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GATE 2024 CE (CE1) – Question 40

Solid Mechanics · Uniform torsion and combined stresses · 2 marks · Multiple choice

A homogeneous shaft PQR with fixed supports at both ends is subjected to a torsional moment T at point Q, as shown in the figure. The polar moments of inertia of the portions PQ and QR of the shaft with circular cross-sections are $J_1$ and $J_2$, respectively. The torsional moment reactions at the supports P and R are $T_P$ and $T_R$, respectively.

[Figure: a shaft fixed at P and R, with the length $L_1$ and the polar moment $J_1$ between P and Q and the length $L_2$ and $J_2$ between Q and R. The torque T acts at Q.]

If $T_P/T_R = 4$ and $J_1/J_2 = 2$, the ratio of the lengths $L_1/L_2$ is

Diagram for GATE 2024 CE (CE1) question 40
  1. 0.50
  2. 0.25
  3. 4.00
  4. 2.00

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Show answer and explanation

Correct answer: (A) 0.50

Explanation

The angle of twist of Q computed from each end must be the same: $\frac{T_PL_1}{GJ_1} = \frac{T_RL_2}{GJ_2}$. So $\frac{T_P}{T_R} = \frac{L_2J_1}{L_1J_2}$. With $\frac{T_P}{T_R} = 4$ and $\frac{J_1}{J_2} = 2$: $4 = 2\frac{L_2}{L_1}$, so $\frac{L_2}{L_1} = 2$ and $\frac{L_1}{L_2} = 0.5$.