Question

Infinite number of masses, each $1 \,kg$ are placed along the $x-$ axis at $x=\pm 1 m, \pm 2 m, \pm 4 m, \pm 8 m, \pm 16 \,m, \ldots$ the magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $(x=0)$ is

• A. G/2
• B. G
• C. -2G
• D. 4G

Answer 1

The Correct Option is D

Given each mass $= 1\,kg$
and are placed in the following way
we know that $V = \frac{GM}{r}$
where $m = $mass , $r = $ distance from origin
Now applying formula.
gravitational potential : $2\left(-\frac{G \times 1}{1}-\frac{G \times 1}{2} \cdots\right)$
$=2\left[\frac{G \times 1}{1}+\frac{G \times 1}{2}+\frac{G \times 1}{4}+\cdots\right]$
$=-2 G\left[1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots\right]$
$=-2 G\left[\frac{1}{1-\frac{1}{2}}\right]$
$=-2 G(2)=-4 G$
in $ \infty $ G.P
$ S_{\infty}=\frac{a}{1-r}$