Question

A uniform plank of length l and mass m is balanced on a fixed semi-circular plate of radius R as shown in the figure. If the plank is tilted slightly from its equilibrium position, and. released, prove that it will execute S.H.M. find the period of its oscillation.

Answer 1

The net torque acting on the bar about P when it is made to till through a small distance $x$ is given as,

$\tau =-mgx\cos \theta$

$-I_P{\omega }^2\theta =-mgx$

Since $\cos \theta =1$ for small $\theta$ so, $x=R\theta$ for small $\theta$.

$-\frac{m{l}^2}{12}{\omega }^2\theta =-mgR\theta$

$\omega =\sqrt{\frac{12gR}{l^2}}$

$T=\frac{2\pi }{\omega }$

$T=\frac{2\pi l}{\sqrt{12gR}}$

It is proved that it will execute S.H.M. with the period of $\frac{2\pi l}{\sqrt{12gR}}$.