Question

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

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

Answer 1

The correct option is D $-4G$

Gravitational potential$=-\frac{GM}{x}$
$V=[-\frac{GM}{1}-\frac{GM}{2}-\frac{GM}{4}-\frac{GM}{8}\dots \dots \infty ]$
$=-GM\frac{[1+\frac{1}{2}+\frac{1}{4}+\dots \dots \infty ]}{Geometricprogression}$
In geometric progression
$S_n=\left[\frac{1-r^n}{1-r}\right]$
$=-2G\left[\frac{1}{1-\frac{1}{2}}\right]=-4G$