Gravitation Part 1

Given, total mass of body is 2M. Potential energy is maximum at $\frac{M}{m} = \frac{x}{1}$ The arrangement of masses to form a square is shown in diagram. The gravitational potential energy of a body is given $U = -\frac{GMm}{r}$ For the given system, the potential energy will be $U_{T} = -\frac{4 Gm(M-m)}{d} - \frac{Gm^{2}}{\sqrt{2} d} - \frac{G(M-m)^{2}}{\sqrt{2} d}$ For maximum potential energy, $\frac{dU_{T}}{dm}=0$ $-\frac{4G}{d}[M-2m] - \frac{G}{\sqrt{2} d}[2m]$ $-\frac{G}{\sqrt{2} d}[2(M-m) \times (-1)]=0$ $\Rightarrow 4M - 8m + \sqrt{2} m = \sqrt{2} (M-m)$ $(4-\sqrt{2})M = (8-2\sqrt{2})m$ $\frac{M}{m} = \frac{2(4-\sqrt{2})}{4-\sqrt{2}} = 2$ Comparing it with the given value, we get x = 2 Thus, potential energy will be maximum when x = 2.