Potential Energy and Conservation of Energy Problems

This section provides 100 problems to test your understanding of potential energy and conservation of energy, including gravitational and elastic potential energy, conservative and non-conservative forces, conservation of mechanical energy, and applications in gravitational systems, springs, and energy transformations. 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 energy conservation, a key topic for JEE/NEET success.

Numerical Problems

1. A $3kg$ block is lifted to $h=4m$ above the ground ($g=9.8m/s^2$). Calculate the gravitational potential energy relative to the ground.

   • (a) $100.0J$
   • (b) $117.6J$
   • (c) $135.2J$
   • (d) $152.8J$

2. A spring with $k=500N/m$ is stretched by $x=0.2m$. Calculate the elastic potential energy.

   • (a) $8.0J$
   • (b) $9.0J$
   • (c) $10.0J$
   • (d) $11.0J$

3. A $2kg$ block is dropped from $h=6m$ above the ground ($g=9.8m/s^2$), with no air resistance. Calculate the speed just before hitting the ground.

   • (a) $9.0m/s$
   • (b) $10.0m/s$
   • (c) $11.0m/s$
   • (d) $12.0m/s$

4. A $0.5kg$ mass on a spring ($k=400N/m$) is stretched to $x=0.3m$ and released. Calculate the speed at $x=0.1m$.

   • (a) $5.0m/s$
   • (b) $6.0m/s$
   • (c) $7.0m/s$
   • (d) $8.0m/s$

5. A $1kg$ block slides down a frictionless incline from $h=5m$ into a spring ($k=200N/m$) ($g=9.8m/s^2$). Calculate the maximum compression of the spring.

   • (a) $0.6m$
   • (b) $0.7m$
   • (c) $0.8m$
   • (d) $0.9m$

6. A $2kg$ block slides down a frictionless incline from $h=8m$, then across a surface with ${\mu }_k=0.2$ for $d=4m$ ($g=9.8m/s^2$). Calculate the final speed.

   • (a) $10.0m/s$
   • (b) $11.0m/s$
   • (c) $12.0m/s$
   • (d) $13.0m/s$

7. A $0.4kg$ pendulum bob is released from $h=1m$ above its lowest point ($g=9.8m/s^2$). Calculate the speed at the lowest point.

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

8. A $1kg$ block at $h=3m$ above a reference level at $h=1m$ ($g=9.8m/s^2$). Calculate the gravitational potential energy relative to the reference.

   • (a) $15.0J$
   • (b) $17.5J$
   • (c) $19.6J$
   • (d) $21.2J$

9. A spring with $k=300N/m$ is compressed by $x=-0.15m$. Calculate the elastic potential energy.

   • (a) $2.5J$
   • (b) $3.0J$
   • (c) $3.375J$
   • (d) $3.75J$

10. A $0.2kg$ block on a vertical spring ($k=150N/m$) is at equilibrium, pulled down $x=0.2m$ and released ($g=9.8m/s^2$). Calculate the speed at equilibrium.

    • (a) $4.0m/s$
    • (b) $5.0m/s$
    • (c) $6.0m/s$
    • (d) $7.0m/s$

11. A $3kg$ block slides down a frictionless track from $h=10m$ into a loop-the-loop of radius $R=2.5m$ ($g=9.8m/s^2$). Calculate the minimum speed at the top to complete the loop.

    • (a) $4.0m/s$
    • (b) $4.5m/s$
    • (c) $5.0m/s$
    • (d) $5.5m/s$

12. A $0.5kg$ pendulum bob swings from $h=0.8m$ above the lowest point, with air resistance doing $W_{\text{air}}=-0.3J$ per swing ($g=9.8m/s^2$). Calculate the speed at the lowest point.

    • (a) $3.0m/s$
    • (b) $3.5m/s$
    • (c) $4.0m/s$
    • (d) $4.5m/s$

13. A $2kg$ block is dropped from $h=7m$ with air resistance doing $W_{\text{air}}=-5J$ ($g=9.8m/s^2$). Calculate the speed just before hitting the ground.

    • (a) $10.0m/s$
    • (b) $11.0m/s$
    • (c) $12.0m/s$
    • (d) $13.0m/s$

14. A $1kg$ block slides down a frictionless incline from $h=4m$, then across a surface with ${\mu }_k=0.25$ for $d=3m$ ($g=9.8m/s^2$). Calculate the final speed.

    • (a) $6.0m/s$
    • (b) $7.0m/s$
    • (c) $8.0m/s$
    • (d) $9.0m/s$

15. A $0.3kg$ mass on a spring ($k=600N/m$) is stretched to $x=0.25m$ and released. Calculate the speed at $x=0$.

    • (a) $10.0m/s$
    • (b) $11.0m/s$
    • (c) $12.0m/s$
    • (d) $13.0m/s$

16. A $4kg$ block is lifted to $h=6m$ above the ground ($g=9.8m/s^2$). Calculate the gravitational potential energy relative to the ground.

    • (a) $200.0J$
    • (b) $225.6J$
    • (c) $235.2J$
    • (d) $250.8J$

17. A spring with $k=800N/m$ is compressed by $x=-0.1m$. Calculate the elastic potential energy.

    • (a) $3.0J$
    • (b) $3.5J$
    • (c) $4.0J$
    • (d) $4.5J$

18. A $1kg$ block slides down a frictionless track from $h=12m$ into a loop-the-loop of radius $R=3m$ ($g=9.8m/s^2$). Calculate the speed at the top of the loop.

    • (a) $10.0m/s$
    • (b) $11.0m/s$
    • (c) $12.0m/s$
    • (d) $13.0m/s$

19. A $0.2kg$ mass on a vertical spring ($k=200N/m$) is at equilibrium, pulled down $x=0.15m$ and released ($g=9.8m/s^2$). Calculate the speed at equilibrium.

    • (a) $4.0m/s$
    • (b) $4.5m/s$
    • (c) $5.0m/s$
    • (d) $5.5m/s$

20. A $2kg$ block is dropped from $h=9m$ with air resistance doing $W_{\text{air}}=-8J$ ($g=9.8m/s^2$). Calculate the speed just before hitting the ground.

    • (a) $12.0m/s$
    • (b) $13.0m/s$
    • (c) $14.0m/s$
    • (d) $15.0m/s$

21. A $0.6kg$ pendulum bob is released from $h=1.5m$ above its lowest point ($g=9.8m/s^2$). Calculate the speed at the lowest point.

    • (a) $4.0m/s$
    • (b) $4.5m/s$
    • (c) $5.0m/s$
    • (d) $5.5m/s$

22. A $3kg$ block slides down a frictionless incline from $h=6m$ into a spring ($k=300N/m$) ($g=9.8m/s^2$). Calculate the maximum compression of the spring.

    • (a) $0.8m$
    • (b) $0.9m$
    • (c) $1.0m$
    • (d) $1.1m$

23. A $0.4kg$ mass on a spring ($k=500N/m$) is stretched to $x=0.4m$ and released. Calculate the speed at $x=0.2m$.

    • (a) $6.0m/s$
    • (b) $7.0m/s$
    • (c) $8.0m/s$
    • (d) $9.0m/s$

24. A $1kg$ block slides down a frictionless incline from $h=8m$, then across a surface with ${\mu }_k=0.15$ for $d=5m$ ($g=9.8m/s^2$). Calculate the final speed.

    • (a) $10.0m/s$
    • (b) $11.0m/s$
    • (c) $12.0m/s$
    • (d) $13.0m/s$

25. A $0.1kg$ mass on a vertical spring ($k=100N/m$) is at equilibrium, pulled down $x=0.1m$ and released ($g=9.8m/s^2$). Calculate the speed at equilibrium.

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

26. A $5kg$ block is lifted to $h=3m$ above a reference level at $h=1m$ ($g=9.8m/s^2$). Calculate the gravitational potential energy relative to the reference.

    • (a) $40.0J$
    • (b) $45.0J$
    • (c) $49.0J$
    • (d) $53.0J$

27. A spring with $k=700N/m$ is stretched by $x=0.25m$. Calculate the elastic potential energy.

    • (a) $20.0J$
    • (b) $21.875J$
    • (c) $23.0J$
    • (d) $24.5J$

28. A $2kg$ block is dropped from $h=10m$ with air resistance doing $W_{\text{air}}=-10J$ ($g=9.8m/s^2$). Calculate the speed just before hitting the ground.

    • (a) $12.0m/s$
    • (b) $13.0m/s$
    • (c) $14.0m/s$
    • (d) $15.0m/s$

29. A $0.5kg$ mass on a spring ($k=800N/m$) is stretched to $x=0.5m$ and released. Calculate the speed at $x=0$.

    • (a) $18.0m/s$
    • (b) $19.0m/s$
    • (c) $20.0m/s$
    • (d) $21.0m/s$

30. A $1kg$ block slides down a frictionless track from $h=15m$ into a loop-the-loop of radius $R=4m$ ($g=9.8m/s^2$). Calculate the speed at the top of the loop.

    • (a) $12.0m/s$
    • (b) $13.0m/s$
    • (c) $14.0m/s$
    • (d) $15.0m/s$

31. A $0.3kg$ pendulum bob swings from $h=1.2m$ above the lowest point, with air resistance doing $W_{\text{air}}=-0.4J$ per swing ($g=9.8m/s^2$). Calculate the speed at the lowest point.

    • (a) $4.0m/s$
    • (b) $4.5m/s$
    • (c) $5.0m/s$
    • (d) $5.5m/s$

32. A $2kg$ block slides down a frictionless incline from $h=7m$ into a spring ($k=400N/m$) ($g=9.8m/s^2$). Calculate the maximum compression of the spring.

    • (a) $0.7m$
    • (b) $0.8m$
    • (c) $0.9m$
    • (d) $1.0m$

33. A $0.4kg$ mass on a spring ($k=300N/m$) is stretched to $x=0.3m$ and released. Calculate the speed at $x=0.15m$.

    • (a) $4.0m/s$
    • (b) $4.5m/s$
    • (c) $5.0m/s$
    • (d) $5.5m/s$

34. A $1kg$ block slides down a frictionless incline from $h=9m$, then across a surface with ${\mu }_k=0.1$ for $d=6m$ ($g=9.8m/s^2$). Calculate the final speed.

    • (a) $12.0m/s$
    • (b) $13.0m/s$
    • (c) $14.0m/s$
    • (d) $15.0m/s$

35. A $0.2kg$ mass on a vertical spring ($k=250N/m$) is at equilibrium, pulled down $x=0.25m$ and released ($g=9.8m/s^2$). Calculate the speed at equilibrium.

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

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Conceptual Problems

36. What does gravitational potential energy depend on?

• (a) Mass only
• (b) Height only
• (c) Mass, gravity, and height
• (d) Mass and velocity

37. What is a conservative force?

• (a) Work depends on the path
• (b) Work is path-independent
• (c) Work is always zero
• (d) Work increases kinetic energy

38. What does conservation of mechanical energy imply?

• (a) Kinetic energy is constant
• (b) Potential energy is constant
• (c) Total mechanical energy is constant with only conservative forces
• (d) Total mechanical energy is constant regardless of forces

39. What happens to potential energy as a spring returns to equilibrium?

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

40. What is the unit of potential energy?

• (a) $N$
• (b) $J$
• (c) $W$
• (d) $kg$

41. What happens to kinetic energy as potential energy increases in a conservative system?

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

42. What does a non-conservative force do to mechanical energy?

• (a) Conserves it
• (b) Increases it
• (c) Decreases it
• (d) Does not affect it

43. What is the role of gravitational potential energy in a falling object?

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

44. How does friction affect energy conservation?

• (a) Conserves mechanical energy
• (b) Dissipates mechanical energy as heat
• (c) Increases mechanical energy
• (d) Does not affect mechanical energy

45. What is the dimension of potential energy?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
• (b) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (c) $[\text{M}\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-3}]$

46. What does a zero mechanical energy imply?

• (a) Object is at rest with no potential energy
• (b) Object is moving with no potential energy
• (c) Object has maximum kinetic energy
• (d) Object has maximum potential energy

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

• (a) Kinetic energy
• (b) Gravitational potential energy
• (c) Elastic potential energy
• (d) Work done

48. What happens to the speed at the lowest point of a pendulum?

• (a) Minimum
• (b) Maximum
• (c) Zero
• (d) Constant

49. What does the work done by a conservative force depend on?

• (a) Path taken
• (b) Initial and final positions
• (c) Speed of the object
• (d) Time taken

50. How does air resistance affect a falling object’s mechanical energy?

• (a) Increases it
• (b) Decreases it
• (c) Conserves it
• (d) Does not affect it

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Derivation Problems

51. Derive the gravitational potential energy near Earth’s surface.

52. Derive the elastic potential energy of a spring.

53. Derive the conservation of mechanical energy for gravity.

54. Derive the speed in a vertical spring system at equilibrium.

55. Derive the energy change with non-conservative forces.

56. Derive the minimum speed at the top of a loop-the-loop.

57. Derive the maximum compression of a spring in a gravitational system.

58. Derive the speed at the lowest point of a pendulum.

59. Derive the speed of a mass on a spring at a given position.

60. Derive the work done by friction in a loop-the-loop system.

61. Derive the speed of a block after sliding down an incline with friction.

62. Derive the potential energy change in a 2D gravitational system.

63. Derive the mechanical energy loss due to air resistance.

64. Derive the speed at the top of a loop-the-loop with friction.

65. Derive the work done by a conservative force in a closed loop.

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NEET-style Conceptual Problems

66. What is the unit of mechanical energy in SI units?

• (a) $N$
• (b) $J$
• (c) $W$
• (d) $kg$

67. What does a zero kinetic energy at the top of a loop-the-loop indicate?

• (a) Object completes the loop
• (b) Object falls off the track
• (c) Object is at rest
• (d) Object has maximum potential energy

68. Which force is conservative?

• (a) Friction
• (b) Air resistance
• (c) Gravity
• (d) Applied force

69. What happens to mechanical energy when friction acts?

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

70. What is the dimension of the spring constant?

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

71. What does elastic potential energy represent?

• (a) Energy of motion
• (b) Energy stored in a stretched spring
• (c) Energy due to height
• (d) Energy due to friction

72. What is the role of potential energy in a spring system?

• (a) Converts to kinetic energy
• (b) Converts to gravitational energy
• (c) Remains constant
• (d) Increases speed

73. What happens to the speed at the top of a pendulum swing?

• (a) Maximum
• (b) Minimum
• (c) Zero
• (d) Constant

74. Why does gravity conserve mechanical energy?

• (a) It is a non-conservative force
• (b) It is a conservative force
• (c) It acts perpendicular to motion
• (d) It acts along the path

75. What is the unit of gravitational potential energy?

• (a) $N$
• (b) $J$
• (c) $W$
• (d) $kg$

76. What does a constant mechanical energy imply?

• (a) Only conservative forces act
• (b) Non-conservative forces act
• (c) Kinetic energy is zero
• (d) Potential energy is zero

77. Which force does work that depends on the path?

• (a) Gravity
• (b) Spring force
• (c) Friction
• (d) Normal force

78. What is the direction of the spring force at maximum stretch?

• (a) Along the displacement
• (b) Opposite to the displacement
• (c) Perpendicular to displacement
• (d) Along the velocity

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

• (a) Conserves mechanical energy
• (b) Affects energy calculations
• (c) Provides centripetal force
• (d) Reduces friction

80. What is the dimension of mechanical energy?

• (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
• (b) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (c) $[\text{M}\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-3}]$

81. What is the role of kinetic energy in a pendulum?

• (a) Maximum at the top
• (b) Maximum at the bottom
• (c) Constant throughout
• (d) Zero at the bottom

82. What happens to potential energy as a block falls?

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

83. Why does air resistance reduce mechanical energy?

• (a) It is a conservative force
• (b) It dissipates energy as heat
• (c) It increases potential energy
• (d) It increases kinetic energy

84. What is the significance of $mgh$?

• (a) Kinetic energy
• (b) Elastic potential energy
• (c) Gravitational potential energy
• (d) Work done

85. What does the spring constant determine?

• (a) Kinetic energy
• (b) Potential energy stored
• (c) Speed of the mass
• (d) Gravitational force

86. What does a zero potential energy at the ground imply?

• (a) Reference level is at the ground
• (b) Object is at the top
• (c) Object is moving
• (d) Object has maximum kinetic energy

87. What is the physical significance of $-\mathrm{\Delta }PE$?

• (a) Work done by a conservative force
• (b) Work done by a non-conservative force
• (c) Kinetic energy
• (d) Power

88. Why does gravity do work on a pendulum?

• (a) It acts perpendicular to motion
• (b) It has a component along displacement
• (c) It acts opposite to motion
• (d) It acts along the normal force

89. What is the dimension of the coefficient of friction?

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

90. How does friction affect a spring system?

• (a) Conserves mechanical energy
• (b) Dissipates mechanical energy
• (c) Increases potential energy
• (d) Increases kinetic energy

91. What is the role of conservation in a loop-the-loop?

• (a) Determines minimum speed at the top
• (b) Determines maximum speed at the top
• (c) Determines potential energy
• (d) Determines kinetic energy

92. What does a zero speed at the top of a loop-the-loop indicate?

• (a) Object completes the loop
• (b) Object falls off the track
• (c) Object is at rest
• (d) Object has maximum potential energy

93. What is the physical significance of $KE+PE$?

• (a) Total mechanical energy
• (b) Kinetic energy
• (c) Potential energy
• (d) Work done

94. What is the dimension of gravitational acceleration?

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

95. Why does potential energy depend on the reference level?

• (a) It is a conservative force
• (b) It is a relative quantity
• (c) It is a non-conservative force
• (d) It is a vector quantity

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NEET-style Numerical Problems

96. A $2kg$ block is dropped from $h=5m$ above the ground ($g=9.8m/s^2$), with no air resistance. What is the speed just before hitting the ground?

• (a) $8.0m/s$
• (b) $9.0m/s$
• (c) $10.0m/s$
• (d) $11.0m/s$

97. A $0.5kg$ mass on a spring ($k=200N/m$) is stretched to $x=0.2m$ and released. What is the speed at $x=0$?

• (a) $3.0m/s$
• (b) $4.0m/s$
• (c) $5.0m/s$
• (d) $6.0m/s$

98. A $1kg$ block slides down a frictionless incline from $h=3m$ into a spring ($k=100N/m$) ($g=9.8m/s^2$). What is the maximum compression of the spring?

• (a) $0.4m$
• (b) $0.5m$
• (c) $0.6m$
• (d) $0.7m$

99. A $0.4kg$ pendulum bob is released from $h=0.5m$ above its lowest point ($g=9.8m/s^2$). What is the speed at the lowest point?

• (a) $2.5m/s$
• (b) $3.0m/s$
• (c) $3.5m/s$
• (d) $4.0m/s$

100. A $2kg$ block slides down a frictionless incline from $h=4m$, then across a surface with ${\mu }_k=0.2$ for $d=2m$ ($g=9.8m/s^2$). What is the final speed?
     - (a) $7.0m/s$
     - (b) $8.0m/s$
     - (c) $9.0m/s$
     - (d) $10.0m/s$

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