Conservation Laws and the Movement of Particles around a Pole

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

First and foremost I am not sure about how the particle moves around the pole .Can it keep moving in horizontal circle .Or do we have to assume this .

Is it something like a conical pendulum ?

I don't have much ideas to solve this problem . Perhaps I need to conserve angular momentum of the particle about the pole . Is that so ?

 

[haruspex]

Perhaps I need to conserve angular momentum of the particle about the pole .

That is one to consider. What conditions are needed for its a.m. about the pole to be conserved? Does the pole have some thickness? Does that matter?
What other conservation law might be applicable? Any reasons that might not be valid?