Centrifugal strain of a thin elastic hoop ($\Delta r/r\simeq \rho {\omega }^2{r}^2/E$)

A small arc of a rotating thin hoop needs inward force per length $\rho A{\omega }^{2}r$. Its circumferential tension supplies this through curvature, giving tension $\rho A{\omega }^2{r}^2$ and tensile stress $\rho {\omega }^2{r}^2$. Hooke's law converts that stress to circumferential strain, which equals fractional radius change. Small strain, a slender cross-section and uniaxial linear elasticity are required; the cross-sectional area cancels at leading order.