A circular ring and a solid sphere having same radius roll down on an inclined plane from rest witho

Question

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{10}}$ where $x=$

Professor PC Thomas Classes · Answer

Applying conservation of mechanical energy,\ $K E_{i}+U_{i}=K E_{f}+U_{f}$\ $0+m g h=\frac{1}{2} m v^{2}\left(1+\frac{k^{2}}{R^{2}}ight)+0$\ $\Rightarrow v=\sqrt{\frac{2 g h}{1+\frac{k^{2}}{R^{2}}}}$\ For ring, $k=1$\ For sphere, $k=\sqrt{\frac{2}{5}}$\ $\frac{v_{	ext {ring }}}{v_{	ext {sphere }}}=\sqrt{\frac{1+\frac{2}{5}}{1+1}}=\sqrt{\frac{7}{10}} \Rightarrow x=7$