What is the oscillation frequency of a marble in a parabolic bowl?

[cheimdal]

Homework Statement

A marble is oscillating in a parabolic bowl ( y = ax^2 ) Assuming mechanical energy is conserved, and that the oscillation amplitude is small, find the oscillation frequency.
(Use the potential energy to find the force acting on the marble as a function of x)

Homework Equations

Well obviously EP = mgy = mgax^2
small oscillations: sin t can be approximated by t.

I can't figure out how to put up the differential equation correctly.

 

[berkeman]

Homework Statement

A marble is oscillating in a parabolic bowl ( y = ax^2 ) Assuming mechanical energy is conserved, and that the oscillation amplitude is small, find the oscillation frequency.
(Use the potential energy to find the force acting on the marble as a function of x)

Homework Equations

Well obviously EP = mgy = mgax^2
small oscillations: sin t can be approximated by t.

I can't figure out how to put up the differential equation correctly.

I'm not sure you can discount the energy in the rotation of the rolling ball. At least at first, I think you should include it in your equations. Then you may be able to neglect it for small angles, but I'm not sure without seeing the equation written out...