Infinite number of bodies, each of mass $2 k g$ are situated on x-axis at distance $1 m, 2 m, 4 m, 8 m,$ .............. respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

- G

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- \frac{8}{3} G

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- \frac{4}{3} G

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- 4 G

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Solution

The correct option is C $- 4 G$
$V = - 2 G [\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} . . . . . . .]$
$= - 2 G [1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} . . . . . . .]$
$= - 2 G \frac{1}{(1 - \frac{1}{2})} = - 4 G$

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