Question

A small ball of mass $m=21g$ suspended by an insulating thread at a height $h=12cm$ from a large horizontal conducting plane performs small oscillation. After a charge $q$ had been imparted to the ball, the oscillation period charged $n=2.0$ times. Find $q$.

Answer 1

In the absence of the change, the oscillation period of the ball
$T=2\pi \sqrt{I/g}$
When we impart the charge $q$ to the ball, it will be influenced by the induced charge on the conducting plane. From The electric image method the electric force on the ball by the plane equals $\frac{q^2}{4\pi {\epsilon }_0(2h{)}^2}$ and is directed downwards. Thus in this case the effective acceleration of the ball
$g^{{}^{\prime }}=g+\frac{q^2}{16\pi {\epsilon }_{0}m{h}^2}$
and the corresponding time period
$T^{{}^{\prime }}=2\pi \sqrt{\frac{1}{g^{{}^{\prime }}}}=2\pi \frac{\sqrt{1}1}{g+\frac{q^2}{16\pi {\epsilon }_{0}m{h}^2}}$
For the condition of the problem
$T=\eta T^{{}^{\prime }}$
So, $T^2={\eta }^2{T}^{{}^{\prime }2}$ or $\frac{1}{g}={\eta }^2\left(\frac{1}{g+\frac{q^2}{16\pi {\epsilon }_{0}m{h}^2}}\right)$
Thus on solving
$q=4h\sqrt{\pi {\epsilon }_{0}mg({\eta }^2-1)}=2\mu C$