Question

An $L-$ shaped system of two identical rods of mass $m$ and length $l$ each, are resting on a peg $P$ as shown in the figure. Find the period of oscillations if the system is displaced slightly in its plane by a small angle $\theta$ ?

• A. $2\pi \frac{\sqrt{2}l}{3g}$
• B. $2\pi \sqrt{\frac{4\sqrt{2}l}{3g}}$
• C. $2\pi \sqrt{\frac{2l}{3g}}$
• D. $3\pi \sqrt{\frac{l}{3g}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{4\sqrt{2}l}{3g}}$


The system is a compound pendulum.
Time period $T=2\pi \sqrt{\frac{I}{mgd}}$
where $d=$ distance of COM of the system from the axis.
$I=$ moment of inertia about axis passing through support
$I=[\frac{m{l}^2}{3}+\frac{m{l}^2}{3}]=\frac{2m{l}^2}{3}$
Distance of COM from the axis $d=\frac{l}{2}\cos {45}^{\circ }=\frac{l}{2\sqrt{2}}$
$\Rightarrow T=2\pi \sqrt{\frac{\frac{2m{l}^2}{3}}{\frac{mgl}{2\sqrt{2}}}}=2\pi \sqrt{\frac{4\sqrt{2}l}{3g}}$
Hence, option $\left(b\right)$ is the correct answer.