= Spring-stabilized chain over a table edge
{title2=$\omega^2=(k-\rho g)/(M+m+\rho L)$}
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.
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