Bifilar Suspension Derivation

Bifilar Suspension:

The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension. The body whose moment of inertia is to be determined. It is suspended by two long parallel flexible strings as shown in fig: When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G. It will vibrate with simple harmonic motion in a horizontal plane.

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 body about a vertical axis through G = m.k2G
l =length of each string
x = Distance of A from G
y = Distance of B from G
θ = Small angular displacement of the body from the equilibrium position in the horizontal plane.
ΦA and ΦB = Corresponding angular displacements of the strings
α = Angular acceleration towards the equilibrium position.