Question

What is a torsional pendulum ? Derive the
expression for the time period of a torsional
pendulum. State one use of such a
pendulum.

Answer 1

A torsional pendulum consists of a disk (or some other rigid body) suspended by a light wire or spring, which is then twisted and released, resulting in an oscillatory motion. 

The oscillatory motion is caused by a restoring torque τ which is proportional to the angular displacement θ:

$$\tau =-k\theta$$

where k>0  is the torque constant of the wire or spring. The minus sign shows that the restoring torque acts in the opposite direction to increasing angular displacement. This equation is a torsional equivalent to Hooke's law. The net torque is equal to the moment of inertia of the disk I times the angular acceleration:

$$I\frac{{{d}^{2}}\theta }{d{{t}^{2}}}=\tau$$

Combining the previous two equations, we obtain

$$I\frac{{{d}^{2}}\theta }{d{{t}^{2}}}=-k\theta$$

or

$$\frac{{{d}^{2}}\theta }{d{{t}^{2}}}+\frac{k}{I}\theta =0$$

This equation is a simple harmonic equation. Therefore, we can immediately write a standard solution.

$$\theta ={{\theta }_{0}}\cos \left( \omega t+\varphi \right)$$

where ${{\theta }_{0}}$ is the maximum angle, $\varphi$ is the phase, $\omega =\sqrt{\frac{k}{I}}$ is the angular velocity. The period of oscillation of the pendulum is

$$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{I}{k}}$$

Torsion pendulum is used in mechanical wristwatch. The balance wheel in a mechanical wristwatch is a torsion pendulum in which the restoring torque is provided by a coiled spring.