Q. 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) $G/2$
B) $G$
C) $2G$
D) $4G$
Explanation
(c) Gravitational potential
$\ \ \ \ \ \ \ \ \ \ \ \ \ V = GM\left( \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} + \ldots \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = G \times 1\left( \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots. \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = G\left( \frac{1}{1 - 1/2} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \left( \therefore sum\ of\ GP = \frac{a}{1 - r} \right)$
$= 2G$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.