Question

A uniform rod of length $L$ is free to rotate in a vertical plane about a fixed horizontal axis through $O$. The rod is allowed to rotate from its unstable vertical position. Find the angular velocity of the rod when it has turned through an angle $\theta$.

Answer 1

Since the reaction force at the height is not doing any work, total mechanical energy is conserved.

$\Delta PE=\frac{mgl}{2}(1-\cos \theta )$

$\Delta K=\frac{1}{2}{I}_{\circ }{\omega }^2$

$=\frac{1}{2}\frac{m{l}^2}{3}{\omega }^2$

$\Rightarrow \frac{1}{2}\frac{m{l}^2}{3}{\omega }^2=mgl{\sin }^2\frac{\theta }{2}$

$\Rightarrow \omega =\sqrt{\frac{6g}{l}}\sin \frac{\theta }{2}$