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.