Question
Two identical heavy spheres are separated by a distance 10 times their radius. Will an object placed at the mid point of the line joining their centres be in stable equilibrium or unstable equilibrium? Give reason for your answer.
✓
Answer
$\text{m}_1=\text{m}_2=\text{M }\text{r}=10\text{R}$
Let mass m is placed at the mid point P of line joining the centres of A and B sphere
$|\text{F}_2|=|\text{F}_1|=\frac{\text{GMm}}{(5\text{R})^2}$
$|\text{F}_1|=|\text{F}_2|=\frac{\text{GMm}}{(25\text{R})^2}$
As the direction of force F1 and F2 are opposite (equal and opposite forces acting on m at P). The net force F1 + F2 = 0, (F1 = -F2). m will be in equilibrium.
If now m is displaced by x slightly from P to A then PA = (5R - x) and PB = (5R + x)
$\text{F}_1=\frac{\text{GMm}}{(5\text{R}-\text{x})^2}$ and $\text{F}_2=\frac{\text{GMm}}{(5\text{R}+\text{x})^2}$
$\therefore\ \text{F}_2<\text{F}_1$
Hence the resultant force acting on P is towards A, resulting in an unstable equilibrium.
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