Question

A simple pendulum having charge +Q, length l, and mass m is hung from a fixed support between the horizontal plates of a charged capacitor. What is the time period of vibration ?

Answer 1

The formula governing the motion of a regular pendulum in a gravitational field is
\[ \tau = I \alpha \]
\[ -mgL \sin(\theta) = mL^2 \ddot{\theta} \]
using the small angle approximation,
\[mL^2 \ddot{\theta} + mgL\theta = 0\]
or
\[\ddot{\theta} + \frac{g}{L} \theta = 0\]
which is the equation of motion for a pendulum with angular frequency
\[ \omega = \sqrt{g/L} \]
and time period
\[T = 2\pi \sqrt{\frac{L}{g}} \]
In this case, everything is the same except for the torque. The torque is no longer mgL sin(theta).