Question

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

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

Answer 1

Answer: (C)

Gravitational potential at a distance r from a mass = $V=\frac{GM}{r}$
Net Gravitational potential = $\sum \frac{GM}{r_i}$
$V=GM\left(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\dots \right)\times 2$
$=G\times 1\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots .\right)\times 2$
$\left(\therefore sumofGP=\frac{a}{1-r}=\frac{1}{1-0.5}=2\right)$
$V=4G$