Torsional pendulum: Definition, Derivation, Formula [with Pdf]

What is torsional pendulum?

Torsional pendulum

The torsional pendulum is the disc suspended to the thin bar which creates twisting oscillations around the axis of the bar. The restoring force developed by twisting or torsional action creates the oscillations in the disc.

If the initial angular displacement θ is given to the disc by applying twisting torque, The thin rod generates the restoring torque, which causes the disc to revolve in the opposite direction.

The continuous twisting and releasing of the string/rod create oscillation in the torsional pendulum.

This mechanism creates simple harmonic motion in the torsional pendulum.

Torsional pendulum formula:

The required equations for the analysis of the torsional pendulum are listed below,

1) Restoring torque (T):

T = – Cθ

Here θ is angular twist and C is torque per unit twist of the pendulum that is given by,

`C=\frac{\pi \times \eta \times r^{4}}{2l}`

Where,
η = Modulus of rigidity
r = Wire radius
l = Length of wire

2) Torsional pendulum period equation:

The equation for the period of the original pendulum is given by,

`T=2\pi \sqrt{\frac{I}{C}}`

3) Equation of modulus of rigidity for torsional pendulum:

The equation for the torsional rigidity of the torsional pendulum is given by,

`\eta =\frac{8\pi I}{r^{4}}(\frac{L}{T^{2}})`

Torsional pendulum period derivation:

The restoring torque is directly proportional to the angle of twist in the wire that is given by,

T= – Cθ —–(1)

Where C = Torsion constant

In angular motion, the equation for torque is,

`T = I\times \alpha`

As, `\alpha =\frac{d^{2}\theta }{dt^{2}}`

Therefore, `T=I\frac{d^{2}\theta }{dt^{2}}` —–(2)

Where, I = Moment of inertia of the disc

Equating the equations 1 and 2 we get,

`I\frac{d^{2}\theta }{dt^{2}}=-C\theta `

`\frac{d^{2}\theta }{dt^{2}}+(\frac{C}{I})\theta =0` —–(3)

The equation of angular simple harmonic motion is given by,

`\frac{d^{2}\theta }{dt^{2}}+\omega^{2} \theta =0` —–(4)

Now by comparing equations 3 and 4 we get,

`\omega ^{2}=\frac{C}{I}`

`\therefore \omega =\sqrt{\frac{C}{I}}` —–(5)

This is the equation for the angular speed of torsional pendulum.

Now the frequency of oscillation is given by,

`\omega =2\pi f`

`f=\frac{\omega }{2\pi }`

`f=\frac{1 }{2\pi }\sqrt{\frac{C}{I}}` —–(6)

This is the equation for frequency of oscillation of the torsional pendulum.

Now the period of torsional pendulum is given by,

`t=\frac{1}{f}`=`\frac{1}{\frac{1}{2\pi }\sqrt{\frac{C}{I}}}`

Period of torsional pendulum = `t=2\pi \sqrt{\frac{I}{C}}` —–(7)

This is the required equation for the period of oscillation of the torsional pendulum.

Derivation for the torsional rigidity of torsional pendulum:

From equation 7, the period of oscillation is,

`t=2\pi \sqrt{\frac{I}{C}}`

Put value of C =`\frac{\pi \eta r^{4}}{2l}`

∴ `t=2\pi \sqrt{\frac{I2l}{\pi \eta r^{4}}}`

`\therefore \eta =\frac{8\pi I}{r^{4}}(\frac{l}{t^{2}})`

This is the equation to determine torsional rigidity of pendulum.

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