Question

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:

• A. \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)m+4\sqrt2M\bigg]\)
• B. \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)M+4\sqrt2m\bigg]\)
• C. \(−\frac{Gm}{d}\bigg[3m^2+4\sqrt2M\bigg]\)
• D. \(−\frac{Gm}{d}\bigg[6m^2+4\sqrt2M\bigg]\)

Answer 1

The Correct Option is A

Total gravitational potential energy

=\(−{\frac{4GMm}{\frac{d}{\sqrt2}}+\frac{4Gm^2}{d}+\frac{2Gm^2}{\sqrt2d}\\}\)

=\(−\frac{Gm}{d}\{{M4\sqrt2+(4+\sqrt2)m}\}\)

=\(−\frac{Gm}{d}\{{4\sqrt2M+(4+\sqrt2)m}\}\)