The following is a useful Mechanics 3 level problem involving a particle in equilibrium whilst undergoing rotational motion. I give my suggested solution below.
I sketch the relevant forces in the following diagram:
Since the particle is in equilibrium, the centripetal and centrifugal forces must be equal so resolving towards the central axis of the cone we must have
Since there is no motion up or down the inner surface of the cone we must also have (resolving downwards along the inner surface)
Using (1) and (3) in (2) we get
Using (5) in (4) gives