(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

[tex]\omega=\sqrt{\frac{2g}{a+z_0}}[/tex]

2. Relevant equations

3. The attempt at a solution

I'm aware that the [tex]\omega[/tex] must come from the equation:

[tex]\ddot{x}+\omega x=0[/tex]

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

The only definition ofggoes to [tex]\frac{MG}{R^2}[/tex], so we must be looking for a force [tex]F=-\frac{mMG}{R^2}*\frac{2}{a+z_0}[/tex]

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

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

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

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# Homework Help: Small oscillations of constrained particle

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