A thin uniform rod of mass m and length l is hinged at one end making an angle $\alpha$ from the horizontal on the ceiling of a room. The other end is supported by a vertical massless string. The angular frequency of small oscillations of this system is

For a small angular displacement $\beta$ from vertical, the figure is: Restoring torque about ‘O’ is:

$\tau =(T\sin \beta )l\approx T\beta l\dots \dots ......\left(i\right)$

where T = tension in the string = ?
when the rod is in equilibrium position.

$\begin{array}{r}\sum \tau =0=mg\left(\frac{l}{2}\cos \alpha \right)-T(l\cos \alpha )\\ \Rightarrow T=\frac{mg}{2}\end{array}$

$\text{ From figure : }p{p}^{\mathrm{\prime }}=\theta l=\beta l\sin \alpha$

$I=\frac{m{l}^2}{3}:\text{ solve for time period. }$