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

Correct answer: D. -4 G The resulting gravitational potential, array ll & V=-2 G[(1)/(1)+(1)/(2)+(1)/(4)+(1)/(8)+ ] & V=-2 G[1+(1)/(2)+(1)/(2^2)+(1)/(2^3) ] & V=-2 G(1+(1)/(2))^ -1 & V=-(2 G)/((1-(1)/(2)))=-(2 G)/(((1)/(2)))=-4 G array

NEET 2013

Exam: NEET
Year: 2013
Subject: Physics
Chapter: Gravitation

NEET 2013 · Physics · Gravitation

Infinite number of bodies, each of mass 2 ~kg are situated on x...

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

Options

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

Correct answer

D. $-4 G$

Step-by-step 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}

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