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A body of mass (2M) splits into four masses [\mathrm{m}, \mathrm{M}-\mathrm{m}, \mathrm{m}, \mathrm{M}-\mathrm{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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Gravitation \: PE\: of \: system = U_{12}+U_{23}+U_{34}+U_{41}+U_{13}+U_{24}
U= \frac{-4Gm\left ( M-m \right )}{d}-\frac{Gm^{2}}{\sqrt{2}d}-\frac{G\left ( M-m \right )^{2}}{\sqrt{2}d}
\frac{dU}{dm}= 0\: for\: U_{max}
\frac{dU}{dm}= 0= \frac{-\Delta G}{d}\left [M-2m-\frac{2m}{\sqrt{2}} -\frac{2\left ( M-m \right )}{\sqrt{2}} \left ( -1 \right )\right ]
0= \left [M-2m-\frac{2m}{\sqrt{2}} +\frac{2\left ( M-m \right )}{\sqrt{2}} \right ]
0= \left [M-2m+\frac{2M-4m}{\sqrt{2}} \right ]
0=\sqrt{2}M-2\sqrt{2}m+2M-4m
4m+2\sqrt{2}m= 2M+\sqrt{2}M
2\sqrt{2}\left (\sqrt{2}+1 \right )m= \sqrt{2}\left ( \sqrt{2}+1 \right )M
\frac{M}{m}= \frac{2}{1}= \frac{X}{1}
\therefore X= 2

Posted by

vishal kumar

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