What is the Minimum Height for a Sphere to Stay on a Loop-the-Loop Track?

[ducnguyen2000]

1. A uniform solid sphere of radius r starts from rest at a height h and rolls without slipping along the loop-the-loop track of radius R as shown.
a) What is the smallest value of h for which the sphere will not leave the track at the top of the loop?


Attempt:

Homework Equations

$$\Delta$$P_E = mgh - 2mgR
$$\Delta$$K_E = 1/2 mv² + 1/2 Iw²
a_c = v²/R
v = rw

The Attempt at a Solution

mgh = 2mgR + 1/2 Iw² + 1/2 mv²
I = 2/5 mr²
mgh = 2mgR + 1/5 mr²w² + 1/2 mv²
N = mg - ma_c = mg - mv²/R = 0
v² = gR
mgh = 2mgR + 1/2 mv² + 1/5 mv² +2mgR = 2.7mgR
h = 2.7R

however, this answer is wrong, and the correct one is h = 2.7R - 1.7r. can anybody correct me on this?

 

[ducnguyen2000]

N is the force of norm exerted by the track on the cart at the top of the loop by the way

 

[ducnguyen2000]

haha, nvrmind, i figured out what i did wrong.