Rings of same gravity center

Question

Using calculus of variations or otherwise, how do we find all non-circular ovals of loop length $ 2\pi $ in the plane with its center of gravity of arc at $ (0,0)? $

Answer 1

If by "oval" you mean something relatively general such as a smooth convex closed curve, there is a large supply of those meeting the conditions. Any curve symmetric around $(0,0)$ with the given perimeter would work.