Question

Four particles of masses $m,2m,3m$ and $4m$ are kept in sequence at the corners of a square of side $a.$ The magnitude of gravitational force acting on a particle of mass $m$ placed at the center of the square will be.

• A. $\frac{24m^{2}G}{a^2}$
• B. $\frac{6m^{2}G}{a^2}$
• C. $\frac{4\sqrt{2}G{m}^2}{a^2}$
• D. $Zero$

Answer 1

The correct option is C $\frac{4\sqrt{2}G{m}^2}{a^2}$

Along the line joining m and 3m, force will be

$F=\frac{G(3m-m)m}{{\left(\frac{a}{\sqrt{2}}\right)}^2}=\frac{4G{m}^2}{a^2}$

Along the line joining 2m and 4m, force will be

$F=\frac{G(4m-2m)m}{{\left(\frac{a}{\sqrt{2}}\right)}^2}=\frac{4G{m}^2}{a^2}$

Since the two forces are at an angle of ${45}^0$ to each other, net force is

$F_{eq}=2\frac{F}{\sqrt{2}}=\sqrt{2}F=\frac{4\sqrt{2}G{m}^2}{a^2}$

Answer is option C.