Question

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

According to the question,

At the top most point $\frac{m{v}^2}{R-r}=mg\Rightarrow v^2=g(R-r)$

Let the sphere is thrown with a velocity $v$

Therefore applying laws of conservation of energy

$\Rightarrow \left(1/2\right)m{v}^2+\left(1/2\right)l{\omega }^2=mg2(R-r)+\left(1/2\right)m{v}^2+\left(1/2\right)l{\omega }^2$$\Rightarrow \frac{7}{10}{v}^2=g2(R-r)+\frac{7}{10}{v}^2$

$\Rightarrow v^2=\frac{20}{7}g(R-r)+g(R-r)$

$\Rightarrow v^{\prime }=\sqrt{\frac{27}{7}g(R-r)}$