Oscillations Problems

This section provides 100 problems to test your understanding of oscillatory motion, including simple harmonic motion (SHM), energy in oscillatory systems, damped and forced oscillations, and resonance. 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 oscillations, a key topic for JEE/NEET success.

Numerical Problems

1. A spring-mass system has $m=0.8\text{kg}$ and $k=320\text{N/m}$. Calculate the period of oscillation.

   • (a) $0.30\text{s}$
   • (b) $0.31\text{s}$
   • (c) $0.32\text{s}$
   • (d) $0.33\text{s}$

2. A simple pendulum of length $1.2\text{m}$ oscillates on Earth ($g=9.8{\text{m/s}}^2$). Calculate the frequency.

   • (a) $0.45\text{Hz}$
   • (b) $0.46\text{Hz}$
   • (c) $0.47\text{Hz}$
   • (d) $0.48\text{Hz}$

3. A mass of $0.5\text{kg}$ on a spring ($k=200\text{N/m}$) has an amplitude $A=0.06\text{m}$. Calculate the maximum velocity.

   • (a) $1.15\text{m/s}$
   • (b) $1.20\text{m/s}$
   • (c) $1.25\text{m/s}$
   • (d) $1.30\text{m/s}$

4. A spring-mass system ($m=0.3\text{kg}$, $k=150\text{N/m}$, $A=0.08\text{m}$) oscillates. Calculate the total energy.

   • (a) $0.45\text{J}$
   • (b) $0.46\text{J}$
   • (c) $0.47\text{J}$
   • (d) $0.48\text{J}$

5. A pendulum ($m=0.2\text{kg}$, $L=0.8\text{m}$, ${\theta }_{\text{max}}=0.1\text{rad}$, $g=9.8{\text{m/s}}^2$) oscillates. Calculate the maximum kinetic energy.

   • (a) $0.015\text{J}$
   • (b) $0.016\text{J}$
   • (c) $0.017\text{J}$
   • (d) $0.018\text{J}$

6. A mass on a spring ($k=100\text{N/m}$, $m=0.4\text{kg}$, $A=0.05\text{m}$) oscillates. Calculate the velocity at $x=0.03\text{m}$.

   • (a) $0.78\text{m/s}$
   • (b) $0.79\text{m/s}$
   • (c) $0.80\text{m/s}$
   • (d) $0.81\text{m/s}$

7. A damped oscillator has $m=0.2\text{kg}$, $k=80\text{N/m}$, and $b=0.4\text{kg/s}$. Calculate the damped frequency.

   • (a) $19.95\text{rad/s}$
   • (b) $19.96\text{rad/s}$
   • (c) $19.97\text{rad/s}$
   • (d) $19.98\text{rad/s}$

8. A forced oscillator ($m=0.6\text{kg}$, $k=240\text{N/m}$, $b=0.3\text{kg/s}$, $F_0=2\text{N}$) is at resonance. Calculate the maximum amplitude.

   • (a) $0.105\text{m}$
   • (b) $0.110\text{m}$
   • (c) $0.115\text{m}$
   • (d) $0.120\text{m}$

9. A spring-mass system has $m=1\text{kg}$ and $k=400\text{N/m}$. Calculate the frequency of oscillation.

   • (a) $3.15\text{Hz}$
   • (b) $3.18\text{Hz}$
   • (c) $3.21\text{Hz}$
   • (d) $3.24\text{Hz}$

10. A simple pendulum of length $0.5\text{m}$ oscillates on Earth ($g=9.8{\text{m/s}}^2$). Calculate the period.

    • (a) $1.40\text{s}$
    • (b) $1.41\text{s}$
    • (c) $1.42\text{s}$
    • (d) $1.43\text{s}$

11. A mass of $0.1\text{kg}$ on a spring ($k=50\text{N/m}$) has $A=0.04\text{m}$. Calculate the maximum acceleration.

    • (a) $19.5{\text{m/s}}^2$
    • (b) $20.0{\text{m/s}}^2$
    • (c) $20.5{\text{m/s}}^2$
    • (d) $21.0{\text{m/s}}^2$

12. A spring-mass system ($m=0.25\text{kg}$, $k=100\text{N/m}$, $A=0.03\text{m}$) oscillates. Calculate the potential energy at $x=0.02\text{m}$.

    • (a) $0.020\text{J}$
    • (b) $0.021\text{J}$
    • (c) $0.022\text{J}$
    • (d) $0.023\text{J}$

13. A pendulum ($m=0.15\text{kg}$, $L=1\text{m}$, ${\theta }_{\text{max}}=0.15\text{rad}$, $g=9.8{\text{m/s}}^2$) oscillates. Calculate the maximum potential energy.

    • (a) $0.016\text{J}$
    • (b) $0.017\text{J}$
    • (c) $0.018\text{J}$
    • (d) $0.019\text{J}$

14. A mass on a spring ($k=300\text{N/m}$, $m=0.6\text{kg}$, $A=0.07\text{m}$) oscillates. Calculate the kinetic energy at $x=0.04\text{m}$.

    • (a) $0.52\text{J}$
    • (b) $0.53\text{J}$
    • (c) $0.54\text{J}$
    • (d) $0.55\text{J}$

15. A damped oscillator has $m=0.4\text{kg}$, $k=160\text{N/m}$, and $b=0.8\text{kg/s}$. Calculate the time for the amplitude to reduce to $1/e$ of its initial value.

    • (a) $0.48\text{s}$
    • (b) $0.49\text{s}$
    • (c) $0.50\text{s}$
    • (d) $0.51\text{s}$

16. A system ($m=0.2\text{kg}$, $k=80\text{N/m}$, $b=0.5\text{kg/s}$, $F_0=1\text{N}$) is at resonance. Calculate the maximum energy.

    • (a) $0.040\text{J}$
    • (b) $0.045\text{J}$
    • (c) $0.050\text{J}$
    • (d) $0.055\text{J}$

17. A spring-mass system has $m=0.9\text{kg}$ and $k=360\text{N/m}$. Calculate the period of oscillation.

    • (a) $0.30\text{s}$
    • (b) $0.31\text{s}$
    • (c) $0.32\text{s}$
    • (d) $0.33\text{s}$

18. A simple pendulum of length $2\text{m}$ oscillates on Earth ($g=9.8{\text{m/s}}^2$). Calculate the frequency.

    • (a) $0.34\text{Hz}$
    • (b) $0.35\text{Hz}$
    • (c) $0.36\text{Hz}$
    • (d) $0.37\text{Hz}$

19. A mass of $0.3\text{kg}$ on a spring ($k=120\text{N/m}$) has $A=0.05\text{m}$. Calculate the maximum velocity.

    • (a) $0.95\text{m/s}$
    • (b) $1.00\text{m/s}$
    • (c) $1.05\text{m/s}$
    • (d) $1.10\text{m/s}$

20. A spring-mass system ($m=0.7\text{kg}$, $k=280\text{N/m}$, $A=0.04\text{m}$) oscillates. Calculate the total energy.

    • (a) $0.22\text{J}$
    • (b) $0.23\text{J}$
    • (c) $0.24\text{J}$
    • (d) $0.25\text{J}$

21. A pendulum ($m=0.1\text{kg}$, $L=0.6\text{m}$, ${\theta }_{\text{max}}=0.2\text{rad}$, $g=9.8{\text{m/s}}^2$) oscillates. Calculate the maximum kinetic energy.

    • (a) $0.011\text{J}$
    • (b) $0.012\text{J}$
    • (c) $0.013\text{J}$
    • (d) $0.014\text{J}$

22. A mass on a spring ($k=150\text{N/m}$, $m=0.3\text{kg}$, $A=0.06\text{m}$) oscillates. Calculate the velocity at $x=0.02\text{m}$.

    • (a) $1.32\text{m/s}$
    • (b) $1.33\text{m/s}$
    • (c) $1.34\text{m/s}$
    • (d) $1.35\text{m/s}$

23. A damped oscillator has $m=0.5\text{kg}$, $k=200\text{N/m}$, and $b=1.0\text{kg/s}$. Calculate the damped frequency.

    • (a) $19.90\text{rad/s}$
    • (b) $19.92\text{rad/s}$
    • (c) $19.94\text{rad/s}$
    • (d) $19.96\text{rad/s}$

24. A forced oscillator ($m=0.4\text{kg}$, $k=160\text{N/m}$, $b=0.2\text{kg/s}$, $F_0=1.5\text{N}$) is at resonance. Calculate the maximum amplitude.

    • (a) $0.185\text{m}$
    • (b) $0.190\text{m}$
    • (c) $0.195\text{m}$
    • (d) $0.200\text{m}$

25. A spring-mass system has $m=0.2\text{kg}$ and $k=80\text{N/m}$. Calculate the frequency.

    • (a) $3.15\text{Hz}$
    • (b) $3.18\text{Hz}$
    • (c) $3.21\text{Hz}$
    • (d) $3.24\text{Hz}$

26. A simple pendulum of length $0.8\text{m}$ oscillates on Earth ($g=9.8{\text{m/s}}^2$). Calculate the period.

    • (a) $1.78\text{s}$
    • (b) $1.79\text{s}$
    • (c) $1.80\text{s}$
    • (d) $1.81\text{s}$

27. A mass of $0.4\text{kg}$ on a spring ($k=160\text{N/m}$) has $A=0.03\text{m}$. Calculate the maximum acceleration.

    • (a) $11.5{\text{m/s}}^2$
    • (b) $12.0{\text{m/s}}^2$
    • (c) $12.5{\text{m/s}}^2$
    • (d) $13.0{\text{m/s}}^2$

28. A spring-mass system ($m=0.6\text{kg}$, $k=240\text{N/m}$, $A=0.05\text{m}$) oscillates. Calculate the potential energy at $x=0.03\text{m}$.

    • (a) $0.100\text{J}$
    • (b) $0.105\text{J}$
    • (c) $0.110\text{J}$
    • (d) $0.115\text{J}$

29. A pendulum ($m=0.05\text{kg}$, $L=0.4\text{m}$, ${\theta }_{\text{max}}=0.1\text{rad}$, $g=9.8{\text{m/s}}^2$) oscillates. Calculate the maximum potential energy.

    • (a) $0.0020\text{J}$
    • (b) $0.0021\text{J}$
    • (c) $0.0022\text{J}$
    • (d) $0.0023\text{J}$

30. A mass on a spring ($k=200\text{N/m}$, $m=0.5\text{kg}$, $A=0.04\text{m}$) oscillates. Calculate the kinetic energy at $x=0.02\text{m}$.

    • (a) $0.060\text{J}$
    • (b) $0.065\text{J}$
    • (c) $0.070\text{J}$
    • (d) $0.075\text{J}$

31. A damped oscillator has $m=0.3\text{kg}$, $k=120\text{N/m}$, and $b=0.6\text{kg/s}$. Calculate the time for the amplitude to reduce to $1/e$ of its initial value.

    • (a) $0.48\text{s}$
    • (b) $0.49\text{s}$
    • (c) $0.50\text{s}$
    • (d) $0.51\text{s}$

32. A system ($m=0.1\text{kg}$, $k=40\text{N/m}$, $b=0.2\text{kg/s}$) has a Q-factor of 10. Calculate $b$.

    • (a) $0.15\text{kg/s}$
    • (b) $0.20\text{kg/s}$
    • (c) $0.25\text{kg/s}$
    • (d) $0.30\text{kg/s}$

33. A rocket component ($m=2\text{kg}$, $k=800\text{N/m}$, $b=4\text{kg/s}$) vibrates. Calculate the resonance frequency.

    • (a) $3.15\text{Hz}$
    • (b) $3.18\text{Hz}$
    • (c) $3.21\text{Hz}$
    • (d) $3.24\text{Hz}$

34. A spring-mass system ($m=0.8\text{kg}$, $k=320\text{N/m}$, $A=0.02\text{m}$) oscillates. Calculate the total energy.

    • (a) $0.060\text{J}$
    • (b) $0.062\text{J}$
    • (c) $0.064\text{J}$
    • (d) $0.066\text{J}$

35. A forced oscillator ($m=0.5\text{kg}$, $k=200\text{N/m}$, $b=0.5\text{kg/s}$, $F_0=2\text{N}$) is at resonance. Calculate the maximum energy.

    • (a) $0.080\text{J}$
    • (b) $0.085\text{J}$
    • (c) $0.090\text{J}$
    • (d) $0.095\text{J}$

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Conceptual Problems

36. What condition defines simple harmonic motion (SHM)?

• (a) Acceleration is constant
• (b) Acceleration is proportional to velocity
• (c) Acceleration is proportional to displacement and opposite in direction
• (d) Acceleration is independent of displacement

37. What does the total energy in SHM depend on?

• (a) Position
• (b) Velocity
• (c) Amplitude and spring constant
• (d) Phase constant

38. What happens to the frequency of a simple pendulum if its length is doubled?

• (a) Increases by a factor of 2
• (b) Decreases by a factor of $\sqrt{2}$
• (c) Decreases by a factor of 2
• (d) Remains the same

39. What does damping do to an oscillatory system?

• (a) Increases amplitude
• (b) Decreases amplitude over time
• (c) Increases frequency
• (d) Decreases total energy to zero instantly

40. What is the unit of angular frequency in SHM?

• (a) $\text{Hz}$
• (b) $\text{rad/s}$
• (c) $\text{s}$
• (d) $\text{J}$

41. What happens to the velocity in SHM at the extreme positions?

• (a) Maximum
• (b) Zero
• (c) Constant
• (d) Minimum but non-zero

42. What does a high Q-factor indicate in a forced oscillator?

• (a) High damping
• (b) Low damping, sharp resonance
• (c) Low amplitude
• (d) High frequency

43. What is the physical significance of $-{\omega }^{2}x$ in SHM?

• (a) Velocity
• (b) Acceleration
• (c) Potential energy
• (d) Kinetic energy

44. What does resonance in a forced oscillator imply?

• (a) Minimum amplitude
• (b) Maximum amplitude at driving frequency equal to natural frequency
• (c) No oscillation
• (d) Constant amplitude

45. What is the dimension of the spring constant $k$?

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

46. What does a zero kinetic energy in SHM indicate?

• (a) System is at equilibrium
• (b) System is at the extreme position
• (c) System is damped
• (d) System is at resonance

47. What is the significance of $A\omega$ in SHM?

• (a) Maximum acceleration
• (b) Maximum velocity
• (c) Total energy
• (d) Period

48. What happens to the period of a spring-mass system if the mass is quadrupled?

• (a) Doubles
• (b) Halves
• (c) Quadruples
• (d) Remains the same

49. What does critical damping imply?

• (a) System oscillates with decreasing amplitude
• (b) System returns to equilibrium without oscillation
• (c) System oscillates with constant amplitude
• (d) System does not return to equilibrium

50. How does the amplitude of a damped oscillator behave over time?

• (a) Increases exponentially
• (b) Decreases exponentially
• (c) Remains constant
• (d) Increases linearly

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Derivation Problems

51. Derive the equation of motion for SHM $a=-{\omega }^{2}x$.

52. Derive the period of a spring-mass system $T=2\pi \sqrt{\frac{m}{k}}$.

53. Derive the total energy in SHM $E=\frac{1}{2}k{A}^2$.

54. Derive the velocity in SHM as a function of position $v=\omega \sqrt{A^2-x^2}$.

55. Derive the period of a simple pendulum $T=2\pi \sqrt{\frac{L}{g}}$ for small angles.

56. Derive the solution to damped SHM $x(t)=A{e}^{-\gamma t}\cos ({\omega }^{\prime }t+\varphi )$.

57. Derive the maximum amplitude at resonance for a forced oscillator.

58. Derive the Q-factor $Q=\frac{{\omega }_0}{2\gamma }$ for a damped oscillator.

59. Derive the velocity and acceleration in SHM from $x(t)=A\cos (\omega t+\varphi )$.

60. Derive the energy oscillation in SHM showing $K+U=\text{constant}$.

61. Derive the damped frequency ${\omega }^{\prime }=\sqrt{{\omega }_0^2-{\gamma }^2}$.

62. Derive the maximum kinetic energy of a pendulum using small-angle approximation.

63. Derive the energy decay in a damped oscillator $E(t)\propto e^{-2\gamma t}$.

64. Derive the resonance frequency for a forced oscillator.

65. Derive the maximum acceleration in SHM $a_{\text{max}}=A{\omega }^2$.

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NEET-style Conceptual Problems

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

• (a) $\text{Hz}$
• (b) $\text{rad/s}$
• (c) $\text{J}$
• (d) $\text{s}$

67. What does a zero velocity in SHM indicate?

• (a) System is at equilibrium
• (b) System is at the extreme position
• (c) System is damped
• (d) System is at resonance

68. Which condition results in resonance in a forced oscillator?

• (a) Driving frequency equals natural frequency
• (b) Driving frequency is zero
• (c) Damping is zero
• (d) Amplitude is zero

69. What happens to the total energy in SHM if amplitude doubles?

• (a) Doubles
• (b) Quadruples
• (c) Halves
• (d) Remains the same

70. What is the dimension of angular frequency?

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

71. What does the phase constant $\varphi$ in SHM determine?

• (a) Amplitude
• (b) Frequency
• (c) Initial position and velocity
• (d) Total energy

72. What is the role of the restoring force in SHM?

• (a) Increases amplitude
• (b) Brings the system back to equilibrium
• (c) Increases frequency
• (d) Reduces damping

73. What happens to the period of a pendulum if $g$ is reduced by a factor of 4?

• (a) Doubles
• (b) Halves
• (c) Quadruples
• (d) Remains the same

74. Why does damping reduce the amplitude of oscillation?

• (a) Increases frequency
• (b) Dissipates energy as heat
• (c) Increases potential energy
• (d) Reduces spring constant

75. What is the unit of the damping coefficient $b$?

• (a) $\text{kg/s}$
• (b) $\text{N/m}$
• (c) $\text{Pa}$
• (d) $\text{J}$

76. What does a constant total energy in SHM imply?

• (a) No damping
• (b) System is at resonance
• (c) System is overdamped
• (d) System is critically damped

77. Which type of motion does a spring-mass system exhibit for small displacements?

• (a) Circular
• (b) Simple harmonic
• (c) Parabolic
• (d) Linear

78. What is the direction of the acceleration in SHM?

• (a) Always toward the equilibrium position
• (b) Away from the equilibrium position
• (c) Along the velocity
• (d) Perpendicular to displacement

79. What does a pseudo-force do in a non-inertial frame for oscillations?

• (a) Maintains SHM
• (b) Affects the effective acceleration
• (c) Provides damping
• (d) Reduces amplitude

80. What is the dimension of total energy in SHM?

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

81. What is the role of resonance in a rocket engine?

• (a) Increases efficiency
• (b) Can cause destructive vibrations if not managed
• (c) Reduces amplitude
• (d) Increases damping

82. What happens to kinetic energy at the equilibrium position in SHM?

• (a) Zero
• (b) Maximum
• (c) Constant
• (d) Minimum but non-zero

83. Why does a simple pendulum exhibit SHM for small angles?

• (a) $\sin \theta \approx \theta$, making the restoring force linear
• (b) Large angles increase frequency
• (c) Damping is negligible
• (d) Gravity is constant

84. What is the significance of $\frac{1}{2}k{A}^2$ in SHM?

• (a) Maximum velocity
• (b) Maximum acceleration
• (c) Total mechanical energy
• (d) Damped frequency

85. What is the unit of the Q-factor?

• (a) Dimensionless
• (b) $\text{Hz}$
• (c) $\text{rad/s}$
• (d) $\text{J}$

86. What does a zero acceleration in SHM indicate?

• (a) System is at the extreme position
• (b) System is at equilibrium
• (c) System is damped
• (d) System is at resonance

87. What is the physical significance of $\sqrt{\frac{k}{m}}$?

• (a) Period
• (b) Angular frequency
• (c) Amplitude
• (d) Damping coefficient

88. Why does a high damping coefficient reduce oscillation amplitude quickly?

• (a) Increases frequency
• (b) Increases energy loss per cycle
• (c) Reduces spring constant
• (d) Increases potential energy

89. What is the dimension of velocity in SHM?

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

90. How does resonance affect a rocket structure during launch?

• (a) Increases stability
• (b) Can amplify vibrations, risking structural failure
• (c) Reduces frequency
• (d) Increases damping

91. What is the role of the spring constant in SHM?

• (a) Determines damping
• (b) Determines the restoring force per unit displacement
• (c) Reduces amplitude
• (d) Increases phase constant

92. What does a negative acceleration in SHM indicate?

• (a) Motion toward the equilibrium position
• (b) Motion away from the equilibrium position
• (c) System is overdamped
• (d) System is at resonance

93. What is the physical significance of $e^{-\gamma t}$ in damped oscillations?

• (a) Frequency decay
• (b) Amplitude decay factor
• (c) Energy increase
• (d) Phase constant

94. What is the dimension of acceleration in SHM?

• (a) $[\text{L}{\text{T}}^{-2}]$
• (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 the frequency of a spring-mass system not depend on amplitude?

• (a) SHM is linear
• (b) Damping is present
• (c) Energy is constant
• (d) Gravity affects it

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NEET-style Numerical Problems

96. A spring-mass system has $m=0.4\text{kg}$ and $k=160\text{N/m}$. What is the period of oscillation?

• (a) $0.31\text{s}$
• (b) $0.32\text{s}$
• (c) $0.33\text{s}$
• (d) $0.34\text{s}$

97. A simple pendulum of length $1.5\text{m}$ oscillates on Earth ($g=9.8{\text{m/s}}^2$). What is the frequency?

• (a) $0.40\text{Hz}$
• (b) $0.41\text{Hz}$
• (c) $0.42\text{Hz}$
• (d) $0.43\text{Hz}$

98. A mass of $0.2\text{kg}$ on a spring ($k=80\text{N/m}$) has $A=0.05\text{m}$. What is the maximum velocity?

• (a) $0.95\text{m/s}$
• (b) $1.00\text{m/s}$
• (c) $1.05\text{m/s}$
• (d) $1.10\text{m/s}$

99. A damped oscillator has $m=0.1\text{kg}$, $k=40\text{N/m}$, and $b=0.2\text{kg/s}$. What is the damped frequency?

• (a) $19.95\text{rad/s}$
• (b) $19.96\text{rad/s}$
• (c) $19.97\text{rad/s}$
• (d) $19.98\text{rad/s}$

100. A forced oscillator ($m=0.3\text{kg}$, $k=120\text{N/m}$, $b=0.3\text{kg/s}$, $F_0=1.2\text{N}$) is at resonance. What is the maximum amplitude?
     - (a) $0.095\text{m}$
     - (b) $0.100\text{m}$
     - (c) $0.105\text{m}$
     - (d) $0.110\text{m}$

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