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
Answer & explanation
Correct answer: option 3
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 = $2 \pi \sqrt{\frac{I}{g'}}=2 \pi\left[\frac{l}{g-\frac{q E}{m}}\right]^{1 / 2}$