Question

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

Answer 1

Net gravitational force experienced by the object placed at the mid point is zero.
Gravitational potential energy at mid point is \(-2.67 \times 10^{-8} j/kg\)
Yes, system is in equilibrium.
System is in an unstable equilibrium.
Explanation:
The situation is represented in the given figure:

Mass of each sphere, \(\text M\) = \(100 \; kg\)
Separation between the spheres, \(r\) = \(1m\)
X is the mid point between the spheres. Gravitational force at point X will be zero. This is because gravitational force exerted by each sphere will act in opposite directions.

Gravitational potential at point X: 
= \(\frac{GM}{\bigg(\frac{r}{2}\bigg)}-\frac{GM}{\bigg(\frac{r}{2}\bigg)}\) = \(-4\frac{GM}{r}\) 

= \(\frac{4 \times 6.67 \times 10 ^{-11} \times 100 }{1}\)

= \(-2.67 \times 10^{-8} j/kg\)

Any object placed at point X will be in equilibrium state, but the equilibrium is unstable. This is because any change in the position of the object will change the effective force in that direction.