What is the Mantra of Polya?
The other day at dinner, someone showed me this video of a slinky dropping. It shoes that the bottom of the slinky stays perfectly stationary for a while after it’s been dropped. (The link goes to the 10-second interesting part).
I spent some time trying to figure out why that happens, but didn’t get it. The next day, I spent half an hour writing down the differential equations that describe the slinky’s motion and staring at them, with no idea how to proceed. Eventually, I watched the video again with sound, and learned the simple answer, which is that the speed of waves traveling in a slinky is very slow—a few meters per second—and the bottom half sits still until a wave can travel down and inform it that the slinky’s been dropped.
The strange thing is that I already knew this, or at least the idea was familiar to me. Also, while at dinner, someone mentioned the “pole-in-the-barn” paradox from special relativity, and mentioned the same speed-of-information-in-materials idea in resolving the paradox, but I still didn’t make the connection to the problem I was considering.
I want a simple phrase, similar to “check consequentialism”, “take the outside view”, or “worth it?” that applies to checking your own thought process while solving problems to stop you from revving your engine in the wrong direction for too long. I realized I’ve read a book about what to do in such situations. It’s George Polya’s How to Solve It. (Amazon Wikipedia Google Books) I don’t have a copy of the book anymore, and I would like to crowdsource creating a short phrase that captures the general mindset endorsed by it. Some questions I remember the book suggesting are
Have you seen a similar problem before?
What are the unknowns?
What information do you have?
Is it obvious that the unknowns are enough information?
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?”
I’m not really sure what you mean by “upward tension”, sorry. Tension in one dimension is just a scalar. The very bottom of the spring is under no tension at all, and the tension increases as the square root of the height for a stationary hanging slinky.
The tension gradient is upward, indicating an upward force per length.
By “upward,” I just meant to emphasize that it was opposing gravity, e.g., positive. But of course, now that I think about it for a minute, I see that I was wrong, it is under no tension at all. Oops.
I think I see what you mean. To clarify, though, tension doesn’t have a direction. In a rope, you can assign a value to the tension at each point. This means that if you cut the rope at that point, you’d have to apply that much force to both ends of the cut to hold the rope together. It’s not upward or downward, though. Instead, the net force on a section of rope depends on the change in the tension from the bottom of that piece to the top. The derivative of the tension is what tells you if the net force is upward or downward. This derivative is a force per unit length.
In general, tension is a rank-two tensor, and is just a name for when the pressure is negative.
Does the top of the slinky accelerate groundwards faster than gravity?
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.
Thanks for the tip.
The center of mass of the slinky accelerates at normal gravitational acceleration. The bottom of the slinky is stationary, so to compensate the top part goes extra-fast. I did a short calculation on the time for the slinky to collapse here http://arcsecond.wordpress.com/2012/07/30/dropping-a-slinky-calculation-12/
Good explanation—another way to think of it is that everything but the top of the slinky is exerting a tension on the top of the slinky which is acting in the direction of gravity (at least at the start). Hence the top of the slinky feels more force than just gravity and does accelerate downwards faster than with just gravity.
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?
Polya’s litany of questions is a toolkit for how to go meta. Is there a pithy or euphonious word that means “the act of going meta”?
Hofstadter sometimes uses JOOTSing for “Jumping Out Of The System”.
”Tear it apart”? “Think [or Solve]; then tear it apart.” “Make a thought baby then rip it to shreds”? Perhaps an implied ‘if you can’ followed by an evil laugh might pair well as a suffix.