Q104 / 180 · NEET 2013 (All India) · Physics

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

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

Solution

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

Hence, the correct answer is option: 'D'.

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