Question

A circular solid shaft of span \( L = 5 \, m \) is fixed at one end and free at the other end. A torque \( T = 100 \, kN.m \) is applied at the free end. The shear modulus and the polar moment of inertia of the section are denoted as \( G \) and \( J \), respectively. The torsional rigidity \( \frac{GJ}{L} \) is \( 50,000 \, kN.m^2/rad \). 
Statement i) The rotation at the free end is \( 0.01 \, rad \). 
Statement ii) The torsional strain energy is \( 1.0 \, kN.m \). 
With reference to the above statements, which of the following is true?

• A. Both the statements are correct
• B. Statement i) is correct, but Statement ii) is wrong
• C. Statement i) is wrong, but Statement ii) is correct
• D. Both the statements are wrong

Hint:

When calculating rotational motion and torsional strain energy, always use the respective formulas. For rotation, use \( \theta = \frac{T \cdot L}{GJ} \), and for strain energy, use \( U = \frac{T^2 L}{2 GJ} \).

Answer 1

The Correct Option is B

Given that \( L = 5 \, m \), \( T = 100 \, kN.m \), and \( \frac{GJ}{L} = 50,000 \, kN.m^2/rad \), we can proceed with the calculations.
1. Rotation at the free end: The rotation \( \theta \) at the free end of a shaft due to an applied torque is given by the formula: \[ \theta = \frac{T \cdot L}{GJ} \] Where: \[ T = 100 \, kN.m = 100 \times 10^3 \, N.m, L = 5 \, m, GJ = 50,000 \, kN.m^2/rad = 50,000 \times 10^3 \, N.m^2/rad \] Substituting the values: \[ \theta = \frac{100 \times 10^3 \times 5}{50,000 \times 10^3} = 0.01 \, rad \] Thus, Statement i) is correct.
2. Torsional Strain Energy: The torsional strain energy \( U \) stored in a shaft due to torque is given by the formula: \[ U = \frac{T^2 L}{2 GJ} \] Substituting the given values: \[ U = \frac{(100 \times 10^3)^2 \times 5}{2 \times 50,000 \times 10^3} \] \[ U = \frac{(10^7) \times 5}{10^6} = 5 \, kN.m \] Thus, Statement ii) is incorrect because the torsional strain energy is \( 5 \, kN.m \), not \( 1.0 \, kN.m \).
Conclusion: The correct option is (B) – Statement i) is correct, but Statement ii) is wrong.