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GATE 2023 CE (CE2) – Question 14

Solid Mechanics · Uniform torsion and combined stresses · 1 mark · Multiple choice

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 polar moment of inertia of the section are denoted as G and J, respectively. The torsional rigidity GJ is 50,000 kN.m$^2$/rad. The following are reported for this shaft: 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?

  1. Both the statements are correct
  2. Statement i) is correct, but Statement ii) is wrong
  3. Statement i) is wrong, but Statement ii) is correct
  4. Both the statements are wrong

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Show answer and explanation

Correct answer: (B) Statement i) is correct, but Statement ii) is wrong

Explanation

**Statement (i): rotation at the free end.** For a fixed-free shaft of length $L$ under end torque $T$,
$$\theta=\frac{TL}{GJ}=\frac{100\times5}{50\,000}=0.01\text{ rad}.$$
Statement (i) is **correct**.

**Statement (ii): strain energy.** The torque is applied gradually, so the work done is
$$U=\tfrac12T\theta=\frac{T^2L}{2GJ}=\tfrac12(100)(0.01)=0.5\text{ kN m}.$$
The statement says 1.0 kN m, which would be $T\theta$ without the factor ½. Statement (ii) is **wrong**.

Answer **B** (i correct, ii wrong).