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Homework Help: Can someone check my Lagrangian ?

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Write down the Lagrangian for a cylinder mass m, radius R an moment of inertia I, that rolls without slipping straight down an inclined plane which is at an angle a from the horizontal. Use as your generalized coordinate the cylinder's distance x measure down the plane from its starting point. Write down the Lagrange equation and solve it for the cylinder's acceleration.





    3. The attempt at a solution
    I tried to find the Lagrangian, this is what i have so far, is it right?
    i said that T=1/2mv2+1/2Iw2
    and U= -mgxsina so my Lagrangian would be
    L=1/2mv2+1/2Iw2+mgxsina.
    I think this is correct, but I'm not positive. If it is right then should i write I and w (omega) differently? and then continue with the partial differentiation?
     
  2. jcsd
  3. Dec 7, 2008 #2
    Everything is correct, except that you must express all the dynamical variables (v, w) in terms of x and its derivatives, as the problem tells you. Also you can explicitly specify the value for I, since you know the shape of the body.
     
  4. Dec 7, 2008 #3
    so if i remember right v=x' but i am not sure what I and w are in terms of x and derivatives. doesn't I have something to do with R, the radius?
     
  5. Dec 7, 2008 #4
    w is the angular velocity, which for non-slipping case is given by

    [tex]\omega = \frac{v}{R} = \frac{\dot{x}}{R}[/tex]

    and I hope you can figure out the moment of inertia I yourself.
     
  6. Dec 7, 2008 #5
    I= 1/2 m R2?
    Just checking before I go on with the partial derivatives.
     
  7. Dec 7, 2008 #6
    Yes, for a uniformly filled cylinder.
     
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