Question

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

• A. $G/2$
• B. $G$
• C. $2G$
• D. $4G$

Answer 1

Answer: (C)

$2G$

Answer 2

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