Solve the following :


Three uniform spheres each having a mass $M$ and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.


Let us assume that the center of mass of spheres are located at point $P, Q$ and $R$. So, re-drawing the figure, it is an equilateral triangle.

Force on mass $M$ at point $P$

$\mathrm{F}_{\mathrm{PR}}=\mathrm{F}_{\mathrm{PQ}} \frac{G M M}{(2 a)^{2}}=\frac{G M^{2}}{4 a^{2}}=\mathrm{k}($ let $)$


$\mathrm{F}_{\mathrm{R}}=\sqrt{F_{P Q}^{2}+F_{P R}^{2}+2 F_{P Q} \cdot F_{P R} \cdot \cos 60^{\circ}}$

$\mathrm{F}_{\mathrm{R}}=\sqrt{k^{2}+k^{2}+2 k^{2} \cdot\left(\frac{1}{2}\right)}$

$\mathrm{F}_{\mathrm{R}}=\sqrt{3} k$

$\mathrm{F}_{\mathrm{R}}=\sqrt{3} \frac{G M^{2}}{4 a^{2}}$

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