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Unit And Measurements, 1., , If the units of length and force are increased four times,, then the unit of energy will:, (A) Increase 8 times, (B) Increase 16 times, (C) Decreases 16 times, (D) Increase 4 times, Answer: [B], Dimensionally,, E ML2 T –2, E(MLT –2 )(L), (4)(MLT –2 )(4L) 16(ML2 T –2 ), , 2., , If velocity, force and time are taken to be fundamental, quantities find dimensions formula for (a) mass:, (A) KV FT, (B) KV FT, (C) KVF T, (D) KV F T, Answer: [B], Let the mass in represented by M then, M = f (V,F,T), Assuming that a function is product of power functions of, V, F and T, –1, , –1, , –1, , –1, , –1, , –1 –1, , M KV xFy T z, , Where k is a dimension less constant of proportionality., The above equation dimensionally becomes., [M] [LT –1 ]x [MLT –2 ]y [T]2, [M] [M ][L, i.e., So equation becomes, y, , xy, , Y – x–2y z ], , 1
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[M] [MyLx y T – x–2y z ], , For dimensionally correct expression,, y 1, x y 0 and –x – 2y z 0, x, , –1, y 1 and z 1, , Therefore M KV FT., Hence correct answer is (B)., –1, , A rectangular plate has length (2 r 0.02) cm and width, (1 r 0.01) cm. The maximum percentage error in the, measurement of its area is:, (A) 1%, (B) 2%, (C) 3%, (D) 5%, Answer: [B], , 3., , Ab, , A, , 'A, u 100, A, , 'A, 'b, u 100 , u 100, A, b, , 4., , In a resonance tube with tuning fork of frequency 512 Hz,, first resonance occurs at water level equal to 30.3 cm and, second resonance occurs at 63.7cm. The maximum possible, error in the speed is:, (A) 51.2 cm/sec., (B) 102.4 cm/sec, (C) 204.8 cm/sec, (D) 153.6 cm/sec, Answer: [C], 2n(A 2 – A 1 ), , v, , 'v, , 2n( 'A 2 'A 1 ), , 2 u 512 u (0.1 0.1) 204.8 cm / sec, , 2
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5., , The density of a cube is measured by measuring its mass and, length of its sides. If the maximum errors in the, measurement of its mass and length are 4% and 3%, respectively, the maximum error in density is, (A) 1%, (B) 7%, (C) 5%, (D) 13%, Answer: [D], d, , Mass, L3, , Or % error in, , d %, , error in mass + 3 [ % error in length], , 4% 3[3%] 13%, , 6., , The dimensions of h/e (h = Planck’s constant and, e = electronic charge) are same as that of:, (A) magnetic flux, (B) electric flux, (C) electric field, (D) magnetic field, Answer: [A], Electric potential V { ddtM, ...(1), (Faraday’s law), and eV hQ (photo-electric effect), or V, , §h·, ¨ e ¸Q, © ¹, , ...(2), , From Eqs. (1) and (2) we can see that magnetic flux, h, have the same dimensions., e, , ( I), , and, , 3
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7., , E2, P0, , has the dimensions (E = electric field,, , of free space), (A) [M L T A ] (B), Answer: [B], 2 3, , –2, , 2, , ª H 0E 2 º, «, », ¬ H0 P0 ¼, , ª E2 º, « », ¬ P0 ¼, , [MLT –4 ], , (C), , [ML3 T –2 ], , P0, , permeability, (D), , [M–1L2 TA –2 ], , ª energy / volume º, « (1/ speed of light)2 », ¬, ¼, , ª energy(speed)2 º, «, », volume, ¬, ¼, ª ML2 T 2L2 T 2 º, «, », L3, ¬, ¼, , [MLT –4 ], , 8., , The dimensions of Vb (V Stefan's constant and b = Wein's, constant) are:, (B) [ML T ], (C) [ML T], (D) [ML T ], (A) [M L T ], Answer: [B], 4, , 0 0, , 0, , Om T b or b4, , and, , 4, , –2, , 6, , –3, , Om4 T 4, , energy, area time, , ? Vb4, , –3, , VT 4 or V, , energy, (area – time) T 4, , § energy · 4, 4, ¨ area time ¸ Om or [V b ], ©, ¹, , [ML2 T 2 ] 4, [L ] [ML4 T –3 ], 2, [L ][T], , 9., , The kilowatt hour is a unit of:, (A) Energy, (B) electric charge, (C) Force, (D) electric power, Answer: [A], Energy, , 4
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10. The dimensional formula of kinetic energy is the same as, that of:, (A) pressure, (B) work, (C) momentum, (D) force, Answer: [B], Work, 11. The frequency of vibration f of a mass m suspended from a, spring of spring constant k is given by relation of the type, f cm k , where c is a dimensionless constant. The values of x, and y are:, (B) –1/ 2, – 1/ 2, (A) 1/ 2, 1/ 2, (C) 1/ 2, – 1/ 2, (D) –1/ 2, 1/ 2, Answer: [D], x, , f, , y, , cmx k y, , [M0L0 T –1 ] [M1 ]x [M0L0 T –2 ]y, M0L0 T –1 Mx yL0 T –2y, xy 0, , ...(1), , –2y, , ...(2), , 12. If, , S, , –1, , 1 3, ft ,, 3, , 'f ' has the dimensions of:, , (A) [M L T ], (C) [M L T ], Answer: [C], f, , 0, , –1, , 0, , 1, , 3, , –3, , (B) [M L T ], (D) [M L T ], 1 1, , 0, , –1, , –3, , –3, , 3S, t3, , 5
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M0 L1 T 0, T3, , [L1T –3 ], , 13. A rectangular plate has length (4 r 0.04) cm and width, (2 r 0.02) cm. The maximum percentage error in the, measurement of its area is:, (A) 2%, (B) 6%, (C) 3%, (D) 4%, Answer: [B], 'A, u 100, A, , 'A, ', u 100 u 100, A, b, , 0.04, 0.02, u 100 , u 100, 4, 2, , 6%, , 14. A wire is of mass (0.3 r .003) gm. the radius is (0.5 r 0.005) cm and, length is (6 r .6) cm. the maximum percentage error in density, is:, (A) 3%, (B) 4%, (C) 8%, (D) 16%, Answer: [B], 'P, u 100, P, 'm, 'A, 2'r, u 100 , u 100 , u 100, m, A, r, , 15. A pressure of 10 dyne cm is equivalent to:, (B) 10 Nm, (C) 10 Nm, (D) 10, (A) 10 Nm, Answer: [A], 6, , 5, , 106, , dyne, cm2, , –2, , –2, , 106 u, , 4, , 10 –5 N, 10 –4 m2, , 105, , –2, , 6, , –2, , 7, , Nm–2, , N, m2, , 16. Dimensional formula for latent heat is:, 6
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(A) M L T, Answer: [A], 0 2, , Q, , (B), , 2, , Q, m, , mL L, , ML2 T 2, M, , MLT 2, , (C), , (D), , ML2 T 2, , ML2 T 1, , (Heat is a form of energy), , [M0L2 T 2 ], , 17. To keep on object moving in a circle at constant speed, requires a force F v m v r . According to dimensional, analysis the a, b, c are:, (A) a 1,b 2,c –1, (B) a 1,b 2,c 1, (D) a 1,b 2,c 0, (C) a 0,b 2,c –1, Answer: [A], a, , b c, , F v ma v b r c, , i.e. [F] [m] [v] [r], or [M L T ] [M ] [L T ] [L], or [M L T ] [M L T ], Comparing LHS and RHS, a, , 1 1, , 1 1, , b, , –2, , c, , a, , –2, , 1, , a, , bc, , a 1, b c 1 & – b, a 1, b 2, c, , –1 b, , c, , –b, , –2, –1, , 18. In the formula X 3YZ , X and Z have dimensions of, capacitance and magnetic induction respectively. The, dimensions of Y in MKSA system are:, (A) M L T A, (B) ML, (C) M L A T, (D) M L A T, Answer: [C], 2, , –3, , –2, , –2, , –3, , –2, , 4, , –4, , 8, , –2, , –3, , –2, , 4, , 4, , 7
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x 3YZ2, [Y], , [X], [Z]2, , [M1L2 T 4 A 2 ], [MT 2 A]2, , [M–3 L–2 T 8 A 4 ], , 19. Which of the following sets cannot enter into the list of, fundamental quantities in any system of units?, (A) length, mass, velocity, (B) length, time, velocity, (C) length, time, mass, (D) mass, time, velocity, Answer: [B], Length, time and velocity cannot enter in to the list of, fundamental quantities because velocity can be expressed, in terms of length and time., 20. In a certain system of units, 1 unit of time is 5 sec, 1 unit of, mass is 20 kg and unit of length is 10 m. In this system, one, unit of power will correspond to:, (C) 25 watts (D) None of, (A) 16 watts (B) 161 watts, these, Answer: [A], [P] ML2 T –3, ? Unit of P 20 kg u (10m)2 u (5 sec)–3, , 16 W, , 21. The time dependence of a physical quantity p is given by, p p e , where D is constant and t is time. The constant D :, – Dt 2, , 0, , 8
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(A) is dimensionless, (B) has dimensions T, (C) has dimensions T, (D) has dimensions of p, Answer: [B], Powers are always dimensionless, –2, , 2, , ? [Dt 2 ] [M0 L0 T 0 ] or [D] [M0 L0 T –2 ], , 22. The force F is given in terms of time t and displacement x, by the equation F A cosBx C sin DT. The dimensional formula, of D/B is, (A) [M L T ], (B) [M L T ], (C) [M L T ], (D) [M L T ], Answer: [D], 0 0, , 0, , 0 0, , Bx, , T (constant); x L–1, , DT, , T (constant); D T –1, , D, B, , T –1, L–1, , –1, , 0 –1, , 0, , 0 1, , –1, , LT –1, , 23. The dimensions of, , e2, ,, 4SH0hc, , where e,, , H0,, , h and c are electronic, , charge, electric permittivity, Planck’s constant and velocity, of light in vacuum respectively is :, (A) [M L T ], (B) [M L T ], (C) [M L T ], (D) [M L T ], Answer: [A], 0, , 0, , 0, , 1, , 0, , 0, , 0, , 1, , 0, , 0, , 0, , 1, , e2, 4SH0hc, , kq2, r2, , F ke2, , Energy E, , Fr 2, , hc, hc, O, , EO, , 9
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Fr 2, hc, , Fr 2, EO, , [M0 L0 T 0 ], , 24. Which of the following groups has different dimensions?, (A) Potential difference, emf, voltage, (B) Pressure, stress, Young’s modulus, (C) Heat, energy, work done, (D) Dipole moment, electric flux, electric field, Answer: [D], (A) Potential difference [M L T A ], = voltage = emf, (B) Pressure, stress, young’s modulus, [M L T ], (C) Heat, energy, work done [M L T ], (D) Dipole moment [L T A ], Electric flux [M L T A ], Electric field [M L T A ], 1, , 2, , –3, , –1, , 1 2, , 1 2, , 1, , 1 3, , –3, , 1 3, , 25. If 1g cm s, (A) 1 u 10, Answer: [C], –1, , x, –1, , –3, , 1, , –2, , –2, , 1, , –1, , –1, , newton-second, then the number x is equal to :, (B) 3.6 u 10, (C) 1 u 10, (D) 6 u 10, –3, , –5, , –4, , 1g cm s–1 1(10 –3 kg) (10 –2 m)s–1, 10 –5 kg, , m s, u, s, s, , 10 –5 N.s, , 26. In a particular system, the units of length, mass and time, are chosen to be 10 cm, 10 g and 0.1 s respectively. The, unit of force in this system will be :, 10
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(A) 0.1 N, Answer: [A], , (B) 1 N, , (C) 10 N, , (D) 100 N, , 1N n2 unit, ª 1kg º, 1«, », ¬ 10 g ¼, , n2, , 1, , 1, , ª 1m º ª 1s º, « 10 cm » « 0.1s », ¬, ¼ ¬, ¼, , –2, , 27. If y represents distance and x-represents time, dimensions, dy, are:, of dx, 2, , 2, , (A) LT, Answer: [D], –1, , d2 y, dx 2, , (B), , L2 T 2, , (C), , (D), , L2 T –1, , LT –2, , d § dy ·, dx ¨© dx ¸¹, , d, (v), dx, , rate of change of speed w.r.t. time, , = acceleration, 28. The dimensional representation of Planck’s constant is, identical to that of, (A) Torque, (B) Power, (C) Linear momentum, (D) angular momentum, (D), As Planck’s constant has dimensions of EQ, ML2 T 2, T 1, , ML2 T –1, , and Dimensions of angular momentum, , r up, , (L u MLT –1 ), , 11
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ML2 T –1, , 29. If the units of M and L are quadrupled, then the units of, torque becomes, (A) 16 times (B) 64 times (C) 8 times, (D) 4 times, Answer: [B], Dimensions of torque ML T, 2, , –2, , (4M) (4L)2 T –2, 64 M L2 T –2, , 30. A radar signal is beamed towards a planet from earth and, its echo is received seven minutes later. If distance between, the planet and earth is 6.3 u 10 m, then velocity of the signal, will be, (A) 3 u 10 m / s (B) 2.97 u 10 m / s (C) 3.10 u 10 m / s (D), 10, , 8, , 6, , 5, , 300m / s, , Answer: [A], Velocity of signal,, , c, , 2x, t, , 2 u 6.3 u 1010, 7 u 60, , 3 u 108 m / s, , 31. The displacement of a particle is given by, denotes time. The unit of k is, (A) htz, (B) metre, (C) radian, Answer: [A], Here, kt is dimensionless. Hence, k = 1/t, , x, , A sin2 kt,, , where t, , (D) second, sec –1, , = htz, , 12
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32. Dimensions of ohm are same as those of (h is Planck’s, constant and e is charge), (A) he, (B) he, (C) eh, (D) he, 2, , 2, , 2, , 2, , Answer: [C], h, e2, , [ML2 T 1 ], [AT]2, , [ML2 T 3 A 2 ] resistance, , 33. Which of the following is not a unit of time, (A) solar year (B) tropical year (C) leap year (D) light, year, Answer: [D], Tropical year is the year in which there is total eclipse., Light year represents distance, 34. Dimensional formula of Stefan’s constant, (A) ª¬ML T T ¼º (B) ª¬ML T T º¼, 2, , 2, , (C)¬ª MLT, q, , 3, , 4, , 3, , 2, , 4, , (D) none of these, , T4 ¼º, , $QVZHU >&@, V, , E, AtT4, , ª¬MLqT 3 T4 º¼, , 35. A cube has side 1.2 u 10, (A) 1.728 u 10 m, (C) 1.7 u 10 m, 6, , 6, , 3, , 3, , –2, , m., , Its volume will be recorded as, (B) 1.72 u 10 m, (D) 0.72 u 10 m, 6, , 6, , 3, , 3, , 13
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Answer: [C], v, , A3, , 1.728 u 106, , Length has two significant figure, , v 1.7 u 106 m3, , 36. An athlete’s coach told his team that muscle times speed, equals power. What dimensions does he view for, “muscle”?, (B) ML T, (C) MLT, (D) L, (A) MLT, Answer: (C), 2, , Power, , 2, , –2, , –2, , force u velcoity, , = muscle times speed, ? muscle represents force, miscle [MLT –2 ], , 37. If force, length and time would have been the fundamental, units what would have been the dimensional formula for, mass?, (B) FL T, (C) FLT, (D) F, (A) FL T, Answer: [B], Let M K F L T, –1, , –2, , –1, , a b, , 2, , –2, , c, , [MLT –2 ]a [Lb ] T c, [MaL(a b) T(–2a c ) ], a 1, a b 0, a 1, b, , &, , –1, c, , –2a c, , 0, , 2, , Hence, (B) is correct., 14
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38. A wave is represented by, y a sin(At – Bx C), , where A, B, C are constants. The Dimensions of A, B, C, are:, (A) T , L, M L T, (B) T , L , M L T, (C) T, L, M, (D) T , L , M, Answer: [B], –1, , y, , 0 0, , 0, , –1, , –1, , 0 0, , –1, , –1, , –1, , 0, , A sin(At – Bx c), , At – Bx c, , is dimension less, , i.e. [At] [Bx] C [M L T ], or [A] [T ], and [B] [L ], and [C] [M L T ], 0 0, , 0, , –1, , –1, , 0, , 0, , 0, , 39. The density of wood is 0.5 g cm in cgs system of units. The, corresponding value of SI units is, (A) 5000, (B) 500, (C) 5, (D) 0.5, Answer: [B], Density = M/V, –3, , 1cm 10 –2 m; 1gm 10 –3 kg, 0.5 g cm–3, , 0.5(10 –3 ) kg (10 –2 )–3 m, , 0.5 u 10 –3 u 106, , 500, , 40. The dimensions of Planck’s constant are, (A) ML2T 2, (B) ML2T 1, 15
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(C) ML2T, (D) ML2T 2, Answer: [B], E, , hv, , Therefore,, >E@, > h@, >v@, , ML2T 2, T 1, , ML2T 1, , 41. The velocity v (in cm/sec) of a particle is given in terms of, time t (in sec) by the relation v at t b c ; the dimensions of a,, b and c are, (B) a LT , b LT, c L, (A) a L , b T, c LT, (C) a LT , b L, c T, (D) a L, b LT, c T, Answer: (C), From the principle of dimensional homogenity, [v] [at] [a] [LT ]. Similarly [b] [L] and [c] [T]., 2, , 2, , 2, , 2, , 2, , 2, , 42. The position of a particle at time t is given by the relation, §v ·, x(t) ¨ ¸ (1 c ), where v is a constant and D ! 0. The, D, Dt, , 0, , ©, , 0, , ¹, , dimensions of v and D are respectively., (A) M L T and T, (B) M L T and T, (C) M L T and LT, (D) M L T and T, Answer: [A], Dimension of Dt [M L T ] ? [D] [T ], 0, , 0 1, , 1, , 0 1, , 1, , –1, , –2, , 0 0, , Again, , ª v0 º, «D», ¬ ¼, , [L], , so, , 0, , 0 1, , 0, , 0 1, , 1, , –1, , –1, , [v 0 ] [LT 1 ], , 16
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43. The dimensions of, , a, b, , in the equation, , P, , pressure, x is distance and t is time, are, (B) M LT, (A)MT, (C) ML T, $QVZHU >$@, [a t ], T, [a] [T ] and [b], [P] [x] [ML T ][L], 2, , 2, , 3, , 2, , 2, , 3, , 1, , a t2, ,, bx, , where, (D), , P, , is, , LT 3, , 2, , 1, , 2, , [b] [M1T 4 ], , So, , ªa º, «b », ¬ ¼, , [T 2 ], [M1T 4 ], , [MT 2 ], , 44. The velocity of a freely falling body changes as g h where g, is acceleration due to gravity and h is the height. The values, of p and q are, (A) 1, 21, (B) 21 , 21, (C) 21 ,1, (D) 1, 1, p q, , Answer: [B], > V @ >g@ >h@, p, , q, , [M0L1T 1 ] [LT 2 ]p [M0K1T 0 ] q M0LP qT 2p, 1 p q and – 1 –2p, ? p, , 1, ,q, 2, , 1, 2, , 45. The density of the material of a cube is measured by, measuring its mass and length of its side. If the maximum, errors in the measurement of mass and the length are 3%, and 2% respectively, the maximum error in the, measurement of density is:, (A) 1%, (B) 5%, (C) 7%, (D) 9%, 17
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Answer: [D], d, , 'd, d, , m, d, v, , m1, L3, , § 'm ·, § 'L ·, 3¨, 1¨, ¸, ¸, © m ¹, © L ¹, , 18
