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Homework Help: Classical mechanics: ball rolling in a hollow sphere

  1. Feb 22, 2008 #1
    [SOLVED] Classical mechanics: ball rolling in a hollow sphere

    1. The problem statement, all variables and given/known data
    This problem is from Gregory:

    A uniform ball of radius a and centre G can roll without slipping on the inside surface of a fixed hollow sphere of (inner) radius b and centre O. The ball undergoes planar motion in a vertical plane through O. Find the energy conservation equation for the ball in terms of the variable [tex]\theta[/tex], the angle between the line OG and the downward vertical. Deduce the period of small oscillations of the ball about the equilibrium position.

    So in summary, we have:
    a: radius of ball
    m: mass of ball
    [tex]\theta[/tex]: angle of the ball's position, relative to the vertical line connecting the center and bottom of the hollow sphere
    I: moment of inertia of ball
    [tex]\omega[/tex]: rotational velocity of ball
    T: kinetic energy of ball
    V: potential energy of ball (V=0 at height [tex]\theta[/tex]=[tex]\pi[/tex]/2, the center of the sphere)
    E: total energy of ball
    g: acceleration due to gravity

    2. Relevant equations
    I = 2/5ma^2

    3. The attempt at a solution
    First of all, I'm assuming that [tex]\omega[/tex]=[tex]\theta[/tex]'. It sounds intuitive, but I could be wrong there.

    I'm given, as a solution, that the period of small oscillation (that is, sin([tex]\theta[/tex])=[tex]\theta[/tex]) is 2[tex]\pi[/tex](7(b-a)/5g)^(1/2), which I'm not getting in my results. I have a very strong hunch that my mistake comes from bad energy equations. So, would you mind taking a look of these?

    T = 1/2mv^2 + 1/2I[tex]\omega[/tex]^2
    v = [tex]\omega[/tex]*(b-a)
    So T = 1/2m([tex]\omega[/tex]*(b-a))^2 + 1/2(2/5ma^2)[tex]\omega[/tex]^2
    T = m[tex]\omega[/tex]^2/10(7a^2-10ab+5b^2)
    V = -(b-a)mgcos([tex]\theta[/tex])

    So E = T + V = that stuff
    Am I correct here?
  2. jcsd
  3. Feb 22, 2008 #2

    Shooting Star

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    Homework Helper

    That is wrong. Draw a simple diagram to figure it out.
  4. Feb 22, 2008 #3
    Hah, it's always the little mistakes in the beginning that steal away an hour of my life.

    That fixed everything. Thank you so much for catching that.
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