A simple pendulum consists of a small sphere of mass m suspended by a thread of length l. The sphere carries a charge q. The pendulum Is placed in a uniform electric fleld of strength E directed vertically upwards.With what perlod will the pendulum oscillate if the electrostatic force acting on the sphere is less than the gravitational force?

Explanation

Let the pendulum be at an angle of `theta` with the vertical. Net force acting on the bob =mg-qE
Torque experienced by the bob about the point of suspension,
`tau=-(mg-qE)lsintheta`
(where, the negative sign indicates that the torque experienced is restoring torque) As the amplitude of oscillation is small,

`sintheta~~theta`
Then, `tau=-(mg-qE)ltheta`
again, `tau=` moment of inertia (l) `xx` angular acceleration `(alpha)`
`thereforealpha=(tau)/I=(-(mg-qE)ltheta)/I=(-(mg-qE)ltheta)/(ml^(2))[becauseI=ml^(2)]`
`=((-(g-(qE)/m)theta)/l`
`thereforealphaprop-theta[because((g-(qE)/m))/l=`constant`]`
`therefore` The pendulum executes an SHM.
`therefore` Time period of the pendulum,
`T=2pisqrt(("angular displacement")/("angular acceleration"))=2pisqrt((theta)/(alpha))`
`thereforeT=2pisqrt(l/(g-(qE)/m))`