Radial spring oscillator on a rotating guide (${\omega }_{\mathrm{osc}}^2=k/m-{\Omega }^2$)

A point mass constrained to a straight guide through the axis of a uniformly rotating reference frame, with signed central displacement $x$ and Hooke's law force $-kx$, satisfies $m\ddot{x}=-kx+m{\Omega }^{2}x$. The transverse Coriolis force is balanced by the guide. The central equilibrium is stable for ${\Omega }^2<k/m$ and unstable above it. At equality the restoring coefficient vanishes and $x(t)=x_0+v_{0}t$: a small initial velocity causes unbounded drift in the ideal guide. The threshold therefore has no positive oscillation frequency. The angular frequency is the positive square root in the title.