Rotation Problems

This section provides 100 problems to test your understanding of rotational motion, including kinematics of rotation, dynamics (torque and moment of inertia), rotational kinetic energy, angular momentum and its conservation, rolling motion, and gyroscopic effects with advanced applications. 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 wheel starts from rest with a constant angular acceleration $\alpha =4{\text{rad/s}}^2$. Calculate the angular velocity after $5s$.

   • (a) $15\text{rad/s}$
   • (b) $20\text{rad/s}$
   • (c) $25\text{rad/s}$
   • (d) $30\text{rad/s}$

2. A fan blade rotates at $\omega =12\text{rad/s}$ and completes 60 revolutions. Calculate the time taken.

   • (a) $28s$
   • (b) $30s$
   • (c) $31s$
   • (d) $32s$

3. A $3kg$ mass is attached to a $1.5m$ rod pivoted at one end. A force $F=15N$ is applied perpendicularly at the other end. Calculate the angular acceleration.

   • (a) $5.0{\text{rad/s}}^2$
   • (b) $6.0{\text{rad/s}}^2$
   • (c) $6.67{\text{rad/s}}^2$
   • (d) $7.5{\text{rad/s}}^2$

4. A uniform disk of mass $5kg$ and radius $0.4m$ has a torque $\tau =10N\cdot m$ applied. Calculate the angular acceleration.

   • (a) $10{\text{rad/s}}^2$
   • (b) $12{\text{rad/s}}^2$
   • (c) $15{\text{rad/s}}^2$
   • (d) $18{\text{rad/s}}^2$

5. A disk of mass $2kg$ and radius $0.3m$ rotates at $\omega =8\text{rad/s}$. Calculate the rotational kinetic energy.

   • (a) $5.0J$
   • (b) $5.76J$
   • (c) $6.0J$
   • (d) $6.5J$

6. A skater with moment of inertia $I=6kg\cdot m^2$ spins at $\omega =4\text{rad/s}$. She reduces $I$ to $3kg\cdot m^2$ by pulling her arms in. Calculate her new angular velocity.

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

7. A solid sphere of mass $4kg$ and radius $0.2m$ rolls without slipping at $v_{\text{CM}}=6m/s$. Calculate the total kinetic energy.

   • (a) $72J$
   • (b) $76J$
   • (c) $80J$
   • (d) $84J$

8. 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) $6.0m/s$
   • (b) $6.5m/s$
   • (c) $7.0m/s$
   • (d) $7.5m/s$

9. A spinning top with $I=0.01kg\cdot m^2$ spins at $\omega =40\text{rad/s}$. Its mass is $0.5kg$, and the distance from pivot to COM is $0.02m$ ($g=9.8m/s^2$). Calculate the precession angular velocity.

   • (a) $0.245\text{rad/s}$
   • (b) $0.3\text{rad/s}$
   • (c) $0.35\text{rad/s}$
   • (d) $0.4\text{rad/s}$

10. A point on a wheel of radius $0.6m$ rotates at $\omega =15\text{rad/s}$. Calculate the centripetal acceleration of the point.

    • (a) $120m/s^2$
    • (b) $135m/s^2$
    • (c) $150m/s^2$
    • (d) $165m/s^2$

11. A disk starts from rest with $\alpha =5{\text{rad/s}}^2$. Calculate the angular displacement after $3s$.

    • (a) $20\text{rad}$
    • (b) $22.5\text{rad}$
    • (c) $25\text{rad}$
    • (d) $27.5\text{rad}$

12. A $1kg$ mass on a $1m$ rod pivoted at the center has a force $F=8N$ applied perpendicularly at the end. Calculate the angular acceleration.

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

13. A rod of mass $2kg$ and length $2m$ pivoted at one end starts from rest. A torque $\tau =6N\cdot m$ is applied for $2s$. Calculate the final angular velocity.

    • (a) $4.5\text{rad/s}$
    • (b) $5.0\text{rad/s}$
    • (c) $5.5\text{rad/s}$
    • (d) $6.0\text{rad/s}$

14. A particle of mass $0.4kg$ moves in a circle of radius $3m$ at $v=12m/s$. Calculate the angular momentum about the center.

    • (a) $12kg\cdot m^2/s$
    • (b) $14.4kg\cdot m^2/s$
    • (c) $16.8kg\cdot m^2/s$
    • (d) $18.0kg\cdot m^2/s$

15. A hoop of mass $2kg$ and radius $0.5m$ rolls down an incline at ${30}^{\circ }$ without slipping ($g=9.8m/s^2$). Calculate the acceleration of the center of mass.

    • (a) $2.0m/s^2$
    • (b) $2.45m/s^2$
    • (c) $2.8m/s^2$
    • (d) $3.0m/s^2$

16. A gyroscope with $I=0.05kg\cdot m^2$ spins at $\omega =80\text{rad/s}$, with mass $0.8kg$ and pivot to COM distance $0.04m$ ($g=9.8m/s^2$). Calculate the precession angular velocity.

    • (a) $0.196\text{rad/s}$
    • (b) $0.22\text{rad/s}$
    • (c) $0.25\text{rad/s}$
    • (d) $0.28\text{rad/s}$

17. A wheel starts from rest with $\alpha =6{\text{rad/s}}^2$. Calculate the angular velocity after $2s$.

    • (a) $10\text{rad/s}$
    • (b) $12\text{rad/s}$
    • (c) $14\text{rad/s}$
    • (d) $16\text{rad/s}$

18. A uniform rod of mass $4kg$ and length $1.5m$ pivoted at one end has a force $F=20N$ 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) $11{\text{rad/s}}^2$

19. A disk of $I=0.15kg\cdot m^2$ rotates at $\omega =10\text{rad/s}$. A torque $\tau =3N\cdot m$ is applied. Calculate the power delivered.

    • (a) $25W$
    • (b) $30W$
    • (c) $35W$
    • (d) $40W$

20. A skater with $I=5kg\cdot m^2$ spins at $\omega =3\text{rad/s}$. She reduces $I$ to $2kg\cdot m^2$. Calculate her new angular velocity.

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

21. A sphere 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) $7.5m/s$
    • (b) $8.0m/s$
    • (c) $8.5m/s$
    • (d) $9.0m/s$

22. A bicycle wheel with $I=0.4kg\cdot m^2$ spins at $\omega =25\text{rad/s}$ and turns at $\mathrm{\Omega }=1.5\text{rad/s}$. Calculate the gyroscopic couple.

    • (a) $12N\cdot m$
    • (b) $15N\cdot m$
    • (c) $18N\cdot m$
    • (d) $20N\cdot m$

23. A point on a wheel of radius $0.8m$ rotates at $\omega =18\text{rad/s}$. Calculate the centripetal acceleration.

    • (a) $220m/s^2$
    • (b) $240m/s^2$
    • (c) $259m/s^2$
    • (d) $280m/s^2$

24. A disk starts from ${\omega }_i=4\text{rad/s}$ with $\alpha =2{\text{rad/s}}^2$. Calculate the angular displacement after $5s$.

    • (a) $40\text{rad}$
    • (b) $45\text{rad}$
    • (c) $50\text{rad}$
    • (d) $55\text{rad}$

25. A $2kg$ mass on a $1.2m$ rod pivoted at the center has a force $F=10N$ 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) $5.9{\text{rad/s}}^2$
    • (d) $6.3{\text{rad/s}}^2$

26. A rod of $I=0.3kg\cdot m^2$ rotates at ${\omega }_i=6\text{rad/s}$. A torque $\tau =-3N\cdot m$ acts for $2s$. Calculate the final angular velocity.

    • (a) $0\text{rad/s}$
    • (b) $2\text{rad/s}$
    • (c) $4\text{rad/s}$
    • (d) $6\text{rad/s}$

27. A particle of mass $0.6kg$ moves in a circle of radius $2m$ at $v=15m/s$. Calculate the angular momentum about the center.

    • (a) $15kg\cdot m^2/s$
    • (b) $18kg\cdot m^2/s$
    • (c) $21kg\cdot m^2/s$
    • (d) $24kg\cdot m^2/s$

28. A hoop of mass $3kg$ and radius $0.4m$ 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) $4.25m/s^2$
    • (d) $4.5m/s^2$

29. A gyroscope with $I=0.08kg\cdot m^2$ spins at $\omega =60\text{rad/s}$, with mass $1kg$ and pivot to COM distance $0.03m$ ($g=9.8m/s^2$). Calculate the precession angular velocity.

    • (a) $0.05\text{rad/s}$
    • (b) $0.06\text{rad/s}$
    • (c) $0.07\text{rad/s}$
    • (d) $0.08\text{rad/s}$

30. A wheel of $I=0.25kg\cdot m^2$ starts from rest. A torque $\tau =5N\cdot m$ is applied for $4s$. Calculate the final angular velocity.

    • (a) $80\text{rad/s}$
    • (b) $85\text{rad/s}$
    • (c) $90\text{rad/s}$
    • (d) $95\text{rad/s}$

31. A disk of mass $3kg$ and radius $0.5m$ rotates at $\omega =12\text{rad/s}$. Calculate the rotational kinetic energy.

    • (a) $27J$
    • (b) $30J$
    • (c) $33J$
    • (d) $36J$

32. 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) $6\text{rad/s}$

33. 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) $5.0m$
    • (b) $5.5m$
    • (c) $6.0m$
    • (d) $6.5m$

34. A bicycle wheel with $I=0.5kg\cdot m^2$ spins at $\omega =30\text{rad/s}$ and turns at $\mathrm{\Omega }=2\text{rad/s}$. Calculate the gyroscopic couple.

    • (a) $25N\cdot m$
    • (b) $30N\cdot m$
    • (c) $35N\cdot m$
    • (d) $40N\cdot m$

35. A rocket with $I=400kg\cdot m^2$ spins at $\omega =12\text{rad/s}$ for stability. A torque $\tau =240N\cdot m$ acts. Calculate the precession rate.

    • (a) $0.04\text{rad/s}$
    • (b) $0.05\text{rad/s}$
    • (c) $0.06\text{rad/s}$
    • (d) $0.07\text{rad/s}$

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Conceptual Problems

36. What does angular velocity represent?

• (a) Rate of change of linear velocity
• (b) Rate of change of angular displacement
• (c) Rate of change of torque
• (d) Rate of change of moment of inertia

37. What causes torque in rotational motion?

• (a) Force applied at the axis
• (b) Force applied perpendicular to the radius
• (c) Force applied parallel to the radius
• (d) Force applied randomly

38. What does the moment of inertia depend on?

• (a) Mass only
• (b) Radius only
• (c) Mass and its distribution from the axis
• (d) Angular velocity

39. 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

40. What is the unit of rotational kinetic energy?

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

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

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

42. 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) $\omega =0$

43. What is the physical significance of precession?

• (a) Change in spin speed
• (b) Change in the axis of rotation’s orientation
• (c) Change in linear velocity
• (d) Change in moment of inertia

44. 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

45. 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) $[\text{L}{\text{T}}^{-1}]$

46. What does a zero angular acceleration imply?

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

47. What is the significance of $\frac{1}{2}I{\omega }^2$?

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

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

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

49. What does the gyroscopic couple do in a turning bicycle?

• (a) Increases speed
• (b) Stabilizes the bike
• (c) Reduces angular velocity
• (d) Increases friction

50. How does angular velocity affect precession rate?

• (a) Higher $\omega$ increases precession
• (b) Higher $\omega$ decreases precession
• (c) No effect
• (d) Depends on torque

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Derivation Problems

51. Derive the rotational kinematic equation ${\omega }_f={\omega }_i+\alpha t$.

52. Derive the moment of inertia of a uniform rod about its end.

53. Derive the rotational kinetic energy formula $K{E}_{\text{rot}}=\frac{1}{2}I{\omega }^2$.

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

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

56. Derive the precession angular velocity $\mathrm{\Omega }=\frac{\tau }{L}$.

57. Derive the moment of inertia of a disk about its center.

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

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

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

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

62. Derive the final angular velocity using the work-energy theorem.

63. Derive the velocity of a rolling object down an incline using energy conservation.

64. Derive the gyroscopic couple in a turning bicycle wheel.

65. Derive the stability condition for a spinning top.

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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 velocity at the top of a rolling object’s path indicate?

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

68. Which quantity is conserved in a system with no external torque?

• (a) Angular momentum
• (b) Linear momentum
• (c) Kinetic energy
• (d) Potential energy

69. What happens to rotational kinetic energy when angular velocity doubles?

• (a) Doubles
• (b) Triples
• (c) Quadruples
• (d) Halves

70. 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) $[\text{L}{\text{T}}^{-2}]$

71. What does angular acceleration represent?

• (a) Rate of change of angular velocity
• (b) Rate of change of linear velocity
• (c) Rate of change of torque
• (d) Rate of change of moment of inertia

72. What is the role of torque in rotational motion?

• (a) Causes linear acceleration
• (b) Causes angular acceleration
• (c) Causes linear displacement
• (d) Causes angular displacement

73. What happens to the moment of inertia when mass is moved closer to the axis?

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

74. Why is angular momentum conserved in a spinning skater?

• (a) Net external force is zero
• (b) Net external torque is zero
• (c) Kinetic energy is conserved
• (d) Linear momentum is conserved

75. What is the unit of angular momentum?

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

76. What does a constant angular velocity imply?

• (a) Zero angular acceleration
• (b) Zero torque
• (c) Zero angular momentum
• (d) Zero kinetic 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 precession in a gyroscope?

• (a) Along the spin axis
• (b) Perpendicular to the torque and spin axis
• (c) Opposite to the torque
• (d) Along the linear 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 rotational kinetic energy in rolling motion?

• (a) Increases linear speed
• (b) Contributes to total kinetic energy
• (c) Reduces angular velocity
• (d) Increases potential energy

82. What happens to angular velocity during precession?

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

83. 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

84. What is the significance of $I\omega$?

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

85. What is the unit of the moment of inertia?

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

86. What does a zero torque imply?

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

87. What is the physical significance of $\tau =I\alpha$?

• (a) Work-energy theorem
• (b) Newton’s second law for rotation
• (c) Angular momentum conservation
• (d) Power in rotation

88. Why does a gyroscope precess instead of falling?

• (a) Linear momentum is conserved
• (b) Angular momentum causes the axis to rotate
• (c) Torque increases spin speed
• (d) Friction prevents falling

89. What is the dimension of torque?

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

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 top?

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

92. 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 precessing

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

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

94. What is the dimension of precession angular velocity?

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

95. Why does moment of inertia depend on the axis of rotation?

• (a) Mass changes with axis
• (b) Distance from the axis affects $r^2$
• (c) Angular velocity changes
• (d) Torque changes

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NEET-style Numerical Problems

96. A wheel starts from rest with $\alpha =3{\text{rad/s}}^2$. What is the angular velocity after $4s$?

• (a) $10\text{rad/s}$
• (b) $12\text{rad/s}$
• (c) $14\text{rad/s}$
• (d) $16\text{rad/s}$

97. A disk of $I=0.2kg\cdot m^2$ rotates at $\omega =10\text{rad/s}$. What is the angular momentum?

• (a) $1.5kg\cdot m^2/s$
• (b) $2.0kg\cdot m^2/s$
• (c) $2.5kg\cdot m^2/s$
• (d) $3.0kg\cdot m^2/s$

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

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

99. A skater with $I=4kg\cdot m^2$ spins at $\omega =5\text{rad/s}$. She reduces $I$ to $2kg\cdot m^2$. What is her new angular velocity?

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

100. A sphere of mass $2kg$ and radius $0.3m$ rolls without slipping at $v_{\text{CM}}=4m/s$. What is the total kinetic energy?
     - (a) $15J$
     - (b) $16J$
     - (c) $17J$
     - (d) $18J$

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