Infinite number of masses, each $2 k g$ , are placed along the $X -$ axis at $x = \pm 1 m, \pm 2 m, \pm 4 m, \pm 8 m, \pm 16 m . . .$
The magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $(x = 0)$ is

- 2 G

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- 4 G

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- 8 G

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- 16 G

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Solution

The correct option is C $- 8 G$

Gravitational potential at 0,
$V_{0} = \frac{- (G \times 2)}{1} (2) - \frac{(G \times 2)}{2} (2) - \frac{(G \times 2)}{4} \times 2... \infty = - 4 G [\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + . . . . \infty] = - 4 G ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{1 - (\frac{1}{2})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = - 8 G$

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The correct option is C −8G Gravitational potential at 0, V0=−(G×2)1(2)−(G×2)2(2)−(G×2)4×2...∞=−4G[11+12+14+....∞]=−4G⎡⎢ ⎢ ⎢ ⎢⎣11−(12)⎤⎥ ⎥ ⎥ ⎥⎦=−8G

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