Title 01. Units and Dimensions hard.pdf ✅

02. UNITS and DIMENSIONS [Hard] 1. The dimensions of physical quantity X in the equation Force Density X = is given by (a) 1 4 −2 M L T (b) 2 −2 −1 M L T (c) 2 −2 −2 M L T (d) 1 −2 −1 M L T 2. Number of particles is given by 2 1 2 1 x x n n n D − − = − crossing a unit area perpendicular to X- axis in unit time, where 1 n and n2 are number of particles per unit volume for the value of x meant to 2 x and . 1 x Find dimensions of D called as diffusion constant (a) 0 2 M LT (b) 0 2 −4 M L T (c) 0 −3 M LT (d) 0 2 −1 M L T 3. E, m, l and G denote energy, mass, angular momentum and gravitational constant respectively, then the dimension of 5 2 2 m G El are (a) Angle (b) Length (c) Mass (d) Time 4. The equation of a wave is given by Y = A sin       − k v x  where  is the angular velocity and v is the linear velocity. The dimension of k is (a) LT (b) T (c) −1 T (d) 2 T 5. The potential energy of a particle varies with distance x from a fixed origin as , 2 x B A x U + = where A and B are dimensional constants then dimensional formula for AB is (a) ML7/2T −2 (b) 11 / 2 −2 ML T (c) 2 9 / 2 −2 M L T (d) 13 / 2 −3 ML T 6. The dimensions of 2 1 2  0E ( 0  = permittivity of free space ; E = electric field ) is (a) −1 MLT (b) ML 2 T −2 (c) ML −1 T −2 (d) 2 −1 ML T 7. You may not know integration. But using dimensional analysis you can check on some results. In the integral        = − − − sin 1 (2 ) 1 2 1 / 2 a x a ax x dx n the value of n is (a) 1 (b) – 1 (c) 0 (d) 2 1 8. A physical quantity m B l P 2 2 = where B= magnetic induction, l= length and m = mass. The dimension of P is (a) −3 MLT (b) 2 −4 ML T I –2 (c) M L T I 2 2 −4 (d) −2 −2 MLT I 9. If the present units of length, time and mass (m, s, kg) are changed to 100m, 100s, and 10 1 kg then (a) The new unit of velocity is increased 10 times (b) The new unit of force is decreased 1000 1 times (c) The new unit of energy is increased 10 times (d) The new unit of pressure is increased 1000 times 10. Suppose we employ a system in which the unit of mass equals 100 kg, the unit of length equals 1 km and the unit of time 100 s and call the unit of energy eluoj (joule written in reverse order), then (a) 1 eluoj = 104 joule (b) 1 eluoj = 10-3 joule (c) 1 eluoj = 10-4 joule (d) 1 joule = 103 eluoj 11. If 1gm cms–1 = x Ns, then number x is equivalent to (a) 1 1 10 −  (b) 2 3 10 −  (c) 4 6 10 −  (d) 5 1 10 −  12. From the dimensional consideration, which of the following equation is correct (a) GM R T 3 = 2 (b) 3 2 R GM T =  (c) 2 2 R GM T =  (d) GM R T 2 = 2 13. A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of the same dimensions and of low modulus of rigidity  such that the lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of the side faces of A. After the force is withdrawn block A executes small oscillations. The time period of which is given by (a) L M 2 (b)   M L 2 (c)   ML 2 (d) L M  2 14. A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity. After some time the velocity of the ball attains a constant value known as terminal velocity . T v The terminal velocity depends on (i) the mass of the ball. (ii)  (iii) r and (iv) acceleration due to gravity g. which of the following relations is dimensionally correct
(a) r mg vT   (b) mg r vT   (c) v rmg T   (d)  mgr vT  15. A dimensionally consistent relation for the volume V of a liquid of coefficient of viscosity  flowing per second through a tube of radius r and length l and having a pressure difference p across its end, is (a) l pr V   8 4 = (b) 4 8 pr l V  = (c) 4 8 r p l V   = (d) 4 8lr p V   = 16. Given that the amplitude A of scattered light is : (A0) of incident light. (ii) Directly proportional to the volume (V) of the scattering particle (iii) Inversely proportional to the distance (r) from the scattered particle (iv) Depend upon the wavelength (  ) of the scattered light. then: (a)  1 A  (b) 2 1  A  (c) 3 1  A  (d) 4 1  A  17. The length, breadth and thickness of a block are measured as 125.5 cm, 5.0 cm and 0.32 cm respectively. Which one of the following measurements is most accurate (a) Length (b) Breadth (c) Thickness (d) Height 18. The mass of a box is 2.3 kg. Two marbles of masses 2.15 g and 12.39 g are added to it. The total mass of the box to the correct number of significant figures is (a) 2.340 kg (b) 2.3145 kg. (c) 2.3 kg (d) 2.31 kg 19. The length of a rectangular sheet is 1.5 cm and breadth is 1.203 cm. The area of the face of rectangular sheet to the correct no. of significant figures is : (a) 1.8045 2 cm (b) 1.804 2 cm (c) 1.805 2 cm (d) 1.8 2 cm 20. Each side of a cube is measured to be 5.402 cm. The total surface area and the volume of the cube in appropriate significant figures are : (a) 175.1 2 cm , 157 2 cm (b) 175.1 2 cm , 157.6 3 cm (c) 175 2 cm , 157 2 cm (d) 175.08 2 cm , 157.639 3 cm 21. Taking into account the significant figures, what is the value of 9.99 m + 0.0099 m (a) 10.00 m (b) 10 m (c) 9.9999 m (d) 10.0 m 22. The value of the multiplication 3.124  4.576 correct to three significant figures is (a) 14.295 (b) 14.3 (c) 14.295424 (d) 14.305 23. The number of the significant figures in 11.118  10 −6 V is (a) 3 (b) 4 (c) 5 (d) 6 24. If the value of resistance is 10.845 ohms and the value of current is 3.23 amperes, the potential difference is 35.02935 volts. Its value in significant number would be (a) 35 V (b) 35.0 V (c) 35.03 V (d) 35.025 V 25. Each side a cube is measured to be 7.203 m. The volume of the cube up to appropriate significant figures is (a) 373.714 (b) 373.71 (c) 373.7 (d) 373 26. A physical parameter a can be determined by measuring the parameters b, c, d and e using the relation a =     b c / d e . If the maximum errors in the measurement of b, c, d and e are 1 b %, 1 c %, 1 d % and 1 e %, then the maximum error in the value of a determined by the experiment is (a) ( 1 1 1 1 b + c + d + e )% (b) ( 1 1 1 1 b + c − d − e )% (c) ( 1 1 1 1 b + c −d − e )% (d) ( 1 1 1 1 b + c + d + e )% 27. Which of the following does not have the same dimension? (a) Electric flux, electric field, electric dipole moment (b) Pressure, stress, Young’s modulus (c) Electromotive force, potential difference, electric voltage (d) Heat, potential energy, work done. 28. The ratio of the dimension of planck’s constant and that of moment of inertia is the dimension of (a) Time (b) Frequency (c) Angular momentum (d) Velocity 29. Out of the following pair, which one does not have the same dimensions? (a) Angular momentum and planck’s constant (b) Impulse and momentum (c) Moment of inertia and moment of force (d) Work and torque. 30. Parsec is the unit of
(a) Time (b) Distance (c) Frequency (d) Angular acceleration 31. A force F is given by F = at + bt2 where t is time, what are the dimension of a and b. (a) MLT-1 , MLT0 (b) MLT-3 , ML2T 4 (c) MLT-4 , MLT-1 (d) ML-3T, MLT-4 32. If M is mass of the earth and R its radius, the ratio of the gravitational acceleration and the gravitational constant is (a) R2 /M (b) M/R2 (c) MR2 (d) M/R 33. The velocity of surface waves depends upon surface tension, coefficient of viscosity and density. The relation is (a) s2 /p (b) s/ (c)/s2 (d) / 34. A chocolate cookie is a circular disk of diameter 8.50  0.02 cm and thickness 0.050  0.005 cm. The average volume in cm3 is (a) 2.83  0.3 (b) 2.38  0.27 (c) 11.35  1.2 (d) 9.31  1.12 35. The fastest commercial airline service is 1450 mi/h. Find the speed in kmh-1 and ms-1 (a) 1938 kmh-1 , 618.3 ms-1 (b) 2030 kmh-1 , 623.1 ms-1 (c) 2334 kmh-1 , 647.5 ms-1 (d) None 36. A spherometer has 20 threads per cm. Its circular scale has 100 divisions. Find the least count of spheometer (a) 5 m (b) 50 m (c) 0.5m (d) 0.05m 37. If force F = Ke-br/r2 varies with distance r. Then write the dimensions of K and b. (a) ML3T -2 (b) M-2LT-3 (c) ML-2T 3 (d) M-2L - 3T 2 38. Given X = anb m/pr . The percent error in measurement of a, b and p is 1%, 0.5% and 0.75% respectively. If n = 2, m = 2 and r = 4 then percent error in x is (a) 0 (b) 6 % (c) 5.25% (d) 0.75 % 39. In a system of units if force F, acceleration A and time T are taken as fundamental units, then the dimensional formula of energy is (a) FA2T (b) FAT2 (c) F2AT (d) FAT 40. Which of the following are dimensionally not correct? (a) h = rg 2Tcos   (b) v =  p (c) l pr t dt dV 4   = (d) T = I mgl 41. In the past, there were attempts to use metric system in the measurement of time, which could not succeed. In this system, whole day was divided into 10 decimal hours, each consisting of 100 decimal minutes and the day starting was at midnight. If one such watch using this system shows 6 decimal hours and 50 decimal minutes then this time in our traditional system is: (a) 3:06 PM (b) 3:30 PM (c) 3:36 PM (d) 4:00 PM. 42. Using a meter-stick, a length is measured by a student and he reported it as 0.3655m. The smallest possible division on the scale is - (a) 0.0001 m (b) 0.001m (c) 0.01m (d) 0.0005 m. 43. A student wants to measure the thickness of a single sheet of paper. He measures the thickness of the stack of 75 sheets with a micrometer and registers the reading as 1.16cm. The thickness of a single sheet should be expressed correctly as- (a) 0.15467mm (b) 0.1547mm (c) 0.155 mm (d) 0.15 mm. 44. 81.6 and 5.1 are two measured quantities and it is desired to find value of 5.1 81.6 . The result should be expressed as: (a) 4.000 (b) 4.00 (c) 4.0 (d) 4 45. A person was weighting 102.1 kg last week and gained 0.28 kg this week. His weight as of now is correctly expressed as: (a) 102.38 kg (b) 102.3 kg (c) 102.4 kg (d) 102 kg 46. When time as shown by a standard watch is 10:25:15AM, two watches A and B are showing times 10:29:37 AM and 10:25 AM respectively. It can be concluded that comparatively: (a) Watch A is accurate and watch B is precise. (b) Watch A is precise and watch B is accurate. (c) Watch A is precise as well as accurate. (d) Watch B is precise as well as accurate. 47. A scientist performs an experiment and takes 50 readings. He repeats the same experiment and now takes 200 readings. By doing so: (a) Both the systematic and random errors remain unchanged. (b) Both the systematic and random errors are reduced by a factor of 1/4. (c) The systematic error remains unchanged and random error is reduced by a factor of 1/4. (d) The systematic is reduced by a factor of 1/4 and random error remains unchanged. 48. A student draws velocity-time curve for the motion of a particle moving along a straight line in four different situations as shown in the figures. Out of these, which situation/situations can not physically occur
(a) Only situation I. (b) Situation III and IV (c) Situation II and IV (d) Situation I and IV 49. A ball thrown from the edge of a building hits the ground at an angle of 60°with the horizontal, 25 m from the building and 2.5s after it is thrown. (take g = 10m/s2 and 3 = 17) The direction of velocity with the horizontal with which the ball was thrown, is (a)  = tan-1 4/5 (b)  = tan-1 3/5 (c)  = tan-1 2/5 (d)  = tan-1 7/10 50. In previous question, the magnitude of velocity at which the ball was thrown, is : (a) 1.2 m/s (b) 9.6 m/s (c) 14.8 m/s (d) 12.8 m/s 51. Suitable unit for universal constant of gravitation is - (a) kg m s–1 (b) N m–1 s (c) N m2 kg–2 (d) kg m s–2 52. The kilowatt hour is a unit of (a) Energy (b) Electric charge (c) Force (d) Electric power 53. The dimensional formula of torque is - (a) [ML T ] 0 -2 (b) [MLT ] -1 (c) [MLT ] -2 (d) [ML T ] 2 -2 54. The dimensional formula of light year is- (a) [M LT ] 0 0 (b) [M L T] 0 0 (c) [M LT] 0 (d) [M LT ] 0 -1 55. The dimensional formula of kinetic energy is the same as that of - (a) Pressure (b) Work (c) Momentum (d) Force 56. Which of the following has the dimensions of pressure? (a) [ML T ] 2 -2 (b) [MLT ] -2 (c) [ML T ] -1 -1 (d) [ML T ] -1 -2 57. 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 = cmxk y , where c is a dimensionless constant. The values of x and y are - (a) 1/2, 1⁄2 (b) – 1/2, – 1⁄2 (c) 1/2, – 1⁄2 (d) – 1/2, 1⁄2 58. If h is height and g is acceleration due to gravity, then the dimensional formula of g 2h is the same as that of - (a) Time (b) Mass (c) Volume (d) Velocity 59. The dimensional formula of [ML–1T –2 ] does not represent the following - (a) Stress (b) Power (c) Pressure (d) Young's modulus 60. The physical quantity that has no dimensions is- (a) Strain (b) Angular velocity (c) Linear momentum (d) Angular momentum 61. If the work done W is represented by the formula kW = m, where m is mass, then the dimensional formula of k is - (a) [L T ] 2 2 (b) [L T ] -2 2 (c) [M L T ] 0 -1 2 (d) [L T ] -2 -2 62. The quantity having dimensions – 2 in the time is - (a) Force (b) Pressure (c) Gravitational constant (d) All of these 63. If S = 3 1 ft3 , 'f ' has the dimensions of - (a) [M L T ] 0 -1 3 (b) [M L T ] 1 1 -3 (c) [M L T ] 0 1 -3 (d) [M L T ] 0 -1 -3 64. If energy E, velocity (V) and time (T) are chosen as the fundamental quantities, then the dimensions of surface tension will be - (a) -2 -1 EV T (b) -1 -2 EV T (c) -2 -2 EV T (d) -2 -1 -3 E V T 65. A pendulum clock, designed to give correct time in planes, gives incorrect time at hill station. Type of error in measurement of time can be -