Newton's Law of Gravitation question from JEE Main 2024 (31 Jan, Shift 1)

Answer: (a)

1. Let the side be $a$. Two adjacent masses each pull with $\frac{Gm^2}{a^2}$ at $90^\circ$: resultant $\frac{\sqrt{2}Gm^2}{a^2}$ along the diagonal.
2. Diagonal mass at $\sqrt{2}a$ pulls along the same diagonal: $\frac{Gm^2}{2a^2}$.
3. Total $F = \left(\sqrt{2} + \frac{1}{2}\right)\frac{Gm^2}{a^2} = \frac{2\sqrt{2}+1}{2}\cdot\frac{Gm^2}{a^2}$
4. Set equal to $\frac{2\sqrt{2}+1}{32}\cdot\frac{Gm^2}{L^2}$: $a^2 = 16L^2 \Rightarrow a = 4L$
5. So the answer is (a) $4L$.