Force and Motion—II Problems

This section provides 100 problems to test your understanding of advanced force and motion concepts, including static and kinetic friction, circular motion dynamics (centripetal force, banking), drag forces and terminal velocity, and applications in complex systems (e.g., pulley systems with friction, non-inertial frames). 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 advanced dynamics, a key topic for JEE/NEET success.

Numerical Problems

1. A $6kg$ block on a surface with ${\mu }_s=0.4$ ($g=9.8m/s^2$). Calculate the force needed to start the block moving.

   • (a) $20.0N$
   • (b) $23.5N$
   • (c) $27.0N$
   • (d) $29.5N$

2. A $4kg$ block on a surface with ${\mu }_k=0.3$ ($g=9.8m/s^2$) is pushed by $F=20N$. Calculate the acceleration.

   • (a) $1.0m/s^2$
   • (b) $1.5m/s^2$
   • (c) $2.0m/s^2$
   • (d) $2.5m/s^2$

3. A $0.2kg$ ball on a string of length $1m$ moves in a horizontal circle at $v=3m/s$. Calculate the tension in the string.

   • (a) $1.96N$
   • (b) $2.50N$
   • (c) $3.04N$
   • (d) $3.58N$

4. A car of mass $1000kg$ on a banked curve of radius $80m$ at $v=20m/s$ ($g=9.8m/s^2$). Find the banking angle.

   • (a) ${20}^{\circ }$
   • (b) ${25}^{\circ }$
   • (c) ${30}^{\circ }$
   • (d) ${35}^{\circ }$

5. A $0.05kg$ raindrop falls with $v_t=8m/s$ ($g=9.8m/s^2$, linear drag $F_d=bv$). Calculate the drag coefficient $b$.

   • (a) $0.049kg/s$
   • (b) $0.061kg/s$
   • (c) $0.073kg/s$
   • (d) $0.085kg/s$

6. A $10kg$ block on an incline at ${30}^{\circ }$ with ${\mu }_k=0.2$ ($g=9.8m/s^2$) is pushed by $F=60N$ parallel to the incline. Calculate the acceleration down the incline.

   • (a) $8.0m/s^2$
   • (b) $9.0m/s^2$
   • (c) $10.0m/s^2$
   • (d) $11.0m/s^2$

7. Masses $m_1=12kg$ (hanging) and $m_2=6kg$ on a surface with ${\mu }_k=0.1$ ($g=9.8m/s^2$) in an Atwood’s setup. Calculate the acceleration.

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

8. A $1500kg$ car on a flat curve of radius $120m$ with ${\mu }_s=0.5$ at $v=25m/s$ ($g=9.8m/s^2$). Will the car skid?

   • (a) Yes
   • (b) No
   • (c) Cannot determine
   • (d) Depends on the car's mass

9. A $0.3kg$ mass in a conical pendulum with $L=0.8m$, angle $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). Calculate the period.

   • (a) $1.2s$
   • (b) $1.4s$
   • (c) $1.6s$
   • (d) $1.8s$

10. A skydiver of mass $80kg$ reaches terminal velocity ($g=9.8m/s^2$, $\rho =1.2kg/m^3$, $C_d=1$, $A=0.9m^2$). Calculate $v_t$.

    • (a) $35m/s$
    • (b) $40m/s$
    • (c) $45m/s$
    • (d) $50m/s$

11. A $5kg$ block on a surface with ${\mu }_s=0.6$ ($g=9.8m/s^2$). Calculate the force needed to start the block moving.

    • (a) $24.5N$
    • (b) $29.4N$
    • (c) $34.3N$
    • (d) $39.2N$

12. A $3kg$ block on a surface with ${\mu }_k=0.2$ ($g=9.8m/s^2$) moves at constant speed with force $F$. Calculate $F$.

    • (a) $5.0N$
    • (b) $5.88N$
    • (c) $6.5N$
    • (d) $7.0N$

13. A $0.1kg$ ball on a string of length $0.5m$ moves in a horizontal circle at $v=2m/s$. Calculate the tension.

    • (a) $0.98N$
    • (b) $1.24N$
    • (c) $1.50N$
    • (d) $1.76N$

14. A $1200kg$ car on a banked curve of radius $60m$ at $v=18m/s$ ($g=9.8m/s^2$). Calculate the banking angle.

    • (a) ${18}^{\circ }$
    • (b) ${23}^{\circ }$
    • (c) ${28}^{\circ }$
    • (d) ${33}^{\circ }$

15. A $0.02kg$ object falls with $v_t=6m/s$ ($g=9.8m/s^2$, linear drag $F_d=bv$). Calculate $v$ after $0.1s$ from rest.

    • (a) $1.0m/s$
    • (b) $1.2m/s$
    • (c) $1.4m/s$
    • (d) $1.6m/s$

16. A $8kg$ block on an incline at ${45}^{\circ }$ with ${\mu }_k=0.3$ ($g=9.8m/s^2$). Calculate the acceleration down the incline.

    • (a) $3.0m/s^2$
    • (b) $3.5m/s^2$
    • (c) $4.0m/s^2$
    • (d) $4.5m/s^2$

17. Masses $m_1=15kg$ (hanging) and $m_2=5kg$ on a surface with ${\mu }_k=0.2$ ($g=9.8m/s^2$) in an Atwood’s setup. Calculate the acceleration.

    • (a) $6.0m/s^2$
    • (b) $6.5m/s^2$
    • (c) $7.0m/s^2$
    • (d) $7.5m/s^2$

18. A $2000kg$ car on a flat curve of radius $150m$ with ${\mu }_s=0.7$ at $v=30m/s$ ($g=9.8m/s^2$). Will the car skid?

    • (a) Yes
    • (b) No
    • (c) Cannot determine
    • (d) Depends on the car's mass

19. A $0.4kg$ mass in a conical pendulum with $L=1m$, angle $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Calculate the period.

    • (a) $1.6s$
    • (b) $1.8s$
    • (c) $2.0s$
    • (d) $2.2s$

20. A skydiver of mass $90kg$ reaches terminal velocity ($g=9.8m/s^2$, $\rho =1.2kg/m^3$, $C_d=1$, $A=1.0m^2$). Calculate $v_t$.

    • (a) $38m/s$
    • (b) $43m/s$
    • (c) $48m/s$
    • (d) $53m/s$

21. A $7kg$ block on a surface with ${\mu }_s=0.3$ ($g=9.8m/s^2$). Calculate the force needed to start the block moving.

    • (a) $15.0N$
    • (b) $17.5N$
    • (c) $20.0N$
    • (d) $22.5N$

22. A $2kg$ block on a surface with ${\mu }_k=0.4$ ($g=9.8m/s^2$) is pushed by $F=15N$. Calculate the acceleration.

    • (a) $1.0m/s^2$
    • (b) $1.5m/s^2$
    • (c) $2.0m/s^2$
    • (d) $2.5m/s^2$

23. A $0.15kg$ ball on a string of length $0.6m$ moves in a horizontal circle at $v=2.5m/s$. Calculate the tension.

    • (a) $1.47N$
    • (b) $1.82N$
    • (c) $2.17N$
    • (d) $2.52N$

24. A $800kg$ car on a banked curve of radius $40m$ at $v=16m/s$ ($g=9.8m/s^2$). Calculate the banking angle.

    • (a) ${22}^{\circ }$
    • (b) ${27}^{\circ }$
    • (c) ${32}^{\circ }$
    • (d) ${37}^{\circ }$

25. A $0.03kg$ object falls with $v_t=12m/s$ ($g=9.8m/s^2$, linear drag $F_d=bv$). Calculate $v$ after $0.15s$ from rest.

    • (a) $1.5m/s$
    • (b) $1.8m/s$
    • (c) $2.1m/s$
    • (d) $2.4m/s$

26. A $6kg$ block on an incline at ${37}^{\circ }$ with ${\mu }_k=0.25$ ($g=9.8m/s^2$). Calculate the acceleration down the incline.

    • (a) $2.5m/s^2$
    • (b) $3.0m/s^2$
    • (c) $3.5m/s^2$
    • (d) $4.0m/s^2$

27. Masses $m_1=20kg$ (hanging) and $m_2=10kg$ on a surface with ${\mu }_k=0.15$ ($g=9.8m/s^2$) in an Atwood’s setup. Calculate the acceleration.

    • (a) $5.5m/s^2$
    • (b) $6.0m/s^2$
    • (c) $6.5m/s^2$
    • (d) $7.0m/s^2$

28. A $1000kg$ car on a flat curve of radius $90m$ with ${\mu }_s=0.8$ at $v=28m/s$ ($g=9.8m/s^2$). Will the car skid?

    • (a) Yes
    • (b) No
    • (c) Cannot determine
    • (d) Depends on the car's mass

29. A $0.25kg$ mass in a conical pendulum with $L=0.7m$, angle $\theta ={60}^{\circ }$ ($g=9.8m/s^2$). Calculate the period.

    • (a) $1.0s$
    • (b) $1.2s$
    • (c) $1.4s$
    • (d) $1.6s$

30. A skydiver of mass $60kg$ reaches terminal velocity ($g=9.8m/s^2$, $\rho =1.2kg/m^3$, $C_d=1$, $A=0.6m^2$). Calculate $v_t$.

    • (a) $35m/s$
    • (b) $40m/s$
    • (c) $45m/s$
    • (d) $50m/s$

31. A $4kg$ block on an incline at ${53}^{\circ }$ with ${\mu }_k=0.3$ ($g=9.8m/s^2$), pulled by $F=50N$ at ${30}^{\circ }$ above the incline. Calculate the acceleration.

    • (a) $12.0m/s^2$
    • (b) $13.0m/s^2$
    • (c) $14.0m/s^2$
    • (d) $15.0m/s^2$

32. A $9kg$ block on an incline at ${30}^{\circ }$ with ${\mu }_s=0.5$ ($g=9.8m/s^2$). Will the block slide?

    • (a) Yes
    • (b) No
    • (c) Cannot determine
    • (d) Depends on the mass

33. A $0.5kg$ mass in a conical pendulum with $L=1.2m$, angle $\theta ={37}^{\circ }$ ($g=9.8m/s^2$). Calculate the tension.

    • (a) $5.0N$
    • (b) $5.5N$
    • (c) $6.0N$
    • (d) $6.5N$

34. A $0.01kg$ object falls with $v_t=4m/s$ ($g=9.8m/s^2$, linear drag $F_d=bv$). Calculate $v$ after $0.05s$ from rest.

    • (a) $0.5m/s$
    • (b) $0.7m/s$
    • (c) $0.9m/s$
    • (d) $1.1m/s$

35. A $3kg$ block on an incline at ${37}^{\circ }$ with ${\mu }_s=0.4$ ($g=9.8m/s^2$), in a frame accelerating at ${\overrightarrow{a}}_{\text{frame}}=1.5\hat{i}m/s^2$ (along the incline). Will the block slide?

    • (a) Yes
    • (b) No
    • (c) Cannot determine
    • (d) Depends on the mass

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Conceptual Problems

36. What is the main difference between static and kinetic friction?

• (a) Static friction acts during motion, kinetic friction prevents motion
• (b) Static friction prevents motion, kinetic friction acts during motion
• (c) Static friction is constant, kinetic friction varies
• (d) Static friction depends on speed, kinetic friction does not

37. What provides the centripetal force for a car on a flat curve?

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

38. What does terminal velocity represent?

• (a) Maximum speed due to drag balancing gravity
• (b) Minimum speed due to drag
• (c) Speed at which drag is zero
• (d) Speed at which acceleration increases

39. What is the purpose of banking a road?

• (a) To increase friction
• (b) To provide centripetal force via the normal force
• (c) To reduce speed
• (d) To increase drag

40. What is the unit of the drag coefficient $b$ in linear drag ($F_d=bv$)?

• (a) $kg/s$
• (b) $kg/m$
• (c) $m/s$
• (d) Unitless

41. What happens to acceleration when friction increases?

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

42. What is the direction of centripetal force?

• (a) Tangent to the circle
• (b) Away from the center
• (c) Toward the center
• (d) Along the velocity

43. What does a drag force depend on for quadratic drag?

• (a) Speed only
• (b) Speed squared, fluid density, area
• (c) Mass only
• (d) Acceleration

44. How does friction affect an Atwood’s machine with one block on a surface?

• (a) Increases acceleration
• (b) Decreases acceleration
• (c) Does not affect acceleration
• (d) Causes constant velocity

45. What is the dimension of centripetal force?

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

46. What does a zero acceleration in circular motion imply?

• (a) No centripetal force
• (b) Straight-line motion
• (c) Constant speed
• (d) Zero velocity

47. What happens to terminal velocity if mass increases?

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

48. What is the role of friction in a complex system on an incline?

• (a) Increases acceleration
• (b) Opposes motion
• (c) Provides centripetal force
• (d) Increases normal force

49. What is the significance of ${\mu }_s$ in friction problems?

• (a) Determines kinetic friction
• (b) Determines maximum static friction
• (c) Determines drag force
• (d) Determines centripetal force

50. How does a pseudo-force affect motion in a non-inertial frame on an incline?

• (a) Acts along the incline
• (b) Acts opposite the frame’s acceleration
• (c) Acts perpendicular to the incline
• (d) Acts along gravity

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Derivation Problems

51. Derive the maximum static friction force for a block on a horizontal surface.

52. Derive the condition for a block to start sliding on an incline with static friction.

53. Derive the centripetal force for a conical pendulum.

54. Derive the banking angle for a car on a curved road without friction.

55. Derive the terminal velocity for an object with quadratic drag.

56. Derive the acceleration of a block on an incline with kinetic friction.

57. Derive the acceleration in an Atwood’s machine with friction on one block.

58. Derive the condition for a car to skid on a flat curve.

59. Derive the period of a conical pendulum.

60. Derive the terminal velocity for an object with linear drag.

61. Derive the acceleration of a block on an incline with friction and an applied force at an angle.

62. Derive the tension in a string for a mass in circular motion in a vertical plane.

63. Derive the velocity as a function of time for linear drag.

64. Derive the pseudo-force effect on a block in a non-inertial frame on an incline.

65. Derive the centripetal force provided by friction for a car on a banked curve with friction.

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NEET-style Conceptual Problems

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

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

67. What does a zero acceleration in circular motion indicate?

• (a) No centripetal force
• (b) Straight-line motion
• (c) Constant speed
• (d) Zero velocity

68. Which force provides centripetal force in a conical pendulum?

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

69. What happens to terminal velocity if the area of an object increases?

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

70. What is the dimension of the drag coefficient $b$ in linear drag?

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

71. What does static friction do?

• (a) Opposes motion once started
• (b) Prevents motion from starting
• (c) Increases acceleration
• (d) Acts during motion

72. What is the role of the normal force in banking of roads?

• (a) Opposes motion
• (b) Provides centripetal force component
• (c) Causes friction
• (d) Reduces speed

73. What happens to the centripetal force if speed doubles?

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

74. Why does friction act opposite to motion?

• (a) To increase acceleration
• (b) To oppose relative motion
• (c) To provide centripetal force
• (d) To increase normal force

75. What is the unit of terminal velocity?

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

76. What does a constant speed in circular motion imply?

• (a) Zero centripetal force
• (b) Constant centripetal force
• (c) Changing centripetal force
• (d) Zero velocity

77. Which force opposes motion in a pulley system with friction?

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

78. What is the direction of drag force on a falling object?

• (a) Downward
• (b) Upward
• (c) Sideways
• (d) Along the velocity

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

• (a) Increases acceleration
• (b) Accounts for the frame’s acceleration
• (c) Provides centripetal force
• (d) Opposes gravity

80. What is the dimension of friction?

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

81. What is the role of friction in circular motion?

• (a) Provides centripetal force
• (b) Increases speed
• (c) Reduces radius
• (d) Opposes gravity

82. What happens to acceleration on an incline if friction increases?

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

83. Why does terminal velocity depend on mass?

• (a) Drag force increases with mass
• (b) Gravity increases with mass
• (c) Area increases with mass
• (d) Density increases with mass

84. What is the significance of $\tan \theta$ in banking of roads?

• (a) Relates speed to radius and gravity
• (b) Determines friction
• (c) Determines drag
• (d) Determines normal force

85. What does kinetic friction do?

• (a) Prevents motion
• (b) Opposes motion once started
• (c) Increases acceleration
• (d) Acts before motion starts

86. What is the unit of centripetal force?

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

87. What does drag force cause in a falling object?

• (a) Constant acceleration
• (b) Terminal velocity
• (c) Circular motion
• (d) Zero velocity

88. Why does friction depend on the normal force?

• (a) Normal force determines speed
• (b) Normal force presses surfaces together
• (c) Normal force determines area
• (d) Normal force determines drag

89. What is the dimension of the drag coefficient in quadratic drag?

• (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 a pseudo-force affect a block on an incline?

• (a) Increases normal force
• (b) Acts along or against motion
• (c) Provides centripetal force
• (d) Reduces friction

91. What is the role of friction in a pulley system with friction?

• (a) Increases tension
• (b) Decreases acceleration
• (c) Provides centripetal force
• (d) Increases normal force

92. What does a zero centripetal force indicate?

• (a) Circular motion
• (b) Straight-line motion
• (c) Zero velocity
• (d) Zero speed

93. What is the physical significance of $v_t$?

• (a) Maximum speed due to drag
• (b) Minimum speed due to drag
• (c) Centripetal speed
• (d) Acceleration

94. What is the dimension of banking angle?

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

95. Why does friction act on a block on an incline?

• (a) To provide centripetal force
• (b) To oppose motion or tendency of motion
• (c) To increase acceleration
• (d) To increase normal force

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NEET-style Numerical Problems

96. A $5kg$ block on a surface with ${\mu }_k=0.2$ ($g=9.8m/s^2$) is pushed by $F=30N$. What is the acceleration?

• (a) $3.0m/s^2$
• (b) $4.0m/s^2$
• (c) $5.0m/s^2$
• (d) $6.0m/s^2$

97. A $1000kg$ car on a banked curve of radius $70m$ at $v=21m/s$ ($g=9.8m/s^2$). What is the banking angle?

• (a) ${25}^{\circ }$
• (b) ${30}^{\circ }$
• (c) ${35}^{\circ }$
• (d) ${40}^{\circ }$

98. Masses $m_1=10kg$ (hanging) and $m_2=5kg$ on a surface with ${\mu }_k=0.2$ ($g=9.8m/s^2$) in an Atwood’s setup. What is the acceleration?

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

99. A skydiver of mass $75kg$ reaches terminal velocity ($g=9.8m/s^2$, $\rho =1.2kg/m^3$, $C_d=1$, $A=0.8m^2$). What is $v_t$?

• (a) $35m/s$
• (b) $40m/s$
• (c) $45m/s$
• (d) $50m/s$

100. A $4kg$ block on an incline at ${45}^{\circ }$ with ${\mu }_k=0.3$ ($g=9.8m/s^2$), pulled by $F=40N$ at ${45}^{\circ }$ above the incline. What is the acceleration?
     - (a) $8.0m/s^2$
     - (b) $9.0m/s^2$
     - (c) $10.0m/s^2$
     - (d) $11.0m/s^2$

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