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Finding the Lagrangian for a wheel-pendulum system

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data

    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

    2qd3c08.jpg

    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 lenght ##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

    2. Relevant equations

    ##L = T- V##

    ##T = \frac{1}{2}mv^2 ##

    ## V = mgh ##

    3. 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 im starting to think that something might be wrong in the energys above, have I missed something?
     
  2. jcsd
  3. Dec 9, 2015 #2

    andrewkirk

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    I can see one thing that seems to be missing, which is the rotational kinetic energy of the wheel, which is ##\frac{1}{2}I\omega^2## where ##I## is the wheel's moment of inertia and ##\omega## is its rotational velocity.
    The problem is incompletely specified because it doesn't tell us the shape of the wheel - eg is it a solid disc or more like a ring? I suggest you assume it is a uniform disk, and state that in your answer.
     
    Last edited: Dec 9, 2015
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