Lifting a chain

A long, uniformly dense chain is lifted from the floor using a mass and a pully.

How will the system behave when the mass is released?

This is problem 28 from Carr (1991).

The parameters and state variables for the system are

p
the density of the chain
m
the mass used to lift the chain
g
acceleration due to gravity
x
the height of the top of the chain
v
the rate at which x changes.

The pulley is frictionless, and the cable between the chain and the mass is weightless. The net lifting force on the system is

F = [m - p x] g.

For a system with varying mass, Newton's law is

F = d[M v] / dt = M v´ + v M´

where M is the total mass of the system. For the chain problem we have

M = m + p x

and

M´ = p x´ = p v

and so

[m - p x] g = [m + p x]v´ + p v².

The system is modelled by the state equations

x´(t) = v
v´(t) = { [m - p x] g - p v² } / [m + p x]

The animation shows the behavior of the system when d = 1, m = 1, g = 9.8, x(0) = 0 and v(0) = 0. The motion of the system was calculated using a Runge-Kutta method.