Question

A wooden sphere rolls down to two different inclined planes of the same height but different inclines. Will it reach the bottom with the same speed in each case?

Answer 1

Yes, both will reach at the bottom with same speed.

As, mass$=m$.

Height of plane$=h$

Velocity at bottom$=v$

Angular Velocity$=\omega$

Using Law of Conservation of energy,

$\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2=mgh$

$I\left(solidsphere\right)=\frac{2}{5}M{R}^2;v=r\omega$

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

$\Rightarrow v^2\left(\frac{7}{10}\right)=gh$

$\Rightarrow v=\sqrt{\frac{10}{7}gh}$

Velocity of sphere depends only on height of plane & $g$. Both are constant. It does not depend on inclination. Hence, velocity at bottom is same.