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

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$- \frac{Gm}{d} [3 m^{2} + 4 \sqrt{2} M]$

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

$- \frac{Gm}{d} [6 m^{2} + 4 \sqrt{2 M}]$

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

answer is A.

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Detailed Solution

$\begin{array}{rcl} PE & = & - \frac{{Gm}^{2}}{d} \times 4 - \frac{{Gm}^{2}}{\sqrt{2} d} \times 2 - \frac{GMm}{d} \times 4 \sqrt{2} \\ = & - \frac{Gm}{d} [(4 + \sqrt{2}) m + 4 \sqrt{2} M] \end{array}$

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