Lagrangian mechanics problem - check my work?

[bcjochim07]

Homework Statement

A cart of mass M is attached to a spring with spring constant k. Also, a T-shaped pendulum is pinned to its center. The pendulum is made up of two bars with length L and mass m. Find Lagrange's equations of motion.

I've attached the figure and my solution as a PDF. I let y=0 be the horizontal line through the center of the cart and let x be the displacement of the spring from equilibrium. I'm pretty sure that I can get Lagrange's equations if I have the correct Lagrangian, so that's the part I'd like you to check. Ah, yes, and I just noticed that the sines in my expression for U should be cosines. Other than that, is my Lagrangian alright? Or do I need to include a translational KE of the center of mass of the upper bar of the pendulum attached to the center of the cart?

Homework Equations

The Attempt at a Solution

 

[jambaugh]

Your pdf file is "pending approval"

WRT your question about the KE, note that if the car moves but the pendulum swings the opposite way you should have less KE than if the car and pendulum swing together. So it is better not to try to naively add the KE component for the pendulum's moment of inertia about the pivot. Rather:

1.) You should have a KE term for the car itself,
2.) A rotational KE term for the rotation of the pendulum about its own center of mass, and
3.) A KE term for the motion of the pendulum's center of mass in both x and y directions as a function of the car position and angle of swing (and of course their rates of change).

If the car were fixed you could take care of 2 and 3 together by taking the moment of inertia about the stationary pivot point instead of about the center of mass. Since the car moves you should work these separately.

Another option is to introduce a Lagrange multiplier term reflecting the constraint that the pendulum is attached to the car, (actually two terms for x and y components). Then you can treat the pendulum as a free rotating T (subject to gravity). This adds some complexity in the method but it does also give you the forces of constraint (if you needed them) as the values of the Lagrange multipliers and you avoid having to worry about issues like whether and how to include translational components of the pendulum.

I'm just now appreciating Lagrange multiplier methods more lately. You just treat each piece of a mechanism like this independently and then tack in Lagrange multipliers time the constraints holding the pieces together. http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html"

 

[bcjochim07]

Ok, so I think anyone should be able to open the PDF now.

Here are the contributions I got for KE:

1) the car's KE

2) the rotational kinetic energy of the upper bar of the pendulum; where
KE = 1/2*I*(thetadot^2) and I = 1/3mL^2.

3) the kinetic energy of the bottom part of the pendulum (the center of mass of this piece traces a circle, so I said KE = 1/2m*l^2)

Is it valid to split the pendulum and analyze its two pieces separately?