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A box on a moving wedge, on an inclined plane

  1. Apr 29, 2012 #1
    1. The problem statement, all variables and given/known data
    The situation is indicated in the diagram below. Block m is at rest, and the block M has an initial velocity upward. Need to find the Lagrangian of this system, and then the Euler-Lagrange equations.

    3. The attempt at a solution

    This is the Lagrange that I came up with from the beginning, but it seems like I am oversimplifying it:

    [itex]T=(1/2)m(\dot{x_{1}}^{2}+\dot{y_{1}}^{2})+(1/2)M(\dot{x_{2}}^{2}+\dot{y_{2}}^{2})[/itex]

    [itex]V= mgy_{1}cos(\beta) + Mgy_{2}cos(\alpha)[/itex]

    [itex]L=(1/2)m(\dot{x_{1}}^{2}+\dot{y_{1}}^{2})+(1/2)M(\dot{x_{2}}^{2}+\dot{y_{2}}^{2}) - mgy_{1}cos(\beta) - Mgy_{2}cos(\alpha)[/itex]

    Like I said, this seems over simplified and I feel like there should be some cross terms in the kinetic energies. The second attempt that I have made is try to come up with velocity vectors. In this case I have the velocity of the wedge with respect to the incline [itex]v_{w}=\left(\partial _tx_2\right)\hat{i}-\left(\partial _ty_2\right)\hat{j}[/itex] and the velocity of the box with respect to the wedge, with respect to the incline [itex]v_{B}=\left(\partial _tx_2\right)\hat{i}-\left(\partial _ty_2\right)\hat{j}+\left(\partial _tx_1\right)\hat{i}'-\left(\partial _ty_1\right)\hat{j}'[/itex].


    Geometry has always been an extreme weakness of mine. It seems like I am approaching this in all of the wrong ways. Any suggestions would be helpful.
     

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    Last edited: Apr 29, 2012
  2. jcsd
  3. May 2, 2012 #2
    You have to write your generalized coordinates at first, then you would have some cos sine dependence in the velocity. That is because the box is constrained to move in a certain way and by not introducing gen. coordinates you would simply get newtons 2nd law.
    Also your potential energies have different zero points?
    Define a coordinate system
     
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