Question

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

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

Answer 1

The correct option is A $-\frac{8}{3}G$

The resulting gravitational potential is

$V=-2G[\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....]=-2G[\frac{1}{1}+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....]=-2G{[1+\frac{1}{2}]}^{-1}=-\frac{2G}{[1+\frac{1}{2}]}=-\frac{2G}{\left[\frac{1}{2}\right]}=-4G$