Question

Two identical rods each of length $l$ and mass $m$ are welded together at right angle 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

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

'C' is combined centre of mass.


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