Spring-stabilized chain over a table edge (${\omega }^2=(k-\rho g)/(M+m+\rho L)$)

Attach the table-end mass to a linear spring of spring constant k. Adding spring potential energy $k(e_0+x{)}^2/2$ gives restoring curvature $k-\rho g$ about equilibrium. The positive gravitational feedback from increasing hanging chain length subtracts from the spring stiffness. Stable oscillations require $k>\rho g$; equality is neutral in this ideal linear model, while $k<\rho g$ gives exponential departure. The frequency is independent of the preload, although the equilibrium position is not.