Question

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

• A. $-4G$
• B. $-G$
• C. $-\frac{8}{3}G$
• D. $-\frac{4}{3}G$

Answer 1

Answer: (A)

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 $\infty$ $\Rightarrow V=-2G\left\{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++to\infty \right\}$
$\Rightarrow \{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}++++$ to
$\infty \}$ is GP series Where $a=1$ and $r=\frac{1}{2}$
$\Rightarrow$ Sum of GP series, $S=\frac{a}{1-r}$
$\Rightarrow V=-2G\times \frac{1}{1-\frac{1}{2}}=-4G$