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 centre of square. The total gravitational potential energy of the system is

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TestHub · Accepted Answer

The total gravitational potential energy of the system is

$U_{s y s t e m} = (U_{12} + U_{23} + U_{34} + U_{41}) + (U_{13} + U_{24}) + (U_{15} + U_{25} + U_{35} + U_{45})$

$= - \frac{4 G m m}{d} - \frac{2 G m m}{d \sqrt{2}} - \frac{4 \sqrt{2} G M m}{d}$

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

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

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