Question

Infinite number of bodies, each of mass 2 kg are situated on X-axis at distance 1m, 2 m, 4 m, 8 m respectively from the origin. The resulting gravitational potential due to this system at the origin will be [NEET 2013]

Moderate

Solution

The resulting gravitational potential,
$V = - 2 G [\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots]$
$\Rightarrow V = - 2 G [1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} \dots]$
Using Binomial expansion, $V = - 2 G {(1 + \frac{1}{2})}^{- 1}$
$\Rightarrow V = - \frac{2 G}{(1 - \frac{1}{2})} = \frac{- 2 G}{(\frac{1}{2})} = - 4 G$

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