The ball in FIGURE $Q23.10$ is suspended from a large, uniformly charged positive plate. It swings with period T. If the ball is discharged, will the period increase, decrease, or stay the same? Explain.

Given a positively charged ball suspended from a uniformly charged positive plate.

Since the positively charged body swings with the time period $T$, we can consider the ball as a pendulum

The time period of a simple pendulum is$T=2\pi \sqrt{\frac{L}{g_{}}}$

$$\begin{gathered} \text{Where}{T}_{}=\text{Newtimeperiod};l=\text{lengthofthependulum}; \\ {g}^{}=\text{accelerationduetogravity} \end{gathered}$$

The restoring force on the charged ball is $\overrightarrow{F}=(mg+qE)\cos \left(45°\right)$. When the ball is discharged the restoring force will decrease. The new restoring force is

$$\begin{gathered} \overrightarrow{F}=(mg+qE)\cos 45° \\ m{g_1}^{}=(mg+qE)\cos 45° \\ {\text{whereg}}_1\text{isthenewaccelerationduetogravity} \\ {g_1}^{}=\left(g+\frac{qE}{m}\right)\cos 45°\cdots \cdots \left(1\right) \end{gathered}$$

From equation $\left(1\right)$we can say that $g_1$is decreases if the ball is discharged. Since the time period is directly proportional to $g$, therefore the time period $T$will increases if the ball is discharged.