A thin cylindrical ring starts from rest at a height h$1=87$ m$.$ The ring has a radius R $=38$ cm and a mass M $=5$ kg$.$

$$ Part $($a$)$

Write an expression for the ring's initial energy at point $1,$ assuming that the gravitational potential energy at point $3$ is zero.

$$ Part $($b$)$

If the ring rolls $($without slipping$)$ all the way to point $2,$ what is the ring's energy at point $2$ in terms of h$2$ and v$2?$

$$ Part $($c$)$

Given h$2=14$ m$,$ what is the velocity of the ring at point $2$ in m$/$s$?$

$$ Part $($d$)$

What is the ring's rotational velocity in rad$/$s at point $2?$

$$ Part $($e$)$

After passing point $2$ the hill becomes frictionless and the ring's rotational velocity remains constant. What is the linear velocity of the ring at point $3$ in m$/$s$?$