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Question
A sphere of mass m rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed \[\nu.\]
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Solution
Let radius of the sphere be R and its angular speed be ω.
Moment of inertia of sphere,
\[I = \frac{2}{5}m R^2\]
Total kinetic energy,
\[K = \frac{1}{2}I \omega^2 + \frac{1}{2}m v^2 \]
\[K = \frac{1}{2} \times \left( \frac{2}{5}m R^2 \right) \times \frac{v^2}{R^2} + \left( \frac{1}{2}m v^2 \right)\]
\[K = \frac{2}{10}m v^2 + \frac{1}{2}m v^2 \]
\[K = \frac{\left( 2 + 5 \right)m v^2}{10} = \frac{7}{10}m v^2\]
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