Clearly the outline of solution for the force due to a ring distribution along the axis can be found in any elementary text on electromagnetism.
In particular, I am interested in the self attraction of the ring. From any point on the ring, all of the mass of the ring is towards the center of the ring. This should provide a central acceleration on the ring. I am wanting to solve for the rotation rate needed to balance this acceleration.
For simplicity, let's assume we are using units where G=1, the radius of the ring is 1 and the mass of the ring is 2pi. This gives a linear density, l, of 1. By symmetry the force must be central.
Let us consider the force felt by a point at an angle of 0 due to the mass at an angle of theta. We also need the distance to that point. I get:
Integrating this over the ring gives us an infinite value. Infinite is much larger than the finite values of a central mass, yet stable rings are found in nature.
Where is my mistake?