Torsional Pendulum Derivation

Torsional Pendulum:

It is also used to find the moment of inertia of a body experimentally. The body say a disc or flywheel whose moment of inertia is to be determined. It is suspended by three long flexible wires A, B, and C as shown in fig. When the body is twisted about its axis through a small angle θ and then released, it will oscillate with simple harmonic motion.

Let m = Mass of the body
W = Weight of the body in newtons = m.g
kG = Radius gyration about an axis through the centre of gravity.
I = Mass moment of inertia of the disc about an axis through O and perpendicular to it = m.k2
l =length of each wire
r = Distance of each wire from the axis of the disc
θ = Small angular displacement of the disc
Φ = Corresponding angular displacement of the wires
α = Angular acceleration towards the equilibrium position.