Question

A disc at rest at the top of an inclined plane of heigh "h' rolls down without slippling and acquires a velocity " on reaching the bottom. If the same disc slides down a smooth incline and acquires the h/3 same velocity on reaching the bottom the height of smooth incline is:

• A. h/3
• B. h/2
• C. $\frac{2h}{3}$
• D. h

Answer 1

The correct option is C $\frac{2h}{3}$

$\begin{array}{c}mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{m{R}^2}{2}\right)\frac{v^2}{R^2}\\ \Rightarrow mgh=\frac{1}{2}m{v}^2+\frac{m{v}^2}{4}\\ \Rightarrow mgh=\frac{3m{v}^2}{4}\\ v^2=\frac{4gh}{3}...............\left(i\right)\\ mg{h}^{\prime }=\frac{1}{2}m{v}^2\\ g{h}^{\prime }=\frac{1}{2}\times \frac{4gh}{3}\\ h^{\prime }=\frac{2h}{3}\end{array}$

$\therefore$ Option $C$ is correct .