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.

Answer 1

A the topmost 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.

$\Rightarrow \frac{1}{2}m{v}^{\prime 2}+1.2I{\omega }^2$

=$m{g}^2(R-r)+\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

$\Rightarrow \frac{7}{10}{v}^{\prime 2}=g^2(R-r)+\frac{7}{10}{v}^2$

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

$\Rightarrow v^{\prime 2}=\frac{27}{7}g(R-r)$

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