Torsional pendulum

TORSIONAL PENDULUM

When a body is fixed at one end and twisted about its axis by means of a torque at the other end, then the body is said to be under torsion. The torsion involves shearing strain and hence the modulus involved is the RIGIDITY MODULUS.

Torsion Pendulum

Principle: When a disc (torsion pendulum) is rotated in a horizontal plane, the disc executes simple harmonic oscillation due to the restoring couple produced in the wire.

Description: A torsion pendulum consists of a wire with one end fixed to a split chuck and the other end fixed to the centre of the circular disc of radius R as shown in Fig. 4.8.

Let ‘L’ be the distance between the chuck end to the disc and 'r' be the radius of the suspended wire.

Working: The circular disc is rotated in horizontal plane so that the wire is twisted through an angle 'θ'. The various elements of the wire will undergo shearing strain and a restoring couple is produced. Now if the disc is released, the disc will produce torsion oscillations.

The couple acting on the disc produces an angular acceleration in it, which is proportional to the angular displacement and is always directed towards its mean position.

Therefore from the law of conservation of energy the total energy of the system is conserved.

Total energy of the torsion pendulum = Potential Energy (P.E) + Kinetic Energy (K.E)

         …………….(1)

The potential energy confined to the wire is equal to the work done in twisting the disc, thereby creating a restoring couple (C).

Total energy. = constant.

r = wire redbus

R = circular disc grediaus

∴ Restoring couple (P.E) through an angle θ = ₀∫^θMoment of couple × dθ

P.E. = ₀∫^θ Cθ.dθ

P.E. = Cθ² / 2            ………..(2)

Let 'ω' the angular velocity with which the disc oscillates, due to the resorting couple, then

The kinetic energy confined to the rotating disc (Deflecting couple) = ½ Iω²

(i.e) K.E. = ½ Iω²                  ……………………..(3)

Here I is the moment of inertia of the circular disc

Total Energy T = Cθ²/2  + Iω²/2 = Constant                ...........(4)

Differentiating equation (4) with respect to time 't' we get,

 Cθ.dθ/dt + Iω.dω/dt = 0

Since the angular velocity ω=dθ/dt and the Angular Acceleration dω/dt = d²θ/dt²

Angular acceleration = d²θ/dt² = Cθ/I          ………..(5)

The negative sign indicates that the couple tends to decrease the twist on the wire.

Period of Oscillation

We know, the time period of oscillation T = 2π√[ Displacement / Acceleration ]

Substituting from Eqn.(5), we have T = 2π√[ θ / Cθ/I ]

(or) Time period of torsion oscillation T = 2π√[I/C]           ……….(6)

 (f=1/T)

∴ Frequency of oscillation f = 1/2π . √[C/I]

Rigidity modulus of the wire

If 'r' is the radius of the wire and ‘L’ is the length of the wire suspended, then we know

The torque per unit twist C = nπr⁴ / 2L                 …………..(7)

Substituting eqn.(7) in eqn.(6) we get,

(or)

Rigidity modulus of the wire (n) =       8πIL / T²r⁴        Nm^‒2

Thus torsion pendulum is used to find the rigidity modulus for various materials.