Question

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.

Short answer

If the ball is discharged, the periodT will increase.

Step-by-step solution

Given information

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

Explanation

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.