Question

A uniform rod of mass m and length L is hinged at one of its end with the ceiling and other end of the rod is attached with a thread which is attached with horizontal ceiling at point P. If one end of the rod is slightly displaced horizontally and perpendicular to the rod and released. If the time period of small oscillation is $2\pi \sqrt{\frac{2lsin\theta }{xg}}$ Find x. ___

Answer 1

Consider a small angular displacement $\beta$
Torque about the rotational axis
$T=-\left[mg\frac{1}{2}sin\theta \right]\left(\beta \right)$
$T=I\propto$
$I=m\frac{(1sin\theta {)}^2}{3}=m\frac{l^{2}si{n}^2\theta }{3}$
$-\left[mg\frac{1}{2}sin\theta \right]\left(\beta \right)=\frac{m{l}^{2}si{n}^2\theta }{3}\propto$
$\propto =-\left[\frac{3g}{2lsin\theta }\right]\beta$
$\therefore T=2\pi \sqrt{\frac{2lsin\theta }{3g}}$