Question

A solid sphere rolls down on two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?

Answer 1

(a)

Total energy of sphere at the top= $mgh$

At the bottom, the sphere has both translational and rotational energy=$\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

Using conservation of energy,

$mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$......(i)

For solid sphere $I=\frac{2}{5}m{r}^2$......(ii)
Also we have the relation, $v=r\omega$.......(iii)

Solving above equations gives $v=\sqrt{\frac{10gh}{7}}$

Clearly the speed of sphere at the bottom does not depend on the angle of inclination, both g and h are independent of the angle of inclination.

(b)

Assuming rolling without slipping,

$I\alpha =Fr$

$\left(7/5\right)m{r}^2\times a/r=mgsin\left(\theta \right)r$

$a=\left(5/7\right)gsin\left(\theta \right)$

Since ${\theta }_1<{\theta }_2$,

(c )Acceleration for the less inclined plane is less.
Thus it takes longer time to reach the bottom along the plane with ${\theta }_1$ inclination.