I have not known that this toy has a name at all. The ‘spring’ that can walk down the stairs is called Slinky:
We all know how the Slinky walks – but how does it fall?
This video might come as a surprise!
The authoritative article on The Falling Slinky is this one: Modelling a Falling Slinky – replacing it with a chain of masses connected by massless springs and calculating the trajectories of bottom, top and center of mass.
Edit: On replying to a comment, I searched for some written explanation by the professor featured in the video: And I found this great scientific paper an arxiv! The wave-like travelling of the “information that tension has collapsed” (as explained in the video) is put into equations.
I believe you could also explain it in the following way:
(1) If you are falling down you feel weightless – gravity is equivalent to acceleration.
You have for sure seen images of cartoon Einstein (the creator of General Relativity) in an elevator falling down in empty space (You find those images close to those cartoon spaceships emitting pulses of light from their cone ends)
(2) So the Slinky is not subject to gravity when it falls but the elastic force will contract it in the same way a Slinky stretched along a table would do.
Now we have to decide on the absolute position of the top or bottom of the Slinky: If it contracts will the bottom move up to the top or the other way round? Or will bottom and top move to the center? I think here we have to resort to considering the center of mass: From the observer’s frame of reference the COM needs to fall down as a point particle with the same mass. The top starting to move will make the COM ‘fall down’ and make the Slinky contract.
Bonus Material: ‘Making of’ This Blog Post
Actually I came to this conclusion after playing with another thought experiment that did not work out well in the end.
I imagined the connection between the individual segments of the slinky becoming weaker and weaker until the Slinky ends up as a pile of separate rings. The rationale for this was that a Slinky is quite a weak spring – but in the explanation given above some restoring force is crucial.
The rings would be connected by strings that just keep the whole thing from falling apart. The strings would not be elastic. Thus there is hardly any elastic force, they wouldn’t be any oscillations when the bottom of the ‘string Slinky’ has been released.
Now all the rings except the top ring are suspended – each ring connected to its superior. When the top ring is released it starts to fall – there would be no restoring force. I expect this thing to fall without any change in shape – in contrast to the Slinky – if the experiment is done in vacuum and everything is balanced carefully. In real live it would rather twist and tumble.
Probably I should test that at Christmas time – connecting ring-shaped cookies with silvery or golden yearn would be both decorative and very close to my mental model.