Two heavy spheres each of mass 100 \mathrm{~kg} and radius 0.10 \mathrm{~m} are placed 1.0 \mathrm{~m} apart on a horizontal table. What is the gravitational force and potential at the midpoint 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?
Two heavy spheres each of mass 100 \mathrm{~kg} and radius 0.10 \mathrm{~m} are placed 1.0 \mathrm{~m} apart on a horizontal table. What is the gravitational force and potential at the midpoint 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?

Radius of spheres is given as R=0.10 \mathrm{~m}

Distance between two spheres is given as r=1.0 \mathrm{~m}

Mass of each sphere is given as M=100 \mathrm{~kg}

We can infer from the figure that ‘A’ is the mid-point. Since each sphere will exert the gravitational force in the opposite direction, so the gravitational force at this point will be zero.

The gravitational potential at the midpoint can be calculated as:

\mathrm{U}=\left[\frac{-G M}{\frac{r}{2}}+\frac{-G M}{\frac{r}{2}}\right]

\mathrm{U}=\left[\frac{-4 G M}{r}\right]

U=4×6.67×1011×(1000)1.0U=2.668×107 J/kg
\mathrm{U}=\left[\frac{-4 \times\left(6.67 \times 10^{-11}\right) \times(1000)}{1.0}\right] \Rightarrow \mathrm{U}=-2.668 \times 10^{-7} \mathrm{~J} / \mathrm{kg}

As a result, the gravitational potential and force at the mid-point of the line connecting the centres of the two spheres is =-2.668 \times 10^{-7} \mathrm{~J} / \mathrm{kg}

The net force on an object, placed at the mid-point is zero. But, even if their is a little displacement towards any of the two bodies, it will not return to its equilibrium position. Therefore, the body is in an unstable equilibrium.