If three uniform spheres, each having mass M and radius R, are kept in such away that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is

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$\frac{{GM}^{2}}{4 r^{2}}$

$\frac{2 {GM}^{2}}{r^{2}}$

$\frac{2 {GM}^{2}}{4 r^{2}}$

$\frac{\sqrt{3} {GM}^{2}}{4 r^{2}}$

answer is D.

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Detailed Solution

$\begin{array}{l} \vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} \\ As |{\vec{F}}_{1}| = |{\vec{F}}_{2}| \\ ∴ | \vec{F} | = 2 F_{1} \cos 30^{\circ} \\ = 2 \frac{{GM}^{2}}{(2 r)^{2}} \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \frac{{GM}^{2}}{r^{2}} \end{array}$

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