Q 11-13-029JEE MainHard
A uniform rod of length $L$ is pivoted at one end and oscillates in a vertical plane as a compound pendulum. The length of a simple pendulum with the same period is
Answer: (C) $\dfrac{2L}{3}$
Concept. Compound pendulum: $T = 2\pi\sqrt{\dfrac{I}{mgd}}$; compare with $2\pi\sqrt{\dfrac{l}{g}}$.
Step 1. Rod about its end. $I = \dfrac{mL^2}{3}$, $d = \dfrac{L}{2}$:
$$T = 2\pi\sqrt{\frac{mL^2/3}{mgL/2}} = 2\pi\sqrt{\frac{2L}{3g}}$$
Step 2. Equivalent length. $\dfrac{2L}{3}$.
Answer. $\dfrac{2L}{3}$ (Option C).
Solution by Sreeraj P, M.Sc Physics