A uniform marble rolls down a symmetric bowl, starting from rest at the top of the left side. The top of each side is a distance h above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. h' is how far up the smooth side the marble will go, measured vertically from the bottom. Find h', express h' in terms of h.
The Attempt at a Solution
mgh = mgh' + 1/2 Iw^2 (w = angular velocity)
mgh = mgh' + 1/2 *(2/5)(mR^2)(w^2)
gh = gh' + v^2/5 -- v is the rotational speed of the marble
h' = h - v^2/g -- I can't seem express h' just in terms of h, what am i doing wrong?