How does circular motion in a cone without gravity work?

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How does circular motion in a cone without gravity work?
If I have an object and I let it circle through the inside of a cone (with gravity) the two forces that act on the object are gravity and the support force. The resultant of the two forces lies on the plane of the circular motion and is therefore the provides the centripetal force to keep the object in a circular motion. This I completely understand

But what if I do the same thing without gravity? would the object fly out of the cone? Would it continue in a circular path? What are the forces that act on the object and how do they affect the resultant?

 

[andrien]

the path under no forces when only the reaction and may be friction is acting then it is a geodesic on a surface of revolution.

 

[Cleonis]

How does circular motion in a cone without gravity work?

Let's use a cone that is infinitely large.

Let the sliding object start in circular motion, perpendicular to the main axis of the cone. The object is released to free sliding motion along the cone.

The object will move away from the central axis of the cone. As the object moves away from the central axis the angular velocity around the central axis decreases all the time, and the velocity component away from the central axis increases all the time. In other words, rotational kinetice energy will convert all the time to kinetic energy of linear motion away from the central axis.

My hunch is that the angular velocity will appoach zero, just as in the case of motion over a flat surface. The sharper the cone the longer it will take to get to the point of approaching zero angular velocity, but I think that for every cone the angular velocity will eventually approach zero.