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Math Help - lagranian mechanics

  1. #1
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    lagranian mechanics

    Hey

    Going to be honest I am struggling to get this question done. Havent been keeping up with my study and I practically have no idea how to do this question. But Im looking through some notes and trying to work it out. But if you are willing to help I would really appreciate it!

    Cheers

    Ed
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  2. #2
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    Quote Originally Posted by edeffect View Post
    Hey

    Going to be honest I am struggling to get this question done. Havent been keeping up with my study and I practically have no idea how to do this question. But Im looking through some notes and trying to work it out. But if you are willing to help I would really appreciate it!

    Cheers

    Ed
    I presume the problem is in setting up the Lagrangian? I will use two generalized coordinates: x and \theta. Take a look at the mass M first. I am setting the 0 point for the GPE to be at the equilibrium position of the spring. So
    V = -Mgx
    and
    T = \frac{1}{2}M \dot{x}^2

    The swinging mass m is a bit more of a problem. The potential energy is easy enough:
    V = -mg(x + y~cos( \theta ))
    where y is the length of the string.

    For the kinetic energy we have the downward motion of the mass m to contend with as well as the radial motion of the swing. So we have two terms:
    T = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}my^2 \dot{ \theta } ^2

    The Lagrangian will be total T minus total V:
    L = \frac{1}{2}M \dot{x}^2 + \frac{1}{2}m \dot{x}^2 + \frac{1}{2}my^2 \dot{ \theta } ^2 + Mgx + mg(x + y~cos( \theta ))

    -Dan
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