Please help! I can\'t figure this out! I know you have to use PE, KE rotational,

Question

Please help! I can't figure this out! I know you have to use PE, KE rotational, and KE translational, but I can't figure it out. Thanks!

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.

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 (radius r = 3.8 m). 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 the top of the loop, the centripetal force is balanced by the weight

mv^2/r = mg

v^2 = gr

v = sqrt(gr)

v = sqrt(9.8)(3.8)

v = 6.10 m/s

Then by conservation of energy

At the height h, PE = mgh

At the top of the loop the mass will have PE = mg(2r)

The KE will be .5mv^2 + .5Iw^2

I for a sphere = 2/5mv^2 and w = v/r, thus KE = .2mv^2

Total KE = .7mv^2

mgh = mg(2r) + .7mv^2 (mass cancels)

(9.8)(h) = (9.8)(2)(3.8) + .7(6.10)^2

h = 10.3 m

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