Question

A uniform rod of length l mass m is fixed at the centre. A spring of spring constant k is connected to rod and wall as shown in figure. The rod is displaced by small angle $\theta$ and released. Find time period of oscillation.

• A. $2\pi \sqrt{\frac{m}{k}}$
• B. $2\pi \sqrt{\frac{m}{2k}}$
• C. $2\pi \sqrt{\frac{3m}{k}}$
• D. $2\pi \sqrt{\frac{m}{3k}}$

Answer 1

Answer: (D)

$2\pi \sqrt{\frac{m}{3k}}$

Let the length of the rod be $l$.
Torque acting on the rod about the center of the rod is given by
$\tau =I\alpha =-\left(kx\right)\frac{l}{2}$
$\frac{m{l}^2}{12}\alpha =-k\frac{l}{2}\theta \left(\frac{l}{2}\right)$
$\alpha =-\frac{3k}{m}\theta$
$\Rightarrow \omega =\sqrt{\frac{3k}{m}}$
$T=2\pi \sqrt{\frac{m}{3k}}$
Option D.