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1. The problem statement, all variables and given/known data
A small solid marble of mass m and radius r will roll without slipping along the looptheloop track, shown below, if it is released from rest somewhere on the straight section of track. For the following answers use m for the mass, r for the radius of the marble, R for the radius of the looptheloop and g for the acceleration due to gravity.
Here's the picture:
From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop?
2. Relevant equations
Centripetal Acceleration = mg
Ki+mgh=Kf+mg(2R)
3. The attempt at a solution
Centripetal Acceleration = mg
Centripetal Acceleration = (mv^2)/R
(mv^2)/R = mg
v^2 = gR
Ki+Ui=Kf+Uf
Ki = 0
Ui = mgh
Kf = (mv^2)/2
Uf = mg(2R)
mgh = (mv^2)/2 + 2mgR
gh = (v^2)/2 + 2gR
gh = gR/2 + 2gR
h = R/2 + 2R
h = (5/2)R
In my attempt at the solution, I calculated (incorrectly) that the initial height needs to be 2.5*R. However, I did visit this old thread here: https://www.physicsforums.com/showthread.php?t=6357
The answer is actually 2.7*R, which is indeed the correct answer. Can someone help me out in identifying what I did wrong?
Thank you. :)
A small solid marble of mass m and radius r will roll without slipping along the looptheloop track, shown below, if it is released from rest somewhere on the straight section of track. For the following answers use m for the mass, r for the radius of the marble, R for the radius of the looptheloop and g for the acceleration due to gravity.
Here's the picture:
From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop?
2. Relevant equations
Centripetal Acceleration = mg
Ki+mgh=Kf+mg(2R)
3. The attempt at a solution
Centripetal Acceleration = mg
Centripetal Acceleration = (mv^2)/R
(mv^2)/R = mg
v^2 = gR
Ki+Ui=Kf+Uf
Ki = 0
Ui = mgh
Kf = (mv^2)/2
Uf = mg(2R)
mgh = (mv^2)/2 + 2mgR
gh = (v^2)/2 + 2gR
gh = gR/2 + 2gR
h = R/2 + 2R
h = (5/2)R
In my attempt at the solution, I calculated (incorrectly) that the initial height needs to be 2.5*R. However, I did visit this old thread here: https://www.physicsforums.com/showthread.php?t=6357
The answer is actually 2.7*R, which is indeed the correct answer. Can someone help me out in identifying what I did wrong?
Thank you. :)
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