Small oscillations of constrained particle

[Math Jeans]

Homework Statement

Consider a particle of mass m constrained to move on the surface of a paraboloid whose equation (in cylindrical coordinates) is r^2=4az. If the particle is subject to a gravitational force, show that the frequency of small oscillations about a cirrcular orbit with radius \rho=\sqrt{4az_0} is

\omega=\sqrt{\frac{2g}{a+z_0}}

Homework Equations

The Attempt at a Solution

I'm aware that the \omega must come from the equation:

\ddot{x}+\omega x=0

This DiffEq comes from F=m\ddot{x}, so we need F=-\frac{2gm}{a+z_0}.

The only definition of g goes to \frac{MG}{R^2}, so we must be looking for a force F=-\frac{mMG}{R^2}*\frac{2}{a+z_0}

My attempt is based on looking at the radial component of a gravitational force pulling out of the plane along \hat{s}, however, every time, I get:

\vec{F}=-\frac{mMG}{S^2}\hat{s}, where the radial component would be \vec{F}\bullet\hat{r}


In a nutshell...my answer keeps turning up wrong.

 

[mukundpa]

The particle is rotating about z axis and in equilibrium at z = z0. First consider equilibrium of particle in the circular orbit and then think of its oscillation on the surface.

 

[Math Jeans]

The particle is rotating about z axis and in equilibrium at z = z0. First consider equilibrium of particle in the circular orbit and then think of its oscillation on the surface.

That's exactly what I'm doing, but I can't get the correct answer.

 

[Math Jeans]

I'm going to make a late bump on this thread because I still haven't solved this problem.