What is the Mantra of Polya?

A way to think about this problem that clarified it for me: The top of the slinky is experiencing tension equal to its whole weight. The bottom is experiencing upward tension equal to its weight, so at the instant when it is dropped, it experiences no net force - until the upper parts of the slinky start traveling downward, removing the upward tension. (The speed-of-information-in-materials wasn't really convincing to my intuition, because it feels like, since the bottom of the slinky is in a gravitational field , it should already "know" it is falling... )

For a mantra: How about something like, "What do you know? What don't you know? How do you connect them?"

Does the top of the slinky accelerate groundwards faster than gravity?

I'm not sure how to pronounce "Polya"

Tip: Use Google Translate to find pretty-good pronunciations of foreign names. Set the source language to the one the name comes from (Hungarian, in this case), type the name with the right accent marks (Pólya), and click the speaker button in the bottom right of the box.

A few thoughts about my own experience with the slinky problem.

I recall that it took me something like an hour to figure out that the video is not a trick and what the underlying physics was. I could be wrong on the time frame, though, as it was about a year ago.

I did not write any differential equations, however, just had to make the connection between the fact that the bottom was not moving and the conclusion that the top was essentially a shock wave. After that all that was left is to verify that the longitudinal slinky waves (not sound waves) are indeed slow enough for the shock to form very quickly.

Now, had I read the Polya book and applied the mantra, would I have found the solution any faster? I doubt that, though there is no way to check. I was quite familiar with shock waves already, as well as with the pole-in-the-barn paradox, having had to explain the latter on IRC many times. However, I did not pay attention to the fact that the contraction of the pole after its front hits the gate is due to a shock wave, not a sound wave, the latter being much too slow in relativistic circumstances. Why did I miss that? Probably because it was not essential to resolving the paradox, as once you realize that there are no rigid bodies in relativity, the paradox goes away.

I suspect that I internalized the "have you seen a similar problem before?" approach as much as I could already, but not expecting to see shock waves on such a slow time scale delayed the realization significantly.

I don't know if it may help develop a helpful phrase, but another thing to keep in mind is that the link between what information you have and the problem you want to solve is often not obvious. You often need to play around with the information before you can figure out how it can be used to solve the problem.

And the complexity of real world problems can confuse the issue even more, so it helps to try to simplify or generalize the problem, so you can see what the core of the problem actually is, first.

Reminds me of a student of Feynman who couldn't tell what happens to the image of a book viewed through a slab of glass kept at an angle, even though a few minutes earlier he had written the equations describing what happens to light passing through a layer of a transparent substance with a given index of refraction.

Polya Principle? Paradigm of Polya?