2-d equations of motion for a Slinky going down stairs?

[benorin]

This problem fascinated me in lower division physics. Find the 2-d equations of motion for a Slinky going down a flight of stairs (assuming the path of the slinky is planar; eg only going up and down and front and back, no side to side). I do confess that whilst I do love physics I’m not terribly good at it, better with math. It’s been 20 years since physics but I still remember $F=-kx$ seems a good place to start, or was it $F=\tfrac{d\vec{p}}{dt}$? We’ll need some initial displacement of the slinky too. I’ll need some clues as to how to proceed... we’re going to wind up with a tweaked cycloid I bet?

 

[phinds]

we’re going to wind up with a tweaked cycloid I bet?

I certainly would not expect that to be the case. Think about it. The center of mass, which is what you have to be looking for the motion of, is going to be static for noticeable periods while the top of the slinky moves forward and then again briefly as the top catches up on the next step down.

 

[klotza]

Slinkies can actually get super complicated, math-wise.
This paper has the most rigorous treatment that I know of, and includes discussions on the shapes formed by a slinky under various boundary conditions: https://arxiv.org/pdf/1403.6809.pdf

This paper from the American Journal of physics uses a simplified model of the slinky and a Lagrangian formalism to look at its stair-hopping behavior: https://aapt.scitation.org/doi/full/10.1119/1.3225921