Question

Four particles having masses m, 2m, 3m, and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the center.

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

Answer 1

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

Four particles of masses $m,2m,3m,4m$ are kept in square $ABCD$ at the corners of a square of side a. a particle of mass $m$ is placed at centre $O$ of a square.

now, force exerted by A on O = Gm²/(a/√2)² = 2Gm²/a²

the force exerted by $B$ on

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

$=\frac{4G{m}^2}{a^2}$

force exerted by $C$ on $O=\frac{3G{m}^2}{(a/\surd 2{)}^2}=\frac{6G{m}^2}{a^2}$

the force exerted by $D$ on

$O=\frac{4G{m}^2}{(\frac{a}{\sqrt{2}}{)}^2}$

$=\frac{8G{m}^2}{a^2}$

Let us consider $F=\frac{2G{m}^2}{a^2}$

then, the force exerted by $A$ on $O=F$

the force exerted by $B$ on $O=2F$

the force exerted by $C$ on $O=3F$

the force exerted by $D$ on $O=4F$

see figure , resultant force of $B$ and $D=4F-2F=2F\text{along CO}$

and resultant force of $A$ and $C=3F-F=2F\text{along DO}$

now, net force acting on $O=\sqrt{(2F{)}^2+(2F{)}^2}=2\sqrt{2}F$

putting $F=\frac{2G{m}^2}{a^2}$

so, the net force acting on $O=\frac{4\sqrt{2}G{m}^2}{a^2}$