Question

Two identical rods each of length $l$ and mass $m$ are welded together at right angles and then suspended from a knife-edge as shown. Angular frequency of small oscillations of the system in its own plane about the point of suspension is

• A. $\sqrt{\frac{3g}{4\sqrt{2}l}}$
• B. $\sqrt{\frac{3g}{2\sqrt{2}l}}$
• C. $\sqrt{\frac{3g}{\sqrt{2}l}}$
• D. None of these

Answer 1

Answer: (B)

$C$ is combined centre of mass.

$h=\frac{l}{2}\cos 45^{\circ }=\frac{l}{2\sqrt{2}}$
Moments of Inertia of rods about O: $I=\frac{2m{l}^2}{3}$
$T=2\pi \sqrt{\frac{I}{2mgh}}$
$=2\pi \sqrt{\frac{2m{l}^{2}2\sqrt{2}}{3[2mg]l}}=2\pi \sqrt{\frac{2\sqrt{2}l}{3g}}$
Angular frequency: $\omega =\frac{2\pi }{T}=\sqrt{\frac{3g}{2\sqrt{2}l}}$