Page 1 : if, , Unit, Dimension and Error Analysis |, , , , , , , 1.1 Units, Fundamental quanti, The physical quantities which are independent of other, quantities are called fundamental quantities., , Example: Mass, length, time ete., , , , , , Derived quan, ‘The physical quantities which are derived from fundamental, , quantities are known as derived quantities., Je: Density, volume, speed, force etc., , , , The SI system of units, , In 1971, General Conference of Weights and Measures, introduced a logical and ration: -d system of units known, as international system of units, abbreviated as SI in all, languages. In this system, there are seven fundamental, , quantities and two supplementary quanti, , , , , , Fundamental quantities and their units, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , S.No. | Physical quantity Unit | Symbol, 1, | Length metre m, 2. | Mass kilogram kg, Time second s, Temperature kelvin K, $.__| Electric current ampere A, | 6 | Luminous intensity candela | od, L_7 | Amount of substance | mole mol, Supplementary quantities and their units, S.No, Unit | Symbol, radian rad, st, , , , , , , , , , , , , , 3. [fany unit is named after a scientist, its symbol should, , capital eter. Ez, N(newton), W(watt),, K(kelvin) etc., , 4. The full name of a unit always begins with a small, letter, even if it is named after a scientist. Eg, $ N or, 5 newton., , 5. Symbols do not take plural form., , , , Some practical units, There are some practical units which are simultaneously used, with SI units., , 1 fermi = 10" m, , 1 angstrom A= 10m, , (ii) 1 nanometer (nm) = 10° m, , (iv) I micron (um) = 107 m, , 1 light year = 9.46 x 10 m, , AU) = 1.496 x 10"'m, , , , , , (vii) 1 parsec = 3.03 x 10° m, (viii) | amu = 1.66 107" kg, (ix) I quintal = 100 kg, , (x) L tonne = 1000 kg, , (xi) I lunar month = 27.3 days, , (xii) 1 shake = 10s, , The two supplementary SI units are defined, , as follows, (i) Radian (rad): 1 radian is the angle subtended at the, centre of a circle by an are equal in length to the radius, , of the circle,, , , , Thus, 9 = AE 88H? rad, radius r, , 2m rad = 360° or mrad = 180°, , , , Steradian (st) ; steradian is the solid angle subtended, at the centre ofa sphere by the area of the sphere which, is equal to the square of the radius of the sphere, , id,

Page 2 : ssics ———— | __ Force/area [ML“T- y, o B12 Pov 7, | Stress., t+ |change in dimension feted, |Change in dimension} _, g, | Strain original dimension 1, an | +, Modulus of Stress/strain Nm? [ML, % {elasticity __|, ‘ Surface Force/length | Nov? | (ML), tension, Surface area _ 1? _ ;, Ths 9°" Fai) yy, | Soxticintof | Forcexistance | NS | weeny, 7 * | viscosity locity, 1.2 Dimensions of a Physical Quanity — ASARIESES,, : 4 ical quantity are the pow’, The dimensions of a physical y hat ; :, which the fundamental quantities ar raised t0 eprese 12, | Latent heat Jket | IMT, quantity., i jons of fundamental quantities Electric Cartearxctine c [MLA], punersee Dimensional 13. | charge, SL.No. Quantity formula Electric Work/charge | 70" | (musa, wu M4 potential orV, 1,_ | Length =, 2. | Mass ) 15. | Resistance | Potential/current Pe [ML-TSA3], 3,_ [Time 7) tas, [M"L?TA],, Electric current tA] 16. | Capacitance ®), 5. | Temperature (k] henry Y, 5 17. | Inductance ———— [MUT#A7), 6. | Luminous intensity [ed] % Currentitime (A), d tof substance mol F i B, 7. | Amount of subst [mol] ae, faceacacia ee ie [MLT2A4], The supplementary quantiti, and solid angle have no dimensions. z, Planck's nergy ur, Dimensional equation 19 oetant Heqency Js IMUT"], The equation obtained by equating a physical quantity with, g q, , its dimensional formula is called dimensional equation of the, given physical quantity., Example: Force = [MLT}, «: Dimensions of force are 1 in mass, 1 in length and—2 in, time. Dimensional formula of force is [MLT=]., , Force = [MLT, , , , called dimensional, , equation,, , Dimensional formulae of some physical quantities, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Relation with Unit Dimensional, other quantities | Uait formula, Mass x :, acceleration iy Mr), Force x, displacement 4 (MLT?], f_Seeiscement|, Forcefarea | Nm? | [MLAT3], j—fecceeneth | wt [tery |, distance® | Nm?, F im, oe Tee [ae | UT, Force xtime | Ns IMT, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Four types of physical quantities, , (1) Dimensional constants: ‘These are the physic, quantities. whose values are constant but they, possess dimensions. Eg., Gravitational constant (9), Stefan’s constant (a) etc. ‘, Dimensional variables: These are the quantities, whose values are variable and they possess dimensions., Eg., Volume, acceleration, force etc. rete, Dimensionless constants: These are the PhYS¥®,, quantities whose values are constant but they 4°", Possess dimensions. Eg., 1, 2, 3, ....-» ® ete, Dimensionless variables: These are quan, Values are variable and they do not have, Eg., Angle, strain, relative density etc., , Q), , G), , , , es whos?, mension’:, , (4), , , , Principle of homogeneity of dimension®, According to this principle, the dimensions of al ie, occurring on both sides of the equations must be same, , ye terms, , , , Uses of Dimensional Analysis, , (1) Conversion of unit of a derived physical quantity from, ‘one system, , , , or O=mu,=nyy,, where 1, ”, are numerical values and u, , , , s of measurement of the physical quantity Q., , (2) To check the correctness of a physical relation. This is, based on the principle of homogeneity of dimensions,, @) D n among the physical quantities: We, , , , can derive an expression of a physical quantity if we, know the various factors on which it depends, by using, the principle of homogeneity., , Let physical quantity X depends on other quant, P, Qand R. Then we can write X= k P* QP Re., where k is a dimensionless constant, whose value can, d by experiment or otherwise, but not, s. By equating dimensions of both sides, of equation, we can get required relation among the, , , , , , , , Limitations of the Dimensional Analysis, , 1. By the method of dimensions, we cannot get any, information about the dimensionless constant., , 2. If a physical quantity depends on more than three, fundamental quantities, the formula cannot be derived., , 3. Wecannot derive the formulae containing trigonometric, , functions, exponential functions, logarithmic functions, etc., , 4. The method of dimensions can’t be used to derive the, relation like Sauron [ie., equations containing, more than one term with + or— symbol on right side]., , 5. It gives no information whether a physical quantity is a, scalar or vector., , , , Order of Magnitude, , It gives an idea about how big or how small a magnitude is., A number N’can be expressed as:, , N=nx 10, , 1f0.5 <n 5, then x will be the order of magnitude of V., , 1.3 Significant Figures, Sig measured value of'a physical quantity, Which we have confidence. Larger, igures obtained in a measurement,, the measurement., , a measurement plus we first, lncertain digit together form significant figures”,, , , , , , Unit, Dimension and Error Analysis 1.3 SSE, , For example, when we measure the length of a straight line, using a meter scale and it lies between 8.6 cm and 8.7 cm,, we may estimate it as 8.64 cm. This expression has three, , icant figures, out of these 8 and 6 are precisely known, but last digit 4 is only approximately known., , , , Rules for counting significant figures, For counting significant figures, we use the following rules:, , Rule 1: All non-zero digits are significant. For example, x = 2567 has four significant figures., , , , Rule 2: The zeroes appearing between two non-zero digits, are counted in significant figures. For example 6.028 has 4, significant figures., , Rule 3 : The zeroes occurring to the left of last non-zero digit, are NOT significant. For example 0.0042 has 2 significant, figures, , Rule 4: In a number without decimal, zeroes to the right of, non-zero digit are NOT significant. However when some, value is recorded on the basis of actual measurement the, zeroes to the right of non-zero digit become significant. For, example L = 20 m has two significant figures but x = 200 has, only one significant igure, , , , Rule 5 : In a number with decimal, zeroes to the right, of last non-zero digit are significant. For example, x= 14.00 has four significant digits., , Rule 6: The powers of ten are NOT counted as significant, digit. For example 1.4 x 10" has only two significant digits, Land 4., , Rule 7: Change in the units of measurement of a quantity does, not change the number of significant figures. For example,, suppose distance between two stations is 4067 m. It has four, significant figures. The same distance can be expressed as, , , , , , 4.067 km or 4.067 x 10° cm. In all these expressions, number, of significant digits is four., , See the following table, , , , , , Rounding off a digit, , Certain rules are applied in order to round off the, measurement., , Rule 1: lr the number lying to the right of digit to be rounded, off is ess than S, then the rounded digit is retained as such., is more than S, thea the digit to be rounded is

Page 3 : pore 3 &, eps ad = 83568, , , , Rake 3:16, , im nd off wo 6.2 wo two significa, = 62 rs olf to 8.36 0 tee SER, , waa be dropped is simply Sor 5 followed, , , , cree For exzxple, x= 6250 or x = 675 becomes x= 62, , air romdng OF, Reke 4: If, , Fp two Simmifican Gets., che Ege ta be cropped is Sor 5 followed by zero,, soz digit is raised by one if is odd, , then the prececng CET, , For exexpie, =, , = 6350 or x = 6.35 becomes x = 6.4 after, , rounding off to two Significant digits., , See the following table, , Messared [After rounding offtothree| Rule |, | significant digits | number, , , , , , Algebraic operations with significant figures, Inaddition, subtraction, multiplication or division, inaccuracy, in the mezsurement of any one variable affects the accuracy of, final result. Hence, in general. the final result has significant, figures according to the rules given below, , wg, , Gy, , Addition and Subtraction: The number of decimal, pieces in the final result (of any of these two operations), bas to be equal to the SMALLEST NUMBER OF, DECIMAL PLACES in any of the terms involved in, calculation., Example: 1.2 +345 + 6.789 = 11.439 = 11.4, Here. the least number of significant digits afier the, decimal is one. Hence the result will be 11.4 (when, rounded off to smallest number of decimal places)., Example: 12.63-102=243=24, Multiplication and Division: In these operations, the, umber of significant figures in the result is same as the, SMALLEST NUMBER OF SIGNIFICANT FIGURES,, in any of the factors. ;, Example: 1.27 1.3=156=16, The least number of significant digits on the measured, values is 2. Hence the result is 1.6 (when rounded off to, smallest number of significant digits), , , , , , , Example: 12 «36.72 = 44.068 = 44, 1100 ms*, : =107.8431=108, Example: 197 ms, , =, 1.4 Errors in measurement “i, , = Resolution” stznds for least count or the minimum reading, which an instrument can read. |, , ‘Accuracy? An instrument is said to be accurate if the physica}, Gonniy measured by it resembles very closely (0 its true, value., , Precision: An instrument is said to have high d, , of precision, if the measured value remains unchanged, howsoever, lage number of times it may have been, , , , There are many causes of errors in measurement. Errors, may be due to instrumental defects, ignoring certain facts,, carelessness of experimenter, random change in temperature,, pressure, humidity, etc. When an experimenter tries to reach, aocurete value of measurement by doing large number, of experiments, the mean of a large number of the results, , | repeated experiments is close to the tue value., , Calculation of Magnitude of Errors, @ Truevalue:1fa,,2..4, +--+ a, are the observed values, of a measurement, then true value of measurement is, the mean of these observed values., G,+0,4+4,+..+4,, Fo Ogg = Ogg = y= AA, , a iy, =-ya, ae, (i) Absolute error: The absolute errors in various, individual measured values are found by subtracting, the observed value from true value. Thus,, Aa, =a,—a,, ha, = a,—a,, Aa, = 0,— Oy v0r----~, =a,—4,, The absolute error may be positive or negative or Zero., (iii) Mean absolute error: The arithmetic mean of the, magnitudes of different values of absolute errors 1S, known as the mean absolute error., ¢. Mean absolute error is, ao = Wail + $Aat + 1A t +, , tec 7, , Aa,, , , , Aa,__ is also represented as Ag, The final result of a measurement can be written 3, a=0,+ha, This implies that value of ‘a’ is likely to lie between, a, + Ba and a,—Za., , (iv) Relative error or fractional error: The ratio of the, mean Value of absolute error to the true value is know?, as the ‘mean relative error’,, , Propagation of Errors, , , , , Mean relative error, , ___Mean absolute error re, , Mean value of measurement ~ g,, , If do and 4b ar, ? are small then 4. =, and thos can be ignored. 7 MP Till be very small, , When expressed in terms of percentaze, relative error is, called “relative percentage error™. Hence, fs es, ee 5109 = 100, , Pree %, , , , , , Percentage error =, Thus the maximum fractional, , cum exer is t i, equal to the sura of factional eners he eas, , , , , , quantities, Suppose we want to get the volume of a sphere, veda (4) Error in quotient (or division), a, This involves multiplication of radius three times. The i, measurement of radius has some error, then what will ois, be error in calculating the volume of sphere? The error in sae rs, final result depends on the individual measurement as weil a, as the mathematical operations involved in calculating the af 122, result. To calculate the net error in the result, we should a 2, study the propagation of errors in the several mathematical =2), operations. é, (1) Error in addition 7, Let x= a+ b. suppose + Aa is absolute error in a and =H-2p-¥), + Abis absolute error in b, then we have os 6, x+Ar =(a+Ab)+(b+Ab) =92S [=F )=, =(a+b)+(Aa+ Ab) ble 4}, Ar=(a+b)+(Aa+Ad)—x ac V,_ 4b), cs Ar=+(Aa+As) [ee x=a+8] aeeraz 122 [= ), Thus, “The maximum possible error in the addition of BOT Ge ae Ap, quantities is equal to the sum of their absolute exrors”. Sale SSS, erate ig SE _ far as ‘, teenorinsis SE x 100 = +[22*2*] « 100 As the terms 22 and © are small, their product, @) Errorin subtraction ‘hod 2 5, Letx=o=b 32.3 can be neplecind Thus masimum facto, 2, , @), , x+Ar =(a+Aa)—(b+ Ab), =(a—5)+(Aa F Ab), Ar=(a—6)+ (Aa F Ab)-x +, “ Axv=+(Aa F Ab) 7, But for maximum possible error Aa and A> must be of And maximum possible percentage error i x is, see ES, Hxto0=-{ B+? 100, wAx= t(Aa + Ab) x a bf, , maximum possible error in subtraction of (5) Exrer in the power of quantity, , is equal to the sum of their absolute errors. Letr=o, , %eervor in.x is Sas092e( at, , , , , , 100, , r :, rearstazauy vets], @, , , , x a-b, Error in product, Letx=ab -o(1272], a+ Ar = (at Aa) (b+ Ab) xe Ar Aa, =ab+ adh t bAat Aa Ab or _ altpe

Page 4 : 1.6 Physics, , ae A?, betes, , Maximum percentage error in x is, , Sx100= so( S00), ‘The fractional error in the quantity raised to power P is, ptimes the fractional error in that quantity., , General case, shal then the maximum possible fractiong}, , a, If x=, ¢ Aa Ab. Ac, «AKL + g—+r—, erorinsis =] p atte =|, , Je percentage error is, d the maximum possibl, , ‘s 100+ % sr00+r x00, %x100= 4 pax q b, , EXERCISES, , , , , , , , SECTION A: TOPICWISE QUESTIONS |, Basic Concept of Unit 9. The unit of angular acceleration in the SI system is,, x =, 1. Which of the following is smallest unit? @) Nig) Oat ae, (a) (b) Angstrom (6) ads (@) mkg, (©) Fermi (@) Meme 10. Temperature can be expressed as a derived quantity in, terms of, 2. Which of the following is not the unit of energy? Geeagiant es & Mieinddne, , 2, , (a) Calorie (b) Joule, (c) Electron volt (d) Watt, , . A watt is:, (a) kg mvs* (o) kgm/s?, (©) kg mvs (@) kg m/s?, , J. Which of the following is not equal to watt?, , (6) Ampere x volt, (@) Ampere/volt, , If the acceleration due to gravity is represented by unity, in a system of units and one second is the unit of time, the, unit of length is:, (@) 98m, , (©) 98m, , (a) Joule/second, (c) (Ampere)? x ohm, , (b) Im, (d) 0.98 m, , ._ Newton-second is the unit of, , (b) Angular momentum, (a) Energy, , (a) Velocity, (©) Momentum, , / Which of the following is not a unit of energy?, , , , (a) Wes (b) kg-nvsee, (c) Nem (a) Joute, , A suitable unit for gravitational constant is, (a) kg msec! (b) Nm’! see, (c) Nm? kg? (d) kg msee, , . The unit of potential energy is, , (c) Length, mass and time (d) None of these, Erg-m'' can be the unit of measure for, , (a) Force (6) Momenturn, (©) Power (a) Acceleration, , , , (b) g(cm/sec)*, , (@) g(emsec*), 7 (d) g(cm/sec), , (©) g{em*isec), , |. Which of the following represents a volf?, , (b) wart’ampere, , (a) joule’second, (@) coulombjoule, , (c) wart’coulomb, , , , If the unit of length and force each becomes four times,, then the unit of energy becomes, (a) 4 times (b) 8 times, (©) 16 times (4) 16 times, . Ampere-hour is a unit of, (a) Quantity, (b) Strength of electric current, (&) Power, (@) Energy, , If u, and uy are the units selected in two systems of, measurement and 7, and n, their numerical values, then, (a) muy = nyu (b) mu; + y= 0, , (©) myn = nyuy (d) (4, +m) = (+)

Page 5 : 1.8 Physics, , , , ermine he Young's modulus of @ WH the formula, fa wires, , , , 7. To det, a pax; where £ = lensth A= area of CrOss, ip =o ay :, de in length of the wire, , = change in len, section ofthe ites Af Sethe conven fo! ©, , when stretched, , , , , , change it from CGS to MKS oe ., @ aa (oo, , 18, Young's modulus of a material i the same unit as, , 19, In CGS systemthe apis ft fa * ta i es In, , another system where, , ae tilogram, metre and minute, tbe magnitude of the, , force is, (2) 0.036 (b) 0.36, (c) 3.6 (a) 36, , jue is found, , |. A physical quantity is measured and its valt, es ens ee unit. Then, , to be nu where = numerical value and u =, which of the following relations is true, (a) new (b) new, , 1, (©) n= Vu @ ner, + b@, where x is the distance travelled by the, , , , 2. Ifx=at, body in kilometres while is the time in seconds, then the, unit of bis, (a) kas (b) km-s,, , (o) kms? (d) km-s?, , 22, In S=a + bt + cf, S is measured in metres and 1 in, seconds. The unit of c is, (a) ms*, (© mst, , (ym, (a) ms?, 23. In the equation (r + 4) (V—b) = constant, the unit of, ais, (a) dyne xem? (b) dyne x cm*, (co) dyne xcm* (d) dyne x cm?, 24. If in a system the force of attraction between two point, , masses of | kg each situated 1 km apart is taken asa unit______physical quantity —__, , force and is called notwen (newton written in reverse, order). If G = 6.67 x 10 N-m? kg? in ST units, the, relation of newion and notwen is:, (a) 1 notwen = 6.67 x 10 newton, (b) 1 newton = 6.67 x 1077 notwen., ~~ (ey L notwen = 6.67 x 107!7 newton, , List-1, List-I, yt, | 1 1Ninnew system A ap?, II. 1 innew system B atply, Cc a! py, , lL, 1 pascal (SI unit of pressure), in new system, TV, @ ACME in watt, , Now, match the List I, choice from given codes., , Codes, (a) I-B, IIA, mI-C, IV—D, , (b) IC, II-B, IID, IV—A, () I-A, -B, IID, IV—C, (@ 1-C, 1-2, IA, IV—-B, , D &py, , ith List Il and mark the correct, , , , Concept of Dimensional Formula, , 26. Select the pair whose dimensions are same, (a) Pressure and stress, (b) Stress and strain, (c) Pressure and force, (d) Power and force, 27. Dimensional formula of magnetic flux is, @ Mer?At (b) ML°T 7A?, (©) META? @ MPT, 28. Inductance L can be dimensionally represented as”, (a) MUTA (bo) MLT*A", (©) ML*T7A?, 29. Dimensional formula for latent heat is, , @) MLT? (b) MLT?, (c) MLT? @ MT", 30. Dimensional formula for angular momentum is, (a) MPT? (o) MLT*, (©) MIT" (@) eT?, 31, Dimensional formula of capacitance is, @ ere (b) M>T'a?, (©) MLT*# (a) ML? TAA?, 32. Dimensional formula ME'T? does not represent the, @ Omani, (b) Stress, (c) Strain, (d) Pressure, , 33, Two quantities A and B have different dimension’., Which mathematical operation given below is physic#!, , (@) 1 newton= 6.67 x 10 notwen meaningful?, 25. Suppose two students are trying tomake anew measurement (a) AIB (b) A+B, system so that they can use it like a code measurement () A-B (d) None, , system and others do not understand it. Instead of taking, I kg, 1 mand 1 sec as basic units they took unit of mass, as akg, the unit of length as im and unit of time as 7, second. They called power in new system as ACME,, , 34. A force F is given by F = ai + 7, where 1 is time. What, are the dimensions of a and 6?, (a) MLT? and M?>T*, , (c) MLT" ff (b) MET? and MLT,*, a and ML:, , () MLT* and MLT', , , , 35. Which pair has the same dimensions?, (a) Work and power, , (b) Density and rel, , (c) Momentum and impulse, (A) Stress and strain, , 36. The dimensional formula for impulse is same as the, dimensional formula for, (a) Momentum, (b) Force, (©) Rate of change of momentum, (d) Torque, 37. Which of the following is dimensionally correct?, (a) Pressure = Energy per unit area, (b) Pressure = Energy per unit volume, (c) Pressure = Force per unit volume, (@) Pressure = Momentum per unit volume per unit time, , 38. The equation of state of some gases can be expressed, , , , is the absolute temperature and a, b, R are, , , , the volume,, , , , constants. The dimensions of ‘a’ are, (@) MT? (b) MET?, © MUP @ MiP, , 39, The frequency of vibration f of a mass m suspended from, a spring of spring constant K is given by a relation of this, type f= Cmi'K’; where Cis a dimensionless quantity. The, values of x and y are, , , , 40. The quantities A and B are related by the relation,, m= AIB, where m is the linear density and A is the force., The dimensions of B are of, (a) Pressure (6) Work, (©) Latent heat (d) None of these, , 41. The velocity of a freely falling body changes as gh?, where g is acceleration due to gravity and / is the height., The values of p and g are, , 1 11, @ 4 $4, 2 o 212, ©}, oda @11, 42. Which one of the following pairs does not have the same, , , , dimensions?, , (@) Work and energy, , (b) Angle and strain, , (©) Relative density and refractive ind, ex, , (@) Planck constant and energy, , , , , , 43. An athletic coach t, , old his tea, equals power, Wh; im that muscle time, (@) Mp2" Simensions does he view for mic, (c) MLP (b) Meer?, , @L, 44. t P represents radiation Pressure, c re, eee Tepresents radiation energy strikin; it, i ond, then non-zero integers aa, that P*Q’C* is dimensionless, are” Ma = SUCH, z=-], =1,2=1, , Presents speed of, , , , (b) x=1,y=-1,2=4, (@ x=1y=, , , , 45. Force F and density dare related as F =, (a) M7L"7T*, M71? respectively, (b) A°7L'77?, M1171? respectively, (c) M?L"?7?, "723? respectively, (@) 71777, 422? respectively, 46. The frequency of vibration of string is given by, ve =fFy. Here pis number of segments, Sn -gments in the sting, and /is the length. The dimensional formula for m will be, , , , (a) (MILT) (b) 7}, © Dor] @ DLP], 47. The dimensions of in the equation P= ant where, =, , Pis pressure, x is distance and / is time, are, @ BFL?) () [MT], © ur) @ WT], , 48. Ifthe time period (7) of vibration ofa liquid drop depends, on surface tension (5), radius (r) of the drop and density, (p) of the liquid, then the expression of Tis, , (a) T=kypris (b) T=ky, , , , , , (©) T=kyY prs? (@) None of these, , 49. Position of a body with acceleration ‘a’ is given by, x= Ka’, here ris time. Find values of m and n., (a) m=ln=l (&) m=1.n=2, (©) m=2,n=1 (d) m=2.n=2, 50. Density ofa liquid in CGS system is 0.625 g/cm, is its magnitude in SI system?, , °. What, , (a) 0.625 (b) 0.0625, (©) 0.00625 (@) 625, 51. The velocity of a body is given by the equation, veered?, 7, The dimensional formula of d is 7, (a) (PLP) () [ME ul, (©) GPL] (@) (MLT"]