Torsion of Shafts MCQ with answer and solution

Concept

Polar MOI of a semicircle about the axis through the centre of its diameter is half that of a full circle.

Formula

J = I_x + I_y; J_circle = πR⁴/2; J_semicircle = πR⁴/4

Step-by-step solution

1. Polar MOI of a full circle about its centre is πR⁴/2. 2. A semicircle is half of the circle, and the axis passes through the same centre. 3. So J = (1/2)(πR⁴/2) = πR⁴/4. 4. Check: I_x = I_y = πR⁴/8, so J = πR⁴/8 + πR⁴/8 = πR⁴/4.

Explanation

A semicircle has half the area of the full circle, and that area sits at the same distances from the centre. So its polar MOI about the centre of the diameter is half of πR⁴/2, which is πR⁴/4. This matches option B.

Why the other options are wrong

A: πR⁴/16 is not a standard semicircle value. C: πR⁴/2 is for the full circle. D: πR⁴/8 is the MOI about one axis (I_x or I_y), not the polar MOI.

Shortcut

Semicircle J about its centre = half of circle J = πR⁴/4.

Common mistake

Taking I_x = πR⁴/8 as the polar moment of inertia.