Page 1 [1] Test 05 (Rotational Motion & Gravitational Motion) 1. 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) 2 g/3 2. Which of the following statements about angular momentum is correct : (a) It is directly proportional to moment of inertia (b) It is a scalar quantity (c) All of these (d) None of these 3. A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The centre of mass of the man plus boat system will remain stationary with respect to water : (a) in all cases (b) only when the man is stationary initially and finally (c) only if the man moves without acceleration on the boat (d) only if the man and the boat have equal masses 4. What is the moment of inertia of a solid sphere of density ? and radius R about its diameter : (a) 176 105 R 5 ? (b) 176 105 R 2 ? (c) 105 176 R 5 ? (d) 105 176 R 2 ? 5. A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed v collides elastically with the stick as shown in fig. If after the collision ball comes to rest, then what should be the mass of the ball : (a) m = 2 M (b) m = M (c) m = M/2 (d) m = M/4 6. A uniform rod of mass m and length ?makes a constant angle ? with an axis of rotation which passes through one end of the rod. Its moment of inertia about this axis is : (a) 3 m 2 ? (b) 3 m 2 ? sin ? (c) 3 m 2 ? sin 2 ? (d) ? 2 2 cos 3 m ? 7. The angular velocity of a body changes from ? 1 to ? 2 without applying torque but by changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is : (a) ? 2 : ? 1 (b) 2 1 2 2 : ? ? (c) 1 2 : ? ? (d) 1/ ? 2 : 1/ ? 1 8. Two particles A and B initially at rest, moves toward each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass is : (a) zero (b) v (c) 1.5 v (d) 3 v 9. In rotatory motion, linear velocities of all the particles of the body are : (a) same (b) different (c) zero (d) cannot say 10. A cylinder of water, is rotating about its own axis with uniform angular velocity ?. The shape of free surface of water will be : (a) parabola (b) elliptical (c) circular (d) spherical 11. A body rolls down an inclined plane, if the K.E. of rotational motion is 40% of its K.E. of translation, then the body is : (a) cylinder (b) ring (c) solid disc (d) solid sphere 12. The moment of inertia of two bodies are I 1 and I 2 . Their geometrical shapes are same, the first made of iron and the second aluminum, then : (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 (d) relation between I 1 and I 2 depends on the actual shape of the bodies 13. Radius of a ring is 4 cm and its mass is 10 gm. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is : (a) 160 g - cm 2 (b) 80 g - cm 2 (c) 16 g - cm 2 (d) None of these 14. A ring and a sphere of the same mass and radius are rolling. Their kinetic energies are equal. The ratio of their velocities is : v Page 2 [1] Test 05 (Rotational Motion & Gravitational Motion) 1. 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) 2 g/3 2. Which of the following statements about angular momentum is correct : (a) It is directly proportional to moment of inertia (b) It is a scalar quantity (c) All of these (d) None of these 3. A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The centre of mass of the man plus boat system will remain stationary with respect to water : (a) in all cases (b) only when the man is stationary initially and finally (c) only if the man moves without acceleration on the boat (d) only if the man and the boat have equal masses 4. What is the moment of inertia of a solid sphere of density ? and radius R about its diameter : (a) 176 105 R 5 ? (b) 176 105 R 2 ? (c) 105 176 R 5 ? (d) 105 176 R 2 ? 5. A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed v collides elastically with the stick as shown in fig. If after the collision ball comes to rest, then what should be the mass of the ball : (a) m = 2 M (b) m = M (c) m = M/2 (d) m = M/4 6. A uniform rod of mass m and length ?makes a constant angle ? with an axis of rotation which passes through one end of the rod. Its moment of inertia about this axis is : (a) 3 m 2 ? (b) 3 m 2 ? sin ? (c) 3 m 2 ? sin 2 ? (d) ? 2 2 cos 3 m ? 7. The angular velocity of a body changes from ? 1 to ? 2 without applying torque but by changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is : (a) ? 2 : ? 1 (b) 2 1 2 2 : ? ? (c) 1 2 : ? ? (d) 1/ ? 2 : 1/ ? 1 8. Two particles A and B initially at rest, moves toward each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass is : (a) zero (b) v (c) 1.5 v (d) 3 v 9. In rotatory motion, linear velocities of all the particles of the body are : (a) same (b) different (c) zero (d) cannot say 10. A cylinder of water, is rotating about its own axis with uniform angular velocity ?. The shape of free surface of water will be : (a) parabola (b) elliptical (c) circular (d) spherical 11. A body rolls down an inclined plane, if the K.E. of rotational motion is 40% of its K.E. of translation, then the body is : (a) cylinder (b) ring (c) solid disc (d) solid sphere 12. The moment of inertia of two bodies are I 1 and I 2 . Their geometrical shapes are same, the first made of iron and the second aluminum, then : (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 (d) relation between I 1 and I 2 depends on the actual shape of the bodies 13. Radius of a ring is 4 cm and its mass is 10 gm. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is : (a) 160 g - cm 2 (b) 80 g - cm 2 (c) 16 g - cm 2 (d) None of these 14. A ring and a sphere of the same mass and radius are rolling. Their kinetic energies are equal. The ratio of their velocities is : v [2] (a) 10 : 7 (b) 7 : 10 (c) 2 : 5 (d) 5 : 2 15. Mass of the earth is 81 times the mass of the moon and the distance between the earth and moon is 60 times the radius of the earth. If R is the radius of the earth, then the distance between the moon and the point on the line joining the moon and earth where the gravitational force becomes zero is : (a) 30R (b) 15R (c) 6R (d) 5R 16. Two particles of equal mass move in a circle of radius r under the action of their mutual gravitational attraction. If the mass of each particle is M, the speed of each particle is : (a) r GM (b) r 2 GM (c) r 4 GM (d) r GM 2 17. Suppose the gravitational force varies inversely as the nth power of the distance. Then the time period of a planet in circular orbit of radius R around the sum will be the proportional to : (a) R n (b) R (n+1)/2 (c) R (n-1)/2 (d) R -n 18. The gravitational potential at height h above earth’s surface is -5.12 ×10 7 J/kg and acceleration due to gravity at this point is 6.4 ms -2 . If radius of earth is 6400 km, the value of h is : (a) 1200 km (b) 1600 km (c) 1800 km (d) 2400 km 19. Two bodies with masses M 1 and M 2 are initially at rest and a distance R apart. They then move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances traveled by M 1 to the distance traveled by M 2 ? (a) 2 1 M M (b) 1 2 M M (c) 1 (d) 2 1 20. The ratio of the kinetic energy required to be given to the sattelite to escape earth’s gravitational field to the kinetic energy required to be given so that the sattelite moves in circular orbit just above earth’s atmosphere is: (a) One (b) Two (c) Half (d) Infinity 21. A tunnel is dug along a diameter of the earth of mass M e and radius R e . The force on a particle of mass m placed in the tunnel at a distance r from the centre is: (a) 3 e R m GM e r (b) r R m GM 3 e e (c) r mR GM 3 e e (d) 2 e e R m GM r 22. If three particles each of mass M are placed at the corners of an equilateral triangle of side a, the potential energy of the system and the work done of the side of the triangle is changed from a to 2a are: (a) 2 a GM 3 , a 2 GM 3 (b) - a GM 3 2 , a 2 GM 3 2 (c) - 2 2 a GM 3 , 2 2 a 4 GM 3 (d) - a GM 3 2 , a 2 GM 3 23. The orbital speed of Jupiter is: (a) equal to the orbital sped of earth (b) greater than the orbital speed of earth (c) less than the orbital speed of earth (d) proportional to the distance from earth 24. If the radius of orbit of a satellite is changed by a factor of 4, then time period is changed by a factor of: (a) 4 (b) 6 (c) 8 (d) none of these 25. The ratio of energy required to raise a satellite to a height h above the earth’s surface to that required to put it into the orbit is: (a) h : R (b) R : h (c) 2h : R (d) h : 2R In every question a statement of ASSERTION is followed by a statement of REASON. Mark the correct answer out of the following choices: If both ASSERTION and REASON are true and reason is the correct explanation of the assertion. (a) If both ASSERTION and REASON are true but reason is not the correct explanation of the assertion. (b) If ASSERTION is true but REASON is false. (c) If ASSERTION is false but REASON is true. 26. Assertion: Moment of inertia is to rotational motion what mass is to translational motion. Reason: Moment of inertia of a body is maximum about the axis passing through its centre of gravity. 27. Assertion: The centre of mass may lie outside the body. Page 3 [1] Test 05 (Rotational Motion & Gravitational Motion) 1. 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) 2 g/3 2. Which of the following statements about angular momentum is correct : (a) It is directly proportional to moment of inertia (b) It is a scalar quantity (c) All of these (d) None of these 3. A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The centre of mass of the man plus boat system will remain stationary with respect to water : (a) in all cases (b) only when the man is stationary initially and finally (c) only if the man moves without acceleration on the boat (d) only if the man and the boat have equal masses 4. What is the moment of inertia of a solid sphere of density ? and radius R about its diameter : (a) 176 105 R 5 ? (b) 176 105 R 2 ? (c) 105 176 R 5 ? (d) 105 176 R 2 ? 5. A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed v collides elastically with the stick as shown in fig. If after the collision ball comes to rest, then what should be the mass of the ball : (a) m = 2 M (b) m = M (c) m = M/2 (d) m = M/4 6. A uniform rod of mass m and length ?makes a constant angle ? with an axis of rotation which passes through one end of the rod. Its moment of inertia about this axis is : (a) 3 m 2 ? (b) 3 m 2 ? sin ? (c) 3 m 2 ? sin 2 ? (d) ? 2 2 cos 3 m ? 7. The angular velocity of a body changes from ? 1 to ? 2 without applying torque but by changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is : (a) ? 2 : ? 1 (b) 2 1 2 2 : ? ? (c) 1 2 : ? ? (d) 1/ ? 2 : 1/ ? 1 8. Two particles A and B initially at rest, moves toward each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass is : (a) zero (b) v (c) 1.5 v (d) 3 v 9. In rotatory motion, linear velocities of all the particles of the body are : (a) same (b) different (c) zero (d) cannot say 10. A cylinder of water, is rotating about its own axis with uniform angular velocity ?. The shape of free surface of water will be : (a) parabola (b) elliptical (c) circular (d) spherical 11. A body rolls down an inclined plane, if the K.E. of rotational motion is 40% of its K.E. of translation, then the body is : (a) cylinder (b) ring (c) solid disc (d) solid sphere 12. The moment of inertia of two bodies are I 1 and I 2 . Their geometrical shapes are same, the first made of iron and the second aluminum, then : (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 (d) relation between I 1 and I 2 depends on the actual shape of the bodies 13. Radius of a ring is 4 cm and its mass is 10 gm. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is : (a) 160 g - cm 2 (b) 80 g - cm 2 (c) 16 g - cm 2 (d) None of these 14. A ring and a sphere of the same mass and radius are rolling. Their kinetic energies are equal. The ratio of their velocities is : v [2] (a) 10 : 7 (b) 7 : 10 (c) 2 : 5 (d) 5 : 2 15. Mass of the earth is 81 times the mass of the moon and the distance between the earth and moon is 60 times the radius of the earth. If R is the radius of the earth, then the distance between the moon and the point on the line joining the moon and earth where the gravitational force becomes zero is : (a) 30R (b) 15R (c) 6R (d) 5R 16. Two particles of equal mass move in a circle of radius r under the action of their mutual gravitational attraction. If the mass of each particle is M, the speed of each particle is : (a) r GM (b) r 2 GM (c) r 4 GM (d) r GM 2 17. Suppose the gravitational force varies inversely as the nth power of the distance. Then the time period of a planet in circular orbit of radius R around the sum will be the proportional to : (a) R n (b) R (n+1)/2 (c) R (n-1)/2 (d) R -n 18. The gravitational potential at height h above earth’s surface is -5.12 ×10 7 J/kg and acceleration due to gravity at this point is 6.4 ms -2 . If radius of earth is 6400 km, the value of h is : (a) 1200 km (b) 1600 km (c) 1800 km (d) 2400 km 19. Two bodies with masses M 1 and M 2 are initially at rest and a distance R apart. They then move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances traveled by M 1 to the distance traveled by M 2 ? (a) 2 1 M M (b) 1 2 M M (c) 1 (d) 2 1 20. The ratio of the kinetic energy required to be given to the sattelite to escape earth’s gravitational field to the kinetic energy required to be given so that the sattelite moves in circular orbit just above earth’s atmosphere is: (a) One (b) Two (c) Half (d) Infinity 21. A tunnel is dug along a diameter of the earth of mass M e and radius R e . The force on a particle of mass m placed in the tunnel at a distance r from the centre is: (a) 3 e R m GM e r (b) r R m GM 3 e e (c) r mR GM 3 e e (d) 2 e e R m GM r 22. If three particles each of mass M are placed at the corners of an equilateral triangle of side a, the potential energy of the system and the work done of the side of the triangle is changed from a to 2a are: (a) 2 a GM 3 , a 2 GM 3 (b) - a GM 3 2 , a 2 GM 3 2 (c) - 2 2 a GM 3 , 2 2 a 4 GM 3 (d) - a GM 3 2 , a 2 GM 3 23. The orbital speed of Jupiter is: (a) equal to the orbital sped of earth (b) greater than the orbital speed of earth (c) less than the orbital speed of earth (d) proportional to the distance from earth 24. If the radius of orbit of a satellite is changed by a factor of 4, then time period is changed by a factor of: (a) 4 (b) 6 (c) 8 (d) none of these 25. The ratio of energy required to raise a satellite to a height h above the earth’s surface to that required to put it into the orbit is: (a) h : R (b) R : h (c) 2h : R (d) h : 2R In every question a statement of ASSERTION is followed by a statement of REASON. Mark the correct answer out of the following choices: If both ASSERTION and REASON are true and reason is the correct explanation of the assertion. (a) If both ASSERTION and REASON are true but reason is not the correct explanation of the assertion. (b) If ASSERTION is true but REASON is false. (c) If ASSERTION is false but REASON is true. 26. Assertion: Moment of inertia is to rotational motion what mass is to translational motion. Reason: Moment of inertia of a body is maximum about the axis passing through its centre of gravity. 27. Assertion: The centre of mass may lie outside the body. [3] Reason: The centre of mass depends only on the size of the body. 28. Assertion: The value of radius of gyration of a body is constant. Reason: Radius of gyration is the root mean square of the sum of the distances of the particles from the axis of rotation for a body having uniform density. 29. Assertion : The orbits of the planets are elliptical. Reason : In planetary motion, the total angular momentum remains constant. 30. Assertion : Moon does not have an atmosphere. Reason : Moon is smaller as compared to earth. Test 03 (Rotational Motion & Gravitational Motion) ANSWER KEY 1 C 6 C 11 D 16 C 21 A 26 C 2 A 7 C 12 C 17 B 22 B 27 C 3 A 8 A 13 A 18 B 23 C 28 D 4 C 9 B 14 B 19 B 24 C 29 A 5 D 10 A 15 C 20 B 25 C 30 A SOLUTIONS 1. a = ) R / K ( 1 sin g 2 2 + ? = 4 / g 1 1 30 sin g = + ° 2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, which is initially at rest, will always be at rest. 4. For solid sphere, I = 2 MR 5 2 = 2 3 R R 3 4 5 2 ? ? ? ? ? ? ? p = ? × 5 R 7 22 15 8 = ? 5 R 105 176 5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum mv ? ? ? ? ? ? 2 L = ? ? ? ? ? ? ? ? ? 12 ML 2 ……(2). As the collision is elastic, we have 2 2 2 I 2 1 MV 2 1 mv 2 1 ? + = ….(3). Substituting the values, we get m = 4 M 6. Mass of the element = dx m ? ? ? ? ? ? ? . M.I. of the element about the axis = ( ) 2 sin X dx m ? ? ? ? ? ? ? ? . I = ? ? ' 0 2 2 dx x sin m ? dx = ? 2 2 sin 3 m ? 7. I 1 ? 1 : I 2 ? 2 or MK 1 2 1 ? : MK 2 2 ? 2 or ? ? ? ? ? ? ? ? ? ? 1 2 2 1 : K K Page 4 [1] Test 05 (Rotational Motion & Gravitational Motion) 1. 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) 2 g/3 2. Which of the following statements about angular momentum is correct : (a) It is directly proportional to moment of inertia (b) It is a scalar quantity (c) All of these (d) None of these 3. A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The centre of mass of the man plus boat system will remain stationary with respect to water : (a) in all cases (b) only when the man is stationary initially and finally (c) only if the man moves without acceleration on the boat (d) only if the man and the boat have equal masses 4. What is the moment of inertia of a solid sphere of density ? and radius R about its diameter : (a) 176 105 R 5 ? (b) 176 105 R 2 ? (c) 105 176 R 5 ? (d) 105 176 R 2 ? 5. A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed v collides elastically with the stick as shown in fig. If after the collision ball comes to rest, then what should be the mass of the ball : (a) m = 2 M (b) m = M (c) m = M/2 (d) m = M/4 6. A uniform rod of mass m and length ?makes a constant angle ? with an axis of rotation which passes through one end of the rod. Its moment of inertia about this axis is : (a) 3 m 2 ? (b) 3 m 2 ? sin ? (c) 3 m 2 ? sin 2 ? (d) ? 2 2 cos 3 m ? 7. The angular velocity of a body changes from ? 1 to ? 2 without applying torque but by changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is : (a) ? 2 : ? 1 (b) 2 1 2 2 : ? ? (c) 1 2 : ? ? (d) 1/ ? 2 : 1/ ? 1 8. Two particles A and B initially at rest, moves toward each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass is : (a) zero (b) v (c) 1.5 v (d) 3 v 9. In rotatory motion, linear velocities of all the particles of the body are : (a) same (b) different (c) zero (d) cannot say 10. A cylinder of water, is rotating about its own axis with uniform angular velocity ?. The shape of free surface of water will be : (a) parabola (b) elliptical (c) circular (d) spherical 11. A body rolls down an inclined plane, if the K.E. of rotational motion is 40% of its K.E. of translation, then the body is : (a) cylinder (b) ring (c) solid disc (d) solid sphere 12. The moment of inertia of two bodies are I 1 and I 2 . Their geometrical shapes are same, the first made of iron and the second aluminum, then : (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 (d) relation between I 1 and I 2 depends on the actual shape of the bodies 13. Radius of a ring is 4 cm and its mass is 10 gm. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is : (a) 160 g - cm 2 (b) 80 g - cm 2 (c) 16 g - cm 2 (d) None of these 14. A ring and a sphere of the same mass and radius are rolling. Their kinetic energies are equal. The ratio of their velocities is : v [2] (a) 10 : 7 (b) 7 : 10 (c) 2 : 5 (d) 5 : 2 15. Mass of the earth is 81 times the mass of the moon and the distance between the earth and moon is 60 times the radius of the earth. If R is the radius of the earth, then the distance between the moon and the point on the line joining the moon and earth where the gravitational force becomes zero is : (a) 30R (b) 15R (c) 6R (d) 5R 16. Two particles of equal mass move in a circle of radius r under the action of their mutual gravitational attraction. If the mass of each particle is M, the speed of each particle is : (a) r GM (b) r 2 GM (c) r 4 GM (d) r GM 2 17. Suppose the gravitational force varies inversely as the nth power of the distance. Then the time period of a planet in circular orbit of radius R around the sum will be the proportional to : (a) R n (b) R (n+1)/2 (c) R (n-1)/2 (d) R -n 18. The gravitational potential at height h above earth’s surface is -5.12 ×10 7 J/kg and acceleration due to gravity at this point is 6.4 ms -2 . If radius of earth is 6400 km, the value of h is : (a) 1200 km (b) 1600 km (c) 1800 km (d) 2400 km 19. Two bodies with masses M 1 and M 2 are initially at rest and a distance R apart. They then move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances traveled by M 1 to the distance traveled by M 2 ? (a) 2 1 M M (b) 1 2 M M (c) 1 (d) 2 1 20. The ratio of the kinetic energy required to be given to the sattelite to escape earth’s gravitational field to the kinetic energy required to be given so that the sattelite moves in circular orbit just above earth’s atmosphere is: (a) One (b) Two (c) Half (d) Infinity 21. A tunnel is dug along a diameter of the earth of mass M e and radius R e . The force on a particle of mass m placed in the tunnel at a distance r from the centre is: (a) 3 e R m GM e r (b) r R m GM 3 e e (c) r mR GM 3 e e (d) 2 e e R m GM r 22. If three particles each of mass M are placed at the corners of an equilateral triangle of side a, the potential energy of the system and the work done of the side of the triangle is changed from a to 2a are: (a) 2 a GM 3 , a 2 GM 3 (b) - a GM 3 2 , a 2 GM 3 2 (c) - 2 2 a GM 3 , 2 2 a 4 GM 3 (d) - a GM 3 2 , a 2 GM 3 23. The orbital speed of Jupiter is: (a) equal to the orbital sped of earth (b) greater than the orbital speed of earth (c) less than the orbital speed of earth (d) proportional to the distance from earth 24. If the radius of orbit of a satellite is changed by a factor of 4, then time period is changed by a factor of: (a) 4 (b) 6 (c) 8 (d) none of these 25. The ratio of energy required to raise a satellite to a height h above the earth’s surface to that required to put it into the orbit is: (a) h : R (b) R : h (c) 2h : R (d) h : 2R In every question a statement of ASSERTION is followed by a statement of REASON. Mark the correct answer out of the following choices: If both ASSERTION and REASON are true and reason is the correct explanation of the assertion. (a) If both ASSERTION and REASON are true but reason is not the correct explanation of the assertion. (b) If ASSERTION is true but REASON is false. (c) If ASSERTION is false but REASON is true. 26. Assertion: Moment of inertia is to rotational motion what mass is to translational motion. Reason: Moment of inertia of a body is maximum about the axis passing through its centre of gravity. 27. Assertion: The centre of mass may lie outside the body. [3] Reason: The centre of mass depends only on the size of the body. 28. Assertion: The value of radius of gyration of a body is constant. Reason: Radius of gyration is the root mean square of the sum of the distances of the particles from the axis of rotation for a body having uniform density. 29. Assertion : The orbits of the planets are elliptical. Reason : In planetary motion, the total angular momentum remains constant. 30. Assertion : Moon does not have an atmosphere. Reason : Moon is smaller as compared to earth. Test 03 (Rotational Motion & Gravitational Motion) ANSWER KEY 1 C 6 C 11 D 16 C 21 A 26 C 2 A 7 C 12 C 17 B 22 B 27 C 3 A 8 A 13 A 18 B 23 C 28 D 4 C 9 B 14 B 19 B 24 C 29 A 5 D 10 A 15 C 20 B 25 C 30 A SOLUTIONS 1. a = ) R / K ( 1 sin g 2 2 + ? = 4 / g 1 1 30 sin g = + ° 2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, which is initially at rest, will always be at rest. 4. For solid sphere, I = 2 MR 5 2 = 2 3 R R 3 4 5 2 ? ? ? ? ? ? ? p = ? × 5 R 7 22 15 8 = ? 5 R 105 176 5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum mv ? ? ? ? ? ? 2 L = ? ? ? ? ? ? ? ? ? 12 ML 2 ……(2). As the collision is elastic, we have 2 2 2 I 2 1 MV 2 1 mv 2 1 ? + = ….(3). Substituting the values, we get m = 4 M 6. Mass of the element = dx m ? ? ? ? ? ? ? . M.I. of the element about the axis = ( ) 2 sin X dx m ? ? ? ? ? ? ? ? . I = ? ? ' 0 2 2 dx x sin m ? dx = ? 2 2 sin 3 m ? 7. I 1 ? 1 : I 2 ? 2 or MK 1 2 1 ? : MK 2 2 ? 2 or ? ? ? ? ? ? ? ? ? ? 1 2 2 1 : K K [4] 8. Force on particle A, F A = m A a A = t m A ? …(1), Similarly, F B = M B a B = t 2 m B ? × ….(2) . Now ) F F ( t 2 m t m B A B A = ? × = ? ? So, m A = 2m B For the centre of mass of the system v = B A B B A A m m v m v m + + or v = 0 m m v 2 m v m 2 B A B B = + - 9. From v = r ?, linear velocities (v) for particles at different distances (r) from the axis of rotation are different. 10. The shape of free surface of water is parabolic, because of difference in centrifugal force (F = m ? 2 , which is proportional to r) 11. 5 2 100 40 n translatio of . E . K rotation of . E . K = = . Now moment of inertia of sphere = 5 2 MR 2 12. ? 1 > ? 2 ? I 1 > I 2 13. I = mr 2 14. Total KE = 2 2 I 2 1 Mv 2 1 ? + = ? ? ? ? ? ? ? ? + 2 2 2 2 R v MK Mv 2 1 KE = ? ? ? ? ? ? ? ? + 2 2 2 R K 1 Mv 2 1 ? 10 7 R 5 R 2 1 1 R R 1 1 v V 2 2 2 2 2 s 2 d = ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? + = 15. Required distance x = 1 1 81 R 60 1 m m d 1 2 + = + = 6R 16. The particles will always remain diametrically opposite so that the force on each particle will be directed along the radius. Considering the circular motion of one particle. 2 2 2 ) r 2 ( Gm r m = ? or ? = r 4 Gm . 17. T = ? pR 2 , E= 1 n 2 R GMm m 2 1 - = ? Or ? = 2 / 1 1 n R GM 2 ? ? ? ? ? ? - ?T = 2 / ) 1 n ( 1 n R GM 2 2 R / GM 2 R 2 + - × p = p ?T ? R (n +1)/2 18. Gravitational potential V = – h R GM + . Also, g '= 2 ) h R ( GM + . Now V = 2 ) h R ( GM + - (R + h) = – g '(R + h) ? – 5.12 ×10 7 = – 6.4 (6400 + h) × 10 3 Solving we get , h = 1600 km 19. Because gravitational force is the mutual force, hence position of centre of mass remains unaffected. M 1 R 1 = M 2 R 2 Or 1 2 2 1 M M R R = 20. 2 R GM R GM 2 m 2 1 m 2 1 ) KE ( ) KE ( 2 0 2 e orbit escape = = ? ? = 21. F = 2 e 2 3 3 e e R m GM r m r 3 4 R 3 4 GM = × p × p r 22. U A = Potential energy of the system = a GMM a GMM a GMM - - - = – a GM 3 2 , (II) U B = – a 2 GMM a 2 GMM a 2 GMM - - = – a 2 GM 3 2 . work done = change in potential energy = U B – U A = a 2 GM 3 2 Page 5 [1] Test 05 (Rotational Motion & Gravitational Motion) 1. 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) 2 g/3 2. Which of the following statements about angular momentum is correct : (a) It is directly proportional to moment of inertia (b) It is a scalar quantity (c) All of these (d) None of these 3. A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The centre of mass of the man plus boat system will remain stationary with respect to water : (a) in all cases (b) only when the man is stationary initially and finally (c) only if the man moves without acceleration on the boat (d) only if the man and the boat have equal masses 4. What is the moment of inertia of a solid sphere of density ? and radius R about its diameter : (a) 176 105 R 5 ? (b) 176 105 R 2 ? (c) 105 176 R 5 ? (d) 105 176 R 2 ? 5. A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed v collides elastically with the stick as shown in fig. If after the collision ball comes to rest, then what should be the mass of the ball : (a) m = 2 M (b) m = M (c) m = M/2 (d) m = M/4 6. A uniform rod of mass m and length ?makes a constant angle ? with an axis of rotation which passes through one end of the rod. Its moment of inertia about this axis is : (a) 3 m 2 ? (b) 3 m 2 ? sin ? (c) 3 m 2 ? sin 2 ? (d) ? 2 2 cos 3 m ? 7. The angular velocity of a body changes from ? 1 to ? 2 without applying torque but by changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is : (a) ? 2 : ? 1 (b) 2 1 2 2 : ? ? (c) 1 2 : ? ? (d) 1/ ? 2 : 1/ ? 1 8. Two particles A and B initially at rest, moves toward each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass is : (a) zero (b) v (c) 1.5 v (d) 3 v 9. In rotatory motion, linear velocities of all the particles of the body are : (a) same (b) different (c) zero (d) cannot say 10. A cylinder of water, is rotating about its own axis with uniform angular velocity ?. The shape of free surface of water will be : (a) parabola (b) elliptical (c) circular (d) spherical 11. A body rolls down an inclined plane, if the K.E. of rotational motion is 40% of its K.E. of translation, then the body is : (a) cylinder (b) ring (c) solid disc (d) solid sphere 12. The moment of inertia of two bodies are I 1 and I 2 . Their geometrical shapes are same, the first made of iron and the second aluminum, then : (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 (d) relation between I 1 and I 2 depends on the actual shape of the bodies 13. Radius of a ring is 4 cm and its mass is 10 gm. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is : (a) 160 g - cm 2 (b) 80 g - cm 2 (c) 16 g - cm 2 (d) None of these 14. A ring and a sphere of the same mass and radius are rolling. Their kinetic energies are equal. The ratio of their velocities is : v [2] (a) 10 : 7 (b) 7 : 10 (c) 2 : 5 (d) 5 : 2 15. Mass of the earth is 81 times the mass of the moon and the distance between the earth and moon is 60 times the radius of the earth. If R is the radius of the earth, then the distance between the moon and the point on the line joining the moon and earth where the gravitational force becomes zero is : (a) 30R (b) 15R (c) 6R (d) 5R 16. Two particles of equal mass move in a circle of radius r under the action of their mutual gravitational attraction. If the mass of each particle is M, the speed of each particle is : (a) r GM (b) r 2 GM (c) r 4 GM (d) r GM 2 17. Suppose the gravitational force varies inversely as the nth power of the distance. Then the time period of a planet in circular orbit of radius R around the sum will be the proportional to : (a) R n (b) R (n+1)/2 (c) R (n-1)/2 (d) R -n 18. The gravitational potential at height h above earth’s surface is -5.12 ×10 7 J/kg and acceleration due to gravity at this point is 6.4 ms -2 . If radius of earth is 6400 km, the value of h is : (a) 1200 km (b) 1600 km (c) 1800 km (d) 2400 km 19. Two bodies with masses M 1 and M 2 are initially at rest and a distance R apart. They then move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances traveled by M 1 to the distance traveled by M 2 ? (a) 2 1 M M (b) 1 2 M M (c) 1 (d) 2 1 20. The ratio of the kinetic energy required to be given to the sattelite to escape earth’s gravitational field to the kinetic energy required to be given so that the sattelite moves in circular orbit just above earth’s atmosphere is: (a) One (b) Two (c) Half (d) Infinity 21. A tunnel is dug along a diameter of the earth of mass M e and radius R e . The force on a particle of mass m placed in the tunnel at a distance r from the centre is: (a) 3 e R m GM e r (b) r R m GM 3 e e (c) r mR GM 3 e e (d) 2 e e R m GM r 22. If three particles each of mass M are placed at the corners of an equilateral triangle of side a, the potential energy of the system and the work done of the side of the triangle is changed from a to 2a are: (a) 2 a GM 3 , a 2 GM 3 (b) - a GM 3 2 , a 2 GM 3 2 (c) - 2 2 a GM 3 , 2 2 a 4 GM 3 (d) - a GM 3 2 , a 2 GM 3 23. The orbital speed of Jupiter is: (a) equal to the orbital sped of earth (b) greater than the orbital speed of earth (c) less than the orbital speed of earth (d) proportional to the distance from earth 24. If the radius of orbit of a satellite is changed by a factor of 4, then time period is changed by a factor of: (a) 4 (b) 6 (c) 8 (d) none of these 25. The ratio of energy required to raise a satellite to a height h above the earth’s surface to that required to put it into the orbit is: (a) h : R (b) R : h (c) 2h : R (d) h : 2R In every question a statement of ASSERTION is followed by a statement of REASON. Mark the correct answer out of the following choices: If both ASSERTION and REASON are true and reason is the correct explanation of the assertion. (a) If both ASSERTION and REASON are true but reason is not the correct explanation of the assertion. (b) If ASSERTION is true but REASON is false. (c) If ASSERTION is false but REASON is true. 26. Assertion: Moment of inertia is to rotational motion what mass is to translational motion. Reason: Moment of inertia of a body is maximum about the axis passing through its centre of gravity. 27. Assertion: The centre of mass may lie outside the body. [3] Reason: The centre of mass depends only on the size of the body. 28. Assertion: The value of radius of gyration of a body is constant. Reason: Radius of gyration is the root mean square of the sum of the distances of the particles from the axis of rotation for a body having uniform density. 29. Assertion : The orbits of the planets are elliptical. Reason : In planetary motion, the total angular momentum remains constant. 30. Assertion : Moon does not have an atmosphere. Reason : Moon is smaller as compared to earth. Test 03 (Rotational Motion & Gravitational Motion) ANSWER KEY 1 C 6 C 11 D 16 C 21 A 26 C 2 A 7 C 12 C 17 B 22 B 27 C 3 A 8 A 13 A 18 B 23 C 28 D 4 C 9 B 14 B 19 B 24 C 29 A 5 D 10 A 15 C 20 B 25 C 30 A SOLUTIONS 1. a = ) R / K ( 1 sin g 2 2 + ? = 4 / g 1 1 30 sin g = + ° 2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, which is initially at rest, will always be at rest. 4. For solid sphere, I = 2 MR 5 2 = 2 3 R R 3 4 5 2 ? ? ? ? ? ? ? p = ? × 5 R 7 22 15 8 = ? 5 R 105 176 5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum mv ? ? ? ? ? ? 2 L = ? ? ? ? ? ? ? ? ? 12 ML 2 ……(2). As the collision is elastic, we have 2 2 2 I 2 1 MV 2 1 mv 2 1 ? + = ….(3). Substituting the values, we get m = 4 M 6. Mass of the element = dx m ? ? ? ? ? ? ? . M.I. of the element about the axis = ( ) 2 sin X dx m ? ? ? ? ? ? ? ? . I = ? ? ' 0 2 2 dx x sin m ? dx = ? 2 2 sin 3 m ? 7. I 1 ? 1 : I 2 ? 2 or MK 1 2 1 ? : MK 2 2 ? 2 or ? ? ? ? ? ? ? ? ? ? 1 2 2 1 : K K [4] 8. Force on particle A, F A = m A a A = t m A ? …(1), Similarly, F B = M B a B = t 2 m B ? × ….(2) . Now ) F F ( t 2 m t m B A B A = ? × = ? ? So, m A = 2m B For the centre of mass of the system v = B A B B A A m m v m v m + + or v = 0 m m v 2 m v m 2 B A B B = + - 9. From v = r ?, linear velocities (v) for particles at different distances (r) from the axis of rotation are different. 10. The shape of free surface of water is parabolic, because of difference in centrifugal force (F = m ? 2 , which is proportional to r) 11. 5 2 100 40 n translatio of . E . K rotation of . E . K = = . Now moment of inertia of sphere = 5 2 MR 2 12. ? 1 > ? 2 ? I 1 > I 2 13. I = mr 2 14. Total KE = 2 2 I 2 1 Mv 2 1 ? + = ? ? ? ? ? ? ? ? + 2 2 2 2 R v MK Mv 2 1 KE = ? ? ? ? ? ? ? ? + 2 2 2 R K 1 Mv 2 1 ? 10 7 R 5 R 2 1 1 R R 1 1 v V 2 2 2 2 2 s 2 d = ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? + = 15. Required distance x = 1 1 81 R 60 1 m m d 1 2 + = + = 6R 16. The particles will always remain diametrically opposite so that the force on each particle will be directed along the radius. Considering the circular motion of one particle. 2 2 2 ) r 2 ( Gm r m = ? or ? = r 4 Gm . 17. T = ? pR 2 , E= 1 n 2 R GMm m 2 1 - = ? Or ? = 2 / 1 1 n R GM 2 ? ? ? ? ? ? - ?T = 2 / ) 1 n ( 1 n R GM 2 2 R / GM 2 R 2 + - × p = p ?T ? R (n +1)/2 18. Gravitational potential V = – h R GM + . Also, g '= 2 ) h R ( GM + . Now V = 2 ) h R ( GM + - (R + h) = – g '(R + h) ? – 5.12 ×10 7 = – 6.4 (6400 + h) × 10 3 Solving we get , h = 1600 km 19. Because gravitational force is the mutual force, hence position of centre of mass remains unaffected. M 1 R 1 = M 2 R 2 Or 1 2 2 1 M M R R = 20. 2 R GM R GM 2 m 2 1 m 2 1 ) KE ( ) KE ( 2 0 2 e orbit escape = = ? ? = 21. F = 2 e 2 3 3 e e R m GM r m r 3 4 R 3 4 GM = × p × p r 22. U A = Potential energy of the system = a GMM a GMM a GMM - - - = – a GM 3 2 , (II) U B = – a 2 GMM a 2 GMM a 2 GMM - - = – a 2 GM 3 2 . work done = change in potential energy = U B – U A = a 2 GM 3 2 [5] 23. orbital velocity ? 0 = r 1 r GM ? (where r is the distance of planet from the sun). Since distance of Jupiter from the sun is more than the distance between the sun and the earth, therefore orbital speed of Jupiter rise less than the orbital speed of earth. 24. T ?R 3/2 2 / 3 2 1 2 1 R R T T ? ? ? ? ? ? ? ? = ? T 2 = T 1 (4) 3/2 = 8 T 1 25. Energy required to raise the satellite to a height h. E 1 = – GMm ) h R ( R GMmh R 1 h R 1 + = ? ? ? ? ? ? - + = h R gmRh ) h R ( R mh gR 2 + = + ? E 2 = 2 m 2 1 ? = h R gR m 2 1 2 + ? R h 2 E E 2 1 = 26. (c) Assertion is true but reason is false. Both mass and moment of inertia oppose change of motion. Due to inertia, a body continues in its state of rest or uniform motion unless external forces act on it to change its state. For translational motion, the magnitude of the external force required depends upon the mass of the body. On the other hand, in rotational motion, an external torque is required to change the rotation about an axis of the body. The magnitude of the torque required depends upon the distribution of mass about the axis of rotation. Moment of inertia of a body depends upon the mass and the axis of rotation. 27. (c) Assertion is true but reason is false. The centre of mass of a body may lie where there is no mass, as for example, the centre of mass of a ring is at its centre. By definition, the position vector ? r of the mass is equal to ? ? ? i i m i r m which depends upon the distribution of mass. It depends upon the shape and size of the body and can lie within or outside the body. 28. (d) Assertion is false but reason is true. Radius of gyration K is a conceptual distance defined by MK 2 = 2 i n 1 i i r m ? = for n particles and . M m i i = ? Only when the particles are of same mass i.e. the body has a uniform density will K = . n r ...... r r 2 n 2 2 2 1 + + + Since r is the distance of the particles from the axis of rotation, the value of K will change according to the axis. The radius of gyration of a body is, therefore, not constant. 29. (a) Both assertion and reason are true but reason is not the correct explanation of the assertion. According to Kepler’s laws, orbits of the planets are elliptical with sun at one of the foci. The speed of the planet and its angular speed keep on changing. Since no torque acts on the planet, its angular momentum remains constant. 30. (a) Both assertion and reason are true but reason is not the correct explanation of the assertion The escape velocity from the surface of moon is only 2.5km/s. The molecules of the atmospheric gases on the moon had much larger thermal velocities. Hence, the molecules of gases have escaped from the moon leaving it without any atmosphere.Read More

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