A small, solid sphere of mass 0.6kg and radius 27 cm rolls without slipping along the track consisting of slope (at an angle of 60degrees from horizontal) and loop-the-loop with radius 2.65m at the end of the slope. It starts from rest near the top of the track at a height, h, where h is large compared to 27 cm. What is the minimum value of h such that the sphere completes the loop?
mgh=1/2mv^2 + 1/2Iw^2
1/2mv(top)^2 + mgr2 = 1/2mv(bottom)^2 + 0
The Attempt at a Solution
I solved for v at the top of the loop to be v=square root of (gR)
Then I used the equation 1/2mv(top)^2 + mgr2 = 1/2mv(bottom)^2 + 0
and solved for v at the bottom.
v(bottom)= 11.4 m/s
Then I plugged this into the equation: mgh=1/2mv(bottom)^2 + 1/2Iw^2
and simplified it to be
gh= v(bottom)^2 + 2/5v(bottom)^2
and solved for h to get 9.27 m.
Where did I mess up?