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

AI Thread Summary
The discussion focuses on determining the minimum height, h, from which a uniform solid sphere must start to ensure it remains on a loop-the-loop track. The initial calculations suggest that h should equal 2.7R, but it is later corrected to h = 2.7R - 1.7r. The equations used include potential energy, kinetic energy, and the moment of inertia of the sphere. The participant clarifies that the normal force at the top of the loop must be zero for the sphere to stay on the track. Ultimately, the correct height accounts for both the radius of the loop and the radius of the sphere.
ducnguyen2000
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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


\DeltaPE = mgh - 2mgR
\DeltaKE = 1/2 mv2 + 1/2 Iw2
ac = 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 - mac = 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?
 
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N is the force of norm exerted by the track on the cart at the top of the loop by the way
 
haha, nvrmind, i figured out what i did wrong.
 

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