Question

Particles each of mass $M$ are placed along $x$-axis at $x=1$m, $x=2$m, $x=4$m, $x=8$m..... etc., to infinity. Gravitational potential at the origin due to this system of particles is :

• A. $2GM$
• B. $\frac{2GM}{3}$
• C. $\frac{4GM}{3}$
• D. $\frac{5GM}{4}$

Answer 1

The correct option is A $2GM$

Gravitation potential is given as

$E_G=\frac{GM}{r}$

For all the masses, it will be

$E_G=GM[\frac{1}{r_1}+\frac{1}{r_2}+....]$

$E_G=GM[\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}....]$

The right hand side is an infinite geometric series with a common ratio, $r=\frac{1}{2}$.

So, the sum is given by

$E_G=GM[\frac{1}{1}-\frac{1}{2}]$$=GM\left[\frac{1}{1/2}\right]$$=2GM$