Question

Two heavy spheres each of mass $100kg$ and radius $0.10m$ are placed $1.0m$ apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centers of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?

Answer 1

Gravitational force at the mid point.

Gravitational force at the midpoint due to two spheres will be equal and opposite.

Therefore, net gravitational force at the mid point of the line joining the centers of the spheres will be zero

Gravitational potential energy at mid point

Here, $G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$

Distance between the two spheres, $d=1.0m$
Mass of sphere, $M=100kg,R=0.1m$

Suppose that the distance of either sphere from the midpoint of the line joining their centre is $r=\frac{d}{2}=0.5m$

Gravitational potential at the mid point $=(-\frac{GM}{r})\times 2$

$=-\frac{6.67\times {10}^{-11}\times 100\times 2}{0.5}=-2.668\times {10}^{-8}Jk{g}^{-1}$

Equilibrium of the object placed at mid point

Since net gravitational force experienced by the object placed at the mid point will be zero, therefore system is in equilibrium.

Also due to the absence of restoring force, system is in an unstable equilibrium, as once displaced object won’t return to the original position.