A sphere rolls down the loop-the-loop track shown in the figure below, starting

Question

A sphere rolls down the loop-the-loop track shown in the figure below, starting from rest at a height h above the bottom. The ball travels around the inside of the circular portion of the track

It is found that the ball is just barely able to travel around the inside of the track without losing contact at the top. Find h.
h =___________ m

A sphere rolls down the loop-the-loop track shown in the figure below, starting from rest at a height h above the bottom. The ball travels around the inside of the circular portion of the track It is found that the ball is just barely able to travel around the inside of the track without losing contact at the top. Find h.

Explanation / Answer

at top N + mg = mv^2/r

so v^2 = gr

conservation of energy

m g h = 1/2 mv^2 + 1/2 I w^2 + m g 2 r

I= 2/5 mR^2
w = v R

so m g h = 1/2 mv^2 + 1/5 mv^2 + 2 mg r

plug in v^2 = gr

m g h = 1/2 m gr + 1/5 m g r + 2 mg r

h = (1/2 + 1/5 + 2) r = (1/2 + 1/5 +2)*4.4= 11.88 m

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