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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?
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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