Question

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

• A. $-4G$
• B. $-G$
• C. $-\frac{8}{3}G$
• D. $-\frac{4}{3}G$

Answer 1

The correct option is A $-4G$

The gravitational potential at a point due to a system of point masses is,
$V=-G(\frac{m_1}{r_1}+\frac{m_2}{r_2}+\frac{m_3}{r_3}+.....)$

For the given system, potential at the origin will be,

$V=-Gm[\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....]$

$V=-G\left(2\right)[\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....]$

$=-2G\left[\frac{1}{1-\left(\frac{1}{2}\right)}\right]=-4G$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(C\right)$ is the correct answer.