A thin cylindrical ring starts from rest at a height h$1=87$ m$.$ The ring has a radius R $=26$ cm and a mass M $=10$ 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.

E$1=$ M$$h$1$g$$Correct$!$

$-$ 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$2?$

E$2=$ M$$h$2$g$+$M$$v$22$Correct$!$

$-$ Part $($c$)$

Given h$2=41$ m$,$ what is the velocity of the ring at point $2$ in m$/$s$?$

v$2=21\mathrm{.}24$Correct$!$

$-$ Part $($d$)$

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

$=81\mathrm{.}7034$

$=81\mathrm{.}70$Correct$!$

$$ 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$?$

v$3=$