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.

Answer 1

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

\[\left( \frac{1}{2}m\nu '^2 + \frac{1}{2}I\omega '^2 \right) = 2mg\left( R - r \right) + \left( \frac{1}{2}m \nu^2 + \frac{1}{2}I \omega^2 \right)\]

\[ \Rightarrow \frac{7}{10}m\nu '^2  = 2mg\left( R - r \right) + \frac{7}{10}m \nu^2 \]

\[ \Rightarrow \frac{7}{10}m\nu '^2  = 2mg\left( R - r \right) + \frac{7}{10}mg(R - r)\]

\[ \Rightarrow \frac{7}{10}\nu '^2  = \frac{27g\left( R - r \right)}{10}\]

\[ \Rightarrow \nu' = \sqrt{\frac{27}{7}  g\left( R - r \right)}\]