Question

Write the differential equation for angular S.H.M.

Hint:

Angular SHM follows the same form as linear SHM; replace linear quantities with angular ones.

Answer 1

Angular simple harmonic motion (SHM) occurs when the restoring torque is proportional to the angular displacement \( \theta \). The torque is \( \tau = -k\theta \), and using Newton’s second law for rotation, \( \tau = I \frac{d^2 \theta}{dt^2} \), where \( I \) is the moment of inertia. Thus: \[ I \frac{d^2 \theta}{dt^2} = -k \theta \quad \Rightarrow \quad \frac{d^2 \theta}{dt^2} = -\frac{k}{I} \theta = -\omega^2 \theta, \] where \( \omega = \sqrt{\frac{k}{I}} \) is the angular frequency.
Answer: \( \frac{d^2 \theta}{dt^2} + \omega^2 \theta = 0 \).