In continuation of the thread:
Analytical expression for varying center of mass
I would like to find the optimal center of mass of a centroid. With this I mean the point where I achieve maximum change in center of mass.
In other words I would like to attain a center of mass, furthest away from origon/center of rotation, where I still obtain a significant change in center of mass by increasing $\theta$.
It seems that you just remove an arc of length $0<\ell<2\pi$ from the given circle of radius $1$. It is obvious that when $\ell$ is almost $2\pi$ then the remaining short arc will have its centroid $c$ near the circumference of the full circle, and in the limit we have $\lim_{\ell\to 2\pi-}|c|=1$. But there is no nondegenerate arc for which $|c|=1$ is realized.