Question

An infinite number of identical point masses each equal to m are placed at point $x=1,x=2,x=4,x=8,...........$
The total gravitational potential at point $x=0$ is

• A. $-Gm$
• B. $-2Gm$
• C. $+2Gm$
• D. $infinite$

Answer 1

The correct option is B $-2Gm$

Let the point masses be $p_1,p_2,p_3......p_a.................(a>0,aisinteger)$
Now potential at $x=0$ due to $p_1$ be
$P_1=\frac{-GM}{1}$
Similarly $P_2=\frac{-GM}{2}$, $P_3=\frac{-GM}{4},........$
So, total potential is given by
$P=P_1+P_2+P_3+.........$
$P=\frac{-GM}{1}+\frac{-GM}{2}+\frac{-GM}{4}+\frac{-GM}{8}+................$
$P=-GM(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+................)$
This is a sum of infinite terms with common ratio $\frac{1}{2}$.
Therefore$,P=(-GM)\frac{1}{(1-\frac{1}{2})}=-2GM$.