Question

A rod of mass $M$ and length $L$ is hinged at its centre of mass so that it can rotate in a vertical plane. Two springs each of stiffness $k$ are connected at its ends, as shown in the figure. The time period of SHM is

• A. $2\pi \sqrt{\frac{M}{6k}}$
• B. $2\pi \sqrt{\frac{M}{3k}}$
• C. $2\pi \sqrt{\frac{ML}{k}}$
• D. $\pi \sqrt{\frac{M}{6k}}$

Answer 1

Answer: (A)

The restoring torque (for small $\theta$ )

${\tau }_{\text{rest }}=-\left[\frac{kL\theta }{2}\times \frac{L}{2}\right]\times 2=\frac{k{L}^2}{2}(-\theta )$
$\therefore \alpha =\frac{{\tau }_{\text{rest }}}{I}=\frac{k{L}^2/2}{M{L}^2/12}(-\theta )=\frac{6k}{M}(-\theta )$
$\therefore T=2\pi \sqrt{\frac{M}{6k}}$.