Infinite number of bodies, each of mass $2 \mathrm{~kg}$ are situated on $x$-axis at distance $1 \mathrm{~m}…

Infinite number of bodies, each of mass $2 \mathrm{~kg}$ are situated on $x$-axis at distance $1 \mathrm{~m}, 2 \mathrm{~m}, 4 \mathrm{~m}, 8 \mathrm{~m}$, respectively from the origin. The resulting gravitational potential due to this system at the origin will be
  1. $-G$
  2. $-\frac{8}{3} G$
  3. $-\frac{4}{3} G$
  4. $-4 G$

Solution

The resulting gravitational potential,
$\begin{array}{ll}
& V=-2 G\left[\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots\right] \\
\Rightarrow & V=-2 G\left[1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3} \ldots\right] \\
\Rightarrow & V=-2 G\left(1+\frac{1}{2}\right)^{-1} \\
\Rightarrow & V=-\frac{2 G}{\left(1-\frac{1}{2}\right)}=-\frac{2 G}{\left(\frac{1}{2}\right)}=-4 G
\end{array}$ .

Asked in: NEET 2013 (All India)

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