Torsion of Shafts MCQ with answer and solution

Concept

For a shaft fixed at both ends, each end resists the torque in inverse ratio of its length, so the nearer end (stiffer part) takes more torque.

Formula

T_A = T × L_B / L, T_B = T × L_A / L

Step-by-step solution

1. L = 9 m. The torque acts 3 m from end A, so L_A = 3 m and L_B = 6 m. 2. Equilibrium: T_A + T_B = 30 t-m. 3. Compatibility: angle of twist is the same for both parts, so T_A × L_A = T_B × L_B. 4. T_A × 3 = T_B × 6, so T_A = 2 T_B. 5. 2 T_B + T_B = 30, so T_B = 10 t-m and T_A = 20 t-m. 6. Reactive torque at the nearer end = 20 t-m.

Explanation

Both ends are fixed, so the twist of the left part equals the twist of the right part. The shorter part is stiffer, so it takes more torque. The nearer end (3 m away) takes 30 × 6/9 = 20 t-m.

Why the other options are wrong

A: Wrong ratio. B: 10 t-m is the reaction at the farther end. C: 15 t-m assumes equal sharing, which is true only if the torque acts at mid-length.

Shortcut

Nearer end torque = T × (far length / total length) = 30 × 6/9 = 20.

Common mistake

Giving more torque to the farther (longer) end, or splitting the torque equally.