- #1
alivedude
- 58
- 5
Homework Statement
Ok so I need to find the Lagrangian ## L ## for this system below, I have drawn some poor sketch in paint but I think its pretty easy to see what i mean
Its a wheel with mass ##m## and radius ##r## that rolls inside a big cylinder with radius ##R## and at the center of the wheel there is a pendulum attached. The pendulum has a length ##l## and a mass ##m## (same ##m##) attached at the end of it and the string can be threated as massless. The angles from equilibirum is ##\theta_1## and ##\theta_2##.
I think that is all that is needed
Homework Equations
##L = T- V##
##T = \frac{1}{2}mv^2 ##
## V = mgh ##
The Attempt at a Solution
OK so this is what I have been thinking so far.
First of I did the substitution ##l_1 = R-r ## and ##l_2=l##
We have calculate the kinetic and potential energy for each mass separatly so for the wheel we have
##
T_1 = \frac{1}{2}ml_1^2 \dot{\theta_1}^2 \\
V_1 = -mgl_1cos\theta
##
and for the pendulum there is a little bit more tricky
##
T_2 = \frac{1}{2}m[\frac{d}{dt}(l_1sin \theta_1+ l_2 sin \theta_2)]^2+\frac{1}{2}m[\frac{d}{dt}(-l_1cos \theta_1- l_2 cos \theta_2)]^2 \\
V_2 =-mg(l_1cos\theta_1+l_2cos \theta_2)
##
After this I form ## L = T_1 + T_2 - V_1 - V_2 ## and just carry out the algebra. But when I use this later on for Lagrange equations of motion I get some factors wrong in the answe. The dimensions and everything else is right and I have checked so many times now so I am starting to think that something might be wrong in the energys above, have I missed something?