Question

Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d(d << l)$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity $v$. Then as a function of distance $x$ between them

• A. $v \propto x$
• B. $v \propto x^{-\frac{1}{2}}$
• C. $v \propto x^{-1}$
• D. $v \propto x^{\frac{1}{2}}$

Answer 1

Answer: (B)

$\tan \theta=\frac{F}{m g}$
or $ \frac{x}{2 l}=\frac{k q^{2}}{m g x^{2}}$
$\frac{x^{3}}{2 l}=\frac{k q^{2}}{m g}$
$\frac{3x^2\frac{dx}{dt}}{2l} = \frac{2kq \frac{dq}{dt}}{mg}$
Also, $q \propto x^{3 / 2}$
$\Rightarrow \frac{d x}{d t} \propto \frac{x^{3 / 2}}{x^{2}},$ i.e., $v \propto x^{-1 / 2}$