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Unit &Dimensions, FUNDAMENTAL UNITS, Fundamental units are those units, which can neither be derived from one another, nor can they be further, resolved into any other units., Example: mass, length ,time Temperature, Electric Current, Luminous intensity, Quantity of matter, DERIVED UNITS, The units of all such physical quantities, which can be expressed in terms of the fundamental units of mass,, length and time, are called derived units., Example: speed, acceleration, momentum, force, work, etc., dis tan ce cov ered, unit of dis tan ce i.e. length, metre, Therefore, unit of speed =, Speed , ,  ms 1, time taken, unit of time, sec ond, S.No., 1., 2., 3., 4., 5., 6., 7., S.No., 1., 2., , Basic physical quantity, Length, Mass, Time, Temperature, Electric Current, Luminous intensity, Quantity of matter, Supplementary physical, quantity, Plane angle, Solid angle, , Unit/dimension, Meter[L], Kilogram[M], Second[T], Kelvin[K] or [θ], Ampere[A] or[I], Candela[cd] or [J], Mole [mol] or[N], , Symbol/abbreviati, on, m, kg, s, K, A, cd, mol, , Unit, , Symbol/abbreviation, , radian, steradian, , rad, sr, , SYSTEMS OF UNITS, 1. cgs system. This system of units was set up in France and it is based on centimeter, gram and second as, the fundamental units of length, mass and time respectively., 2. fps system. This system of units, also known as British system of units, is based on foot, pound and, second as the fundamental units of length, mass and time., 3. MKS system. This system was also set up in France. It makes use of metre, kilogram and second as the, fundamental units of mass, length and time.., Standards of fundamental units, 1. Metre. One metre is defined as to be equal to 1,650,763.3 wavelengths in vacuum of the orange-red, coloured radiation emitted by krypton having mass number 86., In 1983, one metre was defined as the length of the path traveled by light in vacuum during a time interval, 1, of, of a second., 299,792,458, 2. Kilogram. kilogram is the mass of a platinum-iridium cylinder kept in the International Bureau of, Weight and Measures at Sevres, near Paris, France., 3. Second. 9, 192, 631, 770 vibrations hyperfine levels of cesium-133 atom in the ground state., 4. Ampere : It is the current which when flows through two infinity long straight conductors of negligible, cross-section placed at a distance of one metre in air or vacuum produces a force of 2  10 7 N/m between, them., 5 Candela : It is the luminous intensity in a perpendicular direction, of a surface of 1/600,000 square metre, of a black body at the temperature of freezing platinum under a pressure of 1.013  105 N / m 2., 6. Kelvin : It is the 1/273.16 part of the thermodynamic temperature of triple point of water., 7. Mole : It is the amount of substance of a system which contains as many elementary entities as there are, in 1 gm of 6 C 12 ., Dimensional formulae:

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Representation of units in the power of symbols (M,L,,T,θ,A, mol, cd )of fundamental units is the, dimensional formula of the physical quantity, USES OF DIMENSIONAL ANALYSIS, (a) To check the validity of a given equation or formula:, (b) To derive the formula for given physical quantity:, (c) To convert one system of physical units into another:, , See Examples in class, See Examples in class, See Examples in class, , LIMITATIONS OF DIMENSIONAL ANALYSICS, (1) While deriving a formula the proportionality constant cannot be found., (2) The formula for a physical quantity depending on more than three other physical quantities cannot be, derived. It can be checked only., (3) The equations of the type v = u  at cannot be derived. They can be checked only., (4) The equations containing trigonometrical functions sin  , cos  , etc., logarithmic functions, , , , 2, , , , (log x, log x3 etc.) and exponential functions e x , e x etc. cannot be derived. They can be checked only, , Single choice questions (Level 0), 1. The unit of atomic mass is:, (a) O = 16.0000, (b) O16 = 16.0000, (c) C = 12.0000, (d) C12 = 12.0000, 2. One micron is:, (a) 10-9 m, (b) 10-12 m (c) 10-6, (d) 10-15 m, 3. Light year is:, (a) Light emitted by the sun in one year, (b) Time taken by light to travel from the sun to the, earth, (c) The distance traveled by light in one year, in, free space, (d) The time taken by earth to go round the sun once, 4.How many wavelengths of Kr86 are these in one metre:, (a) 1553164.13, (b) 1650763.73, (c) 2348123.73, (d) 652189.63, 5. One sec is defined to be equal to:, (a) 1650763.73 periods of krypton clock, (b) 652189.63 periods of krypton clock, (c) 1650763.73 periods of cesium clock, (d) 9192631770 periods of cesium clock, 6. Which of the following is not the unit of length:, (a) Micron (b) light year (c) angston (d) radian, 7. Indicate which pair of physical quantities given, below has not the same units and dimensions:, (a) momentum and impulse, (b) Torque and angular momentum, (c) Acceleration and gravitational field strength, (d) Pressure and modulus of elasticity, 8. The unit of impulse is the same as that of:, (a) Energy, (b) Force, (c) Angular momentum, (d) Linear momentum, 9. Dyne-sec stands for the unit of:, (a) Force, (b) Work, (c) Momentum, (d) Angular momentum, 10.The joule х s is the unit of:, (a) Energy, (b) Momentum, , (c) Angular momentum, (d) Power, 11. Which one of the following quantities has not, been expressed in power unit:, (A) Stress/Strain = N/m2, (b) Surface tension = N/m, (c) Energy = kg-m/s, (d) Pressure = N/m2, 12.Which of the following physical quantities is, dimensionless?, (a) angle (b) strain (c) specific gravity (d) all of these, 13. Two physical quantities of which one is a victor, and other is a scalar having the same dimensional, formula are:, (a) work and energy, (b) Torque and work, (c) Impulse and momentum (d) Power and pressure, 14. Which of the following physical quantities is, dimensionless?, (a) angle (b) strain (c) specific gravity (d) all of these, 15. The dimensional formula for angular momentum is:, (a) [M0L2T-2] (b) [ML2T-1] (C) [MLT-1] (d) [ML2T-2], 16. Plank’s constant has the dimensions of:, (a) Energy, (b) Momentum, (c) Frequency, (d) Angular momentum, 17. The dimensional formula for Plank’s constant is:, (a) [ML2T-1] (b) [ML2T-3] (c) [ML-1T-2] (d) [MLT-2], 18. The dimensions of gravitational constant G are:, (a) [MLT-2] (b) [ML3T-2] (c) [M-1L3T-2] (d) [M-1LT-2], 19. The dimensional formula for modulus of rigidity is:, (a) [ML-1T-1] (b) [ML-2T2] (c) [M-1LT-1] (d) [ML-1T-2], 20. The time dependence of a physical quantity p is, given by P = Poexp(-αt2) [ where α is a constant and, t is time]. The constant α:, (a) is dimensionless, (b) has dimensionless, (c) has dimensions [T-2] (d) has dimensions of P, 21. The velocity of water (v)waves may depend on their, wavelengths λ , the density of water ρ and the, acceleration due to gravity g. The method of, dimensions gives the relation between these, quantities as:, (a) v2 α g-1λ-1 (b) v2 α gλ (c) v2 α gλρ (d) v2 α g-1λ-3, 22. P represents radiation pressure, c represents speed of, light and S represents radiation energy striking

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unit area per sec. The non zero integers x, y and z, such that PxSycz is dimensionless are:, (a) x = 1, y = 1, z = 1, (b) x = -1, y = 1, z = 1, (c) x = 1, y = -1 z = 1 (d) x = 1, y = 1, z = -1, 23. A system has basic dimensions as density [D],, velocity [V] and [A]. The dimensional, representation of force in this system is:, (a) [AV2D] (b) [A2VD] (c) [AVD2] (d) [AVD], 24. The dimensional formula for calorie is:, (a) [M1L2T-2] (b) [M2L1T-2] (c) [ML-2T2] (d) [M1L1T-1], 25. The dimensional formula for latent heat is:, (a) [M0L2T-2] (b) [ML2T-2] (c) [MLT-2] (d) [ML2T-1], 26. The dimensional formula for coefficient of thermal, conductivity is:, (a) [MLTK] (b) [MLT-2] (c) [MLTK-1](d) [MLT-3K-1], 27. Which of the following quantities can be written in, SI units in kg m2 A-2 s-3:, (a) Resistance, (b) Inductance, (c) Capacitance, (d) Magnetic flux, Level 1, 1., , Let [0 ] denote the dimensional formula of the, permittivity of vacuum. If M = mass, L = length, T, = time and A = electric current, then: [JEE MAINS, 2013], (a) [0 ]  [ M 1L3T 4 A2 ], (b) [0 ]  [ M 1L2T 1 A2 ], 1 2, , 1, , (c) [0 ]  [ M L T A], (d) [0 ]  [ M 1L3T 2 A], 2., , The dimension of magnetic field in M, L, T and C, (Coulomb) is given as, 1, , (a) MLT C, 1, , 1, , 2, , (b) MT C, , 1, , 2, , 2, 1, , (c) MT C, (d) MT C, 3. Which of the following units denotes the, 2, , 2, , dimensions ML / Q , where Q denotes the electric, charge?, (a) Weber (Wb), (b) Wb/m2, (c) Henry(H), (d) H/m2, 4. Out of the following pair, which one does NOT, have identical dimensions is, a., , b., c., d., 5., , impulse and momentum, moment of inertia and moment of a force, work of torque, Which one of the following represents the correct, dimensions of the coefficient of viscosity?, , (a) ML1T 2, , (b) MLT 1, , (c) ML1T 1, , (d) ML2T 2, , 6., , 1, , where symbols have their, 0 0, , Dimensions of, , usual meaning, are, 1, , 2, , 2, , 1, , 2, , (c) [ L T ], 7., , 2, , (b) [ L T ], , (a) [ L T ], , (d) [ LT ], , a., b., c., , The physical quantities not having same dimensions, are, torque and work, momentum and Planck’s constant, stress and Young’s modulus, , d., , speed and  0 0 , , 8., , 1/2, , If the capacitance of a nano-capacitor is measured in, terms of a unit ‘u’ made by combining the electronic, charge ‘e’,Bohr radius ‘a0’, Plank’s constant ‘h’ and, speed of light ‘c’ then:, , (a) u , , e2c, ha0, , (b) u , , hc, e2 a0, , (c) u , , e2 a0, hc, , (d) u , , e2 h, ca0, , 9. The dimensions of angular momentum, latent heat and, capacitance are, respectively., 2 1 2, 2 2, 1 2 2, (a) ML T A , L T M L T, (b) MLT 2 L2T 2 M 1L2T 4 A2, 2, , 1, , 2, , 2, , 2, , 1, , 2, , 2, , 2, , 2, , (c) ML T , L T , ML TA, , 1 2, , 4, , 2, , (d) ML T , L T , M L T A, , ………….by Praveen Gupta, , angular momentum and Planck’s constant, , ERRORS OF MEASURMENTS, MEASUREMENT OF LENGTH (using vernier screw gauge) & ERROR ANALYSIS, VERNIER CALLIPERS:, It is a device used to measure accurately up to (1/10)th of a millimetre. It was designed by a French Mathematician, Vernier, and hence the instrument is named Vernier after the name of its inventor., Vernier Callipers comprises of two scales, the vernier scale V and main scale S. The main scale S is fixed but the, vernier scale, which is also called auxiliary scale, is movable. The vernier scale slides along the main scale as shown

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in the figure. The division of vernier scale are usually a little smaller in size than the smallest division on the main, scale, HOW TO MEASURE, Exp. - If a student measures the outer diameter of cylinder by the errorless (No error like zero error etc.) Vernier, scale as shown in figure . Now by seeing the figure, find the value of diameter of cylinder. [Given : least count = 0.1, mm], , 0 M.S. 1, , Length =, , V.S., , 1.3 +X, {which can you seen easily}, x=, , {, , {can’t measure clearly}, , least number of vernier’s division, , That meets with main scale’s division =, , } {vernier constant}, x, , 6 x 0.1mm = 0.6 mm or 0.06 cm, 1.3 cm + 0.06 cm = 1.36 cm, , [Thus, actual length of diameter is 1.36 cm], Surprisingly,, see in the class, , x = [no. of columns x least count], , HOW?, , IMPERFECTION IN DEVICE:, In your apparatus, if helping scale’s zero line not coincide with main scale’s zero line, means here, some correction is require (i.e. ZERO ERROR). For getting correct value of object, we remove zero error by adding or, subtracting in measured value., As if vernier scale’s zero line right to main scale is positive error.That means it give already a, positive value without using object, so you have to subtract this value after measuring the reading of object to get, actual length.And if zero line is left to main scale, thus negative error is present and you have to add this value for, getting actual length. As shown below,these are the initial conditions when there is no object in between both pins., Negative Error, 0, , Positive Error, , 1, , 0, , M.S., , 1, , M.S., , 0, , 1, , V.S., , V.S., 0, , 1, , Positive Correction, Negative Correction, , SCREW GAUGE, Fundamentally this instrument is designed on the principle of micrometer screw and hence it is known as Screw, gauge. Like Vernier Callipers, this instrument also consists two scales. One is main scale, which is also known as, pitch scale. The other scale is circular scale, also called as head scale. The main scale remains stationary while as the, circular scale moves to end fro over the main scale with the motion of screw. The head scale contains 50 or 100 equal, HEAD, parts., SCREW, , SCALE, , MAIN, SCALE, , PITCH :, , RATCHET, ARRANGEMENT, , The distance covered on main scale by head scale on one complete rotation of Head Scale.

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Pitch , , dis tan ce cov ered (D), No. of rotation (N), , LEAST COUNT:, Least Count , , Pitch length, No. of divisiononHeadScale, , Length Error are measured in similar way as in vernier calliperse, , SIGNIFICANT FIGURES, ‘Significant figures’ in the measured value of a physical quantity tell the number of digits in which we have, confidence. Number of significant figures in a physical quantity depends upon the least count of the instrument used, for its measurement., COMMON RULES FOR COUNTING SIGNIFICANT FIGURES, Rule 1., , :, , All non zero digits are significant., , For example :, , x = 1234 has four significant figures. Again x = 189 has only three significant digits., , Rule 2., , All zeros on the right of non-zero digits are NOT significant., , :, , For example :, , x = 1000 has only one significant figures. Again x = 378000 has three significant figures., , Rule 3., , All zero occurring between two non zero digits are significant., , :, , For example :, , x = 1007 has four significant figures. Again x = 1.0809 has five significant figures., , Rule 4., , :, , In a number less than one, all zeros to the right of decimal point and to the left of a non zero digit, are NOT significant., , For example :, , x = 0.0084 has only two significant figures. Again x = 1.0084 has five significant figures. This is, on account of Rule 2., , Rule 5., , All zeros on the right of the last non zero digit in the decimal part are significant., , :, , For example :, , x = 0.00800 has three significant figures 8, 0, 0. The zeros before 8 are not significant. Again 1.00, has three significant figures., , Rule 6., , All zeros on the right of the last non zero digit become significant, when they come from a, measurement., , :, , ROUNDING OFF, , While rounding off measurements, we use the following rules by convention :, Rule 1., , :, , For example :, , If the digit to be dropped is more than 5, then the preceding digit is raised by one if less, than 5 . preceding digit remains unchanged ., x = 6.87 is rounded off to 6.9. Again x = 12.78 is rounded off to 12.8.

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Rule 2., , :, , If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit, is raised by one., , For example :, , x = 16.351 is rounded off to 16.4. Again x = 6.758 is rounded of to 6.8., , Rule 3., , If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is left, unchanged, if it is even. if it is odd make it even., , :, , For example :, , x = 3.250 becomes 3.2 on rounding off, Again x = 12.650 becomes 12.6 on rounding off, , For example :, , x = 3.750 is rounded off to 3.8. Again x = 16.150 is rounded off 16.2., , ARITHMETICAL OPERATIONS WITH SIGNIFICANT FIGURES, , Any result is calculated by compounding (i.e. adding/subtracting/multiplying/dividing) two or more, variables, which might have been measured with different degrees of accuracy. Hence, in general, the final, result of compounding any number of variables shall have significant figures corresponding to their number, in the least accurate variable involved., (a), , Addition and subtraction, , In addition or subtraction, the number of decimal places in the result should equal the smallest number of, decimal places of terms in the oration., For example, the sum of three measurements of length; 2.1 m, 1.78 m and 2.046 m is 5.926 m,, which is rounded of to 5.9 m (up to smallest number of decimal places)., Similarly, if x = 12.587m and y = 12.5m, then x – y = 12.587 – 12.5 = 0.087 m., Which is rounded of to 0.1 m., Note : In subtraction of quantities of nearly equal magnitudes, accuracy is almost destroyed. For example, if, x = 12.87 m and y = 12.86m, then x – y = 12.87 – 12.86 = 0.01 m, The difference has only one significant, figure, where as x and y have four significant digits each., To avoid this, we try to measure directly the difference between two nearly equal quantities instead of, measuring the quantities and calculating their difference., (b), , Multiplication and Division, , In multiplication and division, the number of significant figures in the product or in the quotient is the same, as the smallest number of significant figures in any of the factors., For example, suppose x = 3.8 and y = 0.125. Therefore, xy = (3.8) (0.125) = 0.475. As least number of, significant figures is 2 (in x = 3.8). Therefore, xy = 0.475 = 0.48 is rounded off to two significant figures., Again, suppose we have to write the quotient to correct number of significant figures, when we divide 9500, by 10.23., Now, 9500 / 10.23 = 928.64125

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9500 has minimum number of significant figures, therefore, the quotient can have only two significant, digits. On rounding off, we obtain the quotient = 930, Note. 1 : If we were to divide 9500 m by 10.23 m, the quotient will have four significant digits, a per Rule, 6. In that case, quotient = 928.6., Note. 2 : In any complex multi-step calculation, we retain in intermediate steps, one digit more than the, significant digits, and round off to proper significant figures at the end of the calculation., Error analysis under measurment, (a), , Absolute error in the measurement of a physical quantity is the magnitude of the difference Between the true, value and the measured value of the quantity., Let a physical quantity be measured n times. Let the measured values be a 1, a2, a3, ……an. The arithmetic, mean of these value is, a  a2 ..... an, am  1, n, ……(1), or, , am , , 1, n, , in, , , , ai, , i1, , ..….(2), , Usually, am is taken as the true value of the quantity, if the same is not know otherwise., By definition, absolute errors in the measured values of the quantity are, a1 = am – a1, a2 = am – a2, , an = am – an, The absolute errors may be positive in certain cases and negative in certain other cases, (b), , Mean absolute error. It is the arithmetic mean of the magnitudes of absolute errors in all the measurements, of the quantity. It is represented by a .Thus, , a , , | a1 |  | a2 | ......| an |, n, , a , , 1, X, n, , in, ,  | ai |, , i1, , …….(3), , Hence the final result of measurement may be written as, , a  am  a, This implies that any measurement of the quantity is likely to lie between (am  a) and (am  a), , (c), , ., , Relative error or Fractional error. The relative error or fractional error of measurement is defined as the ratio, of mean absolute error to the mean value of the quantity measured. Thus, , Relative error or Fractional error , , meanabsoluteerror a, , mean value, am, , When the relative /fractional error is expressed in percentage, we call it percentage error. Thus, Percentage error , , a, X 100%, am

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PROPAGATION OR COMBINATION OF ERORRS, As is known, the result of an experiment is calculated by performing mathematical operations (like addition,, subtraction, multiplication, division, raising to some power etc.) on several measurements, which have different, degrees of accuracy., (a), , Error in sum of the quantities, Suppose, , x=a+b, , Let a = absolute error in measurement of a, b = absolute error in measurement of b, x = absolute error in calculation of x, i.e. sum of a and b, x + x = (a + a) + (b + b), = (a + b) + a + b, =x, + a + b, or + x = + a + b, x = + a, , , , b, , The four possible values of x are (+ a + b), (+ a - b), (- a + b), (- a - b), Therefore , the maximum absolute error in x is, x = + (a + b), , Hence maximum absolute error in sum of the two quantities is equal to sum of the absolute errors in the, individual quantities., (b), , Error in difference of the quantities, Let, x=a–b, , The four possible values of x are (+ a + b), (+ a - b), (- a + b), (- a - b), Therefore the maximum absolute error in x is, x = + (a + b), I.e. maximum absolute error in difference of two quantities is equal to sum of the absolute errors in the, individual quantities., (c.), , Error in product of quantities, Let x = a x b, a=(a+ a ) , b=(b+ b), a x b =(a+ a ) x (b+ b)=ab+ ab+ ba+ ab, , (ab is very small can be taken 0), , max possible absolute error x = ab+ ba, Hence maximum possible value of fractional error, , x,  a b ,    , x, b ,  a, Hence maximum fractional error or relative error in product of quantities is equal sum of the fractional or, relative errors in the individual quantities., (d), , Error in division of quantities, Let, , x=a/b, -1, -1, -1, a x b =(a+ a ) x (b+ b) =(a+ a ) x (b+ b) =(a+ a ) x (b  b), , =ab+ ab  ba+ ab, , (ab is very small can be taken 0), , (using binomial expression)

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Therefore, the maximum value of fractional error, , x,  a b ,  , , , x, b ,  a, , Hence the maximum value of fractional or relative error in division of quantities is equal to sum of the, fraction/relative errors in the individual quantities., (e), , Error in quantity Raised to some power, an, x m, b, Let, Therefore, maximum value of Fractional error or relative error in a quantity raised to power (n) is n times the, fractional/relative error in the quantity itself., , x, b ,  a,   n, m, , x, b ,  a, CLASSIFICATION OF ERRORS, The errors of measurement can be divided into the following three types :, 1., Systematic Errors, 2., Random Errors, 3., Gross Errors, (a), Systematic Errors are the errors whose causes are known. Such errors can, therefore be minimized. For, example, Instrumental errors may be due to imperfection of design and erroneous manufacture of the instruments., Often, there may be zero error in the instrument., Personal error, Error due to imperfection etc. are also some systematic errors., (b), , Random Errors. These errors may arise due to a large variety of factors. The causes of such errors are,, therefore not known precisely. Hence it is not possible to eliminate the random errors., The random errors can be minimized by repeating the observation a large number of times and taking the, arithmetic mean of all the observations. The mean value would be very close to the most accurate reading., , (c), , Gross Errors. These error arise on account of shear carelessness of the observer. For example :, (i), Reading an instrument without setting it properly., (ii), Taking the observations wrongly without caring for the sources of errors and the precautions., These errors can be minimized only when the observer is sincere and mentally alert., , …..By: Praveen Gupta, , Single choice questions (Level 0), 1. The length, breadth and thickness of a strip are (10 ±, 0.1) cm, (1 ± 0.01) cm and (0.100 ± 0.001) cm, respectively. The max percent error in its volume, will be, (a) ± 0.03%, (b) ± 0.111 %, (c) ± 0.012 %, (d) none of these, 2. In an experiment of a simple pendulum, the errors in, the measurement of length of the pendulum (L) and, time period (T) are 3% and 2% respectively. The, maximum percentage error in the value of L/T² is, (a) 5%, (b) 7%, (c) 8%, (d) 1%, 3. The error in the measurement of radius of a sphere is, 0.1%. The max error in calculated value of its, volume will be, (a) 0.3% (b) 0.4% (c) 0.5% (d) 0.6%, 4. What is the maximum percentage error in density of a, body if percentage error in measurement of mass is, 4% and percentage error in measurement of side 5%, , (a) 18%, (b) 19%, (c) 20%, (d) 21%, 5. The number of significant figures in a measurement of, 0.0353 m is:, (a) 5, (b) 4, (c) 3, (d) 2, 6. The number of significant figures in 1.118x10-3 V are:, (a) 7, (b) 4, (c) 3, (d) 2, 7. If 97.52 is divided by 2.54, the correct result in terms, of significant figures is:, (a) 38.4, (b) 38.3937, (c) 38.394, (d) 38.39, 8. In an experiment, the length of cylinder using vernier, calipers is measured s 2.51 cm, 2.49 cm,2.48 cm and, 2.55 cm respectively. The average length of the, cylinder in centimeters is:, (a) 2.5, (b) 2.51, (c) 2.50, (d) 2.518, 9. The radius of a circle is 2.12 m. its area according to, idea of significant figures is:, (a) 14.1124 m2, (b) 14.112 m2, 2, (c) 14.11m, (d) 14.1m2

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10. The volume of a sphere is 1.76 cm3. The volume of, 25 such spheres taking into account the significant, figure is :, (a) 0.44 x 102 cm3, (b) 44.0 cm3, 3, (c) 44 cm, (d) 44.00 cm3, 11. A physical quantity is represented by X = Ma Lb T-c., if percentage error in the measurement of M, L and T, are α%, β% and γ% respectively, then max., percentage error is:, (a) (αa -βb+γc)%, (b) (αa+βb+γc)%, (c)(αa –βb-γc)%, (d) none of these, 12. If Y = ab²/c3, the maximum possible error in the, measurement of Y is,  a 2b 3c , (a)  ,  100%, , b, c ,  a,  a 2b 3c , (b)  ,  100%, , b, c ,  a,  a 2b 3c , (c)  , 100%, , b, c ,  a, (d) None of these, 13. If the value of r is 10.845 ohm and the value of, current is 3.23 amp. then the potential is 35.02935, V. it values in significant number would be, (a) 3.503 V (b) 35.0V (c) 35.029V (d) 35.030V, 14. A certain body weighs 22.4 gm and has measured, volume of 4.7 cc the possible error in the, measurement of mass and volume are 0.01gm and, 0.1 cc then maximum error in the density will be., (a) 2.2% (b ) 2% (c) 0.2 % (d) 0.02 %, 15. The least count of a stop watch of 1/5 sec. the time of, 20 oscillation of a pendulum is measured to be 25, sec. the minimum % error in the measurement of, time will be., (a) 0.1 %, (b) 0.8% (c) 1.8%, (d) 8%, 16. The length of a cylinder is measured with a meter, rod having least count 0.1 cm. Its diameter is, measured with vernier calipers having least count, 0.01 cm. Given that length is 5.0 cm. and radius is, 2.0 cm. The percentage error in the calculated value, of the volume will be, (a) 1%, (b) 2%, (c) 3%, (d) 4%, 17 if copper wire is stretched 0 .1 % longer , the, percentage change in its resistance, (A) 0 .1 % increase, (B) 0.2 % increase, (C) 0 .1 % decrease, (D) 0.2 % decrease, 18. The respective number of significant figures for the, numbers 23.023, 0.0003 and 2.1 x 10-3 are, [JEE MAINS 2010], (a) 5,1,2, (b) 5,1,5, (c) 5,5,2 (d) 4,4,2, 19. In an experiment the angles are required to be, measured using an instrument. 29 divisions of the, main scale exactly coincide with the 30 division of, the vernier scale. If the smallest division of the main, scale is half-a-degree (= 0.50), then the least count, of the instrument is, [JEE MAINS, 2009], (a) one degree, (b) half degree, , (c) one minute, (d) half minute, 20. In vernier calliper N divisions of veriner scale, coincides with N -1 divisions of main scale (in which, length of 1 division is 1 .) the leastcount of the, instrument should be, (a) 1/ N, (b) N-1, (c) 1/10 N (d) 1/N-1, ], , ………….by Praveen Gupta