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

Answer 1

Answer: (C)

Net downward force, $m g^{\prime}=m g+Q E$
$\Rightarrow $ Net acceleration, $g^{\prime}=\left(g+\frac{Q E}{m}\right)$
Hence, time period $T=2 \pi \sqrt{\frac{l}{g^{\prime}}}=2 \pi \sqrt{\frac{l}{\left(g+\frac{Q E}{m}\right)}}$
Required ratio $=\frac{T}{T_{0}}=\left[\frac{g}{g+\frac{Q E}{m}}\right]^{1 / 2}$