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

Hint: In this question, we need to determine the differential equation of the simple harmonic motion of a particle. For this, we will consider a body in the simple harmonic motion (SHM) and derive the differential equation of the same by following the general laws of motion.

Complete step by step solution:
In this question, S.H.M stands for simple harmonic function. We know that angular simple harmonic motion is defined as the to and fro motion of a body about a central point or orientation is called angular simple harmonic motion. When a body is at equilibrium and is disturbed by a small amount of torque then it performs angular simple harmonic motion.

Now differential equation of the simple harmonic function is given as: 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;
\[\tau \propto - \theta \]
\[\therefore \tau = - c\theta ...\left( 1 \right)\]
Where the constant of proportionality \[c\] is the restoring torque per unit angular displacement.

Now, \[I\] is the moment of inertia of the disc and the torque acting on the disc is given by
\[\tau = I\alpha ...\left( 2 \right)\]
Where \[\alpha \]is the angular acceleration
Now from the equation \[\left( 1 \right)\]and \[\left( 2 \right)\], we get
\[I\alpha = - c\theta \]
\[\therefore I\dfrac{{{d^2}\theta }}{{d{t^2}}} + c\theta = 0\]
where \[\alpha = \dfrac{{{d^2}\theta }}{{d{t^2}}}\]

Hence, the differential equation of S.H.M is \[I\dfrac{{{d^2}\theta }}{{d{t^2}}} + c\theta = 0\]where \[\alpha = \dfrac{{{d^2}\theta }}{{d{t^2}}}\]

Note:The acceleration of the particle must be in proportion with the negative displacement governed by the particle. The restoring force in the linear simple harmonic motion is proportional to the negative displacement of the particle.