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jimz
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Homework Statement
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.
Determine the Lagrangian with motion in the z direction included
Homework 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]
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.
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