Rolling, Torque, and Angular Momentum Problems

This section provides 100 problems to test your understanding of rolling, torque, and angular momentum, focusing on torque as the driver of rotational motion, angular momentum and its conservation, and rolling motion as a combination of translation and rotation. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering rotational dynamics, a key topic for JEE/NEET success.

Numerical Problems

1. A uniform rod of mass $3kg$ and length $1.2m$ is pivoted at one end. A force $F=12N$ is applied perpendicularly at the other end. Calculate the angular acceleration.

   • (a) $6.0{\text{rad/s}}^2$
   • (b) $6.5{\text{rad/s}}^2$
   • (c) $7.0{\text{rad/s}}^2$
   • (d) $7.5{\text{rad/s}}^2$

2. A disk of mass $2kg$ and radius $0.4m$ has a torque $\tau =8N\cdot m$ applied about its center. Calculate the angular acceleration.

   • (a) $20{\text{rad/s}}^2$
   • (b) $22{\text{rad/s}}^2$
   • (c) $24{\text{rad/s}}^2$
   • (d) $25{\text{rad/s}}^2$

3. A skater with moment of inertia $I=4kg\cdot m^2$ spins at $\omega =5\text{rad/s}$. She reduces $I$ to $2kg\cdot m^2$ by pulling her arms in. Calculate her new angular velocity.

   • (a) $8\text{rad/s}$
   • (b) $9\text{rad/s}$
   • (c) $10\text{rad/s}$
   • (d) $11\text{rad/s}$

4. A solid sphere of mass $1kg$ and radius $0.3m$ rolls without slipping at $v_{\text{CM}}=4m/s$. Calculate the total kinetic energy.

   • (a) $8.0J$
   • (b) $8.8J$
   • (c) $9.0J$
   • (d) $9.6J$

5. A cylinder rolls down an incline of height $5m$ without slipping ($g=9.8m/s^2$). Calculate the speed of the center of mass at the bottom.

   • (a) $7.0m/s$
   • (b) $7.5m/s$
   • (c) $8.0m/s$
   • (d) $8.5m/s$

6. A particle of mass $0.5kg$ moves in a circle of radius $2m$ at $v=10m/s$. Calculate the angular momentum about the center.

   • (a) $8kg\cdot m^2/s$
   • (b) $9kg\cdot m^2/s$
   • (c) $10kg\cdot m^2/s$
   • (d) $11kg\cdot m^2/s$

7. A hoop of mass $2kg$ and radius $0.5m$ rolls down an incline at ${45}^{\circ }$ without slipping ($g=9.8m/s^2$). Calculate the acceleration of the center of mass.

   • (a) $3.0m/s^2$
   • (b) $3.47m/s^2$
   • (c) $3.8m/s^2$
   • (d) $4.0m/s^2$

8. A rod of $I=0.3kg\cdot m^2$ rotates at ${\omega }_i=8\text{rad/s}$. A torque $\tau =-3N\cdot m$ acts for $2s$. Calculate the final angular velocity.

   • (a) $2\text{rad/s}$
   • (b) $3\text{rad/s}$
   • (c) $4\text{rad/s}$
   • (d) $5\text{rad/s}$

9. A sphere rolls up an incline at ${30}^{\circ }$ with initial $v_{\text{CM}}=6m/s$ ($g=9.8m/s^2$). Calculate the distance it travels before stopping.

   • (a) $3.5m$
   • (b) $4.0m$
   • (c) $4.5m$
   • (d) $5.0m$

10. A disk with $I=0.1kg\cdot m^2$ rotates at $\omega =12\text{rad/s}$. Calculate the angular momentum.

    • (a) $1.0kg\cdot m^2/s$
    • (b) $1.2kg\cdot m^2/s$
    • (c) $1.4kg\cdot m^2/s$
    • (d) $1.6kg\cdot m^2/s$

11. A $1kg$ mass on a $1m$ rod pivoted at the center has a force $F=6N$ applied at ${45}^{\circ }$ at the end. Calculate the angular acceleration.

    • (a) $5.0{\text{rad/s}}^2$
    • (b) $5.5{\text{rad/s}}^2$
    • (c) $6.0{\text{rad/s}}^2$
    • (d) $6.5{\text{rad/s}}^2$

12. A skater with $I=6kg\cdot m^2$ spins at $\omega =3\text{rad/s}$. She reduces $I$ to $3kg\cdot m^2$. Calculate her new angular velocity.

    • (a) $5\text{rad/s}$
    • (b) $6\text{rad/s}$
    • (c) $7\text{rad/s}$
    • (d) $8\text{rad/s}$

13. A cylinder of mass $3kg$ and radius $0.2m$ rolls without slipping at $v_{\text{CM}}=5m/s$. Calculate the total kinetic energy.

    • (a) $37.5J$
    • (b) $40.0J$
    • (c) $42.5J$
    • (d) $45.0J$

14. A hoop rolls down an incline of height $6m$ without slipping ($g=9.8m/s^2$). Calculate the speed of the center of mass at the bottom.

    • (a) $6.0m/s$
    • (b) $6.5m/s$
    • (c) $7.0m/s$
    • (d) $7.5m/s$

15. A particle of mass $0.8kg$ moves in a circle of radius $1.5m$ at $v=15m/s$. Calculate the angular momentum about the center.

    • (a) $15kg\cdot m^2/s$
    • (b) $16kg\cdot m^2/s$
    • (c) $17kg\cdot m^2/s$
    • (d) $18kg\cdot m^2/s$

16. A rod of mass $4kg$ and length $1.5m$ is pivoted at one end. A force $F=20N$ is applied perpendicularly at the other end. Calculate the angular acceleration.

    • (a) $8{\text{rad/s}}^2$
    • (b) $9{\text{rad/s}}^2$
    • (c) $10{\text{rad/s}}^2$
    • (d) $11{\text{rad/s}}^2$

17. A disk with $I=0.15kg\cdot m^2$ rotates at ${\omega }_i=10\text{rad/s}$. A torque $\tau =3N\cdot m$ acts for $4s$. Calculate the final angular velocity.

    • (a) $20\text{rad/s}$
    • (b) $22\text{rad/s}$
    • (c) $24\text{rad/s}$
    • (d) $26\text{rad/s}$

18. A sphere rolls down an incline at ${60}^{\circ }$ without slipping ($g=9.8m/s^2$). Calculate the acceleration of the center of mass.

    • (a) $5.0m/s^2$
    • (b) $5.5m/s^2$
    • (c) $6.0m/s^2$
    • (d) $6.5m/s^2$

19. A skater with $I=8kg\cdot m^2$ spins at $\omega =2\text{rad/s}$. She reduces $I$ to $4kg\cdot m^2$. Calculate her new angular velocity.

    • (a) $3\text{rad/s}$
    • (b) $4\text{rad/s}$
    • (c) $5\text{rad/s}$
    • (d) $6\text{rad/s}$

20. A hoop of mass $1kg$ and radius $0.2m$ rolls without slipping at $v_{\text{CM}}=3m/s$. Calculate the total kinetic energy.

    • (a) $4.0J$
    • (b) $4.5J$
    • (c) $5.0J$
    • (d) $5.5J$

21. A particle of mass $0.6kg$ moves in a circle of radius $1m$ at $v=12m/s$. Calculate the angular momentum about the center.

    • (a) $6.0kg\cdot m^2/s$
    • (b) $6.5kg\cdot m^2/s$
    • (c) $7.0kg\cdot m^2/s$
    • (d) $7.2kg\cdot m^2/s$

22. A rod of $I=0.5kg\cdot m^2$ rotates at ${\omega }_i=4\text{rad/s}$. A torque $\tau =-2N\cdot m$ acts for $3s$. Calculate the final angular velocity.

    • (a) $-2\text{rad/s}$
    • (b) $-1\text{rad/s}$
    • (c) $0\text{rad/s}$
    • (d) $1\text{rad/s}$

23. A sphere rolls up an incline at ${45}^{\circ }$ with initial $v_{\text{CM}}=8m/s$ ($g=9.8m/s^2$). Calculate the distance it travels before stopping.

    • (a) $4.5m$
    • (b) $5.0m$
    • (c) $5.5m$
    • (d) $6.0m$

24. A disk of mass $4kg$ and radius $0.5m$ has a torque $\tau =10N\cdot m$ applied about its center. Calculate the angular acceleration.

    • (a) $8{\text{rad/s}}^2$
    • (b) $9{\text{rad/s}}^2$
    • (c) $10{\text{rad/s}}^2$
    • (d) $11{\text{rad/s}}^2$

25. A cylinder rolls down an incline of height $3m$ without slipping ($g=9.8m/s^2$). Calculate the speed of the center of mass at the bottom.

    • (a) $5.5m/s$
    • (b) $6.0m/s$
    • (c) $6.5m/s$
    • (d) $7.0m/s$

26. A $2kg$ mass on a $1.2m$ rod pivoted at the center has a force $F=15N$ applied perpendicularly at the end. Calculate the angular acceleration.

    • (a) $10{\text{rad/s}}^2$
    • (b) $11{\text{rad/s}}^2$
    • (c) $12{\text{rad/s}}^2$
    • (d) $13{\text{rad/s}}^2$

27. A skater with $I=3kg\cdot m^2$ spins at $\omega =6\text{rad/s}$. She reduces $I$ to $1.5kg\cdot m^2$. Calculate her new angular velocity.

    • (a) $10\text{rad/s}$
    • (b) $11\text{rad/s}$
    • (c) $12\text{rad/s}$
    • (d) $13\text{rad/s}$

28. A hoop rolls down an incline at ${60}^{\circ }$ without slipping ($g=9.8m/s^2$). Calculate the acceleration of the center of mass.

    • (a) $3.5m/s^2$
    • (b) $4.0m/s^2$
    • (c) $4.24m/s^2$
    • (d) $4.5m/s^2$

29. A disk with $I=0.25kg\cdot m^2$ rotates at $\omega =15\text{rad/s}$. Calculate the angular momentum.

    • (a) $3.0kg\cdot m^2/s$
    • (b) $3.5kg\cdot m^2/s$
    • (c) $3.75kg\cdot m^2/s$
    • (d) $4.0kg\cdot m^2/s$

30. A sphere of mass $2kg$ and radius $0.4m$ rolls without slipping at $v_{\text{CM}}=6m/s$. Calculate the total kinetic energy.

    • (a) $36J$
    • (b) $38J$
    • (c) $40J$
    • (d) $42J$

31. A particle of mass $1kg$ moves in a circle of radius $0.5m$ at $v=20m/s$. Calculate the angular momentum about the center.

    • (a) $8kg\cdot m^2/s$
    • (b) $9kg\cdot m^2/s$
    • (c) $10kg\cdot m^2/s$
    • (d) $11kg\cdot m^2/s$

32. A rod of $I=0.2kg\cdot m^2$ rotates at ${\omega }_i=5\text{rad/s}$. A torque $\tau =4N\cdot m$ acts for $2s$. Calculate the final angular velocity.

    • (a) $40\text{rad/s}$
    • (b) $42\text{rad/s}$
    • (c) $44\text{rad/s}$
    • (d) $45\text{rad/s}$

33. A sphere rolls down an incline of height $7m$ without slipping ($g=9.8m/s^2$). Calculate the speed of the center of mass at the bottom.

    • (a) $8.0m/s$
    • (b) $8.5m/s$
    • (c) $9.0m/s$
    • (d) $9.5m/s$

34. A hoop rolls up an incline at ${30}^{\circ }$ with initial $v_{\text{CM}}=5m/s$ ($g=9.8m/s^2$). Calculate the distance it travels before stopping.

    • (a) $2.0m$
    • (b) $2.5m$
    • (c) $3.0m$
    • (d) $3.5m$

35. A disk of mass $5kg$ and radius $0.6m$ has a torque $\tau =12N\cdot m$ applied about its center. Calculate the angular acceleration.

    • (a) $6{\text{rad/s}}^2$
    • (b) $7{\text{rad/s}}^2$
    • (c) $8{\text{rad/s}}^2$
    • (d) $9{\text{rad/s}}^2$

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Conceptual Problems

36. What does torque cause in a rotating system?

• (a) Linear acceleration
• (b) Angular acceleration
• (c) Linear displacement
• (d) Angular velocity

37. When is angular momentum conserved?

• (a) When net external force is zero
• (b) When net external torque is zero
• (c) When kinetic energy is conserved
• (d) When linear momentum is conserved

38. What does rolling without slipping imply?

• (a) $v_{\text{CM}}=R\omega$
• (b) $v_{\text{CM}}\ne R\omega$
• (c) $v_{\text{CM}}=0$
• (d) $\omega =0$

39. What is the unit of angular momentum?

• (a) $kg\cdot m^2/s$
• (b) $N\cdot m$
• (c) $J$
• (d) $kg\cdot m/s$

40. What happens to angular velocity when moment of inertia decreases?

• (a) Decreases
• (b) Increases
• (c) Remains the same
• (d) Becomes zero

41. How does friction affect rolling motion down an incline?

• (a) Provides linear acceleration
• (b) Provides torque for rotation
• (c) Causes slipping
• (d) Reduces angular velocity

42. What is the physical significance of $I\omega$?

• (a) Rotational kinetic energy
• (b) Angular momentum
• (c) Torque
• (d) Moment of inertia

43. What does a negative torque imply?

• (a) Increases angular velocity
• (b) Decreases angular velocity
• (c) No change in angular velocity
• (d) Increases linear velocity

44. What is the dimension of torque?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (d) $[\text{L}{\text{T}}^{-2}]$

45. What happens to the speed of a rolling object compared to a sliding object down an incline?

• (a) Rolling is faster
• (b) Sliding is faster
• (c) Both are the same
• (d) Depends on the object

46. What does a zero total kinetic energy in rolling indicate?

• (a) Object is still rotating
• (b) Object is at rest
• (c) Object is sliding
• (d) Object is spinning

47. What is the significance of $rF\sin \theta$?

• (a) Angular momentum
• (b) Rotational kinetic energy
• (c) Torque
• (d) Moment of inertia

48. What does the conservation of angular momentum imply?

• (a) Constant linear velocity
• (b) Constant angular velocity
• (c) Constant angular momentum when ${\tau }_{\text{net}}=0$
• (d) Constant kinetic energy

49. How does the moment of inertia affect rolling acceleration?

• (a) Larger $I$ increases acceleration
• (b) Larger $I$ decreases acceleration
• (c) No effect
• (d) Depends on the angle

50. What does a zero net torque imply?

• (a) No rotation
• (b) Constant angular velocity
• (c) No angular momentum
• (d) No kinetic energy

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Derivation Problems

51. Derive Newton’s second law for rotation ${\tau }_{\text{net}}=I\alpha$.

52. Derive the conservation of angular momentum for a skater pulling in her arms.

53. Derive the condition for rolling without slipping $v_{\text{CM}}=R\omega$.

54. Derive the angular momentum of a particle $L=mvr\sin \theta$.

55. Derive the total kinetic energy of a rolling object.

56. Derive the acceleration of a cylinder rolling down an incline.

57. Derive the torque due to gravity on a rod pivoted at one end.

58. Derive the angular momentum of a rigid body $L=I\omega$.

59. Derive the speed of a rolling object down an incline using energy conservation.

60. Derive the moment of inertia of a thin hoop about its center.

61. Derive the work done by a constant torque $W=\tau \theta$.

62. Derive the final angular velocity using torque and angular momentum.

63. Derive the distance traveled by a rolling object up an incline.

64. Derive the effect of torque on angular momentum $\frac{dL}{dt}=\tau$.

65. Derive the acceleration of a hoop rolling down an incline.

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NEET-style Conceptual Problems

66. What is the unit of torque in SI units?

• (a) $N\cdot m$
• (b) $J$
• (c) $kg\cdot m^2/s$
• (d) $W$

67. What does a zero angular momentum indicate?

• (a) Object is at rest or not rotating
• (b) Object is sliding
• (c) Object is rolling
• (d) Object has no mass

68. Which quantity is conserved in rolling motion down an incline?

• (a) Angular velocity
• (b) Linear velocity
• (c) Mechanical energy
• (d) Angular acceleration

69. What happens to kinetic energy when an object rolls without slipping?

• (a) Only translational energy
• (b) Only rotational energy
• (c) Both translational and rotational energy
• (d) No kinetic energy

70. What is the dimension of angular momentum?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
• (d) $[\text{L}{\text{T}}^{-1}]$

71. What does torque depend on?

• (a) Force only
• (b) Distance only
• (c) Force, distance, and angle between them
• (d) Mass of the object

72. What is the role of friction in rolling motion?

• (a) Causes slipping
• (b) Provides torque for rotation
• (c) Increases linear speed
• (d) Reduces angular velocity

73. What happens to angular velocity when a skater pulls in her arms?

• (a) Decreases
• (b) Increases
• (c) Remains the same
• (d) Becomes zero

74. Why does a rolling object have less acceleration than a sliding object?

• (a) Rotational energy reduces linear acceleration
• (b) Friction increases acceleration
• (c) Gravity is less for rolling
• (d) Mass distribution changes

75. What is the unit of moment of inertia?

• (a) $kg\cdot m^2$
• (b) $N\cdot m$
• (c) $J$
• (d) $kg\cdot m/s$

76. What does a constant angular momentum imply?

• (a) No net external torque
• (b) Constant linear velocity
• (c) Constant kinetic energy
• (d) Constant potential energy

77. Which motion involves both translation and rotation?

• (a) Pure rotation
• (b) Pure translation
• (c) Rolling motion
• (d) Sliding motion

78. What is the direction of torque in a rotating system?

• (a) Along the force
• (b) Along the radius
• (c) Perpendicular to the plane of $\overrightarrow{r}$ and $\overrightarrow{F}$
• (d) Along the velocity

79. What does a pseudo-force do in a rotating frame?

• (a) Conserves angular momentum
• (b) Affects rotational dynamics
• (c) Provides centripetal force
• (d) Reduces friction

80. What is the dimension of angular velocity?

• (a) $[{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-1}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of angular momentum in a spinning object?

• (a) Increases linear speed
• (b) Maintains rotational motion
• (c) Reduces torque
• (d) Increases potential energy

82. What happens to total kinetic energy when a rolling object stops?

• (a) Becomes zero
• (b) Converts to potential energy
• (c) Remains constant
• (d) Increases

83. Why does torque depend on the angle between force and radius?

• (a) It affects linear acceleration
• (b) It determines the perpendicular component
• (c) It changes the mass
• (d) It affects velocity

84. What is the significance of $\frac{1}{2}M{v}_{\text{CM}}^2+\frac{1}{2}I{\omega }^2$?

• (a) Angular momentum
• (b) Total kinetic energy of rolling
• (c) Torque
• (d) Potential energy

85. What is the unit of angular acceleration?

• (a) ${\text{rad/s}}^2$
• (b) $N\cdot m$
• (c) $J$
• (d) $kg\cdot m^2/s$

86. What does a zero acceleration of the center of mass indicate in rolling?

• (a) Object is sliding
• (b) Object is at rest or moving uniformly
• (c) Object is spinning faster
• (d) Object is precessing

87. What is the physical significance of $\frac{dL}{dt}=\tau$?

• (a) Work-energy theorem
• (b) Torque changes angular momentum
• (c) Angular momentum conservation
• (d) Power in rotation

88. Why does a hoop roll slower than a sphere down an incline?

• (a) Larger moment of inertia
• (b) Smaller mass
• (c) Larger radius
• (d) Smaller friction

89. What is the dimension of moment of inertia?

• (a) $[\text{M}{\text{L}}^2]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (d) $[\text{L}{\text{T}}^{-2}]$

90. How does friction contribute to rolling motion?

• (a) Increases linear speed
• (b) Provides torque for rotation
• (c) Reduces angular momentum
• (d) Increases potential energy

91. What is the role of conservation in a spinning rocket?

• (a) Determines linear speed
• (b) Maintains angular momentum
• (c) Increases kinetic energy
• (d) Reduces torque

92. What does a zero speed of the center of mass indicate?

• (a) Object is still rotating
• (b) Object is at rest
• (c) Object is sliding
• (d) Object is precessing

93. What is the physical significance of $mvr\sin \theta$?

• (a) Angular momentum of a particle
• (b) Rotational kinetic energy
• (c) Torque
• (d) Moment of inertia

94. What is the dimension of linear velocity in rolling?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

95. Why does moment of inertia affect angular momentum?

• (a) It changes linear velocity
• (b) It determines $L=I\omega$
• (c) It changes torque
• (d) It changes kinetic energy

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NEET-style Numerical Problems

96. A rod of mass $2kg$ and length $1m$ is pivoted at one end. A force $F=8N$ is applied perpendicularly at the other end. What is the angular acceleration?

• (a) $10{\text{rad/s}}^2$
• (b) $11{\text{rad/s}}^2$
• (c) $12{\text{rad/s}}^2$
• (d) $13{\text{rad/s}}^2$

97. A disk with $I=0.1kg\cdot m^2$ rotates at $\omega =8\text{rad/s}$. What is the angular momentum?

• (a) $0.6kg\cdot m^2/s$
• (b) $0.7kg\cdot m^2/s$
• (c) $0.8kg\cdot m^2/s$
• (d) $0.9kg\cdot m^2/s$

98. A cylinder rolls down an incline of height $2m$ without slipping ($g=9.8m/s^2$). What is the speed of the center of mass at the bottom?

• (a) $4.5m/s$
• (b) $5.0m/s$
• (c) $5.5m/s$
• (d) $6.0m/s$

99. A skater with $I=3kg\cdot m^2$ spins at $\omega =4\text{rad/s}$. She reduces $I$ to $1kg\cdot m^2$. What is her new angular velocity?

• (a) $10\text{rad/s}$
• (b) $11\text{rad/s}$
• (c) $12\text{rad/s}$
• (d) $13\text{rad/s}$

100. A sphere of mass $1kg$ and radius $0.2m$ rolls without slipping at $v_{\text{CM}}=3m/s$. What is the total kinetic energy?
     - (a) $4.5J$
     - (b) $5.0J$
     - (c) $5.5J$
     - (d) $6.0J$

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