An infinite number of bodies, each of mass $2 ext{ kg}$ are situated on the $x$-axis at distances $1 ext{

Question

An infinite number of bodies, each of mass $2 	ext{ kg}$ are situated on the $x$-axis at distances $1 	ext{ m}, 2 	ext{ m}, 4 	ext{ m}, 8 	ext{ m}, ......$ respectively, from the origin. The resulting gravitational potential due to this system at the origin will be:

Accepted Answer

1. **Principle of Superposition:** Gravitational potential is a scalar quantity. The total potential at the origin is the algebraic sum of the potentials due to each individual mass. The potential due to a mass $m$ at distance $r$ is $V = -\frac{Gm}{r}$ [Section 7.7, Eq. 7.24].
2. **Setup Series:**
   $$ V_{total} = -\frac{Gm_1}{r_1} - \frac{Gm_2}{r_2} - \frac{Gm_3}{r_3} - \dots $$
   Given $m = 2 	ext{ kg}$ for all bodies, and distances are $1, 2, 4, 8, \dots$
   $$ V_{total} = -G(2) \left[ \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots ight] $$
3. **Sum of Geometric Progression:** The term in the brackets is an infinite geometric series with first term $a=1$ and common ratio $r = \frac{1}{2}$.
   $$ S = \frac{a}{1-r} = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2 $$
4. **Final Calculation:**
   $$ V_{total} = -2G 	imes (2) = -4G $$

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