A body of mass (2M) splits into four masses (m, M - m, m, M - m), which are rearranged to form a square as shown in the figure. The ratio of Mm for which, the gravitational potential energy of the system becomes maximum is x : 1. The value of x is _______________.

The gravitational potential energy of systemU=−Gm22d−G(M−m)22d−4Gm(M−m)dFor maximum potential energydUdm=0ddm-Gdm22+(M-m)22+4m(M-m)=0 2m-2(M-m)+4M-8m=0⇒Mm=2

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A body of mass (2M) splits into four masses (m, M - m, m, M - m), which are rearranged to form a square as shown in the figure. The ratio of $\frac{M}{m}$ for which, the gravitational potential energy of the system becomes maximum is x : 1. The value of x is _______________.

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Detailed Solution

The gravitational potential energy of system
$U = - \frac{{Gm}^{2}}{\sqrt{2} d} - \frac{G {(M - m)}^{2}}{\sqrt{2} d} - 4 \frac{Gm (M - m)}{d}$

For maximum potential energy

$\frac{d U}{d m} = 0$

$\frac{d}{dm} (\frac{- G}{d} (\frac{m^{2}}{\sqrt{2}} + \frac{{(M - m)}^{2}}{\sqrt{2}} + 4 m (M - m))) = 0$

$\sqrt{2} m - \sqrt{2} (M - m) + 4 M - 8 m = 0$

$\Rightarrow \frac{M}{m} = 2$

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