First slide

Simple hormonic motion

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

Moderate

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

Solution

C is combined centre of mass.
$h = \frac{l}{2} \cos 45 ° = \frac{l}{2 \sqrt{2}}$
Moments of Inertia of rods about O: $I = \frac{2 {ml}^{2}}{3}$
$T = 2 π \sqrt{\frac{I}{2 mgh}}$
= $2 π \sqrt{\frac{2 {ml}^{2} 2 \sqrt{2}}{3 [2 mg] l}} = 2 π \sqrt{\frac{2 \sqrt{2} l}{3 g}}$
Angular frequency: $ω = \frac{2 π}{T} = \sqrt{\frac{3 g}{2 \sqrt{2} l}}$

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