Question

A large number of identical point masses m are placed along x-axis, at $x=0,2,3,4,.........$The magnitude of gravitational force on a mass $m$ at origin $(x=0),$ will be
$x=0x=1x=2x=4$ and so on

• A. $G{m}^2$
• B. $\frac{4}{3}G{m}^2$
• C. $\frac{2}{3}G{m}^2$
• D. $\frac{5}{4}G{m}^2$

Answer 1

The correct option is B $\frac{4}{3}G{m}^2$

The gravitational force between any point is given by

$F=\frac{G{m}_1{m}_2}{r^2}$

The gravitational force on mass $m$ placed at $x=0$,

$F=\frac{G{m}^2}{(1{)}^2}+\frac{G{m}^2}{(2{)}^2}+\frac{G{m}^2}{(3{)}^2}+\frac{G{m}^2}{(4{)}^2}+...........$

$F=G{m}^2(\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......)$

$F=G{m}^2\left(\frac{1}{1-\frac{1}{2}}\right)$

$F=G{m}^2\frac{4}{3}$

$F=\frac{4G{m}^2}{3}$

The correct option is B.