Tiêu đề 31. Rotational Motion Hard.pdf ✅

1. The distance between the carbon atom and the oxygen atom in a carbon monoxide molecule is 1.1 Å. Given, mass of carbon atom is 12 a.m.u. and mass of oxygen atom is 16a.m.u., calculate the position of the center of mass of the carbon monoxide molecule (a) 6.3 Å from the carbon atom (b) 1 Å from the oxygen atom (c) 0.63 Å from the carbon atom (d) 0.12 Å from the oxygen atom 2. The velocities of three particles of masses 20g, 30g and 50 g are i j k    10 ,10 , and 10 respectively. The velocity of the centre of mass of the three particles is (a) 2i 3 j 5k    + + (b) 10(i j k)    + + (c) i j k    20 + 30 + 5 (d) i j k    2 + 30 + 50 3. Masses 8, 2, 4, 2 kg are placed at the corners A, B, C, D respectively of a square ABCD of diagonal 80 cm . The distance of centre of mass from A will be (a) 20 cm (b) 30 cm (c) 40 cm (d) 60 cm 4. The coordinates of the positions of particles of mass 7, 4 and 10 gm are (1, 5, − 3), (2, 5,7) and (3, 3, − 1) cm respectively. The position of the centre of mass of the system would be (a) cm 7 1 , 17 85 , 7 15       − (b) cm 7 1 , 17 85 , 7 15       − (c) cm 7 1 , 21 85 , 7 15       − (d) cm 3 7 , 21 85 , 7 15       5. The angular velocity of seconds hand of a watch will be (a) rad / sec 60  (b) rad / sec 30  (c) 60  rad / sec (d) 30  rad / sec 6. The wheel of a car is rotating at the rate of 1200 revolutions per minute. On pressing the accelerator for 10 sec it starts rotating at 4500 revolutions per minute. The angular acceleration of the wheel is (a) 30 radians/sec2 (b) 1880 degrees/sec2 (c) 40 radians/sec2 (d) 1980 degrees/sec2 7. Angular displacement () of a flywheel varies with time as 2 3  = at + bt + ct then angular acceleration is given by (a) 2 a + 2bt − 3ct (b) 2b − 6t (c) a + 2b − 6t (d) 2b + 6ct 8. A wheel completes 2000 rotations in covering a distance of 9.5 km . The diameter of the wheel is (a) 1.5 m (b) 1.5 cm (c) 7.5 m (d) 7.5 cm 9. A wheel is at rest. Its angular velocity increases uniformly and becomes 60 rad/sec after 5 sec. The total angular displacement is (a) 600 rad (b) 75 rad (c) 300 rad (d) 150 rad 10. A wheel initially at rest, is rotated with a uniform angular acceleration. The wheel rotates through an angle  1 in first one second and through an additional angle  2 in the next one second. The ratio 1 2   is (a) 4 (b) 2 (c) 3 (d) 1 11. As a part of a maintenance inspection the compressor of a jet engine is made to spin according to the graph as shown. The number of revolutions made by the compressor during the test is (a) 9000 (b) 16570 (c) 12750 (d) 11250 12. Figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting A and B do not slip on the wheels. If x and y be the distances travelled by A and B in the same time interval, then (a) x = 2y (b) x = y (c) y = 2x (d) None of these 13. If the position vector of a particle is ) ˆ 4 ˆ r = (3i + j → meter and its angular velocity is ) ˆ 2 ˆ = (j + k →  rad/sec then its linear velocity is (in m/s) (a) ) ˆ 3 ˆ 6 ˆ (8i − j + k (b) ) ˆ 8 ˆ 6 ˆ (3i + j + k (c) ) ˆ 6 ˆ 6 ˆ − (3i + j + k (d) ) ˆ 3 ˆ 8 ˆ (6i + j + k 14. Five particles of mass = 2 kg are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is (a) 1 kg m2 (b) 0.1 kg m2 (c) 2 kg m2 (d) 0.2 kg m2 A B 0 500 1000 1500 2000 2500 3000 (in rev per min) 1 2 3 4 5 t (in min)
15. A circular disc X of radius R is made from an iron plate of thickness t, and another disc Y of radius 4R is made from an iron plate of thickness 4 t . Then the relation between the moment of inertia IX and IY is (a) IY = 64IX (b) IY = 32IX (c) IY = 16IX (d) IY = IX 16. Moment of inertia of a uniform circular disc about a diameter is I. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be (a) 5 I (b) 6 I (c) 3 I (d) 4 17. IFour thin rods of same mass M and same length l, form a square as shown in figure. Moment of inertia of this system about an axis through centreO and perpendicular to its plane is (a) 2 3 4 Ml (b) 3 2 Ml (c) 6 2 Ml (d) 2 3 2 Ml 18. Three rings each of mass M and radius R are arranged as shown in the figure. The moment of inertia of the system about YY will be (a) 3 MR2 (b) 2 2 3 MR (c) 5 MR2 (d) 2 2 7 MR 19. Let l be the moment of inertia of an uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle  with AB . The moment of inertia of the plate about the axis CD is then equal to (a) l (b)  2 l sin (c)  2 l cos (d) 2 cos 2  l 20. Three rods each of length L and mass M are placed along X, Y and Z-axes in such a way that one end of each of the rod is at the origin. The moment of inertia of this system about Z axis is (a) 3 2 2 ML (b) 3 4 2 ML (c) 3 5 2 ML (d) 3 2 ML 21. Three point masses each of mass m are placed at the corners of an equilateral triangle of side a. Then the moment of inertia of this system about an axis passing along one side of the triangle is (a) 2 ma (b) 2 3ma (c) 2 4 3 ma (d) 2 3 2 ma 22. The moment of inertia of a rod of length l about an axis passing through its centre of mass and perpendicular to rod is I. The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its centre of mass and perpendicular to its plane will be (a) 16I (b) 40 I (c) 60 I (d) 80 I 23. The moment of inertia of HCl molecule about an axis passing through its centre of mass and perpendicular to the line joining the + H and − Cl ions will be, if the interatomic distance is 1 Å (a) 47 2 0.61 10 kg.m −  (b) 47 2 1.61 10 kg.m −  (c) 47 2 0.061 10 kg.m −  (d) 0 24. Four masses are joined to a light circular frame as shown in the figure. The radius of gyration of this system about an axis passing through the centre of the circular frame and perpendicular to its plane would be (a) a / 2 (b) a / 2 (c) a (d) 2a 25. Four spheres, each of mass M and radius r are situated at the four corners of square of side R . The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be (a) (4 5 ) 2 5 2 2 M r + R (b) (4 5 ) 5 2 2 2 M r + R (c) (4 5 ) 5 2 2 2 M r + r (d) (4 5 ) 2 5 2 2 M r + r 26. The moment of inertia of a solid sphere of density  and radius R about its diameter is (a)  5 176 105 R (b)  2 176 105 R O R r 2m m 2m 3m A A O a Y Y  1 2 3 l l O l l P B D C A
(c)  5 105 176 R (d)  2 105 176 R 27. Two circular discs A and B are of equal masses and thickness but made of metals with densities d A and B d ( ) A B d  d . If their moments of inertia about an axis passing through centres and normal to the circular faces be A I and B I , then (a) A B I = I (b) A B I  I (c) A B I  I (d) A B I = I 28. A force of ) ˆ 2 ˆ 4 ˆ (2i − j + k N acts at a point ) ˆ 4 ˆ 2 ˆ (3i + j − k metre from the origin. The magnitude of torque is (a) Zero (b) 24.4 N-m (c) 0.244 N-m (d) 2.444 N-m 29. The resultant of the system in the figure is a force of 8 N parallel to the given force through R . The value of PR equals to (a) 1 4 RQ (b) 3 8 RQ (c) 3 5 RQ (d) 2 5 RQ 30. A horizontal heavy uniform bar of weight W is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to (a) W (b) 2 W (c) 4 3W (d) 4 W 31. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigid rod. If an impulse J = Mv is imparted to the body at one of its ends, what would be its angular velocity (a) v/L (b) 2v/L (c) v/3L (d) v/4L 32. A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity . Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be (a) M m M + 4  (b) M (M + 4m) (c) M m M m 4 ( 4 ) + −  (d) m M 4  33. A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity 0. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform  (t) will vary with time t as (a) (b) (c) (d) 34. The position of a particle is given by : ) ˆ ˆ 2 ˆ r = (i + j − k and momentum ) ˆ 2 ˆ 4 ˆ P = (3i + j − k . The angular momentum is perpendicular to (a) X-axis (b) Y-axis (c) Z-axis (d) Line at equal angles to all the three axes 35. Two discs of moment of inertia I1 and I2 and angular speeds 1 2  and  are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational KE of system will be (a) 2( ) 1 2 1 1 2 2 I I I I +  +  (b) 2 ( )( ) 2 1 + 2 1 +  2 I I (c) 2( ) ( ) 1 2 2 1 1 2 2 I I I I +  +  (d) None of these 36. A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m , which can slide freely along the rod. Initially the two beads are at the centre of the rod and the system is rotating with angular velocity  0 about an axis perpendicular to the rod and passing through the mid point of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is (a)  0 (b) M m M 12 0 +  (c) M m M 2 0 +  (d) M m M 6 0 +  37. Moment of inertia of uniform rod of mass M and length L about an axis through its centre and perpendicular to its length is given by 12 2 ML . Now consider one such rod pivoted at its centre, free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass M moving horizontally at a speed v strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be L/2 L/2 (t) O t (t) O t (t) O t (t) O t R P 5 N 3 N Q
(a) v L (b) 2v L (c) 3v 2L (d) 6v L 38. A solid cylinder of mass 2 kg and radius 0.2 m is rotating about its own axis without friction with angular velocity 3 rad / s . A particle of mass 0.5 kg and moving with a velocity 5 m/s strikes the cylinder and sticks to it as shown in figure. The angular momentum of the cylinder before collision will be (a) 0.12 J-s (b) 12 J-s (c) 1.2 J-s (d) 1.12 J-s 39. In the above problem the angular velocity of the system after the particle sticks to it will be (a) 0.3 rad/s (b) 5.3 rad/s (c) 10.3 rad/s (d) 89.3 rad/s 40. A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with an angular velocity of 20 rad/s. Its kinetic energy is (a) 10 J (b) 100 J (c) 500 J (d) 250 J 41. An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver (a) 350 N-m (b) 440 N-m (c) 531 N-m (d) 628 N- m 42. A body of moment of inertia of 3 kg-m2 rotating with an angular velocity of 2 rad/sec has the same kinetic energy as a mass of 12 kg moving with a velocity of (a) 8 m/s (b) 0.5 m/s (c) 2 m/s (d) 1 m/s 43. A disc and a ring of same mass are rolling and if their kinetic energies are equal, then the ratio of their velocities will be (a) 4 : 3 (b) 3 : 4 (c) 3 : 2 (d) 2 : 3 44. A wheel is rotating with an angular speed of 20 rad / sec . It is stopped to rest by applying a constant torque in 4 s . If the moment of inertia of the wheel about its axis is 0.20 kg-m2 , then the work done by the torque in two seconds will be (a) 10 J (b) 20 J (c) 30 J (d) 40 J 45. If the angular momentum of a rotating body is increased by 200%, then its kinetic energy of rotation will be increased by (a) 400% (b) 800% (c) 200% (d) 100% 46. A ring, a solid sphere and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest (a) Disc (b) Ring (c) Solid sphere (d) All will make same number of rotations 47. The angular velocity of a body is i j k ˆ 4 ˆ 3 ˆ = 2 + + →  and a torque i j k ˆ 3 ˆ 2 ˆ = + + →  acts on it. The rotational power will be (a) 20 W (b) 15 W (c) 17 W (d) 14 W 48. A flywheel of moment of inertia 0.32 kg-m2 is rotated steadily at 120 rad / sec by a 50 W electric motor. The kinetic energy of the flywheel is (a) 4608 J (b) 1152 J (c) 2304 J (d) 6912 J 49. A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom (a) gh 4 3 (b) gh 3 4 (c) 4 gh (d) 2 gh 50. A sphere rolls down on an inclined plane of inclination . What is the acceleration as the sphere reaches bottom (a) sin 7 5 g (b) sin  5 3 g (c) sin 7 2 g (d) sin  5 2 g 51. A ring solid sphere and a disc are rolling down from the top of the same height, then the sequence to reach on surface is (a) Ring, disc, sphere (b) Sphere, disc, ring (c) Disc, ring, sphere (d) Sphere, ring, disc 52. A thin uniform circular ring is rolling down an inclined plane of inclination 30° without slipping. Its linear acceleration along the inclined plane will be (a) g 2 (b) g 3 (c) g 4 (d) 2g 3 53. A solid sphere and a disc of same mass and radius starts rolling down a rough inclined plane, from the same height the ratio of the time taken in the two cases is (a) 15 : 14 (b) 15 : 14 (c) 14 : 15 (d) 14 : 15 54. A solid sphere of mass 0.1 kg and radius 2 cm rolls down an inclined plane 1.4 m in length (slope 1 in 10). Starting from rest its final velocity will be (a) 1.4 m / sec (b) 0.14 m / sec (c) 14 m / sec (d) 0.7 m / sec 55. A solid sphere rolls down an inclined plane and its velocity at the bottom is v1. Then same sphere slides down the plane (without friction) and let its velocity at the bottom be v2. Which of the following relation is correct 3 rad/s