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Question
A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface?
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Solution
If the shell does not slip on the surface, its motion should be pure rolling.
Let the cue hits at a height 'h' above the centre.
Let the centre of shell moves with velocity vc and shell rotates with angular velocity ω after hitting.
For pure rolling,
\[v_c = R\omega\]
On applying the law of conservation of angular momentum at point O, we get
\[m v_c h = I\omega\]
\[m v_c h = \frac{2}{3}m R^2 \left( \frac{v_c}{R} \right)\]
\[h = \frac{2R}{3}\]
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