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## 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 d

^{2}p/dt

^{2}=0 and d

^{2}U/dp

^{2}=0.

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

pw

^{2}-4k

^{2}p(dp/dt)

^{2}-2kgp=(d

^{2}p/dt

^{2})(1+4k

^{2}p

^{2}

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