# Particle in Stable/Unstable Motion, Find Frequency of Oscillation

## 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?

## 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
Homework Helper
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?