1. Tardigrade
  2. Question
  3. Physics
  4. Two identical charged spheres suspended from a common point by two massless strings of lengths l, are initially at a distance d(d < < l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as:

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

BITSATBITSAT 2016

Solution:

$\Rightarrow \tan \theta=\frac{F}{m g} \approx \theta$
$\Rightarrow \frac{ K q ^{2}}{ x ^{2} mg }=\frac{ x }{21} $
$\Rightarrow x ^{3} \propto q ^{2} $
$\Rightarrow x ^{3 / 2} \propto q$
Differentiating wrt time, we get
$\Rightarrow 3 x^{2} \frac{d x}{d t} \propto 2 q \frac{d q}{d t} $
where, $\frac{d q}{d t}$ is constant
$\Rightarrow x ^{2}( v ) \propto q$
On substituting q, we get
$\Rightarrow x ^{2}( v ) \propto x ^{3 / 2}$
$\Rightarrow v \propto x ^{-1 / 2}$

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