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

Answer: (A)

$−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)m+4\sqrt2M\bigg]$

Answer 2

Total gravitational potential energy

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

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

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