Infinite bodies, each of mass $3 k g$ are situated at distances $1 m, 2 m, 4 m, 8 m$ respectively on $x -$ axis. The resultant intensity of gravitational field at the origin will be $x G$ . Find $x$ . (Here $G = Gravitational constant .)$

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Solution

Intensity at the origin

$I = I_{1} + I_{2} + I_{3} + I_{4} + \dots$

$= \frac{G M}{r_{1}^{2}} + \frac{G M}{r_{2}^{2}} + \frac{G M}{r_{3}^{2}} + \frac{G M}{r_{4}^{2}} + \dots$

$= G M [\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{4^{2}} + \frac{1}{8^{2}} + \dots]$

$= G M [1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots]$

$= G M ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{1 - \frac{1}{4}} ⎤ ⎥ ⎥ ⎥ ⎦ [As sum of (G . P . = \frac{a}{1 - r}]$

$= G M \times \frac{4}{3}$

$= G \times 3 \times \frac{4}{3} = 4 G$ [ As $M = 3 k g$ given ]

Comparing it with $x G$ we get $x = 4$

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