Q 11-07-129JEE MainJEE Main 2022 (27 Jun, Shift 2)Medium
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
Answer: (A) $-\dfrac{Gm}{d}\left[\left(4 + \sqrt2\right)m + 4\sqrt2 M\right]$
Add the energy of every pair.
Four sides (separation $d$): $-4\dfrac{Gm^2}{d}$.
Two diagonals (separation $\sqrt2 d$): $-2\dfrac{Gm^2}{\sqrt2 d} = -\sqrt2\dfrac{Gm^2}{d}$.
$M$ with each corner (separation $d/\sqrt2$): $-4\dfrac{GmM}{d/\sqrt2} = -4\sqrt2\dfrac{GmM}{d}$.
$$U = -\frac{Gm}{d}\left[(4 + \sqrt2)m + 4\sqrt2 M\right]$$
Solution by Sreeraj P, M.Sc Physics