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A hollow sphere of mass $m$ filled with a non-viscous liquid of same mass $m$ is released on a slope inclined at angle $q$ with the horizontal. The friction between the sphere and the slope is sufficient to prevent sliding and frictional forces between the inner surface of the sphere and the liquid is negligible. After descending a certain height ratio of translational and rotational kinetic energies is found to be $x:y$, find the numerical value of expression $(x+y)_{min}.$
$4$
$6$
$8$
$10$
Solution
$\frac{(\mathrm{k})_{trans}}{(\mathrm{k})_{\mathrm{rot}}}=\frac{\frac{1}{2} \times 2 \mathrm{mv}^{2}}{\frac{1}{2} \times \frac{2}{3} \mathrm{mR}^{2} \omega^{2}}=\frac{\mathrm{mw}^{2} \mathrm{R}^{2}}{\frac{1}{3} \mathrm{mR}^{2} \omega^{2}}$
$=\frac{3}{1}=\frac{\mathrm{x}}{\mathrm{y}}$
