# Torsion pendulum – Time Period, Rigidity Modulus and Applications

A torsion pendulum consists of an inextensible string clamped at one end and fixed to a rigid body at the other end. This simple apparatus consisting of a clamped wire coupled to a rigid body constitutes a torsion pendulum.

When the disc or body is given a small angular displacement, the rod or wire will twist. Subsequently, a torque is developed in the wire. This causes the body to oscillate until its equilibrium is reached. The torque on the rod responsible for imparting oscillatory motion to the body is called Restoring torque.

In the case of a torsion pendulum, there is a periodic reversal in the direction of the disc owing to torsion. However, the centre of gravity of the body remains constant. The type of oscillation by a torsion pendulum is also called torsion oscillation.

### Time period of oscillation of a Torsion Pendulum

Consider a torsion pendulum, consisting of a body suspended by a clamped wire. Suppose the disc is rotated by an angle θ, then a restoring force τr is developed in the wire or rod.

The objective of the restoring force is to resist the angular displacement and bring the object back to equilibrium. Hence, the relation between θ and τris given by,

τr = α (-θ) ——————————————————–(1)

Negative sign is given because the restoring torque is in a direction opposite to that of displacement.
τr = (-c x θ) —————————————————-(2)

c: Coefficient of stiffness of the wire.

Also, τr =I α —————————————————-(3)

I = Moment of Inertia

α = Angular acceleration

The motion of the body can be considered to be that of an SHM,

Hence, α = -ω2θ

I x -ω2θ = (-c x θ)

This gives the time period of oscillation of a torsion p endulum as;

### Calculating Rigidity Modulus:

Before we actually jump into the calculation, let us analyze the definition of rigidity modulus.

Rigidity modulus is the ratio of shear stress to shear strain. In other words, it is the measure of the rigidity of a body. Rigidity modulus is a material property.

It is possible to calculate the rigidity modulus of the material of the wire using the torsion pendulum experiment.

Suppose we have a small wire, clamped at one end and the other end connected to a heavy disc. The pendulum is made to oscillate by giving an initial displacement to the disc.

Period of a torsional oscillation is given by,

If the mass of the oscillating body is M and radius R, then the Moment of Inertia of the disc about an axis passing through the centre and perpendicular to its plane is given by the equation;

I = (MR2)/2 —————————————————-(5)

Substituting equations (5) and (6) in equation (4), we get

Squaring and rearranging, we get

Modulus of rigidity, n= (4πMR2/r4) (L/T2)

M: Mass of the oscillating body or disc

r: Radius of the wire whose rigidity modulus is to be measured.

L: Length of the wire

T: Time period of oscillation

Experimental procedure to determine the Rigidity Modulus using the torsion pendulum experiment.

1. Measure the radius of the wire using a screw gauge and let it be denoted as ‘r’.
2. The experimental setup is such that the effective length of the hanging wire can be varied. The experiment is conducted by varying length (L).
3. Vary the length of the wire to values like 0.3m, 0.4m, 0.5m till sufficient readings are acquired.
4. The disc is made to oscillate. Find the time period of oscillation by measuring the time for 20 oscillations. Do this at least thrice and obtain the mean value.
5. Calculate the Moment of Inertia using the equation, I = (MR2)/2
6. Find rigidity modulus using the equation given below.

### Applications of Torsion Pendulum

1) To find the Moment of Inertia of irregular bodies

Let us define Moment of Inertia first!
Moment of Inertia is the resistance of a body towards rotation motion. The moment of Inertia of a body depends upon various features like mass, the axis of rotation, and mass distribution about the axis.

It is easy to find the moment of inertia of regular bodies having a specific geometry. But calculating the Moment of Inertia of irregular bodies can be often troublesome at times. In those cases, we employ a torsion pendulum to determine the Moment of Inertia.

2) To find the rigidity modulus of the material of the wire.

3) Apart from its experimental applications, they also find applications as time-keeping devices like a mechanical wrist watch.

### Observations and Calculations

Rigidity Modulus n =  π l I)/(r4 T2)