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AbigailM
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Homework Statement
Consider a bead of mass m constrained to slide without friction along a rigid wire that rotates about the vertical at a fixed angle [itex]\theta[/itex] with constant angular velocity [itex]\omega[/itex]. Write down the Lagrangian in terms of z as the general coordinate. Find the equation of motion of the bead, and determine whether there are positions of equilibrium. If there are equilibrium positions, are they stable?
Homework Equations
[itex]z=rcos\theta[/itex]
[itex]U=mgz[/itex]
The Attempt at a Solution
[itex]\dot{r}=\frac{\dot{z}}{cos\theta}[/itex]
[itex]T=\frac{1}{2}m(\dot{r}^{2}+r^{2}\dot{\theta}^{2})[/itex]
[itex]L=\frac{2}{3}m(\frac{\dot{z}^{2}}{cos^{2}\theta}+(z\omega)^{2}/(cos^{2}\theta}))-mgz[/itex]
[itex]m\ddot{z}=2z\omega^{2}-mgcos^{2}\theta[/itex]
[itex]\ddot{z}=\frac{2z\omega^{2}}{m}-gcos^{2}\theta[/itex]
Just wondering if my equation of motion is correct. Thanks for the help
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