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Sphere rolling inside cylinder - 3 dimensions

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data
    A sphere of radius r and mass m rolls without slipping inside a hollow cylinder of radius R. z direction goes along axis of cylinder.
    2irm5pl.jpg

    Determine the Lagrangian with motion in the z direction included

    2. Relevant equations
    I let θ be the angle of the sphere rotation along the cylinder curve, φ be the angle from the cylinder center to the center of mass of the sphere, and ψ be the angle of rotation in z.

    [tex](R-r)\theta=r\phi[/tex]
    [tex]I=\frac{2}{5}mr^2[/tex]
    [tex]z=r\psi[/tex]


    3. The attempt at a solution
    PE is easy.
    [tex]U=-mg(R-r)cos\phi[/tex]

    KE is harder... I think the translational KE is
    [tex]\frac{1}{2}m[(R-r)^2\dot{\phi}^2+\dot{z}^2][/tex]

    The rotational KE is troubling me... I want to say
    [tex]\frac{1}{2}I(\dot{\theta}^2+\dot{\psi}^2)[/tex]
    but I don't think that is right.

    Any help would be great! Thanks.
     
    Last edited: Sep 22, 2009
  2. jcsd
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