Simple Harmonic (4).

A uniform rod bent in the shape of an arc of a circle is suspended from a horizontal surface by means of a smooth axle connected to the bent rod at its mid-point. The arc of the rod subtends an angle of $2A = \dfrac{3 \pi}{4}$ radians at the centre of its circle (where $A$ is the semi-angle subtended by the rod at its centre) whose radius is $R = \dfrac{8 g}{\pi^2}$ . The rod is currently in the state of equilibrium. Now the bent rod is given a slight angular displacement towards either side such that it executes SHM about the axle. Find the time period of this SHM.

Details: $g = 9.8 \text{ms}^{-2}$ denotes the acceleration due to gravity.