Built on Facts

Last Friday I saw M. Night Shyamalan’s film The Happening. (Don’t worry, no real spoilers follow.) Science types will squirm during a few parts. There’s really very few physics issues to complain about, but the biologists and biochemists will blow a gasket. I wouldn’t worry about it too much - it’s a horror film, and suspension of disbelief is not strained too badly within the internal universe of the film which is really all we can ask in this genre.

I’ll leave the fun dissecting the biology errors to the biology sites (Popular Mechanics has a good article already). Suffice it to say that evolution doesn’t have motives and act suddenly with malice aforethought. And plants don’t work that way anyway. And survival adaptations don’t even come close to working that way either.

Well, I’ll mention my one physics quibble. A character is hit in the chest with a shotgun blast at a range of about one foot. The buckshot pattern on his chest is probably a foot in diameter. Shotguns do not spread like that. At a range of one foot the pattern of buckshot wouldn’t really be all that much bigger than the opening of the barrel. If it were as portrayed on film, the weapon would be essentially useless at ranges of above a yard, which is clearly not the case.

There’s a few philosophy and politics of science issues. You’ll see them. It’s not a subtle film. The old tropes that “Scientists Are Not Omniscient” and “There Are Mysteries Beyond Our Understanding” get taken out for a spin. The first is true and the second is probably true, but it’s also true that it’s possible to type with your toes. After all, science isn’t good at everything but it’s very good at the sort of thing found in the movie. The phenomenon at issue in the film (were it possible, which it’s not) would be thoroughly characterized and counteracted faster than you can say DARPA funding. After all, plant biology is complicated but it’s not nearly an Unsolvable Mystery. The possible larger implication that Humanity Needs To Go is a bit troubling. The New Republic calls that implication The Most Morally Abhorrent Movie Ever Made, but that’s just silly. The TNR reviewer believes that one particular scene juxtaposing of pregnancy with death means Shyamalan thinks new human life is environmentally evil. I think the point is the much simpler idea that a horror film is supposed to frighten, and new life facing death is frightening. It’s a summer scary movie and should be taken nearly that seriously regardless.

All told, I thought it was a much stronger effort than his previous two films. For film quality, I give it a solid B and recommend it as a fun and creepily atmospheric exploration of one of the more horrifying scenarios contemplated on the big screen. Just don’t think too hard about it, draw moral lessons from it, or expect it to be a critical favorite.

On this date in history was one of the earliest known dates in history.

There’s lots of things we can date with certainty. In the last few hundred years most dates can be pegged pretty accurately. December 7, 1941 was the date of the Pearl Harbor attack. Nicholas I of Russia was born on July 6, 1796. So on and so forth, but things tend to get more fuzzy as we go back. Alfred the Great died on October 26, 899 but his birthday cannot be dated to any closer than “about 849″. For very distant events like the founding of Babylon it’s hard to get within centuries. Things like the C-T boundary event have error bars in the hundreds of thousands of years.

The oldest events that can be precisely dated are eclipses. The earth-moon system ticks long like clockwork, gently perturbed by other planets and the tidal friction dissipating energy at a small rate. Though these can’t be perfectly corrected for, eclipses can be accurately calculated quite far into the distant past. When these eclipses are discussed in archaeological inscriptions in conjunction with particular events we can specify their occurrence to the day.

On June 15, 763 BC a total eclipse appeared in Assyria. Mentioned both in Assyrian records and (possibly) the Biblical book of Amos, it’s the oldest specific date of which I’m aware in ancient near eastern history. More spectacularly but later, an eclipse on May 28, 585 BC ended a battle between the Medes and the Lydians by terrifying the combatants into an immediate peace agreement. If you don’t count an eclipse by itself as being a historical event, I believe this is the single oldest event which can be pinned to a specific date.

The Chinese were generally better at both record keeping and astronomy than the nations of the near east, and there are a number of very ancient eclipses which can be dated with certainty. The oldest and most famous of these probably occurred on October 22, 2134 BC. Legend has it that the two royal astronomers had been partying too hard to inform the emperor of the eclipse beforehand and were thus separated from their respective heads. And you thought tenure review was harsh! Though the date of this eclipse is known with certainty, it’s not known for certain that this was the specific eclipse involved in that legend. Several eclipses occurred within the potential time frame of the original writing of the legend and the October 22 date represents the one considered most likely to fit.

Physics is the “hardest” of the hard sciences and history is an often fuzzy social science. You wouldn’t necessary expect there to me many intersections between the two, but as often in science it happens surprisingly often.

Earlier I was killing a little time today by reading about MOND. It’s an ad hoc attempt at a solution so the galaxy rotation problem. Essentially the outer portions of galaxies rotate a lot faster than they should given the their visible mass distribution. The obvious solution is that there’s mass which isn’t easily visible which accounts for the extra gravity. This approach is called dark matter, and there’s various searches underway to detect it.

The MOND proposal is an alternative out of left field. It assumes that in fact it it not true that F = ma for small enough values of a. Instead F = m(a²/a₀) for some constant a₀. This is more or less equivalent to saying that the force of gravity falls off as ~1/r at large distances instead of the usual ~1/r². It’s more than a little far-fetched, and there are observational reasons involving gravitational lensing which make the hypothesis seem quite implausible. Never hurts to try though.

It does however give us the opportunity to think about changes in the behavior of fields at various distances. Take the electric field, for instance. The electric field of a point charge is just

For a continuous charge distribution, you can take the field as being produced by the charge density integrated over the charge region. Consider a finite line charge of length a. The above mentioned integration tells us that the field experienced if you start at the center of that line charge and move perpendicularly outward is

And so for very large a (or equivalently, very small r) this asymptotically approaches

That’s what we call the field of an infinite line charge. But here’s the thing: there’s no such thing as an infinite line charge. It’s a very good approximation when you’re close to a long line, but as you move farther and farther away that line charge starts to look more and more just like a finite little piece. As that happens the field stops falling off as 1/r and begins to fall off as 1/r². We can see this mathematically by taking the variable a to be very small and checking the behavior in r. (If finding “large r” by taking “small a” bothers the mathematicians in the audience, recast the problem in terms of some dimensionless ρ = r/a and take ρ large)

This is a good example of a function changing regimes as some other parameter varies. As in this example, lots of these types of functions tend to die out faster as distance increases. This instance in particular is related to the general idea of multipole moments. Are there any examples of a physical function’s decay changing from faster to slower as distance increases, as hypothetically in MOND? Actually with a tiny bit of modification our charged wire is also an example of this. In real life, a line charge is not a continuous charge distribution but a collection of charged particles strung out at very tiny regular intervals. On a macroscopic level it looks continuous but on an atomic level it’s not even close.

In the immediate neighborhood of each charged particle (much closer to the particle than the particles are to each other) the field will be dominated by the closest particle. It will look essentially like the 1/r² field of that particle. As you move away from that particle and the wire, the field from the closest particle ceases to be the main influence and you experience the field of the wire as a whole - which behaves as 1/r. The decay has entered a slower regime. Assuming the wire is not actually infinite, the field will eventually revert back to the inverse square behavior.

Though inspired by a theory that very probably isn’t true, this kind of thing is fun to think about. In the professional world, the mental effort behind even a theory that turns out to be wrong can end up leading to something better later on.

Is the video with the popcorn being popped by cell phones fake? Despite a bit of controversy about just how much power it takes to pop popcorn, everyone in the physics blogosphere agrees it simply ain’t happening.

But now the mystery of the origin of the video can be laid to rest as well. Here’s the website of the marketing company who produced them.

Here’s my original Pt. 1 post.

Vis consili expers mole ruit sua.
- Quintus Horatius Flaccus

The words of Horace above attain a spare and austere beauty in Latin, but the meaning is carried equally well in English. Force without wisdom falls of its own weight. Two different sets of words carry his millenia-old thoughts to the present time. There are different ways of saying the same thing.

In physics, force permeates the first few semesters of study and continues to be a useful tool in higher physics. But in many respects force is not the most elegant tool in our arsenal. It’s a vector; not only do we have to worry about magnitude but we also have to deal with direction. With some exceptions, vectors are significantly more difficult to deal with than regular numbers. But look at Newton’s First Law, which we all know:

F and a are both vectors. Force in the x-direction produces acceleration in the x-direction, etc. Easy enough for basic problems, but for complicated ones things get rapidly very difficult. Other coordinate systems can make the situation even worse. Non-classical physics is worse still; the concept of a force doesn’t even necessarily have much meaning in quantum mechanics.

But like translating from one language to another, we have alternate ways to work with a given physical situation. Instead of examining the forces, we can look at the energy. This is just a conceptual introduction so we’ll skip all the derivation for now, and present the final result. Define a quantity L, called the Lagrangian. It’s a measure of the energy of the system; specifically it’s the measure of the kinetic energy K minus the potential energy U. Thus by definition L = K - U.

Now pick coordinates: maybe x, y, and z, or θ and φ, dimensionless quantities, or whatever you feel like and in any combination. Pick and choose as many coordinates as it takes to describe the system, and call them q_i, for however many i’s you need. It turns out that the following equation completely describes the classical time evolution of the system:

In this differential equation, you have

1. Energy specified by L
2. Coordinates specified by q. The dot above q just means the time derivative of q, to save space.
3. A time relationship connecting them

And that’s all you need. No forces to be seen. No vectors of any kind. You can specify the coordinate system in whatever way most easily describes the configuration of the system without having to worry about whether that same coordinate system also can easily describe the forces within that system. Now let’s pick the easiest example in the world to test it out. In the future we’ll do harder things that really show the power of Lagrangian dynamics, but since this is just dipping our toes into the water we’ll just use a very simple rock falling straight downward.

The potential energy of that rock is just U = mgy, where y is the height above the ground.

The kinetic energy of that rick is just K = (1/2)mv², where v is the velocity and thus the time derivative of y, which we write as y with a dot over it.

That makes the Lagrangian L = K - U become:

Plug that into the differential equation above with q = y (because as we said before the various q are any coordinate you want), and you get:

Now carry out the y and y-dot differentiation formally, resulting in:

Finally doing the time derivative:

Which lo and behold finally gives us:

The rock has an acceleration g vertically downward, independent of its mass exactly as we expect from Newton’s laws. We never touched a force or a vector. “But Matt,” you say, “isn’t that actually a lot harder than doing things with forces?” Yes it is, for this problem. But this problem is just a toy example to show that this does give us the right answer for a problem we can already do trivially. For instance, try to characterize the motion of a marble placed on top of a basketball using forces. If you can do it at all, you’re a much more patient physicist than me. Using energy methods it’s still not quite a simple calculation but it’s thoroughly tractable. But that’s a story for another day.

The Atlantic has a pointed and somewhat grim article by a pseudonymous Professor X (no, not that one) about assigning failing grades to failing students. It’s called In the Basement of the Ivory Tower. Professor X is an English instructor at a community college teaching largely adult students, so he’s not dealing with an ideally suited class. I myself teach at a well-respected state university in Texas so I have no complaints about my students, but physics is nevertheless not something many of them are prepared to deal with. Now as a TA, I’m also not responsible for directly assigning grades - about 20% of the overall grade comes from me in the form of quizzes and lab reports, with the remaining 80% coming from the professor’s tests. The professor bears most of the burden described in the article. But I feel a bit of it myself. Read the whole article, but I want to comment on a few sections in particular.

The bursting of our collective bubble comes quickly. A few weeks into the semester, the students must start actually writing papers, and I must start grading them. Despite my enthusiasm, despite their thoughtful nods of agreement and what I have interpreted as moments of clarity, it turns out that in many cases it has all come to naught.

Remarkably few of my students can do well in these classes. Students routinely fail; some fail multiple times, and some will never pass, because they cannot write a coherent sentence.

Sentences aren’t generally the issue in physics classes, though there have been a few in lab report conclusions that I’m not happy to have witnessed. “I really enjoyed this lab” has no place in a lab report, but I’ve seen it. Repeatedly. Now though Professor X doesn’t really speculate too much on what differentiates the successful students from the unsuccessful, I find that the only way I myself improve is practice. Correlation is not causation, but doing the homework correlates exceptionally well with good test performance. There is no substitute for practice in physics, writing, or any other academic discipline.

How I envy professors in other disciplines! How appealing seems the straightforwardness of their task! These are the properties of a cell membrane, kid. Memorize ’em, and be ready to spit ’em back at me. The biology teacher also enjoys the psychic ease of grading multiple-choice tests. Answers are right or wrong. The grades cannot be questioned. Quantifying the value of a piece of writing, however, is intensely subjective, and English teachers are burdened with discretion. (My students seem to believe that my discretion is limitless. Some of them come to me at the conclusion of a course and matter-of-factly ask that I change a failing grade because they need to graduate this semester or because they worked really hard in the class or because they need to pass in order to receive tuition reimbursement from their employer.)

Well professor, cheer up. Physics has completely objective answers, but we too have to worry about how we grade the people whose methods are close but not executed entirely properly. No multiple choice for us, except occasionally in very low-point-value concept questions. In fact, even biology isn’t just a set of facts to memorize. “A cell membrane has property Z” is a great thing to know, but the question is rarely “Which of the following properties does a cell membrane have?” That’s not very useful by itself for a biologist to know. The question will instead ask for an extension of that fact, e.g., will that property cause a given molecule to diffuse across the lipid bilayer or does it require active transport through a protein channel?

As for the discresion, I’m rarely asked for it. I really have very little in the first place, and in fact TAs don’t even have access to class grades other than the ones I directly give. This is probably merciful for me, as it prevents me from having to deal with situations like this:

I gave Ms. L. the F and slept poorly that night. Some of the failing grades I issue gnaw at me more than others. In my ears rang her plaintive words, so emblematic of the tough spot in which we both now found ourselves. Ms. L. had done everything that American culture asked of her. She had gone back to school to better herself, and she expected to be rewarded for it, not slapped down. She had failed not, as some students do, by being absent too often or by blowing off assignments. She simply was not qualified for college.

Fortunately I rarely encounter this. If you make it to my class you’re almost certainly qualified for college though you may not be qualified for the major you’ve picked. Failure usually comes from slacking. Those who fail because they simply cannot understand the material are small in number, and will anyway likely be successful in a different major. Most figure this out early and have the sense to drop the class. It’s tough to watch those who don’t, and I don’t envy the professor who has to mark D or F on a grade spreadsheet even though those students will in all probability go on to success in something different. It would be much worse to mark a grade which is a declaration that someone is simply not suited for college in the first place.

In her own mind, Ms. L. had triumphed over adversity. In her own mind, she was a feel-good segment on Oprah. Everyone wants to triumph. But not everyone can—in fact, most can’t. If they could, it wouldn’t be any kind of a triumph at all. Never would I want to cheapen the accomplishments of those who really have conquered college, who were able to get past their deficits and earn a diploma, maybe even climbing onto the college honor roll. That is truly something.

And that same concept makes thousands of physics buildings on thousands of campuses infamous as places of real defeat and real triumph. I’ve personally experienced both. I wouldn’t have it any other way.

There’s a video circulating on the internet that purports to show popcorn being popped by the radio waves emitted from cell phones. It’s been extensively debunked by multiple sources, including Snopes and Wired. Swans on Tea has a quantitative analysis which I highly recommend. Snopes even gives good reasons why it’s probably a viral advertisement.

I have an alternate suggestion. Forget the theoretical reasons why it’s impossible. They won’t convince many people who haven’t gotten their hands dirty in the mathematical details of the theory anyway, and odds are if someone believes this video they haven’t done any of the study necessary to understand the theory. This is not really a fault - probably 99% of people can live life just fine without ever understanding radio waves and microwaves (let alone Maxwell’s equations) in detail.

Besides, what’s this blog called anyway? Built on facts, I think. We get facts by observation.

Finish this post, turn off your computer and get out all the cell phones you’ve got. Put them on a table just like in the video and point them at some popcorn. Pick a phone number (202-762-1401 is the US Naval Observatory master clock, which is interesting in itself), dial it into the phones, and hit send. See what happens.

Now you don’t have to trust me, or another scientist, or an urban legend debunker, or anyone else. You’ll have demonstrated it for yourself. That, my friends, is science.

UPDATE: Here’s my followup post, which has the website of the company which produced the videos.

When I was a young child, my uncle Fred used to try to convince me of some pretty outlandish things. His favorites were to say that the earth was flat and that England was a hoax. I was old enough to know better and I tried to argue as best I could, though with limited success. Looking back, I think uncle Fred was trying to stretch my mind a bit so that I could use critical thinking to defend true propositions in the face of opposition.

In trying to dismantle the idea that the earth was round, he suggested that the people in the southern hemisphere would fall off. I could dispatch that well enough - gravity comes from the earth and there’s nothing “below” it to pull people off. Then he argued that if it were spinning, people would be flung into space, like water droplets from a spinning basketball. I wasn’t sure what to say about that. It took a few days of thinking before I realized the answer: the earth spins pretty dang slowly. It takes a full 24 hours just to make one rotation. That wouldn’t throw water off a basketball, and it probably wouldn’t throw people off the earth.

But it is spinning, so even if the effect is small people will still feel a little less force than they would have if the earth were stationary. The details of this are a classic problem which you can find in many texts and websites, but it’s fun so let’s give it a try. The principle is fairly simple. Circular motion requires an acceleration because the direction of travel and thus the velocity is constantly changing. In this particular case that force is provided by gravity. The force that keeps you from falling to the center of the earth is just the pressure the ground exerts on your feet. Your feet exert an equal and opposite pressure on the earth and so you experience that as your weight. But if you’re in circular motion by virtue of the earth’s rotation there has to be a net force downward, and so you experience this as a decrease in the force from the ground and thus your weight decreases. The acceleration required for uniform circular motion is

Therefore a person on the surface will perceive that much less acceleration, since it’s no longer being matched by upward force of the earth’s surface. Plug in the maximum value for v (at the equator) and the radius of the earth r, and you’ll get a value of about

a = 0.0339 m/s²

This is pretty small compared to the 9.8 m/s² or so that is the standard downward acceleration at the earth’s surface, and so fortunately no one gets thrown off by the rotation. But measuring it would be possible with adequately sensitive equipment. Since F = ma, I can multiply my own mass (about 170 pounds*) and see that I should weigh less on the equator by about 0.94 9.4 ounces (Thanks to Dr. Pion in the comments for catching my careless calculator mistake!) than I would be on a stationary earth or the north pole. Not much of a weight loss solution, but at least I won’t be thrown off the earth.

That’s all we need if you’re standing still, and that’s where most published solutions stop. But if you’re moving - for instance in a boat or airplane - additional complications arise. Your motion in the east-west direction changes v. If you’re traveling with the earth’s rotation, v will be larger. Vice versa for traveling against the earth’s rotation. This produces an additional term that must be taken into account. Finally there’s yet another (generally smaller) term that takes into account the fact that following the curvature of the earth itself requires a centripetal acceleration. These refinements are called the Eötvös effect and are necessary for precision measurements of the earth’s gravitational field.

*The pound is a little ambiguous as to whether it’s a unit of mass or force. Here I take 1 pound as a mass equal to the usual ~0.45359 kg. Why not just use metric and make the units a lot easier to deal with? Because having been raised on them, I find English units more intuitive. Metric readers are invited to take advantage of the much easier math and recalculate for their own weights!

Weddings all around for my family this week. I’m in my hometown with some time to kill this summer before my TA duties resume in July, which gives me the opportunity to catch up with old friends - one of whom was married this weekend. My parents went to a different wedding of some friends of theirs, who had the advantage of a budget roughly an order of magnitude greater than the wedding I attended. But of course marriage is (hopefully!) about love rather than money, and I wish them all the greatest happiness.

My favorite moment of wedding drama is when the audience is asked if anyone has a reason that the couple should not be married. No one ever does, but it’s always a fun moment of suspense. Sadly this tradition is an endangered species, and was not featured at this particular wedding. Presumably few people want to take their chances, though you’d think if a person were really determined to interrupt he (and it’s always a he in the movies) would not be deterred simply by not being asked.

But there’s often another moment of drama which is wholly unintentional: the lighting of the unity candle. If you’re not familiar with this tradition, there’s two candles set alight before the ceremony for the bride and groom. At a specified moment, the bride and groom use their candles to light a larger “unity” candle. Once done, they blow out their own candles to symbolize that they are now joined as one. Or sometimes they don’t, to symbolize that they will continue to love their families. Don’t ask me, I don’t make this stuff up. Here’s what the candles look like:

The unity candle is always brand new, with the delicate equilibrium of capillary action and convection not yet established in the length and saturation of the wick. The new wick is thickly coated, making movement of liquid difficult. Reliably lighting a new candle in one attempt is difficult under the best of circumstances. But it’s a wedding. There’s often a hundred or more people densely packed into the chapel. Each produces a few hundred watts of body heat. The air conditioner is turned on, and vents are often located near the front where the candle ceremony is taking place.

I’ve been to quite a few weddings with a unity candle, and though I haven’t been bringing data notebooks to the ceremonies I’d estimate the failure rate for unity candles is at minimum around the 1 in 4 mark. The symbolism is what really kills things. To light a unity candle and blow out your own only to have the flame symbolizing your love gutter pitifully and die within seconds is always embarrassing. Doubly so when the ancient majesty of the ceremony has to be restored with a kitchen or cigarette lighter. I am at least happy to report that none of this happened at this wedding, though the unity candle dwindled down to alarming levels before rebounding to a vigorous equilibrium.

Do I have suggestions to fix this? Yes I do. Just as a successful wedding treats the people involved with respect and consideration, a successful unity candle will have to treat the physics of the situation in a realistic way.

• First, how about a statistical analysis of the first-light success rates of different brands and varities of candles in different conditions? This is impractical for a single couple, but the weddings are something like a $40 billion industry and so I’m sure somebody has the money to do some tests. Maybe one of the wedding magazines. This suggestion is just Science 101. Repeatability is exactly the thing that’s needed.
• Second, a candle relies on generating enough heat to keep a self-sustaining cycle of melting and capillary action. When it’s first being lit, there is no hot wax pool and thus the new flame hasn’t got a lot of power to waste into a turbulent atmosphere. How about stationing someone to turn off the air conditioner about a minute before the candle is to be lit?
• Third, cheat the system. Physics is just rules, and if you can make the rules work for you then you’re golden. The problem with keeping a new candle alight is almost always excessive heat loss. If you can come up with a way to artificially increase the heat produced then you’re not at as great a risk from power dissipated in air currents. Try this: drill and remove the candle wick. Remove the wick from a trick re-lighting birthday candle and replace it in the unity candle. These trick candles work by including a thin ribbon of magnesium within the wick. It’s more or less shielded from the oxygen in the atmosphere when the candle is properly burning, but when the candle is blown out the magnesium begins to combust from the heat of the remaining ember. The comparatively large amount of heat generated by these sparks is sufficient to relight the candle. I wish I could say I thought of this idea, but in fact I came across this when doing a bit of reading about unity candles for this article. I have no idea why this does not seem to be in common use since it’s a really great idea.

It’s bit off the beaten path, but as we live in nature we’re always subject to its laws and ought to know how to deal with them. For better or for worse, in sickness and in health, ’till death do us part!

A few brief items of interest for the weekend:

One of my favorite things to write about is the interplay between physics and pure mathematics. Beyond the basic idea that physics is the science of the mathematical description of nature, it’s further true that pure math by itself often anticipates physics in ways physicists don’t always expect. Here’s the story of several mathematicians who solved a conjecture about the zeros of harmonic polynomials and by accident revolved an open problem in the cosmology of gravitational lensing.

Here’s more discussion from Swans on Tea on the multi-blog conversation about how to grade physics exams. In short, it’s a lot less complicated in the Navy!

Dr. Pion on a concern that I can definitely relate to - the availability of jobs after grad school. I personally would love a career as a professor, and I’ll try my best to make it happen. Would I be disappointed with work at a national lab or in a suitably challenging research position in the private sector? Nope. Science is science, and I’ve never really felt that stereotypical pure science disinterest toward applied research, even though of course I’m a sucker for a good physics vs engineering joke.

Have a lovely weekend!