The frequency of oscillation of a torsional pendulum is

CONCEPT

• Torsional Pendulum: A torsional pendulum consists of a disk (or some other object) suspended from a wire, which is then twisted and released, resulting in an oscillatory motion. The oscillatory motion is caused by a restoring torque which is proportional to the angular displacement.

EXPLANATION

• If the torsional constant of the wire is k and the body is rotated through an angle θ, then the torque produced is \(\tau = - k\;\theta \)
• If I be the moment of inertia of the body about the verticle axis then the angular acceleration is \(\alpha = \frac{\tau }{I} = - \frac{k}{I}\;\theta =-\omega ^2 \theta \)
• where, \(\omega = \sqrt {\frac{k}{I}} \)

\( ⇒ Time\;period\;\left( T \right) = \frac{{2\pi }}{\omega } = 2\pi \;\sqrt {\frac{I}{k}} \)

\(⇒ Frequency\;\left( {\frac{1}{T}} \right) = \frac{\omega }{{2\pi }} = \frac{1}{{2\pi }}\sqrt {\frac{k}{I}} \)

⇒ Option 4 is correct