Solve LaGrange Problem: Rod of Length L & Disk of Radius R

[don_anon25]

We have a rod of length L and mass M pivoted at a point at the origin. This rod can swing in the vertical plane. The other end of the rod is pivoted to the center of a thin disk of mass M and radius R. Derive the equations of motion for the system.

I have attached a drawing :)
If you can't tell from my drawing, the angle between the vertical and L is theta and the disk has radius R and angle of rotation phi. The rod of length L swings -- the disk swings and rotates. L cos theta would thus be the vertical component...


For the kinetic energy of the system, I have 1/2 M [(L cos theta - R) theta dot]^2 + 1/2 (I of Rod) (theta dot)^2 + 1/2 (I of disk) (phi dot)^2.
Theta dot is the rate of change of angle theta, phi dot is rate of change of angle phi.
Is my generalized term for velocity [(L cos theta - R)*theta dot] correct? I'm a bit uncertain of it. If not, what should it be?

For potential energy of the system, I have U = (L cos theta - (L cos theta - R) * cos theta] mg. Is this correct?

I think I also have an extra constraint: (L Cos theta - R) * theta - R* phi =0. This is when the disc is at its lowest position.

This leads me to believe that I need to use undertermined multipliers to solve this problem. I suppose I will have two Euler-Lagrange equations, one for Theta and one for Phi.

Of course, this may be all wrong...I am terribly confused by this problem. Perhaps someone could refer me to a helpful resource? Or perhaps explain what the kinetic and potential energies should be?

Thanks for help in advance! Everyone on this website is absolutely wonderful when it comes to homework help!

 

[Spectre5]

Look at your equations for the velocity and potential energy, they are incorrect. Look at the disk and rod completely separately for this problem. Assume a homogenius rod and a homogenius disk (unless otherwise specified) and then you the velocity of the center of mass for the rod and disk separatly for the kinetic energy. Also, in your constraint where the disk is at its lowest position, theta = 0.