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Rolling, Forces, and Loops

  1. Nov 2, 2014 #1
    Hi everyone, I've been working on this problem for a while now, and I was hoping someone here could point me in the right direction. Here goes:

    1. The problem statement, all variables and given/known data

    A solid brass ball of mass 0.280 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius R= 14.0, and the ball has radius r <<R. (a) What is h if the ball is on the verge of leaving the track when it reaches the top of the loop?

    I don't have units of R, but I think it's in cm.


    2. Relevant equations
    At the top of the loop, FN + mg = ma
    Fc = mv2 / R where R is radius of the track

    Ui + Ki = Uf + Kf

    3. The attempt at a solution
    First, using FN + mg = ma, I know ma is (mv2)/R, and (mv2)/R = mg if the ball is going to fall at the top, from (mv2)/R - mg = FN.

    I plug in R = .14m, m= .280 X 10-3g, and g to get v2 = 1.37 m/s. I think it's correct up to here...

    However, in Ui + Ki = Uf + Kf, initial U = mgh, final U =mg(2R) because the ball is at the top of the loop, initial K = 0, but final K = 1/2mv2. When I plug everything in and m cancels out,

    gh = 1/2v2 + g(2R)

    (9.8m/s2) = 1/2(1.37m2/s2) + (9.8m/s2)(2 X .14 m)

    and h = .350 m or 35.0 cm. My answer is close to the book's, which gives h = 37.8 cm, but I think I am missing something. Do I need to include rotational inertia, (1/2)Iω2 as part of K final? If I did, it looks like I would need to find I and omega, and I need the radius of the ball to do that (not given). I only have its mass and calculated velocity. Please let me know if I need to give more information and clarify, thank you!
     
  2. jcsd
  3. Nov 2, 2014 #2

    haruspex

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    Yes, that's the error.
    You don't need to know r. Put it in as an unknown and it will cancel out. For a rolling uniform ball, there is a constant fraction of its KE that is in the rotation. Relate omega to linear speed in the usual way.
     
  4. Nov 3, 2014 #3
    Thank you so much! I was able to work it out.
     
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