**[dramatic chord!].**

*The Monkey Problem*Back in my first year of grad school, along with about 20 of my cohort, I was taking the graduate-level course on classical mechanics at Stanford. How hard can mechanics really get, assuming you don't try to do complicated chaos and nonlinear dynamics stuff? I mean, it's just basic stuff - blocks sliding down inclined planes, spinning tops, etc., right? Right?

The course was taught by Michael Peskin, a brilliant and friendly person who was unfailingly polite ("Please excuse me!") while demonstrating how much more physics he knew than us. Prof. Peskin clearly enjoys teaching and is very good at it, though he does have a tendency toward rapid-fire, repeated changes of variables ("Instead of working in terms of \(x\) and \(y\), let's do a transformation and work in terms of \(\xi(x,y)\) and \(\zeta(x,y)\), and then do a transformation and work in terms of their conjugate momenta, \(p_{\xi}\) and \(p_{\zeta}\).") and some unfortunate choices of notation ("Let the initial and final momenta of the particles be \(p\), \(p'\), \(\bar{p}\), and \(\bar{p}'\), respectively."). For the final exam in the class, a no-time-limit (except for the end of exam period) take-home, Prof. Peskin assigned what has since become known among its victims as The Monkey Problem.

For non-specialists, let me explain a bit about rotational motion. You can skip this paragraph if you want, but if you're not a physicist, the horror of what is to come might not come across as well. There are a number of situations in mechanics where we care about extended objects that are rotating. For example, you may want to be able to describe and understand a ball rolling down a ramp, or a flywheel in some machine. The standard situation that crops up in high school and college physics courses is the "rigid body", where you know the axis of rotation, and you know how mass is distributed around that axis. The "rigid" part means that the way the mass is distributed is not changing with time. If no forces are acting to "spin up" or "spin down" the body (no

*torques*), then we say its "angular momentum" \(\mathbf{L}\) is constant. In this simple case, \(\mathbf{L}\) is proportional to the rate at which the object is spinning, \(\mathbf{\omega}\), through the relationship \(\mathbf{L} = \tilde{\mathbf{I}}\cdot \mathbf{\omega}\). Here \(\tilde{\mathbf{I}}\) is called the "inertia tensor" or for simple situations the "moment of inertia", and it is determined by how the mass of the body is arranged around the axis of rotation. If the mass is far from the axis, \(I\) is large; if the mass is close to the axis, \(I\) is small. Sometimes even if we relax the "rigid" constraint things can be simple. For example, when a figure skater pulls in his/her arms (see figure, from here), this reduces \(I\) about the rotational axis, meaning that \(\omega\) must increase to preserve \(L\).

Prof. Peskin posed the following problem:

When you look at the problem as a student, you realize a couple of things. First, you break out in a cold sweat, because this is a

*non*-rigid body problem. That is, the inertia tensor of the (cage+monkey) varies with time, \(\tilde{\mathbf{I}}= \tilde{\mathbf{I}}(t)\), and you are expected to come up with \(\mathbf{\omega}(t)\). However, you realize that there are signs of hope:
1) Thank goodness there is no gravity or other force in the problem, so \(\mathbf{L}\) is constant. That means

*all you have to do*for part (a) is solve \(\mathbf{L} = \tilde{\mathbf{I}}(t)\cdot \mathbf{\omega}(t)\) for \(\mathbf{\omega}(t)\).
2) The monkey at least moves "slowly", so you don't have to worry about the possibility that the monkey moves very far during one spin of the cage. Physicists like this kind of simplification - basically making this a quasi-static problem.

3) Initially the system is rotationally symmetric about \(z\), so \(\mathbf{L}\) and \(\mathbf{\omega}\) both point along \(z\). That's at least simple.

4) Moreover, at the

*end*of the problem, the system is*again*rotationally symmetric about \(z\), meaning that \(\mathbf{\omega}\) at the conclusion of the problem just has to be the same as \(\mathbf{\omega}\) at the beginning.
Unfortunately, that's about it. The situation isn't so bad while the monkey is crawling along the disk. However, once the monkey reaches the edge of the disk and starts climbing toward the north pole of the cage, the problem becomes very very messy. The plane of the disk starts to tilt as the monkey climbs. The whole system looks a lot like it's tumbling. While \(\mathbf{L}\) stays pointed along \(z\), the angular velocity \(\mathbf{\omega}\) moves all over the place. You end up with differential equations describing everything that can only be solved numerically on a computer.

This is a good example of a problem that turned out a wee bit harder than the professor was intending. Frustration among the students was high. At the time, I had two apartment-mates who were also in the class. We couldn't discuss it during the exam week because of the honor code. We'd see each other on the way to brush teeth or eat dinner, and an unspoken "how's it going?" would be met with an unspoken "ughh." Prof. Peskin polled the SLAC theory group for their answers, supposedly. (My answer was apparently at least the same as Prof. Peskin's, so that's something.) Partial credit was generous.

There was also a nice sarcastic response on the part of the students, in the form of "alternate solutions". Examples included "Heisenberg's Monkey" (the uncertainty principle makes it impossible to know \(\mathbf{I}\)....); "Feynman's Monkey" (the monkey picks the lock on the cage and escapes); and the "Gravity Probe B Monkey" (I need 30 years and $143M to come to an approximate solution).

Now I feel like I've done my cultural duty, spreading the tale of the mechanics problem so hard that we all remember it 22+ years later. Professors, take this as a cautionary tale of how easy it can be to write really difficult problems. To my students, at least I've never done anything like this to you on an exam....

## 1 comment:

>> "Let the initial and final momenta of the particles be p, p′, p¯, and p¯′, respectively."

I can't believe you left out the part where someone points out that he's changed notation halfway through, leading to, "Please exchange p′ and p¯ in all of the above and some of the below. Also our West Side Story moment.

:)

I actually HAVE assigned The Monkey Problem, to a particularly obnoxious bunch of AP Physics students who snarked "Rotations aren't that bad."

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