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

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

NEETNEET 2013Gravitation

Solution:

Gravitational potential at distance $x, V=-\frac{ Gm }{ x }$
$\Rightarrow$ Gravitational Potential at origin, $V=-\frac{2 G }{1}-\frac{2 G }{2}-\frac{2 G }{4}-\frac{2 G }{8}----$ to $\infty$ $\Rightarrow V =-2 G \left\{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++ to \infty \right\}$
$\Rightarrow\left\{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++\right.$ to
$\left.\infty\right\}$ is GP series Where $a =1$ and $r =\frac{1}{2}$
$\Rightarrow$ Sum of GP series, $S=\frac{a}{1-r}$
$\Rightarrow V =-2 G \times \frac{1}{1-\frac{1}{2}}=-4 G$