is created in the, A still higher vacuum, of the order of, vessel, connected to /., about 1C- 7 mm. of mercury can be produced with the, help of (he, process of absorption, (see, 279, I) i.e., by placing in the vessel some, loos? coconut charcoal, recently heated under reduced, pressure, whioh., on being cooled externally, with liquid air, absorbs the, gas and the, is, , vapour., N.B., , The modern, , trend is to replace mercury with other liquids, like, Butyl Phthalate or even ten per cent Paraffin with Butyl Phthalate,, 6, as, low, a, baring, for mercury vapour, if it does, vapour pressure as 1Q- mm., once get into the vacuum system, becomes a real source of trouble., , Apiezon, , ,, , oils,, , ;, , 279., Other Methods of Producing Vacua. As we have seen, above, the lowest pressure attainable with the help of a, pumping, device is 10- 6 mm. To produce still lower, pressures, other devices, have to be used. The following is a very brief, description of these, 1, The Sorption Process. This consists in connecting the, to be, :, , system, evacuated to a tube containing some freshly heated coconut charcoal, (preferably, under reduced pressure) and surrounded by liquid air, when, gases, like carbon, dioxide, nitrogen and ammonia, are absorbed by the charcoal., To remove, Hydrogen, palladium black may be used in place of coconut charcoal, The, thus, is, as, low as 10-* mm., which may be even further reduced, obtained,, pressure,, it the method be used in, conjunct ion with a backing pump., 2., The Chemical Process. Known as the chemical process, of 'flashing it, consists in suddenly vaporizing a metal, like, magnesium or calcium, in a vessel, in communication with the, desired, to be evacuated, when most of the, system, vaporized metal condenses back on the walls of the vessel, the rest, forming, with the gases present, a compound of, negligibly small vapour pressure, thus', reducing the pressure in the system to a very large extent., Usually, the vessel, used is a glass bulb, properly fitted with a, tungsten filament, on which the, cnosen metal is placed. A momentary, large current is then passed through the, filament, when the metal burns, put with a flash and gets vaporized, the condenof, it, sing part, forming a bright mirror like deposit on the inner side of the bulb, nis is the method, largely employed in evacuating radio valves., 3., The Thermal Process. This consists in the removal of, gases, like Nitrogen, which disappear slowly in the presence of a, glowing or incandescent, tungsten filament, As can be, easily understood this process plays an important, i
Page 557 :
555, , EXHAUST PUMPS, , part in maintaining the high degree of evacuation in the ordinary incandescent, electric lamps. Langmuir is indisputably the pioneer in this branch of wgn, , vacuum technique., 4. The Electrical Process. This consists in ionizing the atoms or molecules, of the gas to be removed, either by means of a glow discharge or by bombardThe ions,, ing them by means of electrons, obtained by thermionic emmission., thus produced, get deposited on, or adhere to, the walls of the containing vessel,, if it be kept suitably cooled., As can be readily seen, this process finds wide, application in the production and maintenance of high order vacuum in incandescent lamps and radio valves., In actual practice, in most cases, the chemical, the thermal, the electrical processes, although each a completely independent process in itself, operate, conjointly and simultaneously, as, for example, in the case of the evacuation, of radio valves, where the first one produces tbe initial vacuum and the latter, , two augment and maintain it., And, finally, however, , effective the method or the means employed, it is, the whole, imperative for a satisfactory maintenance of a high degree vaccum that, evacuated system should be perfectly leak-proof. Hence it is that it should prebe used,, ferably consist wholly of glass or wholly of metal, but, if both needs must, they must be directly and carefully sealed together, air-tight., , 280., Measurement of Low Pressures. Manometers and Gauges, As mentioned earlier, the production of low pressures, of necessity, Such, led to the development of the proper means to measure them., 'manometers, the, under, fall, or, devices, instruments, heading, measuring, and gauges'. Since we are concerned with the detailed working, , of only the more important or useful ones, here, we shall content, ourselves with only a brief and rapid survey of the rest, not so much, with a view to studying their working in detail as to acquainting ourselves with the different principles on which such devices may possibly, be based., Here, then, are the different measuring devices at a glance., , Low, , Pressure Measuring Devices, , ----, , i, , Manometers, I, , Gauges, I, , Mercury, Manometers, , Mechanical, , Manometers, , Radiometer, Gauges, , Conductivity, , 1, , Viscosity, , Manometers, , lonization, , I, , I, , \, , Effusion, , Gauges, Gauges, Gauges, 1., Mercury Manometers. These are of two important types, viz.,, which are improved modifi(i) Differential and Optical Lever Manometers,, cations of the ordinary mercury manometers, with more sensitive methods of, observation and measurement, suitable for measurement of pressures up to, 10~ 8, , mm., (ii), , McLeod Gauge, which is a standard device of its type, based on the, Law at low pressures, suitable for accurate measurement of, down to 10~ mm. (See pages 558-60)., , validity of Boyle's, , fi, , pressures,, , Mechanical Manometers. These are based on the principle of mechanical deformation produced in a thin wall or diaphragm, due to pressure. They, are calibrated against the McLeod Gauge and their range too does not go below, The two well known manometers of this class are, 10"" 1 mm., 282), and, (i) the Bourdon Spiral Gauge, (see, 2, , (ii), , the Aneroid Barometer type., , Viscosity Manometers, The principle underlying these is that at low, in relative motion is proporpressures, the viscous drag between two surfaces, tional to P/Af, where P is the pressure of the gas, and A/, its molecular weight., 3.
Page 558 :
PROPERTIES OF MATTER, , 656, , They are of two types, viz.,, of which is Coolidge's, (/) The Damping or Decrement type, a good example, Quartz fibre Gou?e, which is suitable for the measurement of pressures, ranging, between 10" f mm. and 10~ 5 mm., a well known example of which is Langrnwr and, (//) The Molecular iype, 7, 1, Dushman's Molecular Gauge, suitable for the pressure range 1Q- mm. to 10 mm., 4., Radiometer Gauges. These are based on the measurement of the rate, of trarsference of momentum from a hot to a cold surface due to molecular, bombardment. Among gauges of this type may be mentioned, t, , (/), , (//), , Crooke*s Radiometer, which, , suitable only for qualitative work., (See Foot note- page 568), , Knudsen's Absolute Gauge* which, , McLeod Gauge in, from 10-* mm. to 10~ 7 mm., , a close rival of, , range,, , is, , 5., , Conductivity Gauges., , is, , a standard gauge of its type and, and accuracy, having a wide, , sensitiveness, 284)., (See, , its, , The underlying, , principle of these, , it, , the effect, , of pressure on the rate of transfer of heat by the process of conduction, their range, 4, 1, being comparatively small, from lO" mm. to 10~ mm. Among gauges of this type, may be mentioned, (/) The thermopile Gauges., 285, and, (//) Pirani-Hall Resistance Gauge, based on linear expansion of metallic wires or strips., These depend for their action on the variation of, lonization Gauges,, Mention may be made here of two, electrical conductivity of a gas with pressure., Found type,, gauges of this t>pe, v/z., (i) Buckley's type, and (//) Dushman and, mm. down, this latter one being suitable for measurement of pressures from 10, (Hi) Gauges,, , 6., , to the lowest attainable pressure. (///) <*-ray ionisat ion gauge (see 288;., Used only for the measurement of vapour pressures, 7., Effusion Gauges., of metals., , viz.,, , of some of these., proceed on with a detailed consideration, , Let us, , now, , 281., , Common Mercury Manometers., , (/) Open and (//") Closed, The Open Manometer., I., , limbs open, one limb being a, , These are of two types,, , It consists of a U-tube, with, , little, , both, , shorter than the other, and bent, at right angles, as shown in Fig., 327. A liquid, of suitable denupon the presssity,, , depending, , ure* to be measured, is poured, into the tube, so as to be above, the bend, and, of course, at the, , same, , level in either, , limb., , The, , shorter limb is then connected to, the gas-supply or the vessel, the, is to be measurpressure of which, Fig, g 327., rises above, or, then, limb, shorter, the, in, of, the, level, The, ed, liquid, the, as, pressure of the, falls below, that in the other limb, according, shown In figs., as, the, of, that, than, atmosphere,, gas ia lower or higher, the levels in the two limbs, of, difference, The, and, (b) respectively., (a), as follows, is then read, and the pressure of the gas calculated, be, limbs, h, and let the, two, the, in, levels, Let the difference of, in case (a) is, barometric height be H. Then, pressure^^f jthe^gas, ----That of the atmosphere by, *If lh"e mes~6ureTo" b^lneasiir^""diffelrs TFom, of level in the two, only a small amount, the difference, the, But, with a lighter liquid, like oil,, if a heavy liquid, like mercury, be used., With mercury as the liquid used, pressures, difference is quite considerable., between i to li atmospheres only can be measured correctly., ., , :, , "^^j^^J*, , ranging
Page 559 :
EXHAUST PCJMPS, (//A), and in case (b),, liquid used be mercury., , 557, , (H+h) cms. of mercury column, if the, In case, however, the liquid be oil or water,, of density p, the pressure in the two cases will be (H ft.p/13 6) and, (//+/f.p/13-6) respectively, (where 13-6 gms.jc.c. is the density of, it is, , mercury)., 2., The Closed Manometer. It is used for the measurement of, high pressures. It is just like the open manometer in construction,, but with the longer limb closed at the top, as shown,, , (Fig. 328), and containing some air, at atmospheric pressure, in the closed space above the liquid, with the, level of the liquid columns in the two limbs the same,, , AIR*, , -i, , to start with., , When the shorter limb is connected to the gas, supply, the level of the liquid column in the shorter, limb is pushed down, and that in the other pushed, Fig. 238., up, so that the air in it gets compressed. The pressure, of this enclosed air being inversely proportional to its volume, it can, be determined by noting its new volume, And, from this pressure, and the difference in the levels of the liquid columns in the two limbs,, the pressure of the gas-supply, in communication with tte shorter, limb, can be easily calculated as follows, :, , Lot original volume of the enclosed air be V c.cs., its pressure, being one atmosphere or 76 cms. of mercury column. Then, if v be, its volume, after the shorter limb is connected to the gas-supply, we, have, by Boyle's law,, , r, 70 \ K, , H<v,, , I, , L, , where His now the pressure of, the Tenclosed air, in terms of, length of mercury column,, , //=76 x K/v., whence,, Thus, knowing the original volume V and the new volume v of, the enclosed air, //, the pressure of the enclosed air can be known., If thereThis, then, is the pressure at B in the longer closed limb., fore, the difference of levels in the two limbs be A, and the liquid, used in the manometer be mercury, the pressure at A, i.e., the pressure of the gas-supply, In case the, (//-f/z) cms. of mercury column., liquid used be oil or water, of density p, we have, , =, , pressure of the gas-supply, , = (H +, , 1, , j.pO, , cms of mercury., -, , The Bourdon Gauge. For the measurewhat is called a, Bourdon Gauge is used. The principle underlying it is the same as that in the case of the, 282., , ment of, , \, , cry high pressures,, , Aneroid barometer, which, of the gauge., , is in, , fact a modification, , It consists of a tube ABC, (Fig. 329), elliptical in section, with the end A closed and the end, (7, open, so that it can be put into communica-, , tion with the gas-supply, the pressure of which, desired to be determined., , Due, , to the high pressure of the gas entering, it tends to become more circular in, section and this results in the end A of the tube, , the tube,, , Fig. 329., , is
Page 560 :
558, , PROPERTIES OF MATTER, , being forced away from C. This movement of A, in its turn, moves, the pointer P over a scale, graduated, directly in atmospheres, by, comparing its indications with a standard gauge, as in the Aneroid, barometer. The instrument is thus a direct readfng one, and can, obviously be used to determine low pressures also., , McLeod Vacuum Gauge., , An, , ordinary manometer, , is, , not, , quite suitable for the measurement of very low pressures, like those, , obtaining in incandescent electric lamps and Z-ray tubes etc., where, the pressure is as low as 10~ 5 mm., For such purposes, the McLeod, Vacuum Gauge, designed by McLeod in the year 1874, is used. The, form of the instrument, shown in Fig. 330 is a, slight modification of, the original, in main due to Gaede, and consists of a, cylindrical or, spherical bulb B, of known volume, ending above in a graduated capillary tube CA, and connected to a reservoir of mercury R and a sidetube EF, which can be put into communication with the vessel or the, pump in which the pressure is desired to bo determined. A side, capillary tube G is attached to it, as shown, whose diameter is the, same as that of CA. Its use is to counteract the, depression of the, , ^zz>, , mercury column, , G/5, , in, , CA, , due to capillarity, , ,, , being of the same diameter as CA, the, depression of the mercury column in it is, for,, , also the, , same as that, , When, , in, , CA., , the reservoir, , R, , is, , lowered until, , the mercury falls below the bend D, the, bulb B and the vessel in which the pressure, is to be measured are, put into communication with each other and the bulb is filled, with the gas, whose pressure P is to be, determined. On raising the reservoir, mercury rises into the bulb as well as the side, tube, thus cutting off EF from B, 'and the, gas enclosed in the bulb is compressed as, rises further and further up into it,, until the whole of it is forced into the capiUary tube CA. The reservoir is raised still, further, until the whole of the bulb B and ^., , mercury, , Fig 330, part of the capillary tube CA are filled with, mercury, and the mercury in the capillary tube G attached to EF,, rises up in a level with the top end A of CA., The depression of the, column CA having been compensated for, as explained above, the, difference of level between the columns 'of mercury in the two, capillary tubes CA and G, or what is the same thing, between the, end A of CA, and the top of mercury column in it (because A is at, the same height as the mercury column in G) gives straightaway the, pressure of the gas in CA. Let it be h cms., t, , Then,, , if, , V, , be the volume of the capillary tube CA and the bulb, D and v, the volume of the gas after the mercury, , to the bend, has risen into it,, , B up, , we have, , P=A.v/K,, , Or,, , whence, , Px V~h X v., , P, the pressure of, , the^as, can be easily calculated.
Page 561 :
EXHAUST PUMPS, , 559, , It will be readily seen that the greater the value of V and the, smaller that of v, the smaller the value of P that can be measured., Thus, the sensitiveness of the gauge depends upon the ratio F/v., , Now, although quite an efficient guage, so much so that, practically all other types of gauges are calibrated with reference to, the performance of the McLeod guage becomes somewhat erratic, it,, This can, however, be, in the presence of easily condewible vapours., easily remedied by introducing a liquid air trap in between the guage, side (i.e., the vessel in which the pressure is to, be determined). In fact, the liquid-air trap must be used even otherwise to prevent any mercury vapours entering the evacuated vessel., Apart from this, there are quite a few other drawbacks in the, lorm of the instrument, discussed above. TLus, for example, // is, rather inconvenient to manipulate the reservoir with such a large amount, and the mercury which remains in contact with the, of mercury in it, rubber of the flexible tube is likely to get contaminated due to the preThis latter trouble has,, sence of sulphur in the composition of rubber., in recent years, been sought to be got over by the use of a tube of, Even so, a better modification of, stainless steel in place of rubber., the gauge is the one described below., , and the high vacuum, , ;, , 284. Improved modification of McLeod Gauge In this improved, version of the gauge, the reservoir, with its attached flexible rubber, tube is dispensed with altogether., is made longer and, Instead, tube, fitted into a rubber, bung in one, mouth of a Wouljf's bottle W, so as, , D, , to dip inside mercury, contained, therein, as shown in Fig. 331. Into, the other mouth of the bottle is, fitted a side-tube, connected, N,, through a stop-cock 5, to (/) a small, soda lime tower T, and (//) a tube L,, leading to some simple form of a, backing pump. The tower T has a, packing of glass wool at either end, , to prevent any particles of sodalime getting into the gauge, and is, , connected at the top to a long capillary tube J, through a small rubber, tubing, provided with a spring-clip,, so as to enable it to be put into, communication with, or cut off from,, the outside, , air,, , as desired., , The procedure, , consists in first, , putting the WoulfFs bottle in communication with the pump, through, Fig- 331., the stop -cock, so that the whole of the mercury in the gauge comes, down into the bottle, the pressure throughout being the same as produced by the backing pump, and, of course, very much lower than, that of the atmosphere. The communication between the bottle and, the pump is now cut off and that between the former and the soda-
Page 562 :
PROPERTIES OF MATTER, , 560, , lime tower partially established by a slight rotation of 5, so that, the air from outside gradually enters the bottle, losing its moisture, during its passage through soda-lime in the tower. This results in an, increase of pressure on the surface of mercury in the Jbottle and it, being forced up into D. The gauge is then used in the manner, already explained. It will easily be seen that the labour involved, in moving the reservoir up and down for necessary adjustment of, the mercury columns in C and G, and the possibility of contamination, of mercury are both thus neatly obviated., , When, , said and done, however, the McLeod gauge still, inherent defects of being rather unwieldy in size, and its inability to give a continuous record of pressure changes in, the vessel, Then, again, the use of the liquid-air trap adversely, affects the rate of pumping and the readings obtained on the gauge, may not be truly indicative of the actual pressure inside the vessel, at any given instant., In any case, its readings in the last lap of its, 4, 5, mm. are really never quite so reliable., range, from 1()~ to 10, all, , is, , continues to have, , its, , 285. The Pirani Resistance Gauge. It is fairly well known that, whereas at high pressures, the thermal conductivity (A') of a gas is, 2, mm. of, quite independent of pressure, at preasures below 10, mercury column, when the mean free path of the gas molecules is of, the same order of magnitude as the diameter of the containing vessel,, it is, , directly proportional to the pressure (/?), oc.p, where a is a constant., , of p. Or,, , /, , .,, , A' is, , a linear function, , K=, , This fact, first made use of by Warburg, in the year 1907, for, the measurement of lo\v pressures, is really the basis of the Pirani, gauge, P.O in Fig. 332(0), which consists of a tungsten or platinum, filament (Fj, enclosed in a small detachable glass bulb (B)>* very, much similar in construction to that of the 'casje-type* incandescent, lamp and maintained at a temperature, higher than that of the, surroundings. The bulb is opan at the lower end which can be, connected to the vessel in which the pressure is to be determined., , With change in the pressure of the gas in between the filament, and the walls of the bulb, the rate of heat conduction across the ga^, change in tho temperature of the filament, Wo measure this change in the resistance, of the filament which gives the change in the thermal conductivity and, also changes, resulting in a, and hence in its resistance., , A calibration curve for tho, hence, indirectly, the pressure of the gas., gauge, is, therefore, plotted by measuring simultaneously the resistance of the filament and the pressure of the gas around it, the, former by means of a Wheastone's bridge, to which a constant potential, difference is applied, sufficient to heat the filament to a temperature, of about 120C, and the latter, by means of a Mcleod gauge. The, pressure, corresponding to any value of tho resistance of the filament,, can then be read directly on this curve., , Now, for the reason already stated, this calibration curve is a, 2, straight line, so long as the pressure of the gas is below 10~, and p no longer, but at higher pressures, the relation between, remains linear and, varies in a somewhat complicated manner with, , mm, , ,, , K, , K, , both the pressure of the gas and the temperature of the surroundings., "There are also other forms of the gauge, , in, , which the bulb, , is, , not detachable.
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EXHAUST PUMPS, , 561, , K, , &nd p can be represented by no simple, fact, the relation between, formula, ^ith the result that the calibration curve now ceases to be, straight. To tide over this difficulty, Campbell suggested that instead, of keeping the voltage across the bridge constant and measuring the, resistance of the filament, the temperature, and hence the resistance,, of the filament must be kept const int. and the potential difference,, required to be applied to tho bridge for the purpose, must be, In, , measured., Accordingly, the gauge (P.O.) is connected in one arm of the, and 2 and the, bridge, [Fig, 3 12 (#)], together with fixed resistances, variable resistance R B in the other three, (BATTERY), arms, as indicated, all these resistances, , ^, , R, , being made of an alley like 'mangcmin*, and 'mina/pha\ having an almost zero, coefficient of temperature. The potential, diffeicnre is applied to the bridge at A and, C, by means of a potential-divider, i.e.,, , through a rheostat included in the battery, circuit, so as to be varied at will, and its, value read on the voltmeter V connected, across the bridge terminals, , A and, , C, as, of the temperature of the surroundings of the filament, the bulb of the gauge is placed in, a thermostat at 0C-, , shown., , To ensure constancy, , Procedure., , (/), , tial difference across, , \, , With a known potenA and C, the bridge, , 6A$ FROM, EXHAUSTED, , p (MMS, (ft), , Fig. 332, , OF MERCURY), , +, , (/?), , balanced in the usual manner by adjusting the variable resistance, so that the deflection in the, jR 3, galvanometer is zero and there is, sufficient current through the filament of the, gaugo to raise its, temperature to about 100C. Now, every time the pressure changes,, the voltage across A and C is, adjusted to restore the balance of the, bridge and make the galvanomstor deflection zero., Then, if 9 be the, excess temperature of the filament over that of the, surroundings, the, he %t-loss along the leads L and L, assumsd small, will be, prop jrtional, to 0, say equal to, j30, where p is a constant., And, if V be the/, is, , ,, , c, , r
Page 564 :
PROPERTIES OF MATTER, , 562, , voltage applied across the bridge, the heat dissipated per second in, the filament is equal to <xF 2 where a is another constant, And,, further, if/7 be the pressure of the gas around the filament, the heat, lost per second by conduction across it is f(p), where f(p) is some, function of pressure. We, therefore, have, ,, , Now, for the same value of 6, but with p, vacuum around the tilament), if the voltaic required, the bridge be, , F, , ,, , clearly,, , Dividing relation, , (/), , aK, , 2, , by, , =, , (ic\,, , ..., , $0., , (//),, , 0,, , therefore,, , with, , to be applied to, (//), , we have, , =, , where /(30, k. a constant, which is almost quite independent of the, material and the length of the filament and varies only with the, nature of the gas., 1, , But, as we have seen, measurement of the pressure of the gas, by means of a McLeod (or any other absolute) gauge shows that,, = yp where y is also a constant So that, f(p), t, , (v*-v*)\vf, i.e.,, , (F, , F 2 )/F, , 2, , 2, , is, , krp,, , directly proportional to pressure., , we plot a graph between/? and (V* -V }\V * for, got a straight line, its inclination with the axis of p, depending rpon7 the nature of the gas in question, as is clear from, the two curves rawn by c ampbell for hydrogen and air, [Fig. 332, (b)],, where p was measured by a McLeod gauge., If,, , therefore,, , we, , the gas,, , <, , In actual practice, it becomes rather tedious to use Campand, therefore, the following simplified procedure is, , (//), , bell's, , met/1 lod, , adopted, , The bridge is first balanced with only vacuum about the filament, and then keeping the voltage across the bridge constant at this very, The balance of the, value, the gas or air is allowed into the gauge., bridge is thus naturally upset, and a current, corresponding to this, upset,, , or the "out of the balance current', as, , it is, , aptly called, passes, , through the galvanometer, the deflections of which, in terms of scale, divisions are noted for various values of pressure of the gas (as, indicated by a, , McLeod, , gauge)., , I, , jV be the number of scale divisions through which the, Then,, galvanometer needle is deflected for a pressure p of the gas surroundA graph between the two is,, ing the filament, we have N czf(p),, therefore, a straight line and gives tho required calibration curve for, the gauge, from which the pressure of the gas for any deflection in, the galvanometer can be read off straightaway., For the success of the gauge as a, Essential points of the gauge., if, , low- press re -measurer, the essential points are that (/) the material, of the filament must have a high coefficient <>j temperature, so that the, change in its resistance must be appreciable for a small change in ts, temperature It is, therefore, made of a tung*t<m or a platinum wire,, of a diameter of about -06, (n) th<* ^lament mu\t throughout be, kept taut, so that the distance between it and tho walls of the erfHos\, , \, , mm, , ;, , t
Page 565 :
563, , EXHAUST PUMPS, , heat losses along the filament(iii) the, ing bulb remains unlatered, support must be as small as possible., To ensure both these conditions (H) and (in), a poor conductor, of heat like a glass rod is used by way of support for the filament and, it is taken round glass beads, as shown, (Fig 332), with its longer, walls of the bulb on either side; and, portions equidistant from the, a high current sensitivity., have, should, the, (iv), galvanometer, ;, , Range, Utility and Drawbacks of thef Gauge. 4 The range of the, to 10~ mm. of mercury,, gauge is rather (small, viz., between 10", but because of its almost instantaneous action, it is extremely useful, , measurement of pressure fluctuations., Its chief drawbacks are that (/) it is wholly unsuitable for use, , for the, , t, , with organic vapours, as its filament gets 'poisoned' by them; (//') it is, rather much too sensitive to sudden or accidental thermal or mechanical, shocks and vibrations, which must, therefore, be avoided as far as, , In fact, as a safeguard against the latter disturbance, it is, possible., usual to provide d bhock-absorbing mounting for it, (Hi) it is not an, absolute gauge and has to be calibrated against a McLeod or some, 3, other absolute gauge; (iv) at pressures below 10~ mm. of mercury column, the heat loss occurs more by radiation than conduction-, (v) pressures, behw 1()~ 4 mm. of mercury cannot be measured with its help, with, ;, , any reasonable amount of certainty., Finally, it may as well be mentioned that, like most other, 5, 3, gauges in th^ pressure range 10~ to 10~ mm. of mercury, this too, be, which are never quite, made,, to, a, manual, some, ljustments, requires, as reliable* as mechanical or automatic ones. This has, however, been, remedied comparatively recently, (in the year 1939) by Scott by his, clever introduction of a Trio le valve hi the gauge circuit, thereby making the working of the gauge at once quick, smooth and automatic., 286. Thermocouple Gauge. It is just a variant of the Pirani Hot, Wire or Resistance Gauge, in which instead of measuring the resistance, of the filament, we measure the temperature, , JO EXHAUSTED, , of the hot junction of a thermocouple,, attached to the> filament, from the thermoThe tempeelectric e m.f developed in it., rature of the hot junction will obviously, depend upon the thermal conductivity /of, the g 4s in- between itself (or the fiUrnenM, and the walls of the containing gla- ulb, the outside of which is maintained at 0C, and which is connected at its upper opm, end <Fig. 33.'{| to the vessel in which the, Th(low) pressure is to be measured, value of K for the gas, at such 1'W pr*sureM, is as we have seen directly propor-, , \, , VESSEL, , T.C., , DO, , \, , f, , tional to its prewsure, , The fila'nent F here, 's a short, ribbon of const ant nn ai>d is heated bv a, current of upto 50 milliamperes and the hot, junction of suitable thermocouple, alumel a <timon bismuth, or any, titr, <, , TC, , -, , Fig. 333., ir, in, , stantan, chrome/attached to the mid, n-ct, , *
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PROPERTIES OF MATTER, , 564, , point of the filament, which is connected through a rheostat R to a, The thermocouple, low-voltage battery and a milli-ammeter M.A., circuit includes a sensitive galvanometer, the deflections of which yive, an indication of the thermo-electric e.m.f. developed and hence that, of the pressure of the gas., , The gauge, , is,, , as usual, calibrated against a, , the same gas in both,, , if strict, , accuracy, , is, , aimed, , McLeod gauge with, at, , 287. lonisation Gauges. Ion Nation, as we know is the process of, knocking out an electron from the outer shell of a gas atom. Amonp;, other ways, this may also be done by means of a fast moving electron, colliding against the gas atom, the process the a being spokon of as, Before an atom can thus ionise a, ionisation by electron collision., a, it, must, certain, minimum amount of energy,, however,, possess, gas,, depending upon the gas in question, and must, therefore, be accelerated through a certain minimum potential difference, calJed the ionisaThe energy thus, tion potential, V, say, for that particular gas., acquired by the electron is measured in terms of electron-volts, where, one electron volt is the energy acquired by an electron on being, accelerated through a potential difference of 1 volt., t, , ,, , Now, when an electron is thus knocked off from a gas atom,, naturally what remains behind is a positive ion, since the atom, as a, Thus, ionisation produces positive ions and, whole, is neutral., If these positive ions be collected on another, electrons (negative)., electrode, (i.e., an electrode in addition to the, positive, auxiliary, electrode or the anode and the negative electrode or the cathode), we, get a positive ion current, or an ionisation current, for a given value, This, V of the accelerating voltage above the ionisation potential V, 3, ionisation, or the positive ion, current, at low pressures, below 10~, of mercury, varies linearly with the pressure of the gas. because at such, pressures, an electron is hardly likely to collide with more than one, atom on its way from the cathode to the auxiliary electrode. Thus, in, the ordinary Triode valve, the grid* may very well act as the auxiliary, electrode, if it be given a, negative potential with respect to the, so that, any Triode valve may be used as an ionisation, filament, gauge. In order to avoid the possibility of electrical leaks between, the electrodes, however, the triodes, meant to be used as ionisation, gauges, are specially constructed with this end in view., ., , t, , mm, , ;, , Since the electrons are emitted on heating the filament or the, cathode, such a gauge is called a hot cathode ionisation gauge and the, A later, first satisfactory form of it was due to Dushman and Found., The, modification, now in common use is the one shown in Fig. 334., usual tungsten filament F is here supported on a glass rod R in the, manner indicated, with a co-axial gridG, (of tungsten or molybdenum), around it, and a nickel or silver coating on the interior of the glass, bulb, enclosing the two, acts as the plate P, with a platinum wire, *In a triode valve, the function of the filament is to emit electrons, when, heated by the current from a low voltage battery, that of the plate (which is a, metal cylinder around the filament and is ordinarily connected to the positive of, a high voltage battery) to attract them, and that of the grid, (which is a spiral, a, wire-gauge or a perforated metal cylinder in-between the filament and the plate) to, control their flow.
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EXHAUST PUMPS, , 565, , w sealed on to, , it, to enable it to be connected to the external electriIn order to prevent deposition of any metal film on it,, (from the filament or the plate) and thus cause inter-electrode leaks,, tiie glass rod is provided with* loose glass collars C, C, as shown., , cal circuit., , TO EXHAUSTED, VESSEL, , F)g. 334, , Now, the positive ions may be collected either on the grid or, on the plate, as desired, the electrical connections necessary for the, purpose being shown in Fig. 334 (a) and (b), \\it\\ the grid being given, a negative potential in the former and a positive potential in the, latter case, with respect to the filament, and with the plate connected, to the positive aiivl the negative poles of the high voltage or high, tension battery (H.T.B.) of about ll'O volts, iu the two cases respecA milliammeter M.A. is included in the plate circuit and a, tively., galvanometer G' in the grid-circuit, in the fir^t case and vice versa in, the second., , Working. (/) First Case, hi this case, the plate being at a positive potential with respect to the filament, electrons emitted by the, latter (when heated by the current from the low voltage battery, L.T.B.) are attracted towards the plate and pass through the meshes, or holes in the grid., Thus, on their way to the plate, they bring, about ionisation of the gas between the grid and the plate. The positive ions formed are collected by the grid which is at a negative potensmall ionisation current, or a positial with respect to the filament., tive ion current, thus flows through the grid- filament circuit and can be, included in the circuit, the usual, oasily read on the galvanometer G, electron current being given by the milliammeter M.A., , A, r, , ,, , Here, since the grid is at a positive and the, (ii) Second Case., plate at a negative potential with respect to the filament, the electrons, emitted by the filament or the hot cathode, are strongly attracted by, the grid (it being nearer to the filament), but quite a number of them
Page 568 :
PROPERTIES OF MATTI, get through it on account of their momentum and thus ionise the gas, in the space between the grid and the plate (in case, of course, the, accelerating voltage applied is higher than the ionisation potential, for the gas). The positive ions thus released are collected at the plate,, any electrons straying into the region being repelled back by it., The positive ion, or the ionisation, current is then read on galvanometer G' included in the plate circuit and the electron current on the, milliam meter, as before., , Of the two arrangements discussed above, the latter, i.e., the, is by far the more sensitive,, but the first one is, simpler to work with., ve plate one,, , As already indicated, the relation between the ionisation, current and pressure is a linear one only at pressures below 10' 3 mm., of mercury., If, however, the emission of electrons from the filament, be relatively small, the relation also becomes linear at higher pressures, but, then, the sensitivity of the gauge is considerably, impaired., , This gauge too is not an absolute one and has to be calibrated, against a McLeod gauge, with the same gas in it as the one, the pressures of which is to be determined by the ionisation gauge., , Once calibrated, the galvanometer G' is replaced by a micro, ammeter, graduated in pressure units., The gauge can measure much lower pressures, in the range, 10~ 3 mm. to 10~ 7 mm of mercury column. It has also other advantages over the McLeod gauge in that (/) it can be used to measure pressures of both vapours and gases and (//) being very much smaller in, size, it can be located in close proximity with the vessel being ex-, , hausted., , Among its drawbacks may be mentioned the fact that Its, manipulation is somewhat complicated and that it requires quiie a lot, of extra electrical equipment with it. Also, its sensitivity depends, upon the nature of the gas, the arrangement of its electrodes and the, electric circuit, , employed. Then, again, organic vapours 'poison' its, reduce the emission of electrons from it. As a necessary, precaution, therefore, a 'cold trap' of carbon dioxide snow or acetone, is arranged in-between the gauge and the exhausted vessel to get rid, of them. And, in case an oil diffusion pump is being used to exhaust, the vessel, some sort of a 'baffle' must be used to prevent any oil, molecules streaming back into the gauge and thus vitiate its working., And, finally, at higher pressures, the life of the gauge is shortened, as, much due to the bombardment of the filament by the ions as to the, possibility of the chemical reaction with the gas around it., filament,, , i.e.,, , 288. a-ray Ionisation Gauge. This is the latest form of an ionisation gauge, in which, as indicated by its very name, the ionisation of, the gas is brought about by means of a-particles from a radio-active, substance., And, since it is not necessary to heat the cathode here,, it, , may, , be called a cold cathode iorisation gauge., , Designed by Downing and Mellen in the year 1946, it consists of, a closed ionisation chamber C, (Fig. 335), inside an outer protective
Page 569 :
EXHAUST PUMPS, , 567, , and perforated at its top and bottom to allow free access, the gas inside it. At the bottom of the chamber] is securely, held in position a small saucershell,, , to, , 2, , shaped plaque P, I cm in area,, surface, and with its upper, made of an alloy of gold and, radium, which is in equilibrium, with its products of decay, viz.,, radon, radium A and radium B,, of which the first one is a gas., To prevent any of this gas escap,, , ing out, the upper surface of the, plaque is electrolytic ally coated, with a layer of nickel which also, serves the additional purpose of, contamination, by, preventing, , TO EXHAUSTED, mercury vapour. The losses from, VESSEL, the plaque (due to radio-active, Fig. 335., emission) are so small that the, instrument needs to be tested only once in a number of years. This, plaque thus forms a highly efficient oc-ray emitter, though with a slow, , emanating power., , The inner electrode or the grid G consists of four wires spread, or stretched out, as shown., This limits the distance to be covered, by the positive ions produced by the ionisation of the gas (by the, oc-particles), thus facilitating their 'capture' before they have time to, cover longer distances. For, in the latter case, the ions may re-unite, and thus the linear relation between the ionisation current and, pressure may no longer remain valid and the whole basis of the, The small ionisation current produced is first, gauge knocked out, amplified and then read on the microarnmeter M.A., the gauge otherwise functioning j ust like the hot cathode one discussed earlier, and, has, likewise, to be calibrated against a McLeod Gauge., relationship between ionisation current and, of, no, the, pressure, gas, longer remains valid beyond a pressure of 10 mm., of mercury, the gauge may be used (in three successive stages) to, measure pressures within a wide range from 10~ 8 mm. to 1000 mm., Further, the gauge is a continuous reading one., , Now, although the, , The one serious drawback of the gauge is that extra precautions, are necessary to work with it, if one is to save oneself from the, hazards of exposure to the radio-active substance used., 289. The Knudsen Gauge This simple and efficient gauge,, which has been used to measure the lowest pressures yet produced*,, depends upon what is called the radiometric effect. It therefore,, becomes necessary to first understind clearly whaothis 'effect' is., Radiometric Effect., At high and m>derately high pressures, a, gas behave* like a Visco is liquid and its flow through narrow tubes is, governed by PoiseuWe's law, the rate of flow being proportional to, , *Up, , 10~ 8 mm-, by Wo^drow. with a temperature difference of 100C, 9, plates- Aad, up to 5 x 12~ m;w., by Shrader and Sherwood who, , to 3 x, , between the two, , used a slight modification of, , it.
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568, , PROPERTIES OF MATTER, , the Jourth power of the radius of the tube and being limited by the, frequency of intra-molecular collisions. At eveiy low pressures, however, the mean free path of the molecules becomes greater than tho, radius of the tube, the frequency of collisions with the containing, walls preponderates over the frequency of ultra-molecular collisions, and considerations of conductivity and viscosity etc cease to be of, any consequence. The tiow of a gas, under those conditions is very, suggestively called 'molecular flow' by Knudsen, whose extensive work, on the behaviour of gases under vacuum conditions has earned its, deserved recognition in a gas, at such pressures, being known as a, 'knudsen gas.", , He assumes that a typical gas- molecule is accelerated by the, pressure gradient along the tube and that all its drift velocity is lost, when it collides against a wall for, according to him, the smoothest, walls are 'molecularly rough', having, here and there, minute projections of molecular proportions, due to the piling up of one or two, atoms above the surrounding ones, so that the directions of motion of, the molecules, before and alter striking the wall, bear no relation to, ;, , each other., , Assuming these knudsen conditions', if a molecule acquires extra, energy on rebounding from a heated surface, it will collide with the, wall a number of times before it reaches the surface, again. It follows,, therefore, from the law of conservation of momentum, that there is a, net force acting normally to the heated surface, which tor moderate, differences of temperature between ihe surface and the wall, is, directly proportional to these differences of temperature., 1, , This mechanical forc^ exerted between two surfaces, very close to, each other and maintained at a difference of temperature, is called, Since it bee ra9s manifest only when the mean, radiometric effect*., free path is longer than the distance between the ueated surface and, the wall, it is a typically Knudsen effect and must not be confused with, that physical effect of radiation we call radiation pressure the latter, being (/) quite independent of the presence or absence of any gas,, a blackened surface, and, (//) about twice as great on a polished as on, (///) of a much smaller order of magnitude., , The Gauge Proper. Knudsen very ingeniously exploited and, harnessed these radiometric forces at low pressures in devising his, absolute gwge^ which enables us to measure pressure in an evacuated, 7, 3, vessel, from 10~ mm. down to 10~ mm., by simply noting the, deflection of a cold plate suspended in the vessel, due to its bombardment by tho molecules rebounding from a nearby hot plate. If the, dimensions of the plates be very large, compared with the distance, between them, (as though we were considering only a very small, *A, , familiar example ol this effect is the Crook's radiometer, so often used, thermal radiations. It consists of a freely pivoted 'spider', fitted with, mica vanes with their planes, parallel to the vertical axis, and coated on one side, with lamp black, the other being polished, the whole instrument being arranged, inside an evacuated glass-bulb. On exposure to thermal radiation; the blackened, sides of the vanes become hotter than the polished ones and radiometric forces, come into play, making the spider rotate, blackened side moving away from the, to detect, , incident radiation., , termed 'abwlute* because, in dynes cm*., , fit is, , units,, , i.e.,, , it, , gives the pressure directly in absolute
Page 571 :
EXHAUST PUMPS, , 569, , portion of an infinite area), so that all edge-effects could be neglected,, and, further, if this distance be smaller than the mean free path of, the molecules, so that the molecules arrive at one, in exactly the same, condition in which they leave the other, the deflecting repulsive force, on the cold plate is found to be proportional to the gas pressure in, the vessel, right up to a pressure of JO" 7 mm., , shows diagcammatically the essentials of the gauge,, and P2 are two fixed plates, kept electrically heated and, arranged on opposite sides of a eold, plate A, in the form of a rectangular, picture-frame, MIS ponied (in the evacuated vessel) by means of a quartzFig. 33(5, , where, , P, , l, , M, , to enable, its, fibre carrying a mirror, deflections to he measured by the usual, , 'lamp and scale* method., Let the temperature- of P, and P ?, be TI and that of A (and hence also, that of the evacuated vessel, in uhich, it is, , suspend* d) be 7\ and, , H 8 be the number, , let, , ard, , //,, , molfiules per cubic, centimetre, travelling from P to A and, from A to P 1? wjtl rcot mean square, <>f, , ^2, '~, , l, , i, , and c 9 respe< i vely. Then, ('^, in the equilibrium state, the rate, of molecular collisions per square centimetre must be the same, we have, velocities, , sin, , -e, , And,, , if, , a be the number of molecules, , S- 336, the vessel, outside the space, A, then, since the number of molecules (lowing out, of the latter into the former and vice versa, must be the same,, , per, , c.c. in, , between Pl and, , we have, n c l -\-n 2 c 2, , ric>>, , So that,, , if, , m, , total pressure, , =, , 2n, , ~, , 2n, , c: 2, , ..., [From relation, (/'/), be the mis-; of e.icb molecule, we have, l, , between, , 1, , cL, , CJ, the., , plates, , /, , fnn, , -, , ^L^, o, , ^,, , >, , whereas pressure on the, , sidev,, , remote from, , A, , =-, , Obviously, therefore, exiw> pressure on the sid^, that, , away from, , it., , is, , given by, , Or, substituting wc 2 /2 for, , we have, excess pressure on, , ^, , AA, , n^, , *c, , ', 1, , and, , +, , for, , /J, , /?//iU 2, , vp, , }miic?, , ], , 2, , /3, , <>f, , A nearer, , ", , Ca, , -, , above., , to, , Plt, , over, , ., , relation, , 2, , (//), , above., , /nnc.c,", ', , ., , Or e \cess pressure on, ,, , n>(, , [, , t, , (/), , A, , =, , rnnl, --, , /, I, , c., , ., , J., , 1, , \, J, , Assuming the molecules striking PL and, pective temperature 7\ and T2 we have, , !,, , to taks, , up, , their res-, , ,, , [See page 537.1
Page 572 :
PROPERTIES OF MATTER, , 570, , And, thus, excess pressure on, , =, , w/ic 2 2 /3, , Now,, , />,, , 2, , A, , (\/ yf-, , 1, , ), , the pressure inside the vessel, , So that, we have excess pressure on A,, , per, , or, force, , cm. on A, say,, , sq., , If the temperature difference (7^, jT2 ) be not very large (i.e.,, not more than 250C), we may consider the flow of molecules in-between the space P^A and the rest of the vessel as a thermal transpira-, , /J1 A-1, , tion* etfect, , outside, , ;, , between the temperature, , so that, if p' be th.3 pressure, , \, 2, , PA, , inside, , (, , J, , between the, , plates,, , and, , J"2 ,, , we hive, , _ A /^+^ _ \ / 2r + ^ ^, , "V, , r2, , V, , '27-,, , *, , ', , 27,, , /1", , So that,, , And,, , /', , =, , F=, , therefore,, , ri -, , P /T,, -J, , T~\, , (-^A),, , showing that the force exerted on plate A, nature of the gas in the vessel., , Now,, , if, , 1, , is, , r From, -..(fr)[, , s, , ,, , nce, , (/''),, , above,, , f = (/ _ p), , ., , quite independent of the, , a be the area of each vertical strip of A,, force acting on each strip = F.a., , we have, , Since these forces act in opposite dirtctions on the two strips,, a couple equal to F.a.Zr, where r is the mean distance, constitute, they, of each strip from the suspension wire., , The frame is thus deflected, gmng rise to a restoring tonional, couple in the suspension wire and, therefore, comes to rest, when the, two couples just balance etch other, say, when it has deflected, through an angle, , 6., , T be the torsional couple, set up in the suspension wire, Then,, per unit twist in it (or per unit deflection of the iracue) the total, torsional couple tending to restore the frame back to its original, position is r9 and we, therefore, have, if, , ;, , T, , =, , 2F a, , r., , Or,, , F, , =, , T0/2ar., , the phenomenon of the flow of a gas from a colder to a hotter, chamber both containing the gas at a low pressure and connected to each other, by means of a capillary tubing, the flow continuing, until a pressure difference,, depending upon tne temperature difference between two chambers, is established., The ratio between the final pressures attained can be shown to be, the same as that between the square root! of ttie absolute temperatures of the two, chambers., *It is
Page 573 :
EXHAUST PUMPS, Substituting the value of, , F in, , relation, , (iv), , 571, , above,, , we have, , 2ar, 41, , Or,, , p-~, , Further, if/ be the moment of inertia of the frame about the, f, period o vibration, we have, suspension wire as axis and t, its //, 2, 2, 47T, T, whence,, 7/f, , =, , Hence, , =, , p, , -&, i, , -, , ar, , ,, , -*, , zr*, , i, , j 2, , whence p 9 the pressure of the gas, , l, , in, , [From, , (v), , above., , can be easily, , the vessel,, , evaluated., It will be readily seen that apart from the ?alue of/? being quite, independent of the molecular weight of the gas, the gauge possesses, the following advantages, It gives a continuous indication of pressure in the vessel., (/), :, , (//), , It is, , unaffected by any outside influences., , and yet very, , (Hi) It is stable, , sensitive, , at low pressures,, , down, , to, , mm., , 70- 7, , (tv), , It, , can be usedio measure the pressure of, , and vapours,, (v), , It, , irrespective, , kinds of gases, , all, , of their mass or condensability., , does not require, , the, , use, , of objectionable, , liquids,, , like, , mercury., , cheap and easy to construct and work wirh,, cannot be used above a pressure of 70~ 3 mm. ; for,, then, the mean free path of the molecules becomes comparable with the, distance between the plates., (v/), , It, , is, , simple,, , though obviously,, , 1., , SOLVED EXAMPLES, that the gas constant R is 8*3 x 10 ergs per 1C,, 7, , Given, , weight of chlorine, molecules at 0C., , We have, Multiplying by, , it, , is, , mean square, , 35*5, find the root, , the relation, P =*= J/nm; 1, the volume of 1 gram-molecule of the gas,, , V, , Now,, , So, , clearly,, , mN, , -, , M,, , nV, , =, , PV, , ^_, , we have, , c\, , N, , is the Avogadro number,., Where, or number of molecules in 1 gramL, tm.N, molecule of the gas., the molecular weight of the gas., , N., , *, , I, , c, , PV = JMc, }Mc* = RT., , that,, , 2, , 2, , Also,, , ., , Or,, , PV -, , =, , c, , RT., , 3RTJM,, , c, , whence,, , \/32iTiMHere, R = 8-3x10' ergslC, T, 0-h273 = 273 Abs, gms., because a chlorine molecule consists of 2 atoms., .'. substituting the values of R, Tand, M, we have, , c=\ /5x8;3xl0^xT73 _, , V, , Or, the root, cms. I sec., , [See page 537), , ., , t, , PV = \mn V, , And,, , and the atomic, (r.m.s.) velocity of the chlorine, , mean, , 3, , ., , 094x, , and, , 1Q4, , M = 35 5x2 =, , cms, , ,, , 1\, , sec, , 71, , square velocity of the, , chlorine molecule, , is, , 3, , '094x10*
Page 574 :
PROPERTJES OF MATTER, , 572, , If the density of nitrogen is 1'25 gms. /litre at normal temperature, 2., pressure, calculate the root mean square velocity of its molecule., , We, , P=, , have the relation, , mn, , =-, , Or,, , pc, , =, , Heie, and, , P -, , But, , P, , p,, , --, , 76c//w., , c =, , for a gas., , =, , 2, , Jpr, , ., , \/3P/p, , ---, , 1, , dynes/mi*.,, '25/1000 -'00125, K./C, , '3x76x1 3 '6^981, , The, , and, , 76x13 6x981, , 1-25 gms-llitre, , v, , ,, , 1, Therefore, I, , the density of the gas., , 3P, whence,, , -, , J/ww, , 2, , =, , :, , 4-933, , x, , c., , 10*, , '00125, , velocity of the nitrogen molecule is,, , therefore,, , equal to 4*933x10*, , cms. I sec., 3., Derive Charles' law and Dalton's law on the basis of the kinetic theory, of gases, as also the Standard Gas Equation, PV, RT, and write a brief note on, Absolute Zero from the standpoint of the theory., P = J mm>* ~ }?*., (i) We have, p, the density of the gas,, [v mn, K is the volume of the, and r 2 CDC r, , Now,, , where, , p, , V, , is, , 1, , [where, , /K,, , gas., , the absolute temperatute uf the gas, 1, , =, , '', , -., , -r., , ,, j,,, , If, , \, , V be constant, we have, , the picssure of a given, is Charles' law., , Or, at constant volume,, , mass of gas varies, , umperatuie, which, , directly as its absolute, , Or,, , 0), , T, , P oc, , wj can put, , relation (1) above, as, , K oc, , stant,, , V oc, , .7, so, , that,, , P, , if, , be con-, , T., , Oi, u/ oust ant preisuie, the volume of a given, ay /fs abwlnti temptsraiuic, which i* also Charted law., , mass of $as, , \attes, , directly, , (ii) Let a number of ga^es or vapours, having no chi.rncal reaction with, each other, and hiving densities and mean square velou.ies, p x p 2 p,... and, J, Then, the, '3, icspCwtively, be mixed together in ihe same volume, i" *V, resultant pressure P will clearly be given h y, J, P iPi'V i-2 /-riP/3 ~r, considering each set of molecules of the, different ga>es 01 vapours, Now, i^^i^, }p/z 2 Jpj'V ^re the individual piessuii\s exerted by the, we have, different gases or vapours, Putting these as/? l PZ+P*, ,, , (, , *, , ?, , i-, , ,, , ,, , -, , I, , P, , --, , Pi, , r, , ^, , rP 3, , ,, , ., , rcsuliant or total pmssuie exerted by the mixture of gas.es is equal to the sum, oj their individual or partial ptessures, winch is Dalton's law of pariidl pretsures., , /.<., //2e, , ^, , We, , P *have, of the gas,, volume, the, both, sides, by K,, Multiplying, (///), , PV =, , Now,, , M,, , pK, , T, PF, , Now,, Or,, , 4, , ., , we have, , Jp.Kc"., , nijis of the gjs-, , PV -, , [, , =, , .', , ;/U5f, , =, , volume x density, , } Me, | J/Wc, 2, oc, E. of the ga< Or, Toe jMc, c>c r, Or, PV - a constant, 3, , K, , 2, , ., , t, , xT, , PF, , Or, , ~-, , /?r,, , where R is a constant, called the gas tonstant, whose va'ue, for, gas, is quite independent of the values of P, V and T., , a given mass of, , Thus, the standard gas equation, (PV**RT), can be deduced from the, kinetic theory of gases, (iv) According to the kinetic interpretation of temperature, the temperature of a gas is, as, we have seen, proportional to the mean K. E. of the mole2, 2, of a molecule --= iwc and c is proportional to T, the, cules, because the K., absolute temperature of the gas., , E, , Clearly,, , or are devoid of, , T, , --, , all, , o* when, motion, , c, , =, , 0, /.#.,, , when, , the molecules have zero velocity*
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EXHAUST PUMPS, , 573, , Hence, on the basis of the kinetic theory, the absolute zero of temperature is, the temperature at which the gas molecules are devoid of all motion. This, obviously,, is not quite correct, lor the above deduction is made on the assumption of the, sogas being an ideal or a perfect one, and n^ actual g*s approaches this ideal, that, even it at the ordinary temperatures and pressures, the actual gases may be, taken to approximate, more or less, to the perfect gas condition, these ideal gas, conditions do not hold down to the absolute zero, ;, , A more satisfactory interpretation of temperature is afforded by thermodynamics, which does not require the cessation of all molecular motion at the, absolute zero, Deduce the, , 4., , = l/T n,, 2, , relation X, , for the, , mean, , free path, , and use it to calculate the diameter of a molecule of benzene,, mol per c.c and X for benzene = 2*2 x 10" G cms., , For answer to, , first, , part, see, , =, , 6, , Ttr, , nr 2, , Or,, , r, , r', , *, , x2'79xl0 19 x2'2xlO- G, , a, , KX 2-79 x, , 2-2, , -, , 2, , *r 2, , Or,, , r=S, , ', , -, , we have, , ', , 19, , x 2-79x1, , 1, , WhenCC, , *10', , of a molecule,, 2 79 x 10 1 *, n, , page 543., , Substituting the values of A and n in the above relation,, ~, , if, , x 2*79 x 10 13 x 2*2, , -, , 1, , V-X2 79x2- 2x10*, 7-201, , xlO~, , 8, , ', , cms., , Or, the radius of the sphere of influence of the benzene molecule, , is, , to, , equal, , xlQ~* cms, , 7-201, , Now,, , the radius, , of the sphere of influence of a molecule, , is, , equal to, , its, , dia-, , meter*, , Hence the diameter of the benzene molecule, , 8, equal to 7 201 x 10 cms., , is, , 5., Find the mean free molecular path in air, taken as a uniform gas ;, 3, 3, and a pressure of, at, given that the density of air *= 1-2 x 10~ gms. /cm, 6, 10 dynes /cm 9 ., and that its coefficient of viscosity = 1-7 X 10~~ 4 dynes/cm 2 per unit, ,, , 0C, , -, , velocity-gradient., , We, , have the relation,, , Also,, , that,, , T), , Or,, , And, , =, , c2, , whence,, , So, , TJ, , Jp?x, for the coefficient, , P=, , Jpc, , Substituting the value of, , TO,, , and, , .'., , -, , Jp.xV3P/P, , =, , X, , P and, X, , ,, , 3P/p,, , X -=, , .-., , of viscosity of a gas., , 2, , =, =, , c, , V> P/3, , ?), , p,, , therefore,, , we have, , 1-7x10, , 10xl-2 x!0~ 3, , mean free molecular path in air = 8-498 x 10~ 6 cms., The mean velocity of a mslecule of nitrogen gas is 4-5 x 10 4 cms. per, , Therefore, the, 6., , sec., , 1*25 x 10~ 3 gms./c c , and its co-efficient of viscosity, 166 x J0~ gms /cm- per sec- Calculate (/) its molecular mean free path (//) the, number of collisions made per second, and (w) its molecular diameter. (Assume, n ^-2'7xl0 19 per c.c.), , Its, , density at N-T.P., , is, , 6, , *Atoms and molecules are really not the simple bodies they were oce, be., To speak of the diameter of a molecule in the geometrical, , imagined to, , sense has, therefore, no meaning, The diameter of a molecule is taken to be the, distance between the centres of two molecules beyond which they do not exert any, influence over each other, which is clearly equal to the radius of the sphere of influence of the molecule.
Page 576 : 574, , PROPERTIES OF MATTER, , We, , <t), , have, , r<, , =, , pr,, , whence,, , 3xi66xln-, , ^, , =, , A, , _, , pc, , 3xl65xlO- 8, , 6, , 1-25x10x45, 125~xUT 3 x 4-5x10*, = 8-853 X 10-3 cmSf, molecular mean free path, i.e., \~^ 8-853 X 10~ 6 cms., , Oi,, , Nu Tiber, , (//), , made per, , of collisions, , second, , A, .-., , =, , number of collisions per second, (///), , Now,, , = -T/, , A, , -, , -, , 5, , 083 x 10 9, , whence,, , r, , ., , 2, , /, , 1, ', , >!2x2-7xl0 10 xnx8-853xlO- 6, , __- =, , 1, , diameter of nitrogen, , .. the molecular, , Calculate the molecular, = 8 3xl0 7 ergs)., , 7., (it), , at, , K, , =, , c/n5., , 068 X 10~ 8 cm?, , E. of 1, , gm. of Hydrogen, , (/), , at, , 0C, , and, , 100C. (R, , We know, Now., K.E. of, , 1, , that the K.E. of, , #., , And,, , =$, , 0C, , at, , 100C, , or 373, , == 16-99, , 100C, , at, , 1, , R.T, , --JR.T, , 12, , ergs, , Abs., , =, , xlOx, , Thus, the K.E. of, , XlWergs, , -, , gram molecule of a gas, , 1, , mol. wt. of Hydrogen = 2 gms., at, or 273* Abs., of, Hydrogen, gm., 3x8'3xl0 7 x273/4 - 16-99 xlO, , =, , 23-21, , x!0- 8, , 3*068, , -g2x2-7xl0 13 xnx8-853, , 23-21, , #m. of Hydrogen, , is, , 16-99, , xlO 9, , 0C,, , ergs at, , .nd, , EXERCISE XIV, 1., Obtain the expression v), i p c.A, for the viscosity of a gas, and use, ^I, to obtain the mean free path lor molecules of benzene vapour, (^ 6 #c) at, 4, - 0*69 x 10, r.m s velocity ot benzene molecules, c.g.s. units, given C that at, 4, O - 2*95 x O cms, I sec-', atomic wt. ot carbon, c cs. in I gm., 12, at, Ans. 2 015 X 10~ 8 cms., molecule - 22,400., it, , T\, , UC, , ;, , I, , 2., , ;, , root mean square velocity of, hydn gen being OU009 gm /c c, , Find the, , N.T.P. the density of, , hydrogen molecule at, , ihe, , Ans, , 1, , 839 x, , 1<), , 6, , cms. I sec., , Calculate the molecular velocity (i.e.. r n, velocity) in the case of a, gas whose density is 1 4 gms- per litre at a pressure of 76 cms of mercury- Den981 cms- see\3-6 gms./c-c., (Manchcs-er, B Sc.), sity of mercury, g, Ans- 4 6 x J O 4 cms. Isec3., , *, , ;, , 4, , Describe he working of a rotary, 1, , pump, , oil, , How, , ft?, , producing low pes, , are these pressures measuied ?, (Allahabad, 1950}, 5Show that the piessureof a gas is equal to two-thirds of the kinetic, energy of translation per unit volume- Calculate the kinetic energy of hydrogen, (Ailahabal, 1910), per gm. moletulezi 0C., sures., , Ans, Give an account, with a neat sketch, <f some, suitable for the attainment of low pressures6-, , fom, , S-67x, , 'O'e^5, , of a gas pump,, , <Pmjahi 1945), , .m above th mercury., An imperfect barometer tub. contained s, the mcrcurv tood at 18 <nches ab ve the utsi r, l- ^cl, ihc air space -*as, "a- c w .s >nly 3 inches the, 6 inches. On pushing the 'ub^ do^v- s> tria' h a, Ans, mercury stood at 26 inches, Find the bdroiiien ic he^hs1, , When, , k, , *, , j, , ,
Page 577 :
EXHAUST PUMPS, , 575, , 8., One hundred litres of a gas at atmospheric pressure are compressed, What is the resulting, into a cylinder, 100 cms. long and 12 cms. in diameter., Ans- 672 cms. of mercury, pressure., , 9., Describe, with a neat diagram, the construction and working of (a), any modern high vacuum pump and (b) a guage which can measure the low, (Madras, 1949), pressure this pump produces., , 10. The reading of a mercury barometer, the brass scale of which was, was 76 69 cms- the room temperature being 178C. If the, correct, when at, coefficient of cubical expansion of mercury be '009180 and the coefficient of, linear expansion of brass be -00019 calculate the reading of the barometerAns. 73'54 cms., reduced to, , 0C, , 0C, , free, , 11., Calculate the number of molecules per c.c- of a gas, taking the mean, 8, 6, path as 1*83 x 10~ cms. and the molecular diameter equal to 2-3 x 10~ cms., = l/^2nr 2 ./*.], Use the moreaccuraie relation,, Ans. 2-3 xlO 19, [Hint., , ., , /), , Describe in detail, with a neat sketch, the construction, of a McLeod Gauge., What is the recent modification effected in, Enumerate the advantages and disadvantages of the gauge., 12., , 13., , diffusion, , and working, working?, , its, , Explain the principle, construction and working of a good form of, , pump with which you are familiar., What are lonisation gauges ? Describe, , 14, of one such gauge, pointing out, , its, , the construction and working, , advantages and disadvantages over a, , McLeod, , Gauge., 15-, , Describe some form of a thermocouple gauge., , Describe in detail the Pirani resistance gauze, illustrating your answer, ^ith a neat diagram and explaining clearly the essential points of the gauge, its, range, utility and drawbacks., 16., , 17., Explain the principle and working of the Knudsen gauge., particular advantages over the other forms of gauges ?, , What, , are, , its
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APPENDICES, APPENDIX, , I, , IMPORTANT TRIGONOMETRICAL RELATIONS, 1., , Trigonometrical Ration ior, -~, , (A+B), , sin, , (/), , ,...., , sin L., , /, , sin 7f, , cos (A, , .,, , _,, , ,, , 2-, , /?-, , B, , tun, >i, , i, , an, , I, , follows from the above lhac, , (vii) sin, , (A, cos (A, , (viii), , co* R-*-sin, , +tan, , I, , It, , A, , ran 4, , (AB) =, , tan, , (vi), , sin, , -, , C'^, , B), , ,, , ,, , (, , ,, , i, , (r), , Angles., sin F., , \, , A, , A, , ,, , (iv), , ,, , + B) = cos, ton A \-ta,i B, T>v, tan (A + B), ta.i A tan t, sin (A, B) = sin A cos B -cos A, ., , ,, , (in), , cor,, , <1, , cos (A, , (it), , Compound, , B \-co*, A cos B -sin, , s:i:, , + B), , sin, , + B), , cos (A, , -=, , (A-B), , =, =, , B), , sin*A-sin 2 B, cos*A-sin*B., , cos*Bsin*A., , Trigonometrical Ratios for Double Angles., 2 sin A cos A, (i) sin 2 A, sin* A, cos* A -sitf-A =, (//) cos 2A, ., , 12, , =, , 2 tan, , [-tan 9 A, , From, , the above,, sin, , Also, that sin, , f\, , ^, , ,, , and, 3., , it, , cos, , =, =, , 2 cos 2 A, , follows that, , 22, ,0., , 2 sin, cos, , 9, , A, , 4 sin, , cos, , A, , cos 2 A., , r, , $, , Q, , cos, , ., , ?, , _, , i, , -^, , 2, , sin*, , }, , -, , Putting 2A =, (i) above,, , in, , Other Important Relations., *, , .i, , i:\, , A, , 2 tan, , A, , ltan A, 2, , (,ii), , cos, , __, , -, , Y, , 2A, , (in) tan, , =, , "*, , -., , ra, , 1, , sin, 1, , 2A, , (w7), , (ix), , ., , si n, , ^, , A, , cot 2 A, , =, , (*, , j, , Products, , in, , (/), , 2 tin, , ^//), , 2 cos, , (/v), , ^4, , ~, , =, , ., , + tan*A, , =, , l+co^2^, I, cos A _, sin A, , f<w, , A, , (vm) cor, , x, z, , ^t, , tan, , 2, , A =, , ', , (JC), , -^l)^, , A, , cos, , B =, , A, , sin, , B=, , =, , :, , sin (sum) 4- sin (difference)., /i, , ^4., , ,, , "", , ', , cot, , ,, , Terms, of Sums and Differences, , i.e.,, , i.e.,, , 2, , __, , cos 2 A, C, , 4., , ], , ', , 2 sin 2 A cos 2 A, , 4A, , =, , A, , (J.-f B), , sin (sum), , +, , 5///, , (^.-B)., , sin (difference),, , - sin (A + B)-sin, , 576, , (A-B)., , 2 co/ 2 A.
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677, , IMPORTANT TRIGONOMETRICAL RELATIONS, 2 cos, , {///), , A, , cos, , B, , 2 sin, , (iv), , cos (sum), , =, , i.e.,, , A, , sin, , B, , Sum, , Sum of two, , cos (sum), , (A + B)., , (A-B)-cos, , Terms of Products., , or Difference in, , (/), , + cos(A-B)., , cos (difference), , cos, , i.e.,, , 5., , cos (difference), , 1-, , cos(A-\- B), , =, , sines, , 2 sin (half sum)- cos (half difference), , s, , Thui,, , Difference of two sines -- 2 cos (half sum). si n(half difference), , (//), , Thus,, , sin, , A-sin B, , Sum of two, , (i/7), , -TU, , cos, , Thus,, (/v), , ^~ B, , =, , 2 cos, , (-~-t).jm (, , -)., , 2 cos (half surri). cos (half difference), , cosines, , A + cos B, , 2 cos, , / -4-B \, , \, / 4-fB, ., i, , J.coyf, , =, , Difference of two cosines, , 2, , W/f, , -, , YS/H, , ), , ^, , (/w// sum). sin (half difference, , reversed)., , cw, , Thus,, 6., , B =, , .4-C0S, , Trigonometrical Relations for, 180)., , =, , Here, because the, , -, , sm ^, , 2, , ~", , T, , \, , the Three Angles of a Triangle,, , sum of any two, , is, , angles, , when, , (i.e.,, , the supplement of the third,, , we have, (0, , sin, , =, , (B + C), , 5m A., , And, since -^, of the, , 7., , sum of, , (ii), , cos, , 4-, , (A+B) = -cos, , f, , H, , A, , =, , -f, , 90,, , the other two half angles, , ;, , i.e.,, , C. (Hi) tan, , (C+A) = -tan, , each half angle, , is, , the, , B., , complement, , so that, we have, , Relations between Sides and Angles of a Triangle :, (i) The sides of a triangle are proportional to the sines of the angles oppob, , fifetothem., (it), , ^-7-, , Thus,, , A, , sin, , In any triangle., , a*, , c*+a*-2ca cos B, Abo, from the above, we have, b* ==, , -o, r, 2bc, , ;, , c^?j, , BD, , 1, , 6 -fc, , and, , ;, , a, , sin, , =, , l, , ~>, C, , =, , B^, , sin, , 26c cos A., c*, , = a*+b*-2ab cos, , f, , 2, -c-ffl -6, v,, 2ca, , ;, , and, cos, , C~, , a, , C., , +^i^ c, , ^2ab,--, , i, , -.
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APPENDIX, , II, , LOGARITHMS, The logarithm of a number to a given base is the index of the power to, which the base must be raised in order to equal that number. Thus, if a 3 - M,, to the base a, and we, then, X is the logarithm of the number, may put it as, = x, really mean, loga^Z ~>x. So that, both the expressions, a* =, and loga, the same thing., There are two systems of logarithms in use, viz.,, 1., , M, , M, , M, , Natural or Napierian Logarithms, invented by Napier. These are to, where the value of e is 2 17828, and are used in Calculus and other, branches of higher mathematics., (i), , the base, , e,, , (ii), , Ihese are, , Common, , to the, , Logarithms, invented by Briggs a contemporary of Napier., and are commonly used in all arithmetical calculations., , base 10,, , The base is usually omitted in writing, in either case, once we know in, our minds which system we are using, We shall here concern ourselves only with, common logarithms, i.e., to the base 10. Thus, expressions like log 2 or, log TT, etc.,, mean Iog 10 2, Iog 10 *, ie logarithm of 2 or TC to the base 10., ,, , Fundamental General Relations., (i) The logarithm of \ is 0, or log 1 =0., (n) The logarithm of the base itself is }, or log 10 = 1., of the product of two or more numbers is equal, (til) The logarithm, logarithms of the individual numbers. Thus,, K., 2., , of the, , (iv), , minus, , The logarithm of a fraction, , the logarithm of the, , is, , denominator., log, , -, , N, , equal, , and, , log, , V ~M, , =, , Mn, , sum, , logarithm of the numerator, , Thus,, =-, , log, , (v) The logarithm of a number, raised, logarithm of that number. Thus,, , log, , to the, , to the, , =, , M, , to the, , n log, , log (M)*, , =, , log N., , power, , M, , n, is, , equal to n times the, , 9, , ilog M., , Characteristic and Mantissa. The integral part of a logarithm is, and may be positive or negative, And, its fractional, part, expressed as a decimal, is called the Mantissa, and is necessarily positive., Determination of the Characteristic of the Logarithm of a Number., 4., 3., , called, , its characteristic,, , characteristic of the logarithm of a number, greater than \,is one, (i) The, than the number of digits in its integral part, and is positive. Thus, the, characteristics of the logarithms of the numbers, 525, 25 and 5 are 2, 1 and, less, , respectively., characteristic of the logarithm of a number, less than, [, is one more, (ii) The, than the number of zeros immediately after the decimal point, and is negative, the, and read as bar\ Thus, the, negative sign being placed above the characteristic,, characteristics of the logarithms of '254, '0254 '00254, and '000254 are ~1, 27 3~, and are read as 'bar one', <bar two\ bar three', etc., and 4, l, , (, , respectively,, , Determination of the Mantissa of the Logarithm of a Number., The mantisme for the logarithms of all numbers, with the same significant, same order or sequence, are the sa,mz Thus, the mantissa for each of, digits in the, the above numbers, *254, *0254, 00254, is the same, and so also for numbers like, 7, 70, 700, 70000 etc., 5., , 578
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LOGARITHMS, , 579, , We sometimes come, , across a logarithm,, so that both the integral part, viz., 3, as, malpart, viz., '5661, are negative. In such cases, the fractional, tissa must be made positive, by subtracting 1 from the integral, arithm and adding 1 to the fractional or decimal part. Thus,, , Caution,, , negative,, , e.g.,, , (a), 3 '5661,, , which, , is, , wholly, , well as the dee^part or the Manpart of the log-, , -3-5661 --3-1, ;, , iVhere the Characteristic, , is, , ==:_4+'4339=*4, 4339,, negative and the Mantissa, positive., , happens during calculation work that we have to add, subnumbers like 4*4339, with a ve characteristic and a, 4-ve Mantissa. In all such cases, the number should be treated as made up of, two parts e.g., as 4 and +'4339, in the above case. Thus,, (b) It also, , tract, multiply or divide, , (,/), , (///), , 44339 + 2*6371+ 3-2567=4-3277., 4-4339 xJ-il-3017., , (if), , 4'4339-2-6780 = 7 7559., , (/, , e.e., we add to the negative Characteristic the least negative number, to make it, completely divisible by the denominator, and add an equal positive number to, the Mantissa, so that the logarithm, as a whole, remains unaltered-, , 6., Logarithmic Tables. We have seen above how the Characteristic of, the logarithm of a number can b3 determined by a mere inspection of the number., It is, therefore, necessary only to tabulate the Ma ntissae* or the decimal parts, of the logarithms, which is done in what are called Logarithmic Tables. Usually,, the Four Figure Logarithm Tables (given at the end) are used, in which the, Mantissae-of the logarithms of all numbers from 10 to 9999 are tabulated. Let, us see how to use their*, *, , 7., Determination of the Logrithm of a Number. To determine the logarithm of ;i number, we proceed as follows, First we put down its Characteristic by inspection, in accordance with, (i), ;, , the rules, mentioned above., , Then, ue proceed to consult the Tables and, ignoring the decimal point,, (11}, */ any, look for the first two digits of the number in the vertical column on the extreme left, and note the figure in tlw horizontal column, against these, under the number at the top, corresponding to the third digit of our number., , we add to it tlw figure given in the same row, in the column of, on tfie extreme right, under the number corresponding to the fourth, number. The sum of the two, with a decimal point prefixed to, digit, it, then gives the Mantissa of the logarithm of that given number, and this, in, its turn added to the Characteristic, gives the logarithm of the given number., (ni) Finally,, , 'mean, , differences', of the given, , Thus, for example, if we desire to find out the logarithm of the number, (v it is greater than 1,, 3,, 3254, we first note that its Characteristic would be, and has four digits). Then, we consult the Tables, and, against 32 on the extreme left, look for the figure under 5, (at the top), in the horizontal column,, and note that it is 5119. We, then, look for the figure, in the same horizontal, row, under 4, in the column of mean differences (on the extreme right), and note, that it is 5. This, when added to 5119 gives 5124. We then prefix a decimal, point to this number, and get the required Mantissa as -5124 which, added to, the Characteristic 3 give* 3-5124 as the logarithm of the given number 3254., , +, , If our number were 32-54, its logaiithm would be, 1^124 and similarly,, the logarithm of the numbers '3254 and -03254, would be F5124 and 2~5124 respectively. It should be noted that the Mantissa remains the same, (the digits, being ihe same, and in the same sequence), and only the Characteristic changes in, accordance with the "position of the decimal point., 8., Determination of the Antilogarithm of a given Logarithm. It is the, reverse of the above process, and we find out here the number from its given, logarithm. This numbsr is called the Antilogziithn of the given logarithm., :, , Thus,, , of a*., , if log, , m=, , #, then, , x, , is, , the logarithm oF m, and w, the antilogarithm
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PROPERTIES OF MATTER, , 580, , There are separate Tables for this purpose, called Antilogarithmic Tables, where the Mantissae of logarithms are tabulated, right up to -9999. These, t, , Tables are used as follows, (i), , We, , :, , ignore the Characteristic of the given logarithm, for the, , moment-, , the decimal, ('?) Then, we look for the first two digits of the Mantissa, (with, point prefixed to them), in the vertmtl column on the extreme left, and note the figure, tn the horizontal row against these, under the number corresponding to the third, digit of our Mantissa., , (in} To the figure, .so obtained, we add the figure,, differences', on the extreme right, in the same horizontal, number corresponding to the fourth digit of the Mantissa., , in the column of 'mean, row as above, under the, , (iv) And, lastly, we put the decimal p^mt in the figure, thus determined, in, proper position, knowing, as we do, the Characteristic of its logarithm. Thus,, if the Characteristic be 2, there should be three significant figures before the, decimal point, and so we put it down after the third digit of the number, obtained, its, , If, on the other hand, the Characteristic be 1, there should be no significant figure before the decimal point, and no zero immediately following it, and, so, we put the decimal point immediately before the first digit., And, again, if, , be 2, there should be no lignificant figures before the decimal, point, but there should bo one zero immediately after it, and hence we put a zero, before the first digit of the number and prefix the decimal point to it- This gives, the Antilogarithm of the given logarithm, />., the number required., , ,the characteristic, , Thus, if the given logarithm be 15124, we look for -51 in the vertical, column, on the extreme left, and note the figure against it in the horizontal, column under 2, at the top. This is found to be 3251. Then, we note the figure, in the same horizontal row, under mean difference 4, on the right., It is found, to be 3, so that, adding the two, we_havc 32514-3 = 3254, And, since the, Characteristic of the given logarithm is 1, there is no significant digit before the, decimal point and no zero immediately after it, and so we fix the decimal point, just before the first digit 3 and thus get -3254 as the Antilogarithm of the given, logarithm 1-5124. In other words, the required number, whose logarithm is, I -51 24 is -3254 or 0*3254, ;, , ,
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APPENDIX, , III, , DIFFERENTIAL CALCULUS, Differential Calculus deals with the way in which one, Introduction., quantity varies with the other, on which it depends, and such other allied, topics., , Before trying to understand the actual process of differentiation, bowmeanings of the following mathematical terms must be clearly grasped., (*) Constants and Variables., Throughout Calculus, we come across two, types of quantities, viz., (a) constants those which retain the same value throughout a set of mathematical investigation, e #., the mass of a body, the value of g at, tliase which take on, a place, the symbol TV etc., (6) variables, different values or, to which any desired values may be given, e.g., the radius of a circle, the side of a, square 01- a cube, for they are not fixed quantities, and any values may be given, to them. The constants are usually denoted by the earlier letters of the alphabet, a, b, c, d, and the variables by the later ones, x, y, z, u, v, t, etc., A variable which can take every numerical, (ii) Continuous Variables., value, (or all numerical values from one given number to another] is called a continuous variable. Thus, if a train, starting from rest, is observed to be moving, with a speed of 20 mjhr, ten minutes later, it must have assumed eveiy possible, and 20 m.jhr. during these ten minutes. Its speed ii, therefore,, velocity between, a continuous variable. We are concerned here only with continuous variables., If a quantity x, (hi} Dependent and Independent Variables Functions., assumes a set of different values and its value does not depend upon that of any, other quantity, it is said to be an independent variable. On the other hand, if a, quantity y bears a certain relation to #, it is said to be a dependent variable., ever, the, , Thus, since the area of a circle depends upon its radius, we say that the, a dependent variable and the radius an independent variable, or, mathematically speaking, that the area of a circle is a function of its radius. Denoting, the area of the circle by ?/, and its radius by #, we express the relationship by, , area, , is, , ,, , ,, , ,, , the expression, , y, , Here, x, , is, , =, , it. a?, , 1, ., , dependent variable, and, , the independent variable, y the, , rr,, , a, , constant., , Thus,, etc.,, , all, , expressions,, , x, e g.,, , containing, , whose values depend upon the value of, , 2#-, , 5,, , #*+ 2^4-3,, , sine x, log #,, , #, are functions of x., , Actually speaking, therefore, o, variable y is said to be a function of the, when its value depends on the values that x assumes., , variable x,, , There are, however, certain functions^ x, where it is not possible to, give all values to x., For, a function like ^/Ax*, can have aojr'meaning only, when x is numerically less than, or equal to, 2- In such a case, we say that the, function is defined for values of x, in the range -2 to +2, brth inclusive. Thus,, the range of values of x for which the function is defined must be clearly indicated., In other words, we must know whether the function is defined for all values, of x, or only for limited values of x., 2., Notations., function of a variable is generally denoted by enclosing the variable in a bracket, and prefixing a letter to it. Thus, f(x), F(x\ <f>(x) t, It must be clearly understood, however, that, etc., all stand for functions of x., f(x) does not mean f into x, but is only a symbolic way of representing 'some function of x'., , A, , if f(x), , =, , We can know the value of /(#),, denotes the function, # a +2a;-f 3,, , if, , we know, , its, , value, , x, 0, being different for different valuesj>f, /(a), if has the value a, and so on-, , 581, , a? ., , the value of x., , is 11,, , when x, , =2, , ,, , For example,, and is 3, when, , Similarly, the value of /,), , is
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PROPERTIES OF MATTER, , 685?, , 3., , Limits, , of, , Functions,, , Meaning ofa?->oo., , (a), , we, , Suppose, , give, , successively increasing values, 1,2, 3, 4,... to x., Then, obviously, x will become, larger and larger, and there will be no end to the extent to which i t may thusbe increased. This continued growth, or increment of x, is expressed by saying that, <x tends to infinity', or that ( x approaches infinity', or, symbolically, as X-+QO, ., , ing., oo is, , This statement means no more than that x goes on continually increasTo think that x will, at some time, be equal to oo, is simply absurd, for, no fixed number. It is something we can only approach but never, , actually attain., , Suppose, now, we give successively decreasing(b) Meaning of x-^-0., values to x. Then, clearly, x will become smaller and smaller, and, if the process be continued, it may become very very small indeed, wry much near, This continued or progressive decrease in the value of, zero, yet not equal to zero., f, x' is expressed by saying that x approaches zero, and, symbolically, as x >0., Again, if we take a finite number a, and, (c) Meaning of x-+a., assume x to take on values such as a-f'l, 04- '01, a-h'OOl, etc., so that as x, assumes these values, the difference between x and a goes on continually decreasing, and x tends to come nearer and nearer to a, and we express it by saying, that 'x tends to a', or that x approaches a\ and, symbolically, as x-+a., 1., Let us now consider a function, such as /(x) = x 2 and see what its, g, , ,, , value approaches to, if x >5., (0 Let us first give to x, continually increasing values, approaching 5., , x, x, , Thus, if, , =, =, , =, , x2, x2, , 4*9,, , 24-01., , and so on., x 2 approaches 25. Or, symbolically, if a?->5, x ? *->25., (ii) Let us now give to x continually decreasing values, approaching 5., x 2 = 26-01., x = 5*1,, Thus, if, x = 5-01,, x 2 = 25-1001, and so on,, So that, again, as x-5, x 2 -25., /.?., the difference between x 2 and 25 becomes, smaller and smaller. And, if, x 2 =25., x=5,, x 2 25, , i.e., , t, , as, , x approaches, , It, , will be seen,, , which, , is, , ^-, , on giving values to, X 2 25, , 5'01, etc.,->5, thatasx->5,, , ,, , 24'90,, , 5,, , Consider another function,, , 2., , -g~, , 4-99,, , meaningless., , ., , ->10., , ,, , as, , x->5., , x, such as, , But, if, , Such an expression, , 49, 4 99 etc.->5, , x=5,, , or, , 5'!,., , the expression becomes., to, , said, , is, , ;, , be 'indeterminate'., , m, , We can now generalise and say tfiat the limit of the function, f(x) is, as 'x tends to a', if the difference between the function f(x) and, can be m,ade a*, small as we please, by taking x sufficiently near a., , m, , All that, , we have, , comes very nearly equal, , to determine in the limit, therefore, is to see, to, as x is made very nearly equal to a., , what f(x), , =, , In the case of the above function, /(x), x f , we may, therefore, say thai, the limit of x 2 is 25, as x-5, or express it symbolically as, , x->5, Important Deductions., , Suppose we have, , <"'), , it, , can be shown that, , *-Lf/, , and, , m,, , x7^ fl /(x), Then,, , ^ ^*, , *, , (/), , W x #*)] - w x, , ^, n., , a, , [/(x), (Hi), , x, , provided x?<*0, (the symbol ^ meaning, 'not equal, Differentiation of Functions., ictual differentiation of a function., 5., , <(x)], , = mn., , ^ fe, , - -~, 1, , ,, , to')., , We are now, , in a position to, , tackle the
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DIFFERENTIAL CALCULUS, Let/(x) be a function of x, and, , Then, , (positive or negative)., , let, , 583, , x be increased by a small quantity h 9, becomes /(x-f h), taking h to be, , function, , the, , positive., , Or, the increase in the function, the increase, , in the function, , to the, , This ratio /*+")""/(*), It is clear that, , function, viz., /(*-t-/0, , is, , given by/(x-f h)f(x), and the ratio o), , increase in x, , js, , is, , -clearly equal to, , -, , V\-/., , ^lled the 'quotient of differences'., , as h becomes smaller and smaller, the increase in the, /(*), also becomes smaller and smaller ; so that, as /i-0., , We, , have seen above, (example 2, page 582), that although the numerain the case of certain fractions approach the limit zero,, the limit of the fraction, as a whole, is a finite one- Similarly, here, the limit, , tor, , and the denominator, , of the expression,^-, , 4, , ~J^, n, , We are concerned, limit of this fraction,, , ' (, , is,, , in general, finite as /t->0., , in Differential Calculus with the determination, , *4, , .---), whca, n, , This limit, , /z-^0., , of the function /(*), with respect to xreferred to as the derivative of the function-, , ential coefficient, , We may, , It is, , of the, , the differ-, , also sometimes, , thus define the differential coefficient (or the derivative) of a func-, , tion, as the limit of the quotient of differences, , h tends, , is called, , ^LJl, , '_""', , h, , v*^ when, , the, , denominator, , to zero., , In actual practice, (/) it is usual to denote the function f(x) by another, variable y, (//) the increment in value of jc by the symbol &c, (read as delta x),, and (i/i) the increment in the function, by S[( fx)] or 8y t (delta y)., , Thus, h of the above expression, corresponds to 8x, and f(x+h) ~f(x), to 8y., , Therefore, the quotient of differences, , / v, , -, , *"r "', , n, , ^, , ;, ,, , is, , given by, , ojc, , Thus, the limit of &y/$x, when Sx~0, is the differential coefficient of y, with respect to x, and is denoted by symbol dy/dx, (read as 'dee y by dee x')., Or, in accordance with our notation, we may say, , dx, N. B-, , Here,, , ,, , is, , purely a notation, and does not, , mean d divided by, , All that it implies is that the expression to which it is appended is to be, dx., It is just like so many other symbols we use;, differentiated with respect to x., as, for instance, -V, log, sine etc- ; thus, >Jx does not mean >ixx, nor does sine x, mean sine x x., , The differentiation of a function is thus merely the process of finding, out the differential coefficient of the function, and consists of the following, steps, , :, , (/), , (ii), , giving a small increment Sx, (positive or negative),, , is, , x,, , and obtaining, , finding 8y, the increase in the function,, , (Hi) obtaining the quotient of differences or Sy/Sx,, (iv) finding the limit of the quotient, as 8x-0., , and, , to, , and, , Ttte result, thus obtained, gives the differential coefficient of y or f(x) f, symbolically represented as, , - it, Ag ain,, and dyjdx,, , it, , '<, , ], , should be carefully noted that Sy/Sx is the quotient of difference*, , the differential coefficient.
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PROPERTIES OF MATTER, , 584, , Important General Results., , 6', , The following are some of the important general, student must make himself familiar., , with which the, , results,, , The differential coefficient of a function is equal to the product of the in(/), dex of the power and the quantity, raised to a power diminished by one. Thus, the, *1, n is, differential coefficient of x, equal to nx*, 1, , ., , ,, , Or,, , -., , ., , For example,, , (/), , (x, , =, , ), , **-!., , the differential coefficient of A 9, , (i'0, , =, , is, , 9^ 9, , ~1, , =, , 9.x, , 8, ,, , = 2x*~, 2x,, I/* or jr-Hs = -i.jt.-i-i, *- 2 = -I/* 2, *2, , ,,, , (W, , l, , is, , ,, , K, , and a function, (ii) The differential coefficient of the product of a constant, times the differential coefficient of the function., equal to, Thus, the differential, a, is, constant, and u, a function of x) is equal to, coefficient of K.u (where, times the differential coefficient of u, i.e., equal K. dufdx., , K, , is, , K, , K, , *, Or,, , K. H, , dx, , ), , =, , d, , "., , K., , dx, , For example, the differential coefficient of nx*. where n, , would be equal, The, , (ii), , of* 8, , to n times the differential coefficient, , ,, , j.e?.,, , =, , is a constant,, n.3x 2 = 3nx*., , ', , =, , differential coefficient of an added constant is zero., For, if y, K,, varies, y still remains equal to K., Thus, there is 'no change, , no matter how x, , dy t corresponding to a change dx, and, therefore,, differential coefficient of a constant, , Or, the, , 'is, , J, , is, , zero, if y be a constant,, , zero., , It follows, therefore, that, the differential coefficient of y, f(x) C is the same as that of>> =/(*), because the differential coefficient of the constant C is zero., Thus, for example, the differential coefficient ofwx-f7is the same as, that ofiix*, v/z., 3/ix 1 , because the differential coefficient of the added, constant 7 is zeroIn other words, the additive constant disappears, during differ-, , =, , +, , entiation., (iv), it, , The, , differential coefficient of the algebraic sum of a number of functions, sum of their individual differential coefficients, i.e.,, , equal to the algebraic, , -, , ...., , dx, , Thus, for example,, , The, , (v), , equal, , if, , y, , -j-.U~, dx, dx, =, , -V-, , -.H'-, , dx, , ~, -,Z., dx, , 5, , 2x*+ 3^*~4A: -f-5^+7 we have, f, , of two functions, u and v ' is, plus uxthe differential coefficient of v., have to do to determine the differential co-, , differential coefficient of the product, , tovx the differential, , coefficient, , of, , u, , In other words, all that we, of the product of two or more functions is to differentiate one, function, at a time, leaving the others unchanged, and then to add, up the resulting expressions together., efficient, , Thus,, , if, , y, , =, , dy, , M.V.W.Z.,, , du, , *+, , dt-d*, For example,, , if v, , we have, , =, , dw, , dv, , ., , 2r '. +, , ,.V., , * 2 (5*-t-3). we have-, , g^ +, , dz, , .V.*.., , (5x-f 3?., , ~ *+** ~, , (5x+3).
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DIFFERENTIAL CALCULUS, , 585, , 3x(5x+ 2)., , -==, , Or,, , 3*(5x+ 2)., , (vi) The differential coefficient of a quotient, Denom. xdiff. coeff. of numerator Numerator x diff., , (Denominator), , Thus,, , if, , y, , =, , w/v,, , Denom., , dv, , -- w, , </M, , v .~., , C/JC, , <if, , of, , we have, fly, , For example,, , coeff., , a, , .y, , jrr, , X*, , =, , -, , ,, , (x+5) 2xlx+5), (VM) The differential, , ____, , ifc, , we have, , coefficient of the, , fu notion of a function, , is, , the differential coefficient of the inner function and, coefficient of the whole function, considering the inner function to be, variable., , equal, , to the, , the differential, , product of, , an independent, ', , Thus,, , if, , y, , --=, , where, , F(z),, , = /(*),, , z, , we have, , d, , ? x, , j-=, , z, , -, , j, , ., , All that we have to do, therefore, is to differentiate the whole function,, the inner function (in this case), (z) were an independent variable, and, then multiply by the differential coefficient of the inner function-, , as, , if, , V, , x +~<** or y, For example, (/) if y =, we differentiate the, (x+a), whole, as though (x-fa) were an independent variable, and get, , Now,, , coefficient, , differential, , *-fO, , (Ixx^-fO), And,, , ,, , of the inner function (x+a), , is, , equal to, , is, , equal to, , 1., , the differential coefficient of y, , therefore,, , =, , <\/ x, , +a, , = (ax+b) n then, differentiating the whole function, (//) If ^, -1, iax+b) were an independent variable, we have n (ax+b), ,, , as, , though, , 11, , And,, , differentiating the inner function, (ax +6),, coefficient of the function y, n~ l, n, , we have oxx-f, , differential, , .-., , =, , axn(ax+ b), , The Second, , 1., , differential, , coefficient, , coefficient, , Differential, is, , ,-y., , of the function y, , is, , = f(x)., , We know, , that if, , what may be called the, dv, For,, , ^, , itself a, , is, , coefficient, , Tfr-f ";r-V, , of y,, , We may, of ^denoted, , denoted by the symbol,, , ** is, , and read as, , f, , dee two, , =, , is, , equal, , y, , - /(*),, , ., , to, , ~, , first, , its, , differential, , function of*, and can,, , therefore, be again differentiated with respect to x, giving us, efficient,, , n, , na(ax+b) ~\, , Coefficient., , It, , (ax-\-b), , t, , the differential co-, , called second differential, , y by dee x square'., , similarly have the 3rd, the 4th, or the nth differential coefficient, 8, dny, flf v, d*y, or, The successive differential coefficients, n~*, , byir, , of y are also denoted by y lt y^^-yVt or by, , D y D\ etc., ,, , Or,, , if, , the function be
Page 588 :
PROPERTIES OJ MATTER, , 586, denoted by, /"(a),, , are represented by, , the differential coefficients, , /(#),, , /'(a?),, , etc., , For example,, , if, , 4, , 4# -f3#, , =, , y, , 2, , >, , the third, the fourth, , we have, , -l-2a -|-#-f 1,, , )> 3, , =, =, =, , >> 4, , == 96,, , the /r* differential coefficient y^, the second ,,, ,,, ;' a, , ,,, , 16.x, , 2, , 1, , 48-c, , f 9;t -M;t-f, , 2, , +0,, , 1, , -fl8x-f 4,, , 96*+ 18,, , =, , and, therefore, also y & = 0, and >> 7, 0, and so onWe are concerned here only with the second differential coefficient, which*, finds a wide application in problems in Physics., , A, , familiar example of the second differential coefficient is the acceleration, so that, if we, of a body, which, as we know, is the rate of change of velocity, is the first differendenote acceleration by o, we have a, ffu/df, i-e>, acceleration, tial coefficient of velocity., But velocity itself is the differential coefficient of distance #, with respect to time, or v, dSfdt., ;, , =, , =, , d / dS\__d*S, -~dt\dt )~ di*', , _, , dv, , '~, , __, , dt, , acceleration is also the second differential coefficient of distance with respect to, , Or,, time., , 8., Differential Coefficients of Logarithmic Functions. Before proceeding, with the differentiation of logarithmic functions, a few important cases of limits^, given below, must bejememberedn, \1, as H_KJO., j_j, , --(1, , At the, w~>oo,, , (l-f, , case is equal to, , the beginner, , glance,, , first, , JL, , Y=f, But, , 1-, , nary finite numbers,, , +, , 1, , may be tempted, , =, , ), , (1, , =, , + 0), , to conclude that if, , 1, i.e.,, , this is not true, because, as we know,, amenable to the laws of algebra., , the limit, in this, , unlike other ordi-, , oo is not, , Taking n to be a finite number, and expanding the expression by a, simple application of the Binomial Theorem, it can be shown that the value of, , (1, 1-f, , .-., , I//?,, , \n, J, , is, , more than 2 and, , than, , less, , 2/netc.-^0, in the limit, ( 14-, , 3-, , ^=, , 1, -, , And, further, that, 2-71828., , as n->oo, , This figure, , is, , taken, , be the base in the natural, or the Napierian system of logarithms, and, noted by the letter e. So that, as, , (1, 1+, , This, , is, , \w, ), , n, , (ii), , is, , to-, , de-, , =., , a very important limit to remember., The following are its simple applications, (*), , and, , :, , (a), , Limit of, , (Hi) Limit of, , t h-i, , as /i->0, , n, , ah, , is, , 1, , ., , \, , ^, , ,, , as, , 1, , ^->0, , is, , ,, , =, , logf a., , Now, although we ordinarily use what are called common logarithms or, logarithms to the base 10, in Calculus, as in all other JD ranches of higher mathematics, natural logarithms, or Napierian logarithms*, (after the name of the inventor of the system), are used, and so we have the following further relations, to remember, :, , 1, , ., , log a x, , =, , log,;, , x x loga, , e., , *Common logarithms to the base 10 can be converted into Napierian, logarithms to the base e by mult ipl> ing them by 2-3026-
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DIFFERENTIAL CALCULUS, Any, , 2., , The following are, mic functions, (i) The, , ,,, , and not, , e,, , d/du (log w), , =, , were, , a,, , d, , The, , y, , a*,, , we, , we have, , being under-, , e, , /*, , 1/w., , du, , )=-*,log*, , 1, , have, , shall, , d, <W, , differential coefficient of, , if, , ax, , >>, , x-, , log, is, , dT, , -, , 1, , du, , M, , .^., , 9, equal to a log f a., B, , log*, , x, , log,, a., , differentiating both sides with respect to x, we have, 1, dy, dy =, Or,, 7 log,, a =0* log,, log, a., ,, , ^, , From, , a very important result follows,, , this,, , x, (Hi) that the differential coefficient of e, ex, if a, we have, , =, , For,, , and, , 9., , (iv), , jy, , y, , =, , =, , ,, , a., , viz.,, , ex ., , ,, , e x log<., , logg e, , e=, , e*, , x, , =, , 1, , *, , Differential Coefficients of Circular Functions., , The, , The, The, , x, , differential coefficient of sin, , The, , (ii), , (Hi), , e,, , dy/dx, , .-., , (\), , lagarith-, , we have, , .., -, , dx, , =, , d/dx (log* x), , Or,, , If the base, , For,, , some important, , of log e x or loj x, (the base, , differential coefficient, , differentiating with respect to x,, , (ii), , x., , :, , Similarly,, , Or,, , a loga x,, , x*=e\ose, , the differential coefficients of, , stood), is equal to \]x., , N-B., , =, , x, , quantity,, , 3-, , 587, , differential coefficient of cos, , differential coefficient of, , x, , tan x, , differential coefficient of cot, , x, , is cos x-, , is, , Or, djdx (sin x), , is sec 8 x., , is, , =, , cos x>, , sin x>, , Or, dldx (cos x), , =sin x,, , Or, dfdx (tan x), , =, , sec* x., , cosec* x., , Or, d/dx, , (cot x), , cosec* x., , Maxima and Minima-, , 10-, , can be shown that for m'tximi and minima, dyldx = 0- To find where, the function is a maximum or a minimum, the procedure is the following, (1) Put dy/dx =0, solve the equation dyjdx = 0, and obtain several, values of x., It, , :, , See for what values of x, dyldx changes sign from positive to negative., the function is a minimum., (Hi) See for what values of x, dy/dx changes sign from negative to positive., For such values of x, the function is a maximum., (ii), , For such values of x,, , N.B., , sometimes happens that for some values of x, obtained from, The function is neither, 0, dy/dx does not change sign., nor a minimum at these points. Such points are called points of in(1) It, , the equation dyjdx, , a, , maximum, , =, , flexion., (2) All points on a curve,, called stationary points., , where the tangent, , is, , parallel to the x-axis, are, , (3) Points, where the function is a maximum or a minimum are called, turning points, and the maximum and minimum values are called turning values., , Example, Let, , it, , :, , be required to find the, , Putting, , y, , *, , x6, , 5x* 4- 5x,, , maximum and minimum, , we have, , values of
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PROPERTIES OF MATTER, , 588, , if, , and, *.e.,, , x be a, , if, , at, , x, , Thus, x, , A-, , =, =, , and, , if, , * be, x, , Thus, at x, , maximum, .'., , And, , ,,, , 3,, , x, , 5x*(x, , =, , 1,, , 1)=0,, , 3)(jc, , x, , =, , 0., , than, , is, , negative, , *,, , little, , minimum, , =, , ,,, , 1,, , value of the function, , than, , 35, , 8, 5x3*4-5 x3, , we have dyjdx =( + )(--)(-), , 1,, , .*.,, , =, , 1, ,,, dyjdx = (+)(-)( + ), .., negative., =, changes, sign from positive to negative, or x, dy/da;, , maximum, x, , =, , ., , value of the function, , 0,, , x be a little, x, greater, , we^have, less than, than, , 0,, , dyjdx, , =, , gives, , =, , 0, c/y/^x, , dyjdx does not change sign at, , Hence, x, , 27., , positive., , greater, , value of the function, , i.e.,, , is, , ;, , is poaieive., , dyjdx, changes sign from negative to positive., minimum, value of the function., , little less, , lastly, at, if, , and, , =, , 3 gives the, , the, , .'', , Now,, , 3,, , x, , = (+)(-)(+), *..,, 3, dy/d x, than, 3, dyjdx = (-f )(4- )( + ),, greater, , little less, , be a, , we have, , to 0,, , And, equating dyjdx, So that,, Taking first, x, 3, we have, , #, , 1, , 5, , + 5=4-1., , (4- )(-)(-), **., positive,, (-f, , )(-)(-), , =, , 0., , 7?>m^ o/ inflexion., , *, , positive, , 9, , 1, , gives the
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APPENDIX, , IV, , INTEGRAL CALCULUS, 1., Introductionthe process of adding or, is called integration., , The word, "integral" simply means 'the whole', and, summing up a large number of little bits of a quantity, , Thus, if A: be supposed to be made up of a large number of tittle bits,, each equal to dv t it is obvious that if we add up all these dx's together, we shall, Mathematically, we put, , get x., , equals x', , ,, , the, , same thing, , symbol, l, , I, , meaning, , as, , it, , sign, , I, , indicates the, , x,, , of\, , 'integral, , and read, , it, , as, , 'integral, , o, , a long'S, and means the, , It is just, , summit ion of., , sum,, as, that other familiar symbol, 2 (stgmi}, latter indicates the summation of a, or, , thv, , =, , dx, , I, , summation of a, , y, , In fact, it resembles in its meaning, with the only difference that whereas the, number of finite quantities, the integral, , number, , large, , minute quan-, , of inflefinitely, , tities,, , 2., We may, Integration The Reverse of the Process of Differentiation., look at this process of summing up small bits to get 'the whole\ from another, point of view, and say that integrating i* really the reverse of differentiating, for,, whereas in differentiation, we are concerned with the problem of finding dyjdx,, that, given y as a function of jc, we have, here, to perform the reverse process, viz, of expressing y as a Junction of x, when dy/dx, is g wan. This is the process,, called 'integration', and y is said to be the integral of dyjdx., ,, , x 2 dy/dx, 2x and so it follows that, For example, we know that if y, we perform the reverse process of determining y, when dyjdx is given to be, =, must, x*., equal to 2x, we, get y, 2, to, Again, if we are given dy/dc = 3x and are asked to find out y, i.e, 3, x 3 dyldx, 3x 2, integrate dy/dx, we shall have y = x, because, if y, ,, , 9, , if, , ,, , ,, , ,, , *, , =, , Similarly, if, , ., , ,, , 4x 8, , y, , ,, , =, , It will thus be seen that the process of integration, the differential coefficients of functions., , (~v, , x*., , if, , y, , =, , x\ -^, , =, , dx*., , depenls upon the recognition of, , There are no, as, , we have, , infallible rules or fool-proof methods here to be guided by,, of differentiation. The first essential step, therefore, to, integration, is to familiarise oneself well with the results of, , in the case, , be successful in, differentiation of simple functions., at the end, for the purpose., , A, , list, , to help, , memorise them, , js, , appended, , We have seen above that if, f, we have y = x 3, dyjdx = 3jc, We may arrange, dy/dx = 3x* as dy, 3x*.dx> which is, what is called, a, differential equation, telling us that a little bit, or, in the language of mathematics, 'on element of x' is equal to the corresponding 'element of x' multiplied, by 3x*. We have now to sum up all these elements dy to get y, or to find out, the 'integral of dy\ which will, obviously, be equal to the integral oflx'.dx;, so that, putting the proper notations, we have, 3., , Integration of Simple Functions., , ., , ,, , =, Or,, , I, , 3x a .dx, (read as, , y, , =*, , x, , 8, ,, , 'integral, , dy equals integral, , 3x*.cfa;')., , as seen above., , The rule to be applied would thus, one, then divide it by this increased power., , seem, , 589, , to be to increase the power, of x by, For, performing this operationf in this
Page 592 :
PROPERTIES OF MATTER, , 590, case,, , we have, , x t+i, *, , 3v, *, J, 3, , Of, y is, , course,, , it is, , obvious that, , broken up, will clearly be, , dy, , =, , y, for the, , sum of ail, , dy's into, , But we can apply the above rule, , y., , in, , which, case, , this, , also-, , Let, , us, , put, , I, , dy as, , y, , I, , Q, , -dy, for, , pression remains unchanged., , y, , =, , 1, , and hence the value of the, , ,, , ff, , f, e! dy, , [dy, , =, , =, , X, , f, , I, I, , y.dy, , ex-, , "*" 1, , ), , == y-, , y., , The student, to the, , little, , dx* or dy, , I, , is, , bits,, , will perhaps feel tempted to ask as to what has happened, dx and dy, at the end. It is enough to remember that this, , symbol of integration, the, , just a part of the, , dy etc., and, , when, , the integration, , is, , full, , notation being, , performed, the symbols,, , I, , I, , dv, , and dx,, , or, , and dy, vanish together., , say that, , We may, , Indenfinite Integials., , 4., , I, , J, , xn .dx, , ^"^, n+1, , J, , ,, , thus generalise the above rule and, , so that the process will seem to be wonderfully easy., , But a snag soon appears, for we ktow that if, 2, y = x, d\\dx = 2*., 2, Also, if, y = .x -f 7,, dyldx = 2x as before, because the, so that, performing the, differential coefficient of an added constant is zero, 2, reverse process of integration, we get y = * in both the cases, which, obviously, is far from correct., Further, the result would still be the same if the added constant weie 9, or 11 instead of 7. It would thus appear that integration is not quite so reliable, a process, after all, and that one has to be guided by the results of diffnrentifition., Therefore, working backwards from dy/dx, allowance must be kept for there, being an added constant., ,, , t, , ;, , Ttus, integrating dyldx, , =, , 2x,, , we, , say, , \, , dy, , =, , 2x.dx, , 2. ~~, , -f, , C., , C, , is a constant, called the, where, y ** x ZjrC,, constant of integration, an abitrary constant, having no particular value ; for,, as we have seen above, it could very well be 7, or 9, or 11, or any other number,, for that matter. The value of the integral is, therefore, not fixed or definite,, and such integrals are, therefore, called indefinite integrals., , Or,, , If, however, we are given the value of the function y, for a particular, value of x, we can determine the value of this constant C. For example, if, 11, when x, 2,, 2x, and y, dyldx, , =, , we have, , =, , =, , 1, , dy, , =, , I, , J*-J, , 2x-dy 9, , Or,, , y, , =, , x*-\-, , C>, , "This dx t originally a part of the differential coefficient, when transferred, to the right hand side, as in 3x*.</x, only indicates that x is the independent, variable with respect to which the operation is to be performed.
Page 593 : 591, , INTEGRAL CALCULUS, Now,, , putting y, , 11,, , So, , and x, , we have, , 2, (as given),, , = 4+C, whence, C, y - * +7., , 11, , 7., , f, , that,, , x'.rfx, , Similarly,, , would be equal to, , i'-f c,, , |, , and, , I, , d* would be equal to, , x-}-, , C, and so on., , need not always be written down,, N.B. The constant of integration, alwayi supposed to be there, in the case of indefinite integrals., When an integral is defined between two limits,, 5., Definite Integrals., The lower value of the limit is called the inferior, it is called a 'definite integral'., or the upper, or, the lower limit and the higher value of the limit, the superior, 9, , but, , it is, , limit,, , themselves being called the 'limits of integration., , -the limits, , For example, if the integral of the function, y = /(*) is to be determined, between the limits, x =* a and x = 6, we represent it symbolically as, , =, Jx =, , b, , P /(*).</*,, , or, simply, as, , f(x).dx,, , [*, , J, and read it as 'integral off(x).dx between the lower limit (x equal to) a, and the, a to b of the function f(x), with, upper limit (x equal to) b, or, <as integral from, respect to je.*, , Now,, , a, , [f(x\dx, , if, , -, , f(x), , =, , + C,, , 'and, , J, , b, the, , value, , = f #U) + C 1, , of ( f(x}.dx, , r, , j'-*", , f <(/>)+C, , =, , j>, , If ^()f C, "j, , -, , <f>(b)-</>(a),, , <he constant of integration having disappeared during the process-, , 27, , f^7, , And,, , in genera.,, , J, Corollary., , It, , ** = Tx*, , Thus, we shall have, , Some, , On, , A^+A, , "!^, , flWf, , 19, , 1, ., , -+, , follow as a necessary corollary from the above, a, b, [, , 6., , "1, , [^ J^*, , fl, , = -, , dx, , f(x), , =, , x*.dx, , Illustrative, , that, , f(x).dx., , [, , ]b, , ]a, , (/), , 1, , ,, , 8, , x*.dx,, , I, , Examples., , Indefinite Integrals., , () The value of, , |, , x~ 1 .dx-, , Here, dyfdx, , dy, , Or,, , =, , =, , x~ l, , ,, , or, dy, , =, , breaks, , down, , x~ l .dx., , \x~ l .dx., , J, , Now, our, , general rule that, , f, , x n .dx, , ., , is, , =, , j, , for that would give the value of the integral, , x- 1, , 1, , jt*, , "*", , 1, , ~, , to be, , #-1+1, -^r =, , in, , ^0, , J, -, , ~Q, , this case,, , -*, , Q, , *By subitituting first b instead of x, in the expression, and then a,, subtracting the Utter from the former., , oo, , ,
Page 594 :
PROPERTIES OF MATTER, , 592, , l, and, surely, we do not get x~, by differentiating oo, as we ought to, if our, integration is correct. It is thus an exception. Once, again, therefore, we have, We, to go back, searching for the function of x, whose dyldx is x~\ or 1 /#., know that it is y, Jog^x., 1, log# x is 1 /x, or, x- , the, Thus, since the differential coefficient of y, reverse process of integrating dy/dx.*** x- 1 , or dy, x~ l .dx should give us, y ss \ogfX. But we must, as usual, be careful to add the constant of integration, C, so that the final result we obtain is y, log*x-f C., , =, , =, , h, , Or,, , ^, , dx, , This brings into bold relief the fact that it is not possible to integrate an, expression unless t)iat expression ^s known io us to Jiave been obtained as a result, We must, therefore, learn up the resultr, of differentiating something else., differentiating as many general functions of x as possible and make sure, this list of ours goes on increasing continually., , The value, , (//), , because,, (, , of, , The value of, , ///), , because, , On, , (2), , f, if,, , ~, , y, , if, , ex ,, , - -, , y, , dy/dx, , J, , And,, , if, , y, , -, , where, , =, , e~ x dx, 1, , =, , ~e~*-\ C., , xe*, , n, g', , 0fofl, , Ja~L, , Functions., , ^, , J", , log *, , F', , We know, , thai, , if, , co# x., , *m, , cos, , Jin, , C is, , I, , r, , -u, , Trigonometrical, , co* x, </y/dx ==, , [sin x.dx, , Hence, , [See page 587, , ->, , s, , S, , L, of, , we have dyldx, , =, , C., , ,, , _r, , fl, , Integration, , mm 8 in x,, , <?*-}-, , Definite Integrals., , p X_, y, , =, , e*., , c^xO, , dy, , \, , - 2 +i, , 7., , =, , The value of, , er^.dx., , I, , value of \c x .d x, , The, , e*.dx, , x., , x+C, and!, , cos x.dx, , =, , nn, , the constant of integration, as usual., , In the, , same manner, we can obtain the integrals of their trigonometrithe end), if we know them to have been obtained, , M, , cal function (see list at, , differentiating oth^r functions., , The value of constant, , C may, , be determined in the same manner as, , in the case of ordinary algebraic functions, discussed above., Further, we may, have definite integrals here also, as elsewhere, whose values are obtained in, , precisely the, , same manner, , as those of algebraic functions., , For example, the value of, , I, , cos x.dx, , can be obtained thus, , :, , JO, I, , cos x.dx, , =, , 8., , sin, , I, , L, , JO, , =, , x, , JO, , sin, , L, , it, , =, , sin, , 0,, , J, , Integral of the Differential Coefficient of a Function., , Since integration, , is,, , by, , its, , very definition, the reverse of the process of, be the differential coefficient of a function, (x) with respect to x, will be/(x)., , differentiation, it follows that if ^ (x}, the integral of, /(x), with respect to x,, , ^
Page 595 :
INTEGKAL CALCHJLUS, Or, putting, , and, , it, , in symbols, if y, , y =, j:, j~.f(*)**$, , (x), , Or, substituting the value of, , </>, , = /<#, , we, , (x) in, , ^, , J, , 1, , (i), , 593, , k, , W^A, , jftv^, , ;w., , ...(/i), , froarffjabove, we have, , (//),, , I,, c, , ,, , the integral of the differential, , coefficient of, , a junction of x, , In other wouls, the operative symbol*, dfdx and, , ..'A*?,, , I, , w, , the, , function, , cancel each, , itself., , other out,, , and quite naturally too, because they represent two inveise operations., Thus, we can straightaway say that, I, , ., , -stn x. dx, , sin A., , Again, if the integral of a function y*U) to /(*),, cient off(x) will be 0(.x), Oi, putting in symbols, i!, (, , K), , =, , dx, , />)., , ., , we, , .(///;, , the differential coeffi-, , shall have, , f(x), , ^, , ---, , (x). ,..(/v), , I, Oi, , t, , So, , ., , substituting the value of/(jt) in (iv), , ,, , the dtjjnential tocffictcnt of the integral of, , that, as, , befoie, the s>mbols, , and, , ., , from, , above, we have, , (//;), , a function of x, |-., , m, , Ifa function itself., , dx cancel out, anJ we can sty, , straightaway that, Product of a Constant and a Function. Just as, so also here, we obtain the integral of the product of a, constant and a fuiction by multiplying th^ integral of that function by that, 9, , of, , Integral, , the, , in Differential Calculus,, , ^tant., , Thus,, , For example,, , (/), , I, , Kf(x).dK, , f, , W.cfx, , -, , =, 4, , A', , f, , I, , where, , f(x).d\,, , x\dx, , -, , 4, , f, , A*, , A' is a, , 1 -, , constant., , A*, , Sum, , 10, of any Finite Number of Functions., Integral of the Algebraic, integral of the algebraic sum of any finite number of functions 1$ equal to th<, bruic stun of their twlteiduul integrals., , Thus,, , //,, , y and z be the functions of x, w: hav*, I, , //, , \-y \-z), , dx, , ==, , I //, , (/A-f, , I, , z.Vv,, , y.d\-\J, , Foi example,, (0, , f, , 3, , f, , 2, , 2, , f, , f, , -J4A ^-hj3A-.^-J2^/A+J, f, f, 2, , 4J*.<fc+3Jx.efx-
Page 596 :
PROPERTIES OF MATTER, , 594, , I, , (ii), , =*, , I, , V, , I (cos, , cos*.dx^-l, .dx^-l, , 2x, , cos, , I*-*'*!, nn 2x, , Or,, 2, , x, , ,, , whence, cos*x, , 4, , (m), , j, , 2x, , cos, , =, , <w* 2 (Me, , I, , -./*-./., 2, , 2, , (w), , tan Q,dQ, , \, , z, , =J, , --, , f, , '/, , O-~l).dQ, , (8ec, , ^C 2 e.dfi-, , j, , I r/g., , f, , dO, , [_, , Other Fundamental Rules (Theorems)., I., If we have a fraction, wliose numerator, Ike, , denominator, then the integral of the fraction, , Thus,, if, , then,, , =, , y, , if, , u be a function of, , log w we have, ,, , ^, , ^"a~~~, , ^8, , =, , u, , i.e.,, , #,, , w), , differential, , r, , ^8, , /^, , 1, , coefficient, , of, , logarithm of the denominator., , f(x) 9 such, , ~, , (7w, , I, , is the, , is tlie, , t, , ,, , that du/dx ~~f'(x),, , u, , ', , )-"j"f, , ., , Or,, /, , f, , /'(), , dx, , JT^Ti.e.,, , of, , the,, , i/w integral of a function whose numerator is the differential, is equal to the logarithm of the denominator., , coefficient, , denominator, , Thus, for example,, f, I, , J, , 00,9, , r, , a, ., , ., , sin, dx = Joe, fo, ,, , -, , *tw, , ,, , x,', , x, , -, , == cos, , a*., , ., , dx, The integral of the product of a function of x, ruised to a power n ana, II., the differential coefficient of the function itself (not raised to the power w), is equa', to the function, raised to the power (n+1) divided by (n+1)., ,, , we have, , For, suppose,, , where / (), Then, dy, , a function like y, , is the differential coefficient, , -, , (n+, , n, l)[f(x)], , .f'(x).dv., , =, , 71 "*" 1, , [/(a?)], , ;, , of /(), , And, , dy, , .-., , =, , j, , Or,, , y, , =, , (n, , + l), , [/, , (x)]*.'f'(x)dx., , so that,, , Or,, , f [}(x}, J, , (n+l)(f(x)].f'(v).<lff., |, , n, , .[^(x)dx], , =-*., w, +i
Page 597 :
595, , INTEGRAL CALCOLUS, , Or,, , [, I, , [, , = Uw*' \, n-\', , /M n, , /'(*).<**, ', , where n * -, , J, , 1, , For example,, I, , (i), , 1, , sin zx.co* x.dx, , 3, , (*' + & J? +c)n.(2, , (), , cos, , *--', , J, , a, , -?,, , a;, , =, , id, , c7 'c, , ;4^*-, , sin x,, , /|M, , J, , ^, , From, , r, , r i, , the above,, , f'( T, ^J', , follows that, , it, , ), , f\, , ., , x, , .-=, , J, , VJ(*), , ?, , a, , | /'(;r), , j, , [, , /(^)], , .^x., , J, , 01, T, , (a.e-j-&), , __, , ", , J, , where (2^-j-26), , r, , ^(2<?^ ^-26), /, , 2, , rra?, J \/, , \/V/J? -t-2ta; j-c, , 8, , + 26j7-[-c, , *, , *, , ', , the differential coefficient of (ax* -\-bx-\-c)., , is, , Hence, , -, , Ivist, , =, , "*, , of Important Integrals, , Algebraic*, , (1), , because, , l.'/j?=r,, , (i), , .\, , /, , 1, r, , ', , ., , (}, , 2*.</x, , -, , ^, , a" dx, , =, , ^L, , 2, , ,, , J, , (, , V), , ?, , J, , (, , where, , n^ -, , because, i, , \, , (v), , f, I, , 1, , x, , J, (2), , 1 ),, , .a;, , ^, , n+1, , v, , <^, , ,, .dx -- log^a?, , because, , ,, , (loggrv), , *, , ,, , =, , (n-\-\)^\, , =, , 1, , x, , Exponential and Logai ithmic, , (0, , J', cind, , ,, , **., , |, , a^.loikft.^, , =, , a jr because, t, , j,, (ii), , e*.dx, \, , J, , *Note, , =, , e* t, , because, , (, -, , dx, , ', , (i, ,, , dx, , ,e, , x, , ., , that/'faj) is the differential coefficient of/(*) f, , and not of [/(a?)]-i
Page 599 :
CONSTANT TABLES, I, , DENSITIES OF, , COMMON SUBSTANCES, , 597
Page 600 :
598, , PBOPEBTIES OF MATTER, II, , COEFFICIENTS OF VISCOSITY, , IV, Liquids, , ELASTIC CONSTANTS, , Gases, , C, , (20 C|, , (15 C), , 1., , Alcohol, , 0-0119, , 1., , Air, , 2., , 0-00649, , 2., , Carbon dioxide, , -000144, , 0-00367, , 3., , Hydrogen, , -000089, , 4., , Benzene, Carbon disulphido, Carbon tetrachloride, , 0-00969, , 4., , Nitrogen, , -000174, , 5., , Chloroform, , 0-00564, , 5., , Oxygen, , -000198, , 6., , Ether, , 0-00234, , 7., , Glycerine, , 8-500, , 1., , Alcohol, , (109C), , S., , Mercury, , 0-016, , 2., , Benzene, , (100*0), , -00009*, , Turpentine, , 0-0149, , 3., , Ether, , '000097, , Water, , 0-01006, , 4., , Mercury, , 5., , Water, , (KXPO), (300C), (1000), , 3., , 9., , 10,, , -000181, , Vapours, -000 HO, , -000532, -000120
Page 601 :
CONSTANT TABLES, , 599,, , V MOLECULAR ELEVATION OF BOILING POINTS OF SOLVENTS, (Elevation per 1 gm.molecule of salt per 100, , c.c., , of solvent), , Solvent, , VI, , 1., , Acetone., , 2., , Alcohol, , 3., , Benzene, , 4., , Chloroform, , 5., , Ether, , <3., , Water, , MOLECULAR DEPRESSION OF FREEZING POINTS OF SOLVENTS, (Depression per 1 gm. molecule of salt per 100, , VII, , c.c., , of solveni), , SURFACE TENSIONS OF IMPORTANT LIQUIDS, (in, , contact with air)
Page 602 :
PBOPEBTIES OF MATTER, , <wo, , VIII=MOLECULAR CONSTANTS, 1., , Number, , 2., , Avogadro Numbz>, , 3., , Mass, , of, , of molecules per, f, , Hydrog n, , number of molecules per gm. molecule, =6-022 x 10 2 3, , or, , vtc, , = 1-67 x 10-*, , m, , Molecu-a r, , Molecular, Diameter, , Gas, , (At N.T.P.), c.c. of a pas =2*75 x 10 19, , Mean, , Velocity, , ?/, , a, , gas, , 4., , Free, , Path, , (r.m.s.), , Collision, , Frequency, , Carbon, 4-32, , dioxide, , x 10- L cm,, , 3-92, , x HH cm. /$*.,', , 240xlO~, , Hydrogen, , 8, , (), , 20, , x 10- 8 cm., , 5-74, , x, , l'^ 9, , per sec, , I, , i, , ,,118 39x10'',,, , 9-255x10, , ,,'l8-3xlO-, , ;, , i, !, , Nitrogen, , Oxygen, , 3-31x10-8, , ,,14-93x104, , '3-11x10-9, , 4-GlxlO 4, , I, , 6, , ,,J9.44xlO-, , I9-95X10- 8, , 4-899, 5-00, , I, , Mean, , molecular velocity, , = -92 1, , r.m.s. velocity., , Collision Frequency =a Mean molecular velocity j Mean free path., , x, , xlO 9, 10, , ,,
Page 603 :
LOGARITHMS
Page 604 :
602, , PROPERTIES OF MATTER, , LOGARITHMS
Page 608 :
INDEX, Astrosuit, 391, Atmolysis, 459, Atrrospheric pressure, Measuremer, of, 359, Atomizer, The, 426, Attack, Angle of, 371, Attracted-Disc Paradox, The, 427, Atwood's machine, The, 193, Attraction, Gravitational, 227, Austen Robert, 454, , Absolute temperature, 537, units, 2, zero, 573, Acceleration, 4, 85, , Angular, 21. 22, Centrifugal, 205, Centripetal, 26, Linear, 21, down an inclined plane. 396, due to gravity, 160, , Anstin, 246, , Automatic Pilot, 99, Average kinetic and potential enerpi, of a particle in S H.M ,137, Avogadro?s Hypothesis, 539, Avogadro number, 540, Axis, Neutral, 307, of rotation, 20, , of a body inS.H M., 115, of a body rolling down an inclined, plane, 87, Adam, 487, Adam's, 247. 248, 249, After-effect, Elastic, 277, Adhesion, Force of, 394, 404, 475, Advective zone, 358-59, Ailerons, 374, Airplane, 367, 368, , B, Balance, The, 146, , Different parts of, and their, functions, 371-377, Atr screw, 373, , The Common, 147, Essentials of a good, 147, Faults in a, 182, Sensitiveness of a, 147-48, Stability (or Quickness) of a, 14, , Air ship, 367, Airy, 233, 246, 247, Altituae, Change of pressure with, 360, Amontons, 394, 403, Amplitude, 112, Aneroid barometer, 208, 359, Angle of attack, 371, , Banking, 28, of contact. 485, 488, Measurement of, 486-88, of friction, 395, of projection for maximum range, of a projectile, 34, of shear, 280,2*2,283, SoUd> Dote on, 259, Stalb'og, 372, Angstrom Unit, 3, 249, i, , 21,, , 22, , ation between Couple and, 22, impulse. 83, momentum, 83-84, Law of conservation of, 84, velocity, 20, Amiclastic surface, 309, Antonow, 512, Archimedes, 353, Principle of 154, 332*53,49, Areal velocity, 226, Arnold. 437, , Ari&tarchus, 224, Artificial horizon, 99, Association, Coert of, 517, Astronomical unit of force* 230, , Truth of, , a, 147-148, Eotvos, 210, 212, 246, , Gravity, 210, tSf trie's), 459, Methods (for the determinate, , Tomon,, , cfO),241,245, Ballistic curve,, Ballistics. 37, , 37, , Band Brake, The, 402, Bank, Turn and, indicators, 99, Banking, 28, of, 28, of railway lines and rotds, 28, Bonerji, 510, Bar pendulum, 169, Owen's modification of the, 172, Barometers, 359, Aneroid, 208, 339, Fortin's, 359, Barometric reading, <?orrectioii of, 31, Baron Eotvos, 2J2, 2M, Bartett Mack, and, 4T>, method, 479, Barton, 302, Heam, 306, Bending, Limitations c: the simple, theory of. 313, Moment of resistance tc 308, Plane of, 306, Bending of, 306, , Angle, , ., , 606
