Question

A small ball of mass $m$ and charge $+q$ tied with a string of length $l$, is rotating in a vertical circle under the influence of gravity and a uniform horizontal electric field $E$, as shown. The tension in the string will be minimum for

• A. $\theta ={\tan }^{-1}\left(\frac{qE}{mg}\right)$
• B. $\theta =\pi$
• C. $\theta =0^{\circ }$
• D. $\theta =\pi +{\tan }^{-1}\left(\frac{qE}{mg}\right)$

Answer 1

The correct option is D $\theta =\pi +{\tan }^{-1}\left(\frac{qE}{mg}\right)$

We can draw the free body diagram for the ball as shown below.


Now, from the free body diagram, we obtain

$\begin{array}{c}T\sin \alpha =qE\\ T\cos \alpha =mg\end{array}]\to \text{Equilibrium conditions}$

On, dividing these two conditions, we get,
$\alpha ={\tan }^{-1}\left(\frac{qE}{mg}\right)$

This angle provide the condition for equilibrium. So directly oppsoite to this point, tension will be minimum for circular motion.

We can observe from the first figure, minimum tension will be obtained at angle $\alpha +\pi$.

So, for minimum tension, $\theta =\pi +{\tan }^{-1}\left(\frac{qE}{mg}\right)$

Thus, option $\left(d\right)$ is the correct answer.