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

Concept:
Total gravitational potential energy = sum of energies of all interacting pairs:

$$U=-\sum_{ }^{ }\frac{Gm_im_j}{r_{ij}}$$

Step 1: Energy between corner masses (m)

• 4 sides of square (distance d):

$$U_1=-4\cdot\frac{Gm^2}{d}$$

• 2 diagonals (distance $$\sqrt{2}d$$):

$$U_2=-2\cdot\frac{Gm^2}{\sqrt{2}d}$$

Step 2: Energy between centre mass M and each corner mass

Distance from centre to corner:

$$r=\frac{d}{\sqrt{2}}$$

For 4 pairs:

$$U_3=-4\cdot\frac{GmM}{d/\sqrt{2}}=-4\sqrt{2}\frac{GmM}{d}$$

Step 3: Total energy

$$U=U_1+U_2+U_3$$

$$U=-\frac{Gm^2}{d}\left(4+\sqrt{2}\right)-\frac{4\sqrt{2}GmM}{d}$$

Final Answer:

$$U=-\frac{Gm^2}{d}\left(4+\sqrt{2}\right)-\frac{4\sqrt{2}GmM}{d}$$