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(A) The total gravitational potential energy $U$ of the system is the sum of the potential energies of all pairs of spheres.$1$. Potential energy of t

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$-\frac{ Gm }{ d }[(4+\sqrt{2}) m +4 \sqrt{2} M]$

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(A) The total gravitational potential energy $U$ of the system is the sum of the potential energies of all pairs of spheres.$1$. Potential energy of the four spheres of mass $m$ at the corners of the square:There are $4$ sides of length $d$ and $2$ diagonals of length $\sqrt{2}d$.$U_{m-m} = -\frac{G m^2}{d} 	imes 4 - \frac{G m^2}{\sqrt{2}d} 	imes 2 = -\frac{G m^2}{d} (4 + \sqrt{2})$.$2$. Potential energy of the central sphere of mass $M$ with the four spheres of mass $m$:The distance of each corner from the centre is $r = \frac{\sqrt{2}d}{2} = \frac{d}{\sqrt{2}}$.$U_{M-m} = -\frac{G M m}{r} 	imes 4 = -\frac{G M m}{d/\sqrt{2}} 	imes 4 = -\frac{4\sqrt{2} G M m}{d}$.$3$. Total potential energy $U = U_{m-m} + U_{M-m} = -\frac{G m}{d} [(4 + \sqrt{2})m + 4\sqrt{2}M]$.

Four spheres each of mass $m$ form a square of side $d$ (as shown in the figure). $A$ fifth sphere of mass $M$ is situated at the centre of the square. The total gravitational potential energy of the system is

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