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

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

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


Answers (1)

best_answer

For gravitational potential energy of the system, total no.of pair to write energy equation is

\mathrm{^{n}C_{2}=\, ^{5}C_{2}= \frac{5\times9}{2} = 10}


\mathrm{U_{system} =\left [ 4\times\left ( \frac{-Gm^{2}}{d} \right )\right ] }
                     \mathrm{+\left [ 2\times ( \frac{-Gm^{2}}{\sqrt{2}d} \right ] }
                      \mathrm{+\left [ 4\times ( \frac{-GMm}{\left ( \frac{d}{\sqrt{2}} \right )} \right ] }

  \mathrm{= \frac{-Gm}{d}\left [ 4m+\sqrt{2}m+4\sqrt{2}M \right ] }

The correct answer is (1)

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Pankaj

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