Two identical heavy spheres are separated by a distance of 10 times their radius. will an object placed at the midpoint of the line joining their centers be in stable equilibrium or unstable equilibrium ? Give reason for your answer.

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Solution

$m_{1} = m_{2} = M r = 10 R$
Let mass $m$ is placed at the mid point $P$ of line joining the centres of $A$ and $B$ sphere
$| F_{2} | = | F_{1} | = \frac{G M m}{(5 R)^{2}}$
$| F_{1} | = | F_{2} | = \frac{G M m}{25 R^{2}}$

As the direction of force $F_{1}$ and $F_{2}$ are opposite (equal and opposite forces acting on m at $P$ ) the net force $F_{1} + F_{2} = 0, (F_{1} = - F_{2})$ . m will be in equilibrium.
If now m in displace by $x$ slightly from $P$ to $A$ then $P A = (5 R - x)$ and $P B = (5 R + x)$

$F_{1} = \frac{G M m}{(5 R - x)^{2}}$ and $F_{2} = \frac{G M m}{(5 R + x)^{2}}$
$∴ F_{2} < F_{1}$
Hence the resultant force acting on $P$ is towards A, resulting in an unstable equilibrium.

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