Question

A thin uniform spherical shell and a uniform solid cylinder of the same mass and radius are allowed to roll down a fixed incline without slipping, starting from rest. The times taken by them to roll down the same distance are $t_{1}and{t}_2$ respectively. Then $\frac{t_1}{t_2}$ is :

• A. $\sqrt{\frac{14}{15}}$
• B. $\sqrt{\frac{15}{14}}$
• C. $\frac{3}{\sqrt{10}}$
• D. $\frac{\sqrt{10}}{3}$

Answer 1

The correct option is D $\frac{\sqrt{10}}{3}$

We have $k^2$ for hollow sphere as $\frac{2}{3}{r}^2$ and that for solid cylinder as $\frac{r^2}{2}$.

And time taken by the body to reach the bottom of the inclined plane is given as- $\sqrt{\frac{2l(1+k^2/r^2)}{gsin\theta }}$.
Thus we get the ratio of the times as $\frac{\sqrt{1+\frac{2}{3}}}{\sqrt{1+\frac{1}{2}}}$
Which reduces to $\frac{\sqrt{10}}{3}$.