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  4. Infinite number of masses, each 1 kg , are placed along the x-axis at x=± text 1 text m,± 2m,± text 4 m,± text 8 text m,± text 16m.... The magnitude of the resultant gravitational potential in terms of gravitational constant G at the origin (x=0) is

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Q. Infinite number of masses, each $ 1 \,kg $ , are placed along the $ x-axis $ at $ x=\pm \text{ }1\text{ }m,\pm 2m,\pm \text{ }4 $ $ m,\pm \text{ }8\text{ }m,\pm \text{ }16m.... $ The magnitude of the resultant gravitational potential in terms of gravitational constant $ G $ at the origin $ (x=0) $ is

KEAMKEAM 2008Gravitation

Solution:

Gravitational potential
$ V=GM\left( \frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}}+\frac{1}{{{r}_{3}}}+...... \right) $
$ =G\times 1\left( \frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+.... \right) $
$ =G\left( \frac{1}{1-1/2} \right) $
$ \left( \because sum\,of\,GP=\frac{a}{1-r} \right) $