# ∫ A Real Stumper of a Puzzle

December 10, 2010 2 Comments

Here’s your chance to try to outsmart the vasy majority of the Stony Brook physics department (including a Dirac Medalist, several Guggenheim fellows, and so on). At our colloquium on Tuesday, a speaker from UMass-Boston, Dr. Arthur Eisenkraft, gave a talk about physics education strategies, and along the way posed the following problem to the audience: consider a pendulum which consists of a large bathtub (filled with water) swinging back and forth. Now imagine the pendulum is positioned next to a faucet, such that every time the tub swings back to the right, it comes under the faucet and a little water is added to the tub before it swings away:

Ignore the fact that the force of the water pouring in will disturb the oscillations a little bit, and the fact that the water will slosh back and forth. Imagine perhaps it’s a very thick liquid like honey instead so that it doesn’t move much while it is in the tub: the point is that the volume of fluid in the tub (and therefore the mass of the pendulum) is gradually increasing a little bit with each swing.*

The question is, as time goes on, does the pendulum speed up, slow down, or stay the same?

Dr. Eisenkraft asked us to vote on what we thought the answer was by show of hands, and I would say about 70% of those of us in the audience (self included!) turned out to have gotten it wrong. Granted, we were working on the fly and with slightly more time to think about it, I’m sure some of them would have worked out the right answer (and I’m pleased to say, two of the three fellows who I think of as potential thesis advisors got it right right off the bat!). Nevertheless, it’s a bit harder than it looks. Make your guess, then check the answer below the fold:

So if you said “it stays the same,” then you voted with my physics department. But you are also certainly wrong. If you’re not particularly mathematical, you were probably thinking of the fact that one of the reasons pendulums were used to run clocks is that they run in a very precise way that doesn’t much depend on the mass of the pendulum– this is true. If you’re more mathematical, you probably had the same thought, but without the cool historical context and in a slightly more formal way. For example, I guarantee that the 70% of us who voted “stays the same” were all picturing this equation when we said it:

That is to say, for a pendulum undergoing small oscillations, the period “*T*” (the time it takes to swing back and forth) depends only on the length of the pendulum *L *and the gravitational acceleration *g*, which is also constant so long as you stay on the Earth’s surface. The mass of the pendulum does not show up at all (this remains true even for “large” oscillations, although for large oscillations the period will also depend on the initial angle the pendulum is displaced by).

This is definitely true: the period of the pendulum does not depend on the mass of the pendulum. So why were we wrong? Because we are too used to working with mathematical abstractions of pendulums and not with real ones. When a physicist pictures a pendulum, he pictures a point mass on the end of a massless string; in other words, all of the mass is concentrated at a single point at the end of the rod. Of course, real pendulums** are not like that, and therein lies the subtlety. If you like, I encourage you to stop here for a second and use that as a hint, if you haven’t already figured out the answer.

…

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Got it yet? The key is that in a real pendulum, the “length” which shows up in the equation above is the distance between the pivot-point at the top and the center-of-mass of the pendulum. And as you add water to the tub, the water level rises and brings the center-of-mass with it! Thus, the length *L *shrinks as time goes by, and therefore the period of the pendulum decreases, i.e. it swings faster because it is taking less time to go back and forth. Who would have thought? Even if you weren’t tricked by the simplicity of the equation above, I bet your intuition told you that maybe, with more mass to drag back and forth the thing would go slower. But you would have been wrong!***

Man, I just love that puzzle; very devious, but without feeling particularly unfair. What fun. I love puzzles.

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*Actually, the way Dr. Eisenkraft presented the puzzle, he wanted us to imagine a hose at one end sucking the water *out, *which avoids some of the problems of the water from the faucet hitting the tub and affecting it’s motion, but which was harder for me to illustrate!

** I keep typing “pendulums” because my spell-checker doesn’t recognize “pendula.” Perhaps it’s trying to tell me not to be a snob.

*** Actually, if you said that the pendulum would slow down, there is a small sense in which you would be right! The fact is, if you also consider the mass of the *tub*, and if you assume that the tub starts out empty, then for at the beginning of the pendulum’s motion then the center-of mass will be up above the bottom of the tub (because of the mass of the sides of the tub) and for a brief period, the water you add will all be below this center of mass, meaning that you will be lowering the center of mass and increasing L! of course, this situation quickly reverses itself and the pendulum starts to speed up again, as soon as the water level has risen above this initial center of mass.

I love the exclamation marks in this post! Here’s a little tidbit about the plural of “pendulum”:

http://www.phy.mtu.edu/alumni/history/Pendula_ums.html

In retrospect they look maybe a bit excessive… but I was really excited when I was writing it!

Thanks for that link. It confirms somewhat my theory that “pendula” is in widespread use among physicists but not so much everywhere else, which explains why it sounds so correct to me and yet isn’t favored by my spell-check! I am going to continue to use it, I think; I don’t buy the argument that just because it’s only

basedon a Latin word it doesn’t deserve, shall we say, full Latin honors.