Title 4-rotational-motion-and-gravitation-.pdf ✅

Rotational Motion and Gravitation 1. A thin wire is bent in form of a helix of radius R and height H. The pitch of helix is H 2 and mass per unit length of wire is λ. Moment of inertia of wire about the axis of helix is: (A) λHR2 (B) 2λR 2√H2 + 4πR2 (C) λR 2√H2 + 16π 2R2 (D) None of these 2. Consider a uniform rectangular plate of mass M and dimensions (a × b). Its moment of inertia about one of the diagonal is: (A) Ma 2b 2 6(a2+b2) (B) Ma 2b 2 3(a2+b2) (C) Ma 2b 2 12(a2+b2) (D) None of these 3. A uniform smooth rod of mass m = 1 kg and length L is balanced in the vertical position as shown in the figure. When a horizontal force F is applied at end A, the acceleration of top point Bat this instant is: (A) 2 m/s 2 to right (B) 1 m/s 2 to left (C) 1 m/s 2 to right (D) None of these 4. A body is rolling without slipping on a horizontal plane. If the rotational energy of the body is 40% of the total kinetic energy, then the body might be: (A) Cylinder (B) Hollow sphere (C) Solid cylinder (D) Ring 5. A sphere of mass m is given some angular velocity about a horizontal axis through its centre, and gently placed on a plank of mass m as shown in the figure. The coefficient of friction between the two is μ. The plank rests on a smooth horizontal surface. The initial acceleration of centre of mass of the sphere relative to the plank will be (A) zero (B) μg (C) 7 5 μg (D) 2μg 6. A rod of length L is held vertically on a smooth horizontal surface. The top end of the rod is given a gentle push. At a certain instant of time, when the rod makes an angle θ with horizontal the velocity of COM of the rod is v0. The velocity of the end of the rod in contact with the surface at that instant is : (A) v0cot θ (B) v0cos θ (C) v0sin θ (D) v0tan θ 7. A uniform bar AB of mass m and a ball of the same mass are released from rest from the same horizontal position. The bar is hinged at end A. There is gravity downwards. What is the distance of the point from point B that has the same acceleration as the ball, immediately after release? (A) 2L 3 (B) L 3 (C) L 2 (D) 3L 4
8. A uniform bar of mass M and length L collides with a horizontal surface. Before collision velocity of centre of mass was v0 and no angular velocity. Just after collision velocity of centre of mass of bar becomes v in upward direction as shown in the figure. Angular velocity ω of the bar just after impact is : (A)6(v0+v)cos θ L (b) (v0+v)cos θ L (c) 6(v0−v)cos θ L (D) (v0−v)cos θ 6L 9. A light thread is wound on a disk of mass m and other end of thread is connected to a block of mass m, which is placed on a rough ground as shown in the diagram. Find the minimum value of coefficient of friction for which block remain at rest: (A) 1 3 (B) 1 4 (C) 1 2 (D) 1 5 10. A satellite is projected with velocity √5/6 times its escape speed from earth's surface. The initial velocity of the satellite is parallel to the surface of earth. The maximum distance of the satellite from the centre of earth will be : (A) 3R (B) 4R (C) 5R (D) 8R 11. Two satellites S1 and S2 revolve around a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of S1 is 104 km. When S1 is closest to S2, the angular speed of S2 as observed by an astronaut in S1 is : (A) πrad/hr (B) 2πrad/hr (C) π/3rad/hr (D) π/2rad/hr 12. A thin uniform annular disc of mass M has outer radius 4R and inner radius 3R as shown in the figure. The work required to take a unit mass from point P on its axis to infinity is : (A) 2GM 7R (4√2 − 5) (B) 2GM 7R (4√2 + 5) (C) GM 4R (D) 2GM 5R (√2 − 1) 13. Two point masses of 0.3 kg and 0.7 kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of : (A) 0.42 m from mass of 0.3 kg (B) 0.70 m from mass of 0.7 kg (C) 0.98 m from mass of 0.3 kg (D) 0.98 m from mass of 0.7 kg 14. At what distance from the centre of the moon is the point at which the strength of the resultant field of earth's and moon's gravitational fields equal to zero ? The earth's mass is 81 times that of moon and the distance between centres of these planets is 60R where R is the radius of earth. (A) 6R (B) 4R (C) 3R (D) 5R
15. When a satellite in a circular orbit around the earth enters the atmospheric region, it encounters air resistance to its motion. Then which of the following is incorrect : (A) it loses mechanical energy (B) its kinetic energy increases (C) its kinetic energy decreases (D) its angular momentum about the earth decreases 16. Gravitational field due to uniform thin hemispherical shell at point P is I, then the magnitude of gravitational field at Q is : (mass of hemisphere is M. radius R ). (A) GM 2R2 − I (B) GM 2R2 + I (C) GM 4R − I (D) 2I − GM 2R2 17. A particle is projected form the earth's surface with an initial speed of 4 km/sec. What will be the maximum height attained by the particle in km : (A) 382.6 (B) 914.3 (C) 435.2 (D) 637.6 18. A body placed in a capsule will be weightless with respect to the capsule, when the capsule, in the gravitational field of the Earth is : (A) free falling vertically (B) a projectile near the surface of the earth (C) falling with a constant acceleration (D) projected from the surface of the earth with an escape velocity 19. A satellite is orbiting around the earth in a circular orbit of radius r. A particle of mass m is projected from the satellite in a forward direction with a velocity v = √2/3 times the orbital velocity (this velocity is given w.r.t. earth). During subsequent motion of the particle, its minimum distance from the centre of earth is : (A) r 2 (B) r (C) 2r 3 (D) 4r 5 20. A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is 3 × 105 times heavier than the Earth and is at a distance 2.5 × 104 times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is ve = 11.2kms−1 . The minimum initial velocity (vs ) required for the rocket to be able to leave the Sun- Earth system is closest to (Ignore the rotation and revolution of the Earth and the presence of any other planet) (A) vs = 62kms−1 (B) vs = 42kms−1 (C) vs = 72kms−1 (D) vs = 22kms−1 MULTIPLE CORRECT ANSWERS TYPE Each of the following Question has 4 choices A, B, C & D, out of which ONE or MORE Choices may be Correct: 21. A tunnel is dug along a chord of earth at perpendicular distance R/2 from the earth's centre. The wall of the tunnel may be assumed to be frictionless. A particle is released from one end of the tunnel. The pressing force by the particle on the wall, and the acceleration of the particle vary with x (distance of the particle from the centre) according to : (A) (B)
(C) (D) 22. An orbiting satellite will escape if (A) Its speed is increased by 41% (B) Its speed in the orbit is made √(1.5) times of its initial value (C) Its KE is doubled (D) It stops moving in the orbit 23. Which of the following statements is(are) true (A) Work done by kinetic friction on a rigid body may be positive (B) A uniform sphere rolls up an inclined plane without sliding. The friction force on it will be up the incline. (only contact force and gravitational force is acting) (C) A uniform sphere rolls down an inclined plane without sliding. The friction force on it will be up the incline. (only contact force and gravitational force is acting) (D) A uniform sphere is left from rest from the top of a rough inclined plane. It moves down the plane with slipping. The friction force on it will be up the incline always. 24. A thin bar of mass M and length L is free to rotate about a fixed horizontal axis through a point at its end. The bar is brought to a horizontal position and then released. The angular velocity when it reaches the lowest point is (A) Directly proportional to its length and inversely proportional to its mass (B) Independent of mass and inversely proportional to the square root of its length (C) Dependent only upon the acceleration due to gravity and the length of the bar (D) Directly proportional to its length and inversely proportional to the acceleration due to gravity 25. A rod bent at right angle along its centre line, is placed on a rough horizontal fixed cylinder of radius R as shown in the figure. Mass of rod is 2 m and rod is in equilibrium. Assume that frictional force on rod at A and B are equal in magnitude. (A) Normal force applied by cylinder on rod at A is 3mg/2. (B) Normal force applied by cylinder on rod at B must be zero. (C) Frictional force acting on rod at B is upward. (D) Normal force applied by cylinder on rod at A is mg. 26. A disc of mass m and radius r is gently placed on another disc of mass 2 m& radius r. The disc of mass 2 m is rotating with angular velocity ω0 initially. The disc is placed such that axis of both are coincident. The coefficient of friction is μ for surfaces in contact. Assume that pressure on disc is uniformly distributed. Find the correct statement. (A) Loss in kinetic energy of system ΔK = 1 3 mr2ω0 2 . (B) Loss in kinetic energy of system ΔK = 1 6 mr2ω0 2 . (C) The common angular velocity is 2 3 ω0. (D) The common angular velocity is 4 3 ω0