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Using Lagrange to solve rotating parabolic motion and equilibrium

  1. Mar 4, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be z = kp2. Write down the Lagrangian in terms of p as the generalized coordinate. Find the equation of motion of the bead and determine whether there are positions of equilibrium, that is, values of p at which the bead can remain fixed, without sliding up or down the spinning wire. Discuss the stability of any equilibrium positions you find.




    2. Relevant equations


    L=E-U
    v1=pw
    v2=dp/dt
    v3=2kp(dp/dt)
    U=mgkp^2
    E=mv^2/2



    3. The attempt at a solution

    I found the equation of motion using the lagrangian, which matched the answer in the back of the book. I just don't know how to use Lagrance to find equilibrium, and I've tried several things such as setting d2p/dt2=0 and d2U/dp2=0.


    The equation of motion is: (g=accel. due to gravity)

    pw2-4k2p(dp/dt)2-2kgp=(d2p/dt2)(1+4k2p2
     
    Last edited: Mar 4, 2014
  2. jcsd
  3. Mar 4, 2014 #2
    If the bead remains fixed, what can be said about the derivatives of its polar radius?
     
  4. Mar 4, 2014 #3
    dp/dt=0, d2p/dt2=0

    so equil. occurs when w2=2kg, or when p=0. Is that right? And then how can I find which is stable/unstable?
     
  5. Mar 4, 2014 #4
    Have you studied any equilibrium conditions/criteria?
     
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