Two torsional pendulums with wires of the same material are oscillated. Find the ratio of the period of oscillation if the radius and length are in the ratio 1:2.


Torsional Pendulum

A torsional pendulum consists of a wire suspended from a rigid support. A rod or a light disc is attached to the lower end of the wire. If the rod is twisted and released, it oscillates about its equilibrium position. The motion is simple harmonic if the angle through which the wire is twisted is small.

Period of Oscillation of Torsional Pendulum

The period of oscillation of a torsional pendulum is given by the formula:

T = 2π√(I/C)
where T is the period of oscillation, I is the moment of inertia of the suspended body and C is the torsional constant of the wire.

Ratio of Period of Oscillation

Let the radius of the first torsional pendulum be r and its length be l. The moment of inertia of the suspended body is given by:

I₁ = (1/2)mr²
where m is the mass of the suspended body.

Let the radius of the second torsional pendulum be 2r and its length be 2l. The moment of inertia of the suspended body is given by:

I₂ = (1/2)m(2r)² = 2mr²

The torsional constant of the wire is the same for both the pendulums as they are made of the same material. Thus, the ratio of the period of oscillation of the two torsional pendulums is given by:

T₁/T₂ = √(I₁/I₂) = √(1/2)/√2 = 1/√4 = 1/2

Hence, the ratio of the period of oscillation of the two torsional pendulums is 1:2.