Physics - Rotation Question with Solution | TestHub

The rolling sphere has rotational as well as translational kinetic energy.

$\therefore$Kinetic energy$=\frac{1}{2}m{u}^2+\frac{1}{2}I{\omega }^2$

$=\frac{1}{2}m{u}^2+\frac{1}{2}\left(\frac{2}{5}m{r}^2\right){\omega }^2$

$=\frac{1}{2}m{u}^2+\frac{1}{5}m{u}^2=\frac{7}{10}m{u}^2 (\because u=r\omega )$

From conservation of energy

$ie$,$m\mathrm{g}h=\frac{7}{10}m{u}^2$

or$h=\frac{7u^2}{10\mathrm{g}}$