Simple harmonic motion can also be angular. In this case, the restoring torque required for producing SHM is directly proportional to the angular displacement and is directed towards the mean position.

**Angular harmonic oscillator**

*Simple harmonic motion can also be angular. In
this case, the restoring torque required for producing SHM is directly
proportional to the angular displacement and is directed towards the mean
position.*

Consider a wire
suspended vertically from a rigid support. Let some weight be suspended from
the lower end of the wire. When the wire is twisted through an angle θ from the mean
position, a restoring torque acts on it tending to return it to the mean
position. Here restoring torque is proportional to angular displacement θ.

Hence r = -C θ ???..(1)

where C is
called torque constant.

It is equal to
the moment of the couple required to produce unit angular displacement. Its
unit is N m rad−1.

The negative
sign shows that torque is acting in the opposite direction to the angular
displacement. This is the case of angular simple harmonic motion.

Examples :
Torsional pendulum, balance wheel of a watch.

But τ = I α
...(2)

where τ is
torque, I is the moment of inertia and α is angular acceleration

∴
Angular acceleration,

Α = τ /I = - C θ / I

This is similar
to a = −ω^{2} y

Replacing y by
θ, and a by α we get

α = −ω^{2}θ
= - (C/I) θ

ω = rt(C/I)

Period of SHM T = 2π rt(I/C)

Frequency n = 1/T = 1/2 π rt(C/I)

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