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

Answer 1

Answer: (A)

The restoring torque (for small $\theta$ )

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