# Homework Help: Sphere rolling inside cylinder - 3 dimensions

1. Sep 22, 2009

### jimz

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.

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.

$$(R-r)\theta=r\phi$$
$$I=\frac{2}{5}mr^2$$
$$z=r\psi$$

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

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

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

Any help would be great! Thanks.

Last edited: Sep 22, 2009