A simple pendulum consists of a small sphere of mass m suspended by a thread of length l. The sphere carries a positive charge q. The pendulum is placed in a uniform electric field of strength E directed vertically upwards. With what period will the pendulum oscillate if the electrostatic force acting on the sphere is less that the gravitational force? Assume the oscillations to be small

$T=2 \pi\left[\frac{l}{\left(g-\frac{q E}{m}\right)}\right]^{1 / 2}$

Let x be the small displacement given to the pendulum such that the angle θ is small. The forces acting at A are

(i) tension T along the thread

(ii) weight mg acting vertically downwards

(iii) electrical force qE vertically upwards

The resultant force vertically down wards is (mg – qE). Therefore

Net acceleration g' = $g - \frac{qE}{m}$

Time period T = $2 \pi \sqrt{\frac{I}{g'}}=2 \pi\left[\frac{l}{g-\frac{q E}{m}}\right]^{1 / 2}$