Question

A weightless rigid rod with a small iron bob at the end is hinged at point A to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible string of length 20 cm, fixed at its midpoint. The bob is displaced slightly perpendicular to the plane of the rod and the string. If the period of small oscillations of the system( in seconds) is $\frac{\mathrm{\pi X}}{10}$, then the value of X is

Answer 1

The bob will execute SHM about a stationary axis passing through AB. If its effective length is l' then $T=2\pi \sqrt{\frac{l}{g}}$
$l^{\prime }=\frac{l}{\sin \theta }=\sqrt{2}l(\because \theta ={45}^{\circ }$)
$g^{\prime }=g\cos \theta =\frac{g}{\sqrt{2}}$
$T=2\pi \sqrt{\frac{2l}{g}}=2\pi \sqrt{\frac{2\times 0.2}{10}}=\frac{2\pi }{5}s$