First slide

Simple harmonic motion

Question

The angular frequency of oscillation for a particle executing angular oscillations with couple per unit twist of c is

Moderate

A

$ω = \sqrt{\frac{c}{I^{2}}}$
B

$ω = \sqrt{\frac{I}{c}}$
C

$ω = \sqrt{\frac{c}{I}}$
D

$ω = \sqrt{\frac{c^{2}}{I}}$

Solution

For a body executing simple harmonic motion, Restoring torque acting on it after a small displacement $θ$ , about an axis,
$τ = - c θ$
We know that this results in angular oscillations which can be also related by the equation $τ = I α$
Equating the above two we get $- c θ = I α$
Where for angular oscillations $α = - ω^{2} θ$
Hence $- c θ = - I ω^{2} θ$
Simplifying this $c = I ω^{2}$
Hence $ω = \sqrt{\frac{c}{I}}$

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