Question

A solid sphere rolls down two different inclined planes of the same height but of different inclinations

• A. the speeds will be same but time of descent will be different
• B. in both cases, the speeds and time of descent will be same
• C. speeds and time of descent both will be different
• D. the speeds will be different but time of descent will be same

Answer 1

The correct option is A the speeds will be same but time of descent will be different

Work-energy theorem, $mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{w}^2$

$mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\times \frac{2}{5}m{r}^2{w}^2$

$mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\times \frac{2}{5}m{v}^2$

$⟹v^2=\frac{10gh}{7}$

Hence speed in both cases is same.

Acceleration of the body rolling without slipping on the inclined plane, $a=\frac{gsin\theta }{1+\frac{I}{m{r}^2}}$

Now $v=at⟹t=\frac{v(1+\frac{I}{m{r}^2})}{gsin\theta }$

Thus for different inclinations i.e different $\theta$, time of descent will be different.