 Circular-orbit energy ratio in a power-law potential

A circular orbit in a central potential $U(r)=A{r}^s$ requires inward force $U^{\prime }(r)=m{v}^2/r>0$. Thus $2K=r{U}^{\prime }(r)=sU$, giving $K/U=s/2$ for the specified zero of potential. The ratio depends on that choice of zero; adding a constant to $U$ leaves the orbit unchanged but changes $K/U$. This is the circular special case of the virial theorem.