Using Lagrange to solve rotating parabolic motion and equilibrium

  • #1

Homework Statement



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.




Homework Equations




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



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:

Answers and Replies

  • #2
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If the bead remains fixed, what can be said about the derivatives of its polar radius?
 
  • #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?
 
  • #4
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390
Have you studied any equilibrium conditions/criteria?
 

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