Question

Figure (10-E14) 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.

Figure

Answer 1

Let the sphere be thrown with velocity $v\text{'}$ and its velocity becomes v at the top-most point.
From the free body diagram of the sphere, at the topmost point, we have:
$$\begin{gathered} \frac{m{v}^2}{R-r}=mg \\ \Rightarrow v^2=g\left(R-r\right) \end{gathered}$$

On applying the law of conservation of energy, we have:
$$\begin{gathered} \left(\frac{1}{2}m\nu {\text{'}}^2+\frac{1}{2}I\omega {\text{'}}^2\right)=2mg\left(R-r\right)+\left(\frac{1}{2}m{\nu }^2+\frac{1}{2}I{\omega }^2\right) \\ \Rightarrow \frac{7}{10}m\nu {\text{'}}^2=2mg\left(R-r\right)+\frac{7}{10}m{\nu }^2 \\ \Rightarrow \frac{7}{10}m\nu {\text{'}}^2=2mg\left(R-r\right)+\frac{7}{10}mg(R-r) \\ \Rightarrow \frac{7}{10}\nu {\text{'}}^2=\frac{27g\left(R-r\right)}{10} \\ \Rightarrow \nu \text{'}=\sqrt{\frac{27}{7}g\left(R-r\right)} \end{gathered}$$