Particle in gravitational field

[Math Jeans]

1. The problem statement, all variables and given/known data

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 circular orbit with radius \rho=\sqrt{4az_0} is

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

2. Relevant equations



3. The attempt at a solution

The problem that I'm having is that I don't understand the wording of the question?

How do I draw out this scenario?

[tiny-tim]

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 circular orbit with radius \rho=\sqrt{4az_0} is

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


Hi Math Jeans! :smile:

It means that the ball is freely rotating (I don't know why they call it oscillating) around a horizontal circle at height z0 (so the radius is √4az0).

I think the word "small" means that you can pretend that eg sinx = x.

Use Newton's second law to find the angular velocity, and therefore the frequency of the rotation. :smile: