Let’s say, for easy of discussion, the particle in question is on the x axis, at positive position “r”. We will arbitrarily have it orbit counter-clockwise.

The direction of orbital velocity, at this moment, will be straight up, parallel to the y axis. The magnitude of this velocity is what we need to determine.

We can cut the disk along this velocity direction line into two sections; the larger section (containing the center) will exert a net force towards the center “Fleft”. The smaller section will exert a net force directly away from the center, “Fright”.

We can get the total force towards the center Fc = Fleft-Fright (since Fleft contains the center, we know Fleft>Fright)

Once we have Fc, we can compute the orbital velocity.

Fleft will consist of a sum of all the forces for every point in that section of the disk.

Fright will consist of a sum of all the forces for every point in that section of the disk.

I suspect that computing the sum of forces for those points will involve integration, but I’m just not sure how to proceed.