Question

Sol. [Key Concept: Energy conservation $mgh = \frac{1}{2}{mv}^{2} + \frac{1}{2}\ I\omega^{2}$; the given $h = \frac{7v_{0}^{2}}{10g}$ forces $I = \frac{2}{5}\ {mR}^{2}$ — a solid sphere.]

$$h = \frac{7v_{0}^{2}}{10\text{ }g} $$Applying work energy theorem

$${- mgh = 0 - \frac{1}{2}{mv}^{2} - \frac{1}{2}I\omega^{2} }{- mg\left( \frac{7v_{0}^{2}}{10\text{ }g} \right) = - \frac{1}{2}{mv}_{0}^{2} - \frac{1}{2}\frac{{Iv}_{0}^{2}}{R^{2}} }{\frac{7{mv}_{0}^{2}}{10} = \frac{{mv}_{0}^{2}}{2} + \frac{{Iv}_{0}^{2}}{2R^{2}} }{\frac{{mv}_{0}^{2}}{5} = \frac{{Iv}_{0}^{2}}{2R^{2}} }{I = \frac{2{mR}^{2}}{5} }$$