Question

Infinite number of masses, each 1 kg, are placed along the $x$ -axis at $x=\pm1m, \, \pm2m, \, \pm4m, \, \pm8m, \, \pm16m$ ..... The magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $\left(\right.x=0\left.\right)$ is

• A. $G$
• B. $3G$
• C. $4G$
• D. $8G$

Answer 1

Answer: (C)

Gravitational potential at a distance r from a mass = $V=\frac{G M}{r}$
Net Gravitational potential = $\displaystyle \sum \frac{G M}{r_{i}}$
$ \, V=GM\left(\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} + \ldots \right)\times 2$
$ \, \, =G\times 1\left(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots .\right)\times 2$
$ \, \, \left(\therefore s u m \, o f \, G P = \frac{a}{1 - r} = \frac{1}{1 - 0 . 5} = 2\right)$
$V=4G$