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Question
A sphere starts rolling down an incline of inclination θ. Find the speed of its centre when it has covered a distance l.
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Solution
Let radius of the sphere be r. Let r be negligible w.r.t. l.
Potential energy of the sphere, P.E. = \[mgl\sin\theta\]
Total kinetic energy of the sphere of mass m rolling with speed v = \[\frac{7}{10}m v^2\]
On applying the law of conservation of energy, we get
\[mgl\sin\theta = \frac{7}{10}m v^2 \]
\[ \Rightarrow gl\sin\theta = \frac{7}{10} \nu^2 \]
\[ \Rightarrow \nu = \sqrt{\left( \frac{10}{7}gl\sin\theta \right)}\]
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