Particle in Stable/Unstable Motion, Find Frequency of Oscillation

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Homework Statement


A particle moves around the surface of an upside-down cone, in a horizontal circular path, in equilibrium. The particle is given a small radial kick. Use the Lagrangian equation for motion (found in a previous section of this problem):

m[tex]\ddot{r}[/tex] = (ml[tex]_{z}[/tex][tex]^{2}[/tex])/(r[tex]^{3}[/tex]cos[tex]^{2}[/tex]([tex]\alpha[/tex])sin[tex]^{2}[/tex]([tex]\alpha[/tex])) - cos([tex]\alpha[/tex])mg

to decide whether the circular path is stable. If so, with what frequency does r oscillate about the equilibrium?


Homework Equations





The Attempt at a Solution



If I can put in r[tex]_{o}[/tex] ± [tex]\epsilon[/tex] for r in that equation and show that the right side is positive when epsilon is negative and negative when epsilon is positive, then I will have shown that it is stable.

But I don't know how to do that. Also, when I can show that it is stable, how should I go about knowing the frequency of oscillation?
 

Answers and Replies

  • #2
ideasrule
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You know that at some value of r, the left side is equal to 0. What happens to the right side if you make r bigger? Does the second derivative of r become negative or positive?
 

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