Question

Three uniform spheres, each having mass $m$ and radius $r$, are kept in such a way that each touches the other two. The magnitude of the gravitational force on any sphere due to the other two is

• A. $\frac{Gm^2}{r^2}$
• B. $\frac{Gm^2}{4r^2}$
• C. $\sqrt{2} \frac{Gm^2}{4r^2}$
• D. $\sqrt{3} \frac{Gm^2}{4r^2}$

Answer 1

Answer: (D)

The given system may be regarded as a system of three particles located at the three vertices of an equilateral triangle of side $2r$.
Now, $F_A = F_B$
$ = \frac{Gm^2}{(2r)^2} = \frac{Gm^2}{4r^2}$
$F_A$ and $F_B$ are inclined to each other at an angle of $60^{\circ}$.
If $F$ is the resultant of $F_A$ and $F_B$, then
$F = \sqrt{3} \times \frac{Gm^2}{4r^2}$