Question

A small sphere carrying a charge $Q$ is hanging in between two parallel plates by a string of length $L$. Time period of pendulum is $T_0$. When parallel plates are charged, the time period changes to $T$. The ratio $T/T_0$ is equal to

• A. ${\left(\frac{g+\frac{QE}{m}}{g}\right)}^{1/2}$
• B. ${\left(\frac{g}{g+\frac{QE}{m}}\right)}^{3/2}$
• C. ${\left(\frac{g}{g+\frac{QE}{m}}\right)}^{1/2}$
• D. none of these

Answer 1

Answer: (C)

Net downward force, $m{g}^{\prime }=mg+QE$
$\Rightarrow$ Net acceleration, $g^{\prime }=\left(g+\frac{QE}{m}\right)$
Hence, time period $T=2\pi \sqrt{\frac{l}{g^{\prime }}}=2\pi \sqrt{\frac{l}{\left(g+\frac{QE}{m}\right)}}$
Required ratio $=\frac{T}{T_0}={\left[\frac{g}{g+\frac{QE}{m}}\right]}^{1/2}$