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Question
The following figure shows a rough track, a portion of which is in the form of a cylinder of radius R. With what minimum linear speed should a sphere of radius r be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part.
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Solution
Let the sphere be thrown with velocity \[v'\] and its velocity becomes v at the top-most point.
From the free body diagram of the sphere, at the topmost point, we have
\[\frac{m v^2}{R - r} = mg\]
\[ \Rightarrow v^2 = g\left( R - r \right)\]
On applying the law of conservation of energy, we have
\[\left( \frac{1}{2}m\nu '^2 + \frac{1}{2}I\omega '^2 \right) = 2mg\left( R - r \right) + \left( \frac{1}{2}m \nu^2 + \frac{1}{2}I \omega^2 \right)\]
\[ \Rightarrow \frac{7}{10}m\nu '^2 = 2mg\left( R - r \right) + \frac{7}{10}m \nu^2 \]
\[ \Rightarrow \frac{7}{10}m\nu '^2 = 2mg\left( R - r \right) + \frac{7}{10}mg(R - r)\]
\[ \Rightarrow \frac{7}{10}\nu '^2 = \frac{27g\left( R - r \right)}{10}\]
\[ \Rightarrow \nu' = \sqrt{\frac{27}{7} g\left( R - r \right)}\]
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