As I’ve written before, I was a math and physics major in college but I switched to statistics because math seemed pointless if you weren’t the best (and I knew there were people better than me), and I just didn’t feel like I had a good physical understanding.

My lack of physical understanding comes up from time to time. An example occurred the other day. I was viewing a demonstration of Foucault’s pendulum and the guide said the period was something like 35 hours. I was surprised, having always thought it had a 24-hour period. Sure, I can understand it in words, but, even after reflecting on it for a minute, I couldn’t see my way to even an approximate derivation. If I wanted to, I think I could go through the math (for example here), but I feel that if I really had the intuition, I wouldn’t need to.

I’m sure I have a lot more physical understanding than the average person but I don’t think I have enough to be a real physicist.

I started out to be a mathematician, and I was derailed by the same problem: people in college who were obviously better at it than I was. That led me to economics (through a slightly odd route) where everything seemed to click into place — my intuition matched the material and my math was good enough to show that my instincts were correct, which is what economics was about back then, and arguably still is. That’s what I like about your hostility to utility theory — it’s not that you’re wrong, exactly, but it tells me immediately when I’m talking to a smart person who will never think like an economist.

The only useful intuition I have to offer (and it isn’t very useful) is that *if* the period were 24 hours at all latitudes, the pendulum would have to discontinuously switch from clockwise to counter-clockwise right at the equator. That isn’t consistent with the “local forcing” aspect of the non-inertial forces (the idea that they are small forces acting locally on the string and bob).

…and that those small forces vary continuously with latitude.

Not bad and explains why it is 24 hr at the poles and 0 at the equator.

“Right. Must be zero at equator and must be 24h at the poles, and direction must reverse at equator. ”

It’s infinity at the equator (the pendulum doesn’t rotate at all so the period is infinity).

Direction reverses at equator so period has to be represented by an odd function (negative sign rotates in opposite direction). Period has to be infinity at equator and 24 hours at the poles. Has to be a circular function since latitude is an angle. So,

P=24/sin(latitude),

which has all the right properties.

Bill: It’s infinity at the equator (the pendulum doesn’t rotate at all so the period is infinity)

That’s not my intuition. The pendulum is assumed to be suspended by a frictionless mechanism, so it swings in a fixed plane. The earth rotates under it, with period 24h.

I think the argument that the period has to change as you move away from the poles since otherwise something strange happens at the equator is flawed (at least without further elaboration). It could instead be that the amplitude goes to zero at the equator, in which case it can smoothly go up again with an opposite sign as you move to the other pole.

The only way one could possibly think this is if one wasn’t even attempting to visualize a Foucault pendulum setup in different locations (e.g., at a pole, somewhere on the equator) on the Earth’s surface as viewed from a frame of reference at rest with respect to Earth’s center. It’s like the epitome of the absence of physical intuition…

Ahh… I somehow misread Bill’s clearly written word “equator” as “pole”… perhaps thinking he was disputing Lord’s claim it was 24h at the poles…

So forget my comment…

Huh. My prior probability for you suffering a brain-fart was so low that the evidence didn’t promote it into my conscious awareness.

I wasn’t disputing that the period is 24 hours at the poles (really 23 h 56 m because that’s the actual rotation period of the Earth relative to the stars. 24 hours comes in because of the motion of the Earth around the Sun, but if you are talking about when the pendulum will trace out the same line in the sand, it’s the sidereal period of 23 h 56 m).

If you look at the formula I wrote, P=24/sin(latitude), this is 24 at the poles and infinity at the equator, as required.

Radford, your comment is insightful (re amplitude going to zero)! All I have in response (without going to math, which is not permitted, as per the rules of this conversation) is that the pendulum amplitude is set by the experimenter, not the Earth, so there is no way for the Earth to ensure that amplitude vanishes at the equator.

But you *have* identified a serious flaw in my (weak) intuition: You could have the period stay constant and have something about the *shape* of the “orbit” change with latitude, like that it collapses to an elliptical precession which collapses to a line. This would be hard to support under energy conservation, so it can probably be ruled out at intuition level, but it shows how bad my intuition-based argument was.

If we’re on the pole, the pendulum is swinging back and forth in space, and the earth is rotating under it, clearly it rotates once every ~24 hrs. The Period is 24 hours. The main physical intuition here is that the attachment point of the string can’t really change the direction of the swinging relative to the “fixed stars” because the attachment point itself doesn’t move relative to the fixed stars (we’re going to ignore the orbit around the sun etc).

If we’re on the equator, the earth doesn’t “turn under” the pendulum, it constantly pulls the pendulum directly west (sun rises in the east and sets in the west, so the earth is rotating “west”) so the pendulum doesn’t change its direction (relative to the earth surface) hence the period is infinity (you have to wait FOREVER to see a full orbit)

The real question is, how can we get intuition about some intermediate location, like 45 north latitude, or just slightly away from the poles or slightly away from the equator?

The only thing I can offer here is that the effect is obviously very different if you have a nice simple hinge at the top and a stiff stick with a weight at the bottom. It NEVER changes directions then, at any latitude. With the hinge in place, the swinging is always along a given line. With just a string, the swinging is really not along a line. Its deviation from a line is very small so we can’t see it at each swing, but it obviously accumulates the deviation in time. This deviation comes about because of a perturbation in the direction of the force on the string caused by the earth moving. Clearly just slightly away from the pole this effect of the hinge moving will be small, and will therefore cause the period to be just a little different from 24 hours (or as Bill points out 23h 56m) and just away from the equator the effect is also small, which will cause the period to be something…. We can imagine being at the south pole, and we’ll see that the pendulum “goes the other way” from at the north pole. So the sign has to switch at the equator.

Is it reasonable for the period to switch signs at the equator by going through 0? That would mean that it would take NEGLIGIBLE time for the pendulum to do an orbit… whirling round and round in some unreasonable way. So if the sign is going to switch it has to switch by going “through” infinity.

P ~ 24/sin(latitude) as Bill mentions above has all the right properties

That’s a great point. I think, Feynmann repeatedly brings it up in his writing. Some of the best minds in Physics could literally “see” the result much before they could actually, rigorously prove it.

To an extent though, one has to do a lot of Physics before such intuition is acquired. @Andrew: Maybe you gave up too early. :)

I agree, physical intuition is acquired, not innate. People thought long and hard about physics problems before Newton developed Newtonian mechanics, and even the smartest of the smart had intuitions that turned out to be completely wrong. And even Newton didn’t intuit the answers to most of his problems. Einstein and Dirac were famously delighted and astounded by a “tippe top.” You develop intuition about something by working with it. One thing great physicists have is a way of developing that intuition once they’ve seen a few similar problems: they think of a way to think about the problem so that they won’t just get the right answer to the problem they’ve already solved, but to a whole class of problems with related features.

You certainly could have been a good physicist. You’re probably right that you couldn’t have been a great physicist. I had the same realization while I was still in grad school in physics, but somehow momentum carried me through and I ended up with a PhD in physics, then switched fields. It might have been better to switch fields earlier, although I’m not really sure that’s the case. It has worked out OK.

> One thing great physicists have is a way of developing that intuition once they’ve seen a few similar problems: they think of a way to think about the problem so that they won’t just get the right answer to the problem they’ve already solved, but to a whole class of problems with related features.

+1

Given that Dr. Gelman’s sister is a very successful (NAS member) psychologist, perhaps he also inherited a strong interest in social science topics and reasoning and being a statistician does readily enables him to moonlight and contribute effectively to understanding people and institutions.

Professor Gelman seemed to make wise choices in considering his skills and pursuing statistics, but maybe he benefitted from both choice and chance. Statistics really seems to be an awesome fit for him.

Physicist or statistician? Why not both?

So here’s my story. I decided early on (age 12, I think) to go into astronomy, which is what I did. As an undergraduate I was introduced to maximum likelihood as a useful tool for solving astronomical data analysis problems, and used this for decades (I had no need for hypothesis testing, we were fitting models to data and this was a good tool).

But at about the time I became the PI on one of the Hubble Telescope’s instruments, I also began to get wind of Bayesian methods. As the project developed, I became interested, and learned as much as I could. Read an article by Jim Berger and Robert Wolpert in “American Scientist” on defects in statistical hypothesis testing (though I wasn’t using p-values and such I did know about them and had to teach them on occasion in our course on Mathematical Methods for Astronomy).

This was all completely new to me. Made contact with Jim and he pointed me to some interesting articles…and we began to correspond. Wrote some papers with Jim before I actually met him after I became department chair in 1994. Decided to do a sabbatical with Jim after I stepped down as department chair. After retirement, I moved to Vermont and offered to teach a Bayesian course at UVM (similar to one I had taught four or five times at Texas…an outgrowth of the earlier Mathematical Methods course that I had taught earlier). They didn’t have such a course, so I taught it a number of times in Vermont.

Now they have people to teach this subject (much younger than I so this will continue now that I have retired for a second time).

So yes, it is possible to be an (astro)physicist and a statistician at the same time. I was fortunate to get into a field in physical sciences that was appropriate to my talents. So luck plays a role as well.

PS: Harold Jeffreys (no relation except maybe in the distant past) was simultaneously an outstanding geophysicist, astronomer, and statistician.

They had one at the science museum at my high school. (Yeah, there was a science museum.) They also had a display of the planets on arms showing how far apart they were relatively, though I think they stopped at Saturn due to the era of the device. Really cool. They showed on a model why the pendulum took more or less time to go around given where you were on the planet versus the pole or the equator. I think at one point they even had a drawing of the sine calculation.

I took a few minutes and looked this up. I was amazed at the poor quality of the basic explanations of why; I had to dig through 10 links to find even one that discussed the location issue in a specific way. It’s that kind of thing which makes understanding hard: a bunch of copies of not very good explanations and, in this case, a copied visualization that only has meaning if you know what it means.

I guess if I were Foucault, I would have stuck up my device in a few locations and noticed it took more or less time … and then realized it was location-based and calculated that out. I think you would have done the same. In other words, how you are presented with the experimental data draws the picture. It’s not a flaw to miss the picture when you’re given one slice and the explanation without the rest of the data.

I also started out as a physics major, but soon realized that what I liked most about physics was the real-world problem solving I found in kinematics, dynamics, and even optics. Thus, in my sophomore year, I switched to applied mathematics and never looked back.

If your goal was to learn a trade, then I guess you did switch trades, but if your original intent was to be what Newton and his like would have called a “Natural Philosopher”, you didn’t switch at all. Plus you picked the best 50 year stretch in the last 400 years not to do physics.

Sure, the Silly Stringers command too much clout and resources but you can’t tell me this isn’t top notch – http://opli.net/opli_magazine/eo/2013/scientists-discover-how-to-turn-light-into-matter-after-80-year-quest-may/ The results of the experiment will be more interesting than the proposal but I like the fact that they’ve suggested an experiment to test a novel hypothesis – as opposed to spending time and energy developing untestable theories.

(Also, I like this a lot – https://www.youtube.com/watch?feature=player_embedded&v=wGkx1MUw2TU – but I admit it’s not in the same league as creating matter from light.)

In high school I wanted to be a golf course architect. I was fine at diagramming 2-d golf holes, like you’d see from an airplane. But real golf course architects need to be good at imagining in 3-d, a skill I just didn’t have.

3-d cognition is less correlated with the general factor of intelligence than most other subtraits.

How much is the elevation change over a typical course?

The important physical question is always expressible best in dimensionless form. How big is the elevation change over a typical course, as a fraction of the diameter of a golf ball?

whoops, hit enter too quickly accidentally.

I think in this form, it’s clear that the topography matters a lot on even small scales. A putting green that places the hole at the bottom of a large parabolic bowl with the edges 2 inches higher than the center… is still a LOT easier to putt into than one which places the hole at the top of a 2 inch high parabolic mound for example.

I honed my physics intuition by considering extreme cases. So here you go:

http://www.dailymail.co.uk/news/article-2573646/The-worlds-toughest-Par-3-golf-hole-players-Legend-Golf-Resort-South-Africa-tee-MOUNTAIN-green-400metres-below.html

Objects moving horizontally near the surface of the earth are deflected to the right by the coriolis force in a coordinate system rotating around the earth with an observer. The coriolis force is small, less than 0.1% of the gravitational force for objects moving at 5 mph or less (Symon, Mechanics, 3d edition, § 7.4) . If you think of the pendulum bob’s v vector being nudged rightward a tiny bit on every oscillation as it swings to and fro, then that explains the precession of its plane of motion.

Think of taking a diameter of a circle and stretching it oh so slightly so that it becomes a very elongated ellipse tangential to a circle and fitting inside it. Now let the points of contact move slowly around the circle with constant Ω. This spirograph-like figure generated thereby approximates the motion of the Foucault pendulum.

This is the same force that causes artillery shots to be deflected from the initial v determined by the gun barrel.

addendum: deflected to the right in the Northern hemisphere.

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math seemed pointless if you weren’t the best

You proved Rabin’s calibration theorem two years before he did. You proved it in a way that was simple, clear and elegant – the exact opposite of what he did. He’s got an endowed chair at Harvard now because of the theorem. p>.95321 you would have been a great mathematician.