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A ball rolling on a bowl

  1. Sep 2, 2007 #1
    Ok, I've given this question already soooooo much time and I simply cannot solve it.
    1. The problem statement, all variables and given/known data
    There's a circular surface that's holding still (like a bowl), with radius R, and a ball, with radius r, on it.
    The ball is rolling without sliding.
    The mass of the ball and it's moment of intertia are given.
    1. First question - express the relation between W, which is the angular velocity of the ball, and "d(theta)/dt", where "theta" is the angle formed in any time between the "main axis of the bowl" and the radius streaching to the ball.
    2. Second question - find the relation between W and theta(not theta dot as before), given that the ball starts it's movement at hight "h" above the surface.
    3. Third and last - given that the ball is oscillating in small values of theta, what is the period time of the harmonic movement? (they give a hint: differentiate (with t) the function we found in the last questions - W(theta), find a "movement equation", and compare it to the "harmonic oscillation" classic equation.


    3. The attempt at a solution
    1. I Think I did that - w(t)* r = (theta dot) * R.
    2.Ok, here I used mechanical energy cons. and after some effort found a pretty complex relation between W and theta. I won't specify it here but it has square root and all :)
    3. Here's the real trouble:
    From their hint I think I need to find d(W(t))/dt ? So I can do that using the "chain law" (I'm not sure if that's the name) and take the derivitive of t like this:
    d(W(t))/dt = d(W(theta))/d(theta) * d(theta)/d(t)
    After doing that, using the relations I got in the previous questions, I get a non-linear, second-order diffrential equation. Not solvable of course.

    The excercise is a pretty classic one, just a plate with a ball rolling on it - but I still find it very complicated! Maybe I'm over-complicating things?

    I'm desprate for help :-\

    Thank you very much for reading.
  2. jcsd
  3. Sep 2, 2007 #2


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    Homework Helper

    Since they are referring to small [tex]\theta[/tex], approximate [tex]sin\theta\approx\theta[/tex] and [tex]cos\theta\approx1[/tex].

    Then you get a linear diff. equation. It should the equation for a simple harmonic oscillator.
    Last edited: Sep 2, 2007
  4. Sep 2, 2007 #3
    I know that. I've tried that. Still not a linear diff. equation.

    If I must, I'll post my calculations...
    However, it's so hard to write them in here...

    Thanks anyway.
  5. Sep 2, 2007 #4


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    Homework Helper

    I'm getting a linear differential equation... there's a major cancellation (simplification) that happens when you substitute in [tex]\frac{d\theta}{dt}[/tex] into your equation for [tex]\frac{d^2\theta}{dt^2}[/tex]

    You can get [tex]\frac{d\theta}{dt}[/tex] in terms of [tex]\theta[/tex] using your 2 equations for [tex]\omega[/tex]... the one you get in the first part, and the one you get in the conservation of energy part.
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