Equilibrium and Elasticity Problems

This section provides 100 problems to test your understanding of equilibrium and elasticity, including static and dynamic equilibrium, center of gravity and stability, stress, strain, elastic moduli, and applications in beams and structures. 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 these topics, which are crucial for JEE/NEET success.

Numerical Problems

1. A uniform rod of mass $4kg$ and length $2m$ is pivoted at one end and held horizontal by a vertical rope at the other end. Calculate the tension in the rope ($g=9.8m/s^2$).

   • (a) $18.0N$
   • (b) $19.6N$
   • (c) $20.0N$
   • (d) $21.5N$

2. A 15 kg sign hangs from a 3 m rod attached to a wall with a hinge, supported by a wire at ${30}^{\circ }$ to the horizontal at the end. Calculate the tension in the wire ($g=9.8m/s^2$).

   • (a) $220N$
   • (b) $235N$
   • (c) $255N$
   • (d) $270N$

3. A uniform block of mass $20kg$, height $1.5m$, and base $0.6m$ is on a flat surface. Calculate the critical angle for tipping.

   • (a) ${21.8}^{\circ }$
   • (b) ${22.5}^{\circ }$
   • (c) ${23.6}^{\circ }$
   • (d) ${24.5}^{\circ }$

4. A steel wire of length $1.5m$ and diameter $0.8mm$ is under a force $F=80N$. Given Young’s modulus $Y=2\times {10}^{11}\text{Pa}$, calculate the extension.

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

5. A simply supported beam of length $5m$ has a uniform load $w=400\text{N/m}$. Calculate the maximum bending moment.

   • (a) $2400N\cdot m$
   • (b) $2500N\cdot m$
   • (c) $2600N\cdot m$
   • (d) $2700N\cdot m$

6. A cantilever beam of length $3m$, with $Y=1.5\times {10}^{11}\text{Pa}$ and $I=1\times {10}^{-6}{m}^4$, has a load $F=500N$ at the free end. Calculate the deflection at the end.

   • (a) $15.0\text{mm}$
   • (b) $16.5\text{mm}$
   • (c) $18.0\text{mm}$
   • (d) $19.5\text{mm}$

7. A ladder of mass $25kg$ and length $6m$ leans at ${60}^{\circ }$ against a frictionless wall, with friction at the floor ($\mu =0.3$, $g=9.8m/s^2$). Calculate the friction force to prevent slipping.

   • (a) $60N$
   • (b) $65N$
   • (c) $70N$
   • (d) $75N$

8. An L-shaped object has a horizontal arm of mass $3kg$, length $1m$, and a vertical arm of mass $4kg$, length $2m$. Calculate the x-coordinate of the center of gravity.

   • (a) $0.80m$
   • (b) $0.85m$
   • (c) $0.90m$
   • (d) $0.95m$

9. A rubber block under shear force $F=40N$ over area $A=0.02m^2$ deforms by angle $\varphi =0.05\text{rad}$. Calculate the shear modulus.

   • (a) $35\text{kPa}$
   • (b) $40\text{kPa}$
   • (c) $45\text{kPa}$
   • (d) $50\text{kPa}$

10. A material under pressure $\mathrm{\Delta }P=2\times {10}^6\text{Pa}$ has a fractional volume change $\mathrm{\Delta }V/V_0=-0.001$. Calculate the bulk modulus.

    • (a) $1.8\times {10}^9\text{Pa}$
    • (b) $2.0\times {10}^9\text{Pa}$
    • (c) $2.2\times {10}^9\text{Pa}$
    • (d) $2.4\times {10}^9\text{Pa}$

11. A uniform rod of mass $6kg$ and length $1.5m$ is pivoted at one end, held horizontal by a rope at the other end. Calculate the tension in the rope ($g=9.8m/s^2$).

    • (a) $40.0N$
    • (b) $42.5N$
    • (c) $44.1N$
    • (d) $46.0N$

12. A 12 kg sign hangs from a 2.5 m rod attached to a wall with a hinge, supported by a wire at ${45}^{\circ }$ at the end. Calculate the tension in the wire ($g=9.8m/s^2$).

    • (a) $160N$
    • (b) $166N$
    • (c) $172N$
    • (d) $178N$

13. A block of mass $30kg$, height $2m$, and base $0.8m$ is on an incline. Calculate the angle at which it tips.

    • (a) ${21.8}^{\circ }$
    • (b) ${22.5}^{\circ }$
    • (c) ${23.6}^{\circ }$
    • (d) ${24.5}^{\circ }$

14. A copper wire of length $1m$ and diameter $1.2mm$ is under a force $F=120N$. Given $Y=1.1\times {10}^{11}\text{Pa}$, calculate the extension.

    • (a) $0.90\text{mm}$
    • (b) $0.96\text{mm}$
    • (c) $1.00\text{mm}$
    • (d) $1.05\text{mm}$

15. A simply supported beam of length $6m$ has a uniform load $w=300\text{N/m}$. Calculate the maximum bending moment.

    • (a) $2600N\cdot m$
    • (b) $2700N\cdot m$
    • (c) $2800N\cdot m$
    • (d) $2900N\cdot m$

16. A cantilever beam of length $2.5m$, with $Y=2\times {10}^{11}\text{Pa}$ and $I=5\times {10}^{-7}{m}^4$, has a load $F=800N$ at the free end. Calculate the deflection at the end.

    • (a) $8.0\text{mm}$
    • (b) $8.5\text{mm}$
    • (c) $9.0\text{mm}$
    • (d) $9.5\text{mm}$

17. A ladder of mass $15kg$ and length $4m$ leans at ${45}^{\circ }$ against a frictionless wall, with $\mu =0.5$ at the floor ($g=9.8m/s^2$). Calculate the friction force to prevent slipping.

    • (a) $70N$
    • (b) $72N$
    • (c) $74N$
    • (d) $76N$

18. An L-shaped object has a horizontal arm of mass $5kg$, length $1.2m$, and a vertical arm of mass $2kg$, length $1.8m$. Calculate the y-coordinate of the center of gravity.

    • (a) $0.50m$
    • (b) $0.55m$
    • (c) $0.60m$
    • (d) $0.65m$

19. A material under shear force $F=60N$ over area $A=0.03m^2$ deforms by angle $\varphi =0.02\text{rad}$. Calculate the shear modulus.

    • (a) $90\text{kPa}$
    • (b) $95\text{kPa}$
    • (c) $100\text{kPa}$
    • (d) $105\text{kPa}$

20. A material under pressure $\mathrm{\Delta }P=3\times {10}^6\text{Pa}$ has $\mathrm{\Delta }V/V_0=-0.003$. Calculate the bulk modulus.

    • (a) $0.8\times {10}^9\text{Pa}$
    • (b) $0.9\times {10}^9\text{Pa}$
    • (c) $1.0\times {10}^9\text{Pa}$
    • (d) $1.1\times {10}^9\text{Pa}$

21. A uniform rod of mass $8kg$ and length $3m$ is pivoted at one end, held horizontal by a rope at the other end. Calculate the tension in the rope ($g=9.8m/s^2$).

    • (a) $110N$
    • (b) $115N$
    • (c) $117N$
    • (d) $120N$

22. A 5 kg sign hangs from a 1 m rod attached to a wall with a hinge, supported by a wire at ${60}^{\circ }$ at the end. Calculate the tension in the wire ($g=9.8m/s^2$).

    • (a) $50N$
    • (b) $55N$
    • (c) $56.6N$
    • (d) $60N$

23. A block of mass $40kg$, height $1.8m$, and base $0.9m$ is on an incline. Calculate the angle at which it tips.

    • (a) ${25.0}^{\circ }$
    • (b) ${26.0}^{\circ }$
    • (c) ${26.6}^{\circ }$
    • (d) ${27.5}^{\circ }$

24. A steel rod of length $2.5m$ and diameter $1.5mm$ is under a force $F=150N$. Given $Y=2\times {10}^{11}\text{Pa}$, calculate the extension.

    • (a) $1.00\text{mm}$
    • (b) $1.05\text{mm}$
    • (c) $1.06\text{mm}$
    • (d) $1.10\text{mm}$

25. A simply supported beam of length $8m$ has a uniform load $w=200\text{N/m}$. Calculate the maximum bending moment.

    • (a) $3100N\cdot m$
    • (b) $3200N\cdot m$
    • (c) $3300N\cdot m$
    • (d) $3400N\cdot m$

26. A cantilever beam of length $4m$, with $Y=1\times {10}^{11}\text{Pa}$ and $I=2\times {10}^{-6}{m}^4$, has a load $F=600N$ at the free end. Calculate the deflection at the end.

    • (a) $30\text{mm}$
    • (b) $32\text{mm}$
    • (c) $34\text{mm}$
    • (d) $36\text{mm}$

27. A ladder of mass $10kg$ and length $3m$ leans at ${60}^{\circ }$ against a frictionless wall, with $\mu =0.4$ at the floor ($g=9.8m/s^2$). Calculate the friction force to prevent slipping.

    • (a) $25N$
    • (b) $27N$
    • (c) $28N$
    • (d) $30N$

28. An L-shaped object has a horizontal arm of mass $2kg$, length $0.8m$, and a vertical arm of mass $3kg$, length $1.2m$. Calculate the x-coordinate of the center of gravity.

    • (a) $0.60m$
    • (b) $0.64m$
    • (c) $0.68m$
    • (d) $0.72m$

29. A material under shear force $F=100N$ over area $A=0.04m^2$ deforms by angle $\varphi =0.01\text{rad}$. Calculate the shear modulus.

    • (a) $240\text{kPa}$
    • (b) $250\text{kPa}$
    • (c) $260\text{kPa}$
    • (d) $270\text{kPa}$

30. A material under pressure $\mathrm{\Delta }P=4\times {10}^6\text{Pa}$ has $\mathrm{\Delta }V/V_0=-0.002$. Calculate the bulk modulus.

    • (a) $1.8\times {10}^9\text{Pa}$
    • (b) $2.0\times {10}^9\text{Pa}$
    • (c) $2.2\times {10}^9\text{Pa}$
    • (d) $2.4\times {10}^9\text{Pa}$

31. A rocket strut (length $1.5m$, diameter $0.02m$, $Y=2\times {10}^{11}\text{Pa}$) is under compression. Calculate the critical buckling load.

    • (a) $6500N$
    • (b) $6800N$
    • (c) $7000N$
    • (d) $7200N$

32. A rectangular beam under shear force $V=3000N$ has a cross-section (width $0.04m$, height $0.08m$). Calculate the maximum shear stress.

    • (a) $0.90\text{MPa}$
    • (b) $0.95\text{MPa}$
    • (c) $1.00\text{MPa}$
    • (d) $1.05\text{MPa}$

33. A seesaw has a 50 kg child at 2 m left of the pivot and a 40 kg child at 2.5 m right. Calculate the distance a 60 kg person must sit on the left to balance it ($g=9.8m/s^2$).

    • (a) $0.80m$
    • (b) $0.85m$
    • (c) $0.90m$
    • (d) $0.95m$

34. A steel wire of length $3m$ and diameter $2mm$ is under a force $F=200N$. Given $Y=2\times {10}^{11}\text{Pa}$, calculate the extension.

    • (a) $0.95\text{mm}$
    • (b) $0.96\text{mm}$
    • (c) $0.97\text{mm}$
    • (d) $0.98\text{mm}$

35. A simply supported beam of length $10m$ has a uniform load $w=150\text{N/m}$. Calculate the maximum bending moment.

    • (a) $3600N\cdot m$
    • (b) $3700N\cdot m$
    • (c) $3750N\cdot m$
    • (d) $3800N\cdot m$

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Conceptual Problems

36. What is required for an object to be in static equilibrium?

• (a) Net force is zero
• (b) Net torque is zero
• (c) Both net force and net torque are zero
• (d) Object must be moving

37. What does the center of gravity represent?

• (a) The geometric center of an object
• (b) The point where the weight acts
• (c) The point of maximum mass
• (d) The point of minimum stability

38. What is stress in the context of elasticity?

• (a) Deformation per unit length
• (b) Force per unit area
• (c) Change in volume
• (d) Change in shape

39. What does a high Young’s modulus indicate?

• (a) High flexibility
• (b) High stiffness
• (c) High compressibility
• (d) High shear resistance

40. What is the unit of stress?

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

41. When does an object tip on an incline?

• (a) When the incline angle exceeds the critical angle
• (b) When the mass increases
• (c) When the base widens
• (d) When the height decreases

42. What does Hooke’s law state in elasticity?

• (a) Stress is proportional to strain
• (b) Stress is inversely proportional to strain
• (c) Stress equals strain
• (d) Stress is independent of strain

43. What is the physical significance of the bulk modulus?

• (a) Resistance to shear
• (b) Resistance to volume change
• (c) Resistance to length change
• (d) Resistance to bending

44. What happens to a beam under bending?

• (a) Uniform stress throughout
• (b) Tensile stress on one side, compressive on the other
• (c) Only shear stress
• (d) No stress at the ends

45. What is the dimension of strain?

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

46. What does a zero net torque imply?

• (a) No linear motion
• (b) No rotational acceleration
• (c) No linear acceleration
• (d) No deformation

47. What is the significance of $\frac{F}{A}$?

• (a) Strain
• (b) Stress
• (c) Young’s modulus
• (d) Shear modulus

48. What does a high shear modulus indicate?

• (a) Easy to compress
• (b) Easy to stretch
• (c) Resistance to shear deformation
• (d) Resistance to bending

49. What is the role of the center of gravity in stability?

• (a) Determines linear acceleration
• (b) Determines tipping behavior
• (c) Determines shear stress
• (d) Determines strain

50. How does friction contribute to equilibrium in a ladder problem?

• (a) Increases normal force
• (b) Prevents slipping by providing torque
• (c) Causes tipping
• (d) Reduces weight

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Derivation Problems

51. Derive the conditions for equilibrium using Newton’s laws.

52. Derive the tipping condition for a block on an incline.

53. Derive Young’s modulus $Y=\frac{F{L}_0}{A\mathrm{\Delta }L}$.

54. Derive the center of gravity for a system of particles.

55. Derive the maximum bending moment for a simply supported beam with uniform load.

56. Derive the shear modulus $G=\frac{F/A}{\varphi }$.

57. Derive the deflection of a cantilever beam under a point load.

58. Derive the torque balance for a seesaw.

59. Derive the bulk modulus $K=-\frac{\mathrm{\Delta }P}{\mathrm{\Delta }V/V_0}$.

60. Derive the equilibrium of a hanging sign supported by a wire.

61. Derive the maximum shear stress in a rectangular beam.

62. Derive the critical buckling load for a column using Euler’s formula.

63. Derive the center of gravity for an L-shaped object.

64. Derive the stability condition for a ladder against a wall.

65. Derive the stress in a wire under tension using Hooke’s law.

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NEET-style Conceptual Problems

66. What is the unit of Young’s modulus in SI units?

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

67. What does a zero net force indicate?

• (a) Object is rotating
• (b) Object has no linear acceleration
• (c) Object is deforming
• (d) Object is tipping

68. Which quantity is dimensionless in elasticity?

• (a) Stress
• (b) Strain
• (c) Young’s modulus
• (d) Shear modulus

69. What happens to a material when stress exceeds the elastic limit?

• (a) Returns to original shape
• (b) Permanently deforms
• (c) Increases elasticity
• (d) Decreases strain

70. What is the dimension of shear modulus?

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

71. What does the center of gravity determine?

• (a) Point of maximum stress
• (b) Point where weight acts
• (c) Point of maximum strain
• (d) Point of shear deformation

72. What is the role of torque in equilibrium?

• (a) Causes linear acceleration
• (b) Must sum to zero for no rotation
• (c) Causes deformation
• (d) Increases stability

73. What happens to a block’s stability when its base widens?

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

74. Why does a material deform elastically?

• (a) Stress exceeds elastic limit
• (b) Stress is within elastic limit
• (c) Strain is zero
• (d) Modulus is zero

75. What is the unit of strain?

• (a) Dimensionless
• (b) $Pa$
• (c) $N\cdot m$
• (d) $J$

76. What does a constant bending moment imply?

• (a) No bending
• (b) Uniform bending
• (c) Maximum shear stress
• (d) No deflection

77. Which type of stress is involved in bending a beam?

• (a) Shear stress only
• (b) Tensile and compressive stress
• (c) Bulk stress
• (d) No stress

78. What is the direction of the normal force in equilibrium?

• (a) Along the force
• (b) Perpendicular to the surface
• (c) Along the torque
• (d) Along the displacement

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

• (a) Maintains equilibrium
• (b) Affects force balance
• (c) Provides torque
• (d) Reduces friction

80. What is the dimension of the center of gravity position?

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

81. What is the role of friction in ladder equilibrium?

• (a) Increases weight
• (b) Prevents slipping
• (c) Causes tipping
• (d) Reduces stability

82. What happens to strain when stress increases within the elastic limit?

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

83. Why does a high bulk modulus indicate low compressibility?

• (a) Less volume change under pressure
• (b) More volume change under pressure
• (c) No volume change
• (d) Increases shear

84. What is the significance of $\frac{F{L}_0}{A\mathrm{\Delta }L}$?

• (a) Shear modulus
• (b) Bulk modulus
• (c) Young’s modulus
• (d) Stress

85. What is the unit of the bending moment?

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

86. What does a zero deflection at the end of a cantilever beam indicate?

• (a) No load applied
• (b) Maximum load applied
• (c) No bending moment
• (d) Maximum shear stress

87. What is the physical significance of $\sum \overrightarrow{F}=0$?

• (a) Rotational equilibrium
• (b) Translational equilibrium
• (c) Elastic deformation
• (d) Structural failure

88. Why does a wider base improve stability?

• (a) Lowers the center of gravity
• (b) Keeps COG over the base longer
• (c) Increases weight
• (d) Reduces friction

89. What is the dimension of bulk modulus?

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

90. How does shear stress affect a material?

• (a) Changes its length
• (b) Changes its volume
• (c) Causes angular deformation
• (d) Causes tipping

91. What is the role of equilibrium in a rocket launch platform?

• (a) Determines deformation
• (b) Ensures stability before launch
• (c) Increases stress
• (d) Reduces strain

92. What does a zero shear stress in a beam indicate?

• (a) No bending
• (b) No transverse force
• (c) Maximum deflection
• (d) Maximum bending moment

93. What is the physical significance of $-\frac{\mathrm{\Delta }P}{\mathrm{\Delta }V/V_0}$?

• (a) Young’s modulus
• (b) Shear modulus
• (c) Bulk modulus
• (d) Strain

94. What is the dimension of deflection in a beam?

• (a) $[\text{L}]$
• (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 a high center of gravity reduce stability?

• (a) Increases weight
• (b) Easier for COG to move outside the base
• (c) Reduces friction
• (d) Increases strain

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NEET-style Numerical Problems

96. A uniform rod of mass $5kg$ and length $1.5m$ is pivoted at one end, held horizontal by a rope at the other end. What is the tension in the rope ($g=9.8m/s^2$)?

• (a) $34.0N$
• (b) $36.0N$
• (c) $36.75N$
• (d) $38.0N$

97. A steel wire of length $2m$ and diameter $1mm$ is under a force $F=100N$. Given $Y=2\times {10}^{11}\text{Pa}$, what is the extension?

• (a) $1.25\text{mm}$
• (b) $1.27\text{mm}$
• (c) $1.30\text{mm}$
• (d) $1.35\text{mm}$

98. A simply supported beam of length $4m$ has a uniform load $w=500\text{N/m}$. What is the maximum bending moment?

• (a) $1900N\cdot m$
• (b) $2000N\cdot m$
• (c) $2100N\cdot m$
• (d) $2200N\cdot m$

99. A block of mass $10kg$, height $1m$, and base $0.5m$ is on an incline. What is the angle at which it tips?

• (a) ${25.0}^{\circ }$
• (b) ${26.0}^{\circ }$
• (c) ${26.6}^{\circ }$
• (d) ${27.5}^{\circ }$

100. A cantilever beam of length $2m$, with $Y=1\times {10}^{11}\text{Pa}$ and $I=1\times {10}^{-6}{m}^4$, has a load $F=1000N$ at the free end. What is the deflection at the end?
     - (a) $25.0\text{mm}$
     - (b) $26.0\text{mm}$
     - (c) $26.7\text{mm}$
     - (d) $27.5\text{mm}$

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