 Chain sliding over a table edge

An inextensible uniform chain of length L and linear mass density $\rho$ links a mass m sliding on a frictionless table to a hanging mass M. If the initial hanging chain length is zero and x is its subsequent hanging length, all moving material has speed $\dot{x}$. The kinetic energy is $(M+m+\rho L){\dot{x}}^2/2$, while the gravitational potential energy is $-Mgx-\rho g{x}^2/2$. The Euler-Lagrange equation gives the displayed equation until a mass hits the floor or reaches the edge. Increasing hanging chain weight produces a hyperbolic cosine displacement from rest, rather than uniform acceleration.