A thin hollow spherical shell of mass m and radius r is set into pure rolling on a rough horizontal surface with its centre of mass having velocity $v$ initially as shown. Work done by the upper hemispherical portion(shaded in the figure shown) on the lower hemispherical portion, as the sphere displaces by, $π r$ is

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$\frac{1}{2} m V^{2} + \frac{m g r}{2}$

$\frac{1}{2} m V^{2} + m g r$

$\frac{1}{3} m V^{2} + m g r$

$\frac{1}{3} m V^{2} - m g r$

answer is A.

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Detailed Solution

$W_{g r a v i t y} + W_{u p p e r o n l o w e r h a l f} =$ of lower half.
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$Δ K = \frac{1}{2} (\frac{m}{2}) [\frac{9 V^{2}}{4} - \frac{V^{2}}{4}]$ $= \frac{1}{2} m V^{2} \Rightarrow - (\frac{m}{2}) g r + W = \frac{1}{2} m V^{2}$ $\Rightarrow W = \frac{m}{2} g r + \frac{1}{2} m V^{2}$