First slide

Gravitaional potential at a point

Question

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distances 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

Moderate

A

$\frac{- 4}{3} G$
B

$- 4 G$
C

$- G$
D

$\frac{- 8}{3} G$

Solution

The resulting gravitational potential at the origin O due to each of mass 2 kg located at positions as shown in figure is
$V = \frac{G \times 2}{1} - \frac{G \times 2}{2} - \frac{G \times 2}{4} - \frac{G \times 2}{8} - .....$
$= - 2 G [1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ......] = - 2 G [\frac{1}{1 - \frac{1}{2}}]$
$= - 2 G [\frac{2}{1}] = - 4 G$

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