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. $\sqrt{\frac{3 g}{4 \sqrt{2} l}}$
• B. $\sqrt{\frac{3 g}{2 \sqrt{2} l}}$
• C. $\sqrt{\frac{3 g}{\sqrt{2} l}}$
• 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: $I=\frac{2 m l^{2}}{3}$
$T =2 \pi \sqrt{\frac{I}{2 m g h}} $
$=2 \pi \sqrt{\frac{2 m l^{2} 2 \sqrt{2}}{3[2 m g] l}}=2 \pi \sqrt{\frac{2 \sqrt{2} l}{3 g}}$
Angular frequency: $\omega=\frac{2 \pi}{T}=\sqrt{\frac{3 g}{2 \sqrt{2} l}}$