Two identical charged spheres suspended from a common point

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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:(a) (b) v ∝ x(c) (d) v ∝ x⁻¹

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Two identical charged spheres are suspended from a common point by two massless strings of length l, are initially at a distance d (d << l ) apart because of their mutual repulsion. at an instant of time, distance between them is x as shown on figure.

in figure, Tsin\theta=\frac{kq^2}{x^2}

and Tcos\theta=mg

then, tan\theta=\frac{kq^2}{x^2mg}

from ∆ABC,

tan\theta=\frac{x/2}{\sqrt{l^2-\frac{x^2}{4}}}

so,\frac{x/2}{\sqrt{l^2-\frac{x^2}{4}}}=\frac{kq^2}{x^2mg}

as l >> x => l² >> x²/4

so, l² - x²/4 ≈ l²

then, \frac{x/2}{\sqrt{l^2}}=\frac{kq^2}{x^2mg}

or, \frac{x}{2l}=\frac{kq^2}{x^2mg}

or, x^3=\frac{2kq^2l}{mg}

here it is clear that, q^2\propto x^3

so, q\propto x^{3/2}

differentiating with respect to time,

\frac{dq}{dt}\propto\frac{3}{2}x^{1/2}\frac{dx}{dt}

or, x^{-1/2}\frac{dq}{dt}\propto v

hence, it is clear that, v\propto x^{-1/2}


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