Torsion Pendulum

TORSION PENDULUM

A circular metallic disc suspended using a thin wire that executes torsional oscillation is called torsional pendulum.

• It executes torsional oscillations, whereas a simple pendulum executes linear oscillations.

Description 

A torsional pendulum consists of a metal wire suspended vertically with the upper end fixed. The lower end of the wire is connected to the centre of a heavy circular disc (Fig. 1.45). 

Expression for the period of Oscillation of a Torsion Pendulum

When the disc is rotated by applying a twist, the wire is twisted through an angle θ. Then, the restoring couple set up in the wire

= Cθ             ... (1)

where C - couple per unit twist.

If the disc is released, it oscillates with angular velocity dθ/dt in the horizontal plane about the axis of the wire. These oscillations are known as torsional oscillations.

If d2θ/dt2 is the angular acceleration produced in the disc and I its moment of inertia of the disc about the axis of the wire then,

Applied couple = I d2θ/dt2            ... (2)

At equilibrium, applied couple = restoring couple

I d2θ/dt2 =  -Cθ            

Here - ve sign signifies that the restoring couple is opposite to applied couple (torque)

d2θ/dt2  =  -C/I θ                      ... (3)

This equation represents simple harmonic motion which shows that angular acceleration (d2θ/dt2) is proportional to angular displacement θ and is always directed towards the mean position.

Hence, the motion of the disc being simple harmonic motion, the time period of the oscillation is given by

Uses of Torsional Pendulum

Torsional pendulum is used to determine

1. Rigidity modulus of the wire,

2. Moment of inertia of the disc

3. Moment of inertia of an irregular body.

Determination of Rigidity Modulus of the Wire

The rigidity modulus of the wire is determined by the following equation

Experiment

A circular disc is suspended by a thin wire, whose rigidity modulus is to be determined. The top end of the wire is fixed firmly in a vertical support.

The disc is then rotated about its centre through a small angle and set free. It executes torsional oscillations. 

The time taken for 20 complete oscillations is noted. The experiment is repeated and the mean time period (T) of oscillation is determined.

The length l of the wire is measured. This length is then changed by about 10 cm and the experiment is repeated. The readings for five or six different lengths of wire are measured.

The disc is removed and its mass and diameter are measured.

The time period of oscillation is

Squaring on both sides, we have

Substituting couple per unit twist in eqn (4),

rearranging the equation (5), 

The rigidity modulus of the material of the wire

I - moment of inertia of circular disc = MR2/2

where M - Mass of the circular disc

R- Radius of the disc