$(40$ points$)$ A hollow sphere $(I=\frac{2}{3}m{r}^2)$ sits at the top of a $5$ m tall ramp that makes an angle of $30$ degrees above the horizontal. The mass of the sphere is $20$ kg and the radius is $2$ meters.

m

$r$

a$)$ If there is no friction between the ramp and the sphere, what will the final velocity of the sphere be at the bottom of the ramp? Assume the sphere started from rest.

b$)$ Assume that there is friction between the ramp and the sphere, and the sphere rolls without slipping. What will the final velocity of the sphere be at the bottom of the ramp?