Particle in Stable/Unstable Motion, Find Frequency of Oscillation

[Oijl]

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$$\ddot{r}$$ = (ml$$_{z}$$$$^{2}$$)/(r$$^{3}$$cos$$^{2}$$($$\alpha$$)sin$$^{2}$$($$\alpha$$)) - cos($$\alpha$$)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$$_{o}$$ ± $$\epsilon$$ 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?

 

[ideasrule]

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?