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A hemisspherical shell of mass $2M$ and radius $6R$ and a point mass $M$ are performing circular motion due to their mutual gravitational interaetion Their positions are shown in figure at any moment of time during motion. If $r_1$ and $r_2$ are the radii of circular path of hemispherical shell and point mass respectively and ${\omega _1}$ and ${\omega _2}$ are the angular speeds of hemi-spherical shell and point mass respectively, then choose the correct option
${\omega _1} = \sqrt {\frac{{GM}}{{18{R^3}}}} $
${\omega _2} = \sqrt {\frac{{GM}}{{24{R^3}}}} $
${\omega _1} = \sqrt {\frac{{GM}}{{36{R^3}}}} $
${\omega _2} = \sqrt {\frac{{GM}}{{72{R^3}}}} $
Solution
The system is equivalent to a binary system as shown in the diagram.
$r_{1}=\begin{array}{l}{M \times 3 R} \\ {M+2 M}\end{array}=R$
$r_{2}=2 R$
Considering the circular motion of point mass $m$
$\mathrm{M} \omega_{2}^{2} \mathrm{r}_{2}=\frac{\mathrm{GM}^{2}}{36 \mathrm{R}^{2}}$
$\omega_{2}=\sqrt{\frac{\mathrm{GM}}{72 \mathrm{R}^{3}}}=\omega_{1}$
