Four spheres each of mass m form a square of side d a fifth sphere of mass M is situated at the center of the square. The total gravitational potential energy of the system is

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

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

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

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

answer is A.

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Detailed Solution

potential energy for square

$U_{s q u a r e} = \frac{- G m^{2}}{d} [4 + \sqrt{2}]$

Potential energy for center of mass to each corner

$U_{c m} = \frac{- G M m}{d} [4 \sqrt{2}]$

Total PE $U_{t o t a l} = U_{s q u a r e} + U_{c m}$

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

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

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