The falling slinky

It is an interesting and counterintuitive fact that a Slinky released from a hanging position does not begin to fall all at once but rather each part of the Slinky only starts to fall when the collapsed part above it reaches its level. The analyses published so far have given physical arguments to explain this property of Slinkies. In particular, they have relied on the fact that a perturbation to a Slinky travels through the Slinky as a wave and therefore has a certain propagation speed. Releasing a Slinky that was being held at the top is a perturbation at the top, and it takes time for that perturbation to propagate downward. This "high-level" analysis is, of course, correct. But, it is also interesting to analyze the dynamics from a purely mathematical perspective. We present such a careful mathematical analysis. It turns out that we can derive an explicit formula for the solution to the differential equation, and from that solution, we see that the effect of gravity exactly counteracts the tension in the Slinky. The mathematical analysis turns out to be as interesting as the physics.