Question

Infinite number of bodies, each of mass $2kg$, 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

Answer 1

Gravitational potential at distance $x$, $V=-\frac{Gm}{x}$

$\Rightarrow$ Gravitational Potential at origin, $V=-\frac{2G}{1}-\frac{2G}{2}-\frac{2G}{4}-\frac{2G}{8}----to$ $\infty$

$\Rightarrow V=-2G\{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++to\infty \}$

$\Rightarrow \{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++to\infty \}$ is GP series Where $a=1$ and $r=\frac{1}{2}$

$\Rightarrow$ Sum of GP series, $S=\frac{a}{1-r}$

$\Rightarrow V=-2G\times \frac{1}{1-\frac{1}{2}}=-4G$