Question

A uniform rod of length $l$ and mass $m$ is fixed at the centre. A spring of spring constant $k$ is connected to rod and wall as shown in the 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

The correct option is D $2\pi \sqrt{\frac{m}{3k}}$

Applying the rotational form of Newton's law
$\tau =I\alpha =-\left(kx\right)\frac{l}{2}$
or $\frac{m{l}^2}{12}\alpha =-\left(k\frac{l}{2}\theta \right)\frac{l}{2}$
or $\alpha =-\frac{3k}{m}\theta$
Comparing the above expression with
$\alpha ={\omega }^2\theta$
$\omega =\sqrt{\frac{3k}{m}}$
or $T=2\pi \sqrt{\frac{m}{3k}}$