Define angular S.H.M. and obtain its differential equation. - Vidyadip

Question

Define angular S.H.M. and obtain its differential equation.

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Answer

Angular S.H.M. is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement.
i. Consider a metallic disc hanging from rigid support, when twisted, it performs an oscillatory motion for which the restoring torque acting upon it, for angular displacement $\theta$ is,
$ \tau \propto-\theta$
$\therefore \tau=-c \theta . $
ii. The constant of proportionality (c) is the restoring torque per unit angular displacement.
iii. If I is the moment of inertia of the disc, the torque acting on the disc is given by,
$\tau=\mid \alpha \text {.....(2) }$
Where $\alpha$ is the angular acceleration.
iv. From equations ( 1 ) and (2),
$ l \alpha=- c \theta$
$\therefore I \frac{ d ^2 \theta}{ dt ^2}+ c \theta=0 \ldots \ldots . .\left(\because \alpha=\frac{ d ^2 \theta}{ dt ^2}\right) $
This is the differential equation for angular S.H.M.

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