Infinite number of bodies, each of mass $2 k g$ , are situated on $x - a x i s$ 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

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Answer 1

Gravitational potential at distance $x$ , $V = - \frac{G m}{x}$
$\Rightarrow$ Gravitational Potential at origin, $V = - \frac{2 G}{1} - \frac{2 G}{2} - \frac{2 G}{4} - \frac{2 G}{8} - - - - t o$ $\infty$
$\Rightarrow V = - 2 G {\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + + + + t o \infty}$
$\Rightarrow {\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + + + + t o \infty}$ is GP series Where $a = 1$ and $r = \frac{1}{2}$
$\Rightarrow$ Sum of GP series, $S = \frac{a}{1 - r}$
$\Rightarrow V = - 2 G \times \frac{1}{1 - \frac{1}{2}} = - 4 G$

Answer 2

Gravitational potential at distance

x

,

V = - \frac{G m}{x}

\Rightarrow

Gravitational Potential at origin,

V = - \frac{2 G}{1} - \frac{2 G}{2} - \frac{2 G}{4} - \frac{2 G}{8} - - - - t o

\infty

\Rightarrow V = - 2 G {\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + + + + t o \infty}

\Rightarrow {\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + + + + t o \infty}

is GP series Where

a = 1

and

r = \frac{1}{2}

\Rightarrow

Sum of GP series,

S = \frac{a}{1 - r}

\Rightarrow V = - 2 G \times \frac{1}{1 - \frac{1}{2}} = - 4 G