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Solved Examples 1. Calculate the dimensional formula of energy from the equation E=1/2 mv2.
2. What are the dimensions of : (a) volume of a cube of edge a, (b) volume of a sphere of radius a, (c) the ratio of the volume of a cube of edge a to the volume of a sphere of radius a? 3. Suppose you are told that the linear size of everything in the unverse has been doubled overnight. Can you test this statement by measuring sizes with a metre stick? Can you test it by using the fact that the speed of light is a universal constant and has not changed? What will happen if all the clocks in the universe also start running at half the speed? 4. If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equaiton have same dimensions, is it neccessary that they have same units? 5. If two quantities have same dimensions, do they represent same physical content? 6. It is desirable that the standards of units be easily available, invariable, indestructible and easily reproducible. If we use foot of a person as a standard unit of length, which of the above features are present and which are not? 7. Suggest a way to measure : (a) the thickness of a sheet of paper. (b) the distance between the sun and the moon. |
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5. If velocity, time and force were chosen as basic quantities, find the dimensions of mass. Sol. Dimensionally, Force = mass × acceleration.
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[2ax] = [a] [x] = (L/T)2 L = L2 T-2 Thus, the equation may be correct. 7. The distance covered by a particle in time t is given by x = a + bt + ct2; find the dimensions of a, b, c and d. Sol. The equation cotains five terms. All of them should have the same dimensions. Since [x] = length, each of the remaining four must have the dimension of length. Thus, [a] = length = L [bt] = L, or, [b] = LT-1 [ct2] = L, or, [c] = Lt-2 and [dt3] = L, or, [d] = L-2
8. When a solid sphere moves through a liquid, the liquid opposes the motion with a force F. The magnitude of F depends on the coefficient of viscosity h of the liquid, the radius r of the sphere and the speed v the sphere. Assuming that F is proportional to different powers of these quantities, guess a formula for F using the method of dimensions. Sol. Suppose the formula is F = k ha rb vc Then, MLT-2 = [ML-1 T-1]a Lb (L / T)c = Ma L-a + b + c T-a - c Equating the exponents of M, L and T from both sides, a = 1 - a + b + c = 1 - a - c = - 2, Solving these, a = 1, b = 1, and c = 1 Thus, the formula for F is F - khrv. 10. The heat produced in a wire carrying and electric current depends on the current, the resistance and the time. Assuming that the dependence is of the product of powers type, guess on equation between these quantities using dimensional analysis. The dimensional formula of resistance is ML2I-2T-3 and heat is a form of energy. Sol. Let the heat produced be H, the current through the wire be I, the resistance be R and the time be t. Since heat is a form of energy, its dimensional formula is ML2T-2. Let us assume that the required equation is H + k Ia Rb tc, where k is a dimensionless constant. Writing dimensions of both sides, ML2T-2 = Ia (ML2 I-3)b Tc = Mb L2b T-3b + c I a - 2b Equation the exponents, b = 2 2b = 2 - 3b + c = - 2 a - 2b = 0. Solving these, we get a = 2, b =1 and c = 1. Thus, the required equation is H = kI2 Rt. Exercise 1. Find the dimensions of HCV-Ch-1-Ex.-1 (a) linear momentum, (b) frequency and (c) pressure. 2 Find the dimensions of HCV-Ch-1-Ex.-2 (a) angular speed w, (b) angular acceleration a, (c) torque T and (d) momentum of inertia I. Some of the equations involving these quantities are |
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4. Find the dimension of HCV-Ch-1-Ex.-4 (a) electric dipole moment p and (b) magnetic dipole moment M The defining equations are p = q.d and M = IA where d is distance, A is area, q is charge and I is current. Ans. (a) LTI (b) L2I 5. Find the dimensions of Planck's constant h from the equation E = hn where E is the energy and n is the frequency. HCV-Ch-1-Ex-5 Ans. ML2 T-1 6. Find the dimensions of HCV-Ch-1-Ex.-6 (a) the specific heat capacity c, (b) the coefficient of linear expansion a and (c) the gas constant R. Some of the equations involving these quantities are Q = mc (T2 - T1), lt = l0 [1 + a (T2 - T1)] and PV = nRT. Ans. (a) L2 T-2 K-1 (b) K-1 (c) M L2 T-2 K-1 (mol)-1 7. Taking force, length and time to be the fundamental quantities find the dimensions of (a) density, (b) pressure, (c) momentum and (d) energy. HCV-Ch-1-Ex.-7 Ans. (a) F L-4 T2 (b) F L-2 (c) FT (d) FL 8. Suppose the acceleration due to gravity at a place is 10 m/s2. Find its value in cm/(minute)2.HCV-Ch-1-Ex.-8 Ans. 36 × 105 cm/(minute)2 9. The average speed of a snail is 0.020 miles/hour and that of a leopard is 70 miles/hour. convert these speeds in SI units. HCV-Ch-1-Ex.-9 Ans. 0.0089 m/s, 31 m/s. 10. The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercury = 13.6, Density of water = 103 kg/m3, g = 9.8 m/s2 at Calcutta. Pressure = h r g in usual symbols. HCV-Ch-1-Ex.-10 Ans. 10 × 104 N/m2 , 10 × 105 dyne/cm2 11. Express the power of a 100 watt bulb in CGS unit. HCV-Ch-1-Ex.-11 Ans. 109 erg/s 12. The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in microcenturies. 1 microcentury = 10-6 × 100 years. How many microcenturies did you sleep yesterday ? HCV-Ch-1-Ex.-12 |
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Ans. 1.9 microcenturies 13. The surface tension of water is 72 dyne/cm. Convert it in SI unit. HCV-Ch-1-Ex.-13 Ans. 0.072 N/m 14. The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed w. Assuming the relation to be K = kIa wb where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is Ans. a = 1, b = 2 15. Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions. HCV-Ch-1-Ex.-15 Ans. E = kmc2 16. Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are ML2 I -2 T-3 and ML2 T-3 I-1 . HCV-Ch-1-Ex.-16 Ans. V = I R 17. The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for tis frequency from dimensional analysis.
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Mr2. HCV-Ch-1-Ex.-14