Built on Facts

In Which Your Host Witnesses a High-Speed Chase

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Last night I saw a classic conservation of momentum problem in person. It was about midnight, and I was on a service road beside west Houston's Beltway 8 (avoiding the tolls) when I slowed down to stop at a red light. In my rear view mirror I saw the red and blue flash of emergency lights approaching, so with the room I had left I crept over a bit to the right to let them by. Whoosh! A car blew by my left side at high speed, swerving in front of me and speeding into the intersection heedless of the light. My neurons barely had time to start cooking up some surprise when a black pickup truck coming from the cross street entered the intersection from the right and neatly intersected the path of the speeding car with a tremendous and violent bang.

Conservation of momentum happened. The collision was not entirely inelastic nor was it entirely elastic. The truck struck the car very solidly on its right side behind the center of mass of the car, sending the car into a spin more or less along its original trajectory. The collision reduced the speed of the truck but didn't change its trajectory much either, and the truck skidded to a halt. (A bonus friction problem!)

"Holy crap! This is a serious accident, I should pull over and try to help!", I'd have thought to myself, if I had had time to form the mental impression into words. I didn't have time, because within a second or two probably four or five police cars blazed past as well, surrounding the stopped and thoroughly totaled car.

"Holy crap! This wasn't an accident, it's a crime in progress! I might have to be ready to react to real danger," I'd have thought to myself, if I had had time to form the mental impression into words. I didn't have time, because the police jumped out with guns drawn and raced toward the car. I was in the process of pulling my car into the service station just to my right when the arrest happened, and though I didn't see it clearly it looked like the driver was pulled from his car without appearing to be injured, arrested, and put into the back of one of the police cars.

Within a minute or two another dozen squad cars were on the scene, along with an ambulance and two firetrucks. The driver of the pickup truck seemed to be uninjured as well, and the ambulances and firetrucks seemed to be a precautionary measure. The police processed the scene and photographed everything, cleaned up the road with the help of the fire department, and later took a statement from me. They seemed pretty nonchalant about it, which I expect is because a witness isn't really needed when there's a bunch of dashboard cameras, two new-looking red light cameras on the intersection itself, and the driver of the truck as a more direct witness, aside from whatever other evidence they might have had for whatever caused the suspect to run in the first place.

It's not really all that directly related to the practice of physics, but then again it's not so often you see a classic Physics 101 problem instantiated in front of you in such dramatic fashion. In any event, I'm back in College Station now, tomorrow will have a quick July 4 holiday post, and then back to our regular schedule with a Sunday Function the next day. Enjoy your weekend, and remember that fireworks are physics too!

Posted by Matt Springer at 3:11 PM • 2 Comments • 0 TrackBacks

You Spin Me Right Round

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The National Weather Service does a very useful thing for people who live in an area expexted to experience severe weather danger. I have a little Firefox app in my browser that links to the NWS and advises me of the current conditions and forecast for the next few days, and as part of its mechanical duties it advises me of the various severe weather alerts that happen. They're popping up at the rate of several times per week now, when the sky is a beautiful crystal blue. Why? Severe heat. Welcome to the southern summer!

I regret to say that I have an advisory of my own: a travel alert. I'm going to be on the road until Friday(ish) and thus posting might be sporadic or nonexistent. The latter is unlikely, I should be able to get at least a couple days in this week. But if not don't worry. I'm not joining the ranks of the celebrity death outbreak.* Well I'm not a celebrity either, but I'm going to try to avoid the death bit too.

And since I'm on the road, I want to kick off the trip with a driving-related physics fact that few in my classes believe when I tell them. On a smoothly rolling wheel, the point in contact with the ground is stationary. It ain't moving, no matter how fast the wheel is turning.

Why? The easy answer is that if it were, it would be skidding. A more meaningful answer is that a rolling wheel is really the combination of two motions: circular spinning and linear translation. For a wheel whose center is moving forward at velocity v, the wheel clearly must be spinning such that the points on its rim are also moving at v. The rolling itself makes the points on the top of the wheel spin in the forward direction at speed v, and the points on the bottom are obviously going in the opposite direction, velocity -v.

Add the overall forward speed v, and -v + v = 0 for the bottom of the wheel. You might not believe me, so let's do the experimentalist thing and actually watch one in action. This nice and much more mathematically complete web explanation has the video.

Not surprising when you think about it, but maybe a little surprising before you think about it.

*Honestly it's Billy Mays whose death actually managed to make me legitimately a bit sad. Go figure.

Posted by Matt Springer at 10:00 AM • 9 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

I've been away from campus visiting family during the first part of the summer, in one of those rare confluences of events where research, classes, and teaching all find themselves on temporary hiatus. While it's pretty rare, I'm glad to take it. It is giving me Mathematica withdrawals though, especially with being able to easily make nice graphical plots and do some of the number-crunching I sometimes need for the more technical posts around here. Fortunately this is the last Sunday I'll have that problem, as I'll be back in Texas soon (and all those responsibilities will come roaring back). I'm looking forward to it.

For the moment I'd like to take this particular Sunday to kick off a Sunday Function project I've been thinking about for a while. Regular readers might recall the Greatest Physicists series here, and I'd like to do something similar with functions. Now obviously "Greatest Functions" is pretty much impossible. How could you even decide what counts as "one" function - are the exponential function and the trig functions different, or are they really different aspects of the same thing in complex analysis? What about polynomials and roots? I don't think a ranking of importance going to work.

But there's nothing stopping us from making a sort of Wonders of the World list for functions, unranked in any sense of "importance" but instead as a guide for tourists of mathematics to admire. Call it "10 Coolest Functions". So just like the Greatest Physicists, it's time to take nominations. Here's the guidelines:

1. If it's already been covered in Sunday Function, it's still perfectly fine. I could probably do dozens of posts on (say) the sine function alone.
2. Anything an undergraduate in any major might encounter is fine. Try to avoid very obscure and/or technically difficult functions.
3. Relevance in mathematical physics is not required, but preferred.
4. Sets of functions are fine, in contexts like Bessel functions or Legendre polynomials.
5. Pick your favorite 10 (or less), and rank them if you'd like.

Remember that this is a loose and fun exercise; a listing of favorites and subjective coolness rather than by "importance". You have lots of latitude, so if for instance you want to pick exponential/logarithm as one of your favorites that's fine.

Well it's a difficult task. But I know you're up to it!

Posted by Matt Springer at 1:04 PM • 20 Comments • 0 TrackBacks

Waxman-Markey is probably crap.

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There's a lot not to like about the Waxman-Markey cap-and-trade bill that passed the House this last week. You'd expect the right not to like it, but this bill has many people of all political opinions unhappy.

From the left: The bill is a huge 1300ish page monstrosity developed behind closed doors. What we do know about what's in the bill is not promising. Greenpeace opposes it and lists several reasons. The "cap" is weak, flexible, and full of loopholes. The "trade" part is shot full of offsets and concessions to the dirtiest power generation coal plants. Even if everything goes as planned, it will mean increases in fossil fuel generation and the result goal is hilariously far short of the IPCC recommendation. The bill just doesn't do much of anything for the environment.

From the right: It's going to be expensive. Really expensive. The CBO gives low numbers for the operating cost of the bill amounting to a few hundred bucks annually per family, but this doesn't take into account changes in the GDP resulting from the bill's provisions. Those could reach into the thousands annually per family, and in a regressive way since energy is not exactly a luxury good (it's usually my second biggest expense, behind rent).

From good-government advocates of any stripe: Literally no one has read the entire bill. The house voted on the bill before a complete copy had even been printed, and 300 pages worth of amendments were passed without having been read. Conservative schemes to foil Captain Planet and poison the air inside day-care centers? Liberal plots to ban air-conditioning and make us all wear birkenstocks? Who knows? Not anyone who voted on it, that's for sure.

From Built On Facts' personal list of hobby horses: What could actually have great effects for the environment, the economy, and international geopolitics is if the US actually did develop energy technologies that were honestly cheaper than coal, displacing dirty technologies the old fashioned way - by being better. Nuclear power, offshore wind power, geothermal, space solar, etc. Some of this is almost legitimately competitive with cheap coal now, but is smothered in regulations having nothing to do with safety or the environment and everything to do with politics and NIMBYism. So far as I can tell this bill doesn't do a whole lot to help get those kinds of technologies closer to a true competitive advantage, and that's a real shame.

The bill is likely to die in the Senate anyway. Too many Democratic senators are from coal industry states. It's probably for the best. Congress ought to take a deep breath, clear their heads, and start over. Maybe then they'll come up with something that, I dunno, actually does something useful.

I'm not going to hold my breath.

Posted by Matt Springer at 4:27 PM • 8 Comments • 0 TrackBacks

Blackbodies and Black Bodies

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Alas for Michael Jackson. Talented musician, deeply broken human being. Most of my knowledge of him was through cultural osmosis rather than his actual music, and I'm young enough so that I can't really remember the time before he was a punchline about changing skin color, disastrous plastic surgery, and child molestation. Well, de mortuis nil nisi bonum.

His death reminds me of my undergrad thermodynamics book, of all things. As you know, an object at a particular temperature will radiate light with a particular spectrum depending on the material. Heat something up enough, and it will glow red, then orange, then white, and so on. Cooler objects radiate in the infrared and lower wavelengths, and extremely hot objects like plasmas can radiate into the ultraviolet and x-rays. All other things being equal this radiation will follow Planck's law and emit light of certain frequencies in certain proportions. All other things are generally not equal, but Planck's law describes a very frequently encountered type of emission called blackbody radiation. A blackbody is an object that absorbs all light hitting it, without reflection.

The book gave a problem about calculating the blackbody radiation emitted by a human being. This immediately raises a question: are people blackbodies? Does it depend on what color the skin is? The fact that the term is essentially "black body" and thus somewhat evocative of race makes the problem scenario more awkward-sounding than it might have otherwise been. Had the author been a crass 8th-grader instead of a classy and professional physicist it would have been a good opportunity for a Michael Jackson joke.

I don't have the book handy and so I can't tell you the exact way the book gamely but slightly self-conscious way phrases the answer. But there is an answer, and it's pretty easy.

The visible light we see has energies on the order of an electron-volt. This is a very typical sort of number for the energy gaps between the different energy levels of electrons in atoms and molecules. Light might hit an atom, bounce the electron up a level, and be re-emitted as the electron falls back down to its previous position. We see this all the time in phenomena like fluorescence. But this isn't thermal energy. You might have guessed something from the fact that things have to be really hot before they start glowing visibly: an electron-volt is much more energetic than the energy being transferred by the random thermal motions of atoms as they jiggle around. The mechanisms of visible light emission and thermal light emission at human body temperatures are completely different.

Skin color is a product of melanin content - a visible light phenomenon of electron levels. More melanin affects those incoming visible photons and results in the different shades of human skin that we see. But thermally that melanin matters not a whit, as it's just another molecule vibrating and jostling thermally throughout our bulk. As a result, human skin of all races has effectively identical thermal emission characteristics. And those characteristics happen to be pretty much those of an ideal blackbody for that far-infrared light resulting from thermal radiation. Pretty much everything is a blackbody at those wavelengths, and humans are no exception.

"We all look the same under a FLIR scan" is not likely to be the next anti-racism PSA. But maybe it should be. There's not enough statistical mechanics in public service announcements these days...

Posted by Matt Springer at 10:00 AM • 2 Comments • 0 TrackBacks

Technology & Middle Earth

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Chad Orzel's got a great post up about the physics of Lord of the Rings. It's about Legolas the elf and his excellent eyesight. His eyes are so good that in fact they're probably operating well beyond the physical diffraction limits of any optical device with a human-sized pupil. Some speculation was discussed about how his eyes might plausibly be so good without magic: maybe he can see in the short-wavelength UV, maybe he can do interferometry(!), maybe elf pupils are bigger than we think, maybe the Middle Earth "league" is shorter than our identically-named unit of distance, along with a few other interesting suggestions. "It's magic" is probably the right answer, but then we miss the fun of a physics discussion.

I have another thought. Why doesn't anyone just use a telescope or binoculars? Ok sure, it's ancient middle earth and presumably a Galileo hasn't been born yet. But I see no reason that this ought to stop them. Middle Earth is not a completely pre-technological environment, there's science of some sophistication. Let's see:

Glassworking: Good enough to produce Palantir, which are crystal spheres used for communication and surveillance. There's also very considerable skill in gem cutting, which is important in The Hobbit.

Metallurgy: At least to the medieval level given all the sword-reforging and such, and probably much better since they can work with the practically indestructible Mithril. We're leaving aside the whole ring-forging thing since that seems to be closer to magic than science.

Chemistry: It's unclear how much theory they actually understand, but as a practical matter they've at least got gunpowder figured out. Nobody uses it but Saruman, which is a shame. Mining the Pelanor Fields might have saved our heroes a lot of trouble. I don't remember if the metal of the swords is mentioned specifically, but if it's steel as opposed to iron then that probably adds to the sophistication of the people of Middle Earth with respect to both chemistry and metallurgy.

Biology: Given that the flora and fauna of Middle Earth is so far removed from our experience it's hard to be definitive about this. The orcs and Uruk-Hai are both arguably bioengineered, but this seems to have been done with magic rather than purely genetic means.

So despite the fantasy setting I'd say the characters have access to an early and little-developed but still considerable quantity of scientific knowledge. If Gondor had spent some of its money on an ancient Los Alamos the whole thing might have been a lot easier.

Posted by Matt Springer at 2:38 PM • 7 Comments • 0 TrackBacks

Electric Charge and Sharp Points

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Yesterday we dumped a bucket of electrons on the Statue of Liberty and watched what happened. The most important thing we saw is that all the charge immediately distributes itself over the outside surface of the statue in such a way so that the electric field within the statue is zero. We also noted that the field was high on sharp points like the spikes in the crown, but we left that without further explanation. Today we fix that.

Fields can be funny things. In the case of electric fields and gravitational fields and numerous others, the fields are vector fields. This means that a given point will have an electric field with a strength and a direction for it to point. These vector fields can be a pain in the neck to deal with and thus physics often uses a concept called potential to simplify the situation. A potential tells you how fields are set up without reference to vectors. To use gravity as a quick example: if you're sitting on a piece of cardboard on a hill, the field is the direction gravity makes you slide along with the strength of the pull. The potential is the height of the hill at every given point. If the potential is changing rapidly the hill is steep and the field is high along the slope. And you're pulled along rapidly by gravity. In short, a field is just the rate of change of potential with distance.

In that spirit let's talk about the charges on the Statue in terms of potentials instead of fields. There being no field in the statue means the potential is constant - if potential isn't changing with distance, there's no field. That's true within the statue. It's also going to be true along the surface. If there were a potential gradient, there would be a field. If there were a field, the charges would be moving, which they're not or they wouldn't be at their equilibrium positions. From that we have to conclude that the potential along the surface is constant. The only thing left is to deduce what charge configuration produces a constant potential.

To answer that question, let's break out an analogy we first used with black holes: the physics choir. Let's give them some music: I suggest "Battle Hymn of the Republic" on account of the reference to lightning. Now each member of the choir represents an electron. The volume of their voices represents the potential the charged electrons produce. It's high in magnitude as you're close to the choir, and low in magnitude as you stand far away.

A statue is a complicated shape, so let's simplify the problem by replacing it with the simplest charge configuration we can: a piece of straight wire. So take the choir out to a field, lay a long line of tape down to represent the wire, and have the choir distribute themselves evenly along it. Is the potential (the volume of the singing) the same everywhere along the wire? Heck no. If you stand at the center of the wire and listen, no choir member is more than half the length of the wire away from you. If you stand at the end of the wire, exactly half of the choir is farther from you than the farthest member was when you were at the center. As a result the choir is going to sound a lot quieter to you when you're at the end of the wire. Which means the wire isn't at a constant potential, which means that it doesn't represent the way electrons actually distribute themselves.

So how will they distribute themselves? You might think that if the choir arranges itself so that it's more spaced out near the center and more closely packed near the ends, the volume might be constant everywhere in the wire. And you'd be right. The fact that the volume has to be constant necessarily concentrates more singers near the ends. In two and three dimensions, electrons do the same thing. To produce a constant potential, they have to densely pack in to the places where geometry doesn't give them a lot of room to produce the same potential by the expedient of low density over a large area.

If we wanted to argue this in a mathematically specific way we'd have a pretty hard time of it. Even very simple geometries quickly become to difficult to handle exactly. Nonetheless it's possible to show that the sharper the point the higher the charge density. This is responsible for everything from the way sparks fly off a Tesla coil to the shape of lightning rods.

And that's why electric charge tends to collect at sharp points.

Posted by Matt Springer at 11:37 AM • 3 Comments • 0 TrackBacks

Statue of Charge

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Today we need an example of something weirdly-shaped and electrically conductive. There's no shortage of such things, so we might as well go with the iconic. This is the Statue of Liberty:

It's made out of copper, which over the years has taken on a decidedly not-copper color due to chemical reactions between the copper surface and the surrounding atmosphere. But it's still copper and thus a very good conductor of electricity. Unfortunately for our purposes here the statue is also hollow, and in fact the copper is only a few millimeters thick. This isn't unusual, almost all metal statues of any size are hollow. Metal is very expensive and very heavy. But for the moment, go ahead and pretend the Statue of Liberty is in fact a solid mass of copper metal.

Now apply an electric charge to it. Dump a bucket of electrons on it, fire up a Van de Graff generator, rub a balloon on your head and hold it close to the statue, whatever. Now there's an excess of charge on the statue. How does it distribute itself? We know that charges experience a force when they're placed in an electric field. Conversely an electric field is generated by the presence of charge. Thus the charges are going to be pushed around by their mutual repulsion until they reach a stable configuration. With a little thought we can figure out what the stable configuration is, even for something so complicated as the Statue.

It's not possible to quite do this in reality, but imagine that you want to very sensitively probe the electric field at a particular point. You do this by taking a single electron as a test charge, placing it at the location you want to test, and seeing which way it moves. In that way you can see what direction the electric field points.

But at this point we can also extrapolate backwards. Once we charge up the statue, there can't be an electric field anywhere inside. If there were, our test charge and all the other charges would be moving, which means they wouldn't have found their equilibrium positions yet. Once they find their equilibrium positions, they aren't moving anymore. Which means there's no field. But if there's no field, that means there's no net charge - because charge generates an electric field. So is there no charge in the statue, despite the fact that we just put it there? The answer is that there's no charge in the statue. Our argument shows that the equilibrium position of all the charge is on the surface of the statue. Any charged conductor will have all of its charge on its exterior surface.

This argument only works for conductors, since it requires that the electrons be free to move. In an insulator they can't, and so they'll mostly stay wherever they're put.

Back to the Statue. All of the charge is on the surface, but we have no guarantee that's it's evenly distributed on the surface. In fact it's not. In general the distribution on the surface will be a complicated function of the geometry. In particular it will tend to be highest at sharp points like the spikes on the crown, or (more saliently for other types of structures) lightning rods. And that's a story for another day - though I'm pretty sure that other day will be tomorrow!

Posted by Matt Springer at 12:55 PM • 7 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

If you want to kill some time, try to think of all the different definitions of the word "set". You have chess sets, sets in tennis, sets of dishes, sets musicians play, sets as abstract mathematical entities, television sets. You can set a table, set a clock, set someone straight, set a price, set out on a journey. That's just scratching the surface. The Oxford English Dictionary lists literally hundreds of different senses of the word. And yet if I set out (ha!) a particular instance of the word, you'll process its meaning instantly.

You'd think mathematics could avoid the ambiguity of multiple meanings for the same symbol, but I regret to say it's not so. There's much less ambiguity to be sure, but there is some out there. Even the venerable π symbol has more than one common meaning. So you'll not be surprised that when physicists talk about gamma as a function, it's not necessarily the gamma function. The one we'll talk about today is generally just called gamma even when treated as a function to avoid confusion. On paper it's more clear because our gamma today is lowercase and the gamma of the gamma function is upper case in the Greek. Not that a lower-case gamma is definitive either. When in doubt today's Sunday Function has its own proper name: the Lorentz factor.

Our gamma is a property of a moving object. It's a dimensionless function of velocity, and it's defined like this:

It's not a very complicated function, which is nice. Lorentz was the first person to do much with it, and when Einstein put together his theory of special relativity this function turned out to be a central feature permeating the mathematical structure of his theory. There's only two constants in the equation: v is the relative velocity between two non-accelerating frames of reference (we call these inertial frames), and c is the speed of light. Relativity predicts that time and space aren't the same for two observers in different inertial frames, but instead time and space are scaled by a factor of gamma via the Lorentz transformation. Notice two things. First, if v is very small than the fraction under the square root is very tiny. Thus the denominator is pretty much 1, and so gamma itself is pretty much one. Second, if v gets close to c, then the ratio in the denominator becomes close to 1, leaving the denominator as a whole close to 0. And this makes gamma huge. We can graph the rest of the behavior, with units in meters per second. Roughly speaking, gamma close to 1 means that the classical non-relativistic treatment is a very close approximation to reality. When gamma gets starts to get much higher than one, classical physics is a bad approximation and you need relativity:

Let's do an example. It turns out that in relativity the kinetic energy of an object in special relativity is not the usual one-half mass times velocity squared. Instead it's:

So the kinetic energy of an object isn't just the usual gently sloping parabola as a function of velocity*, instead at velocity near the speed of light the kinetic energy will look pretty much like the graph above. Without going into the why of the Lorentz factor, this is why it's not possible to go faster than the speed of light. The graph goes rapidly to infinity, and thus the energy you need to approach the speed of light goes rapidly to infinity as your speed gets close to c.

There are some subtleties to thinking about it this way. There's no absolute reference frame, so it's not as though your spaceship suddenly gets near the speed of light and refuses to accelerate further. Instead, time and space are simply differently scaled in your reference frame and the reference frame of the galaxy around you. You need more and more energy to go faster in the galaxy's frame, but in your frame you keep accelerating fine while the galaxy around you is distorted due to the gamma-factor in the scaling of space and time.

It's hard to explain precisely without doing the math. Which we in fact will do at some point in the future. For now, just be glad we had Lorentz and Einstein thought this stuff up so we wouldn't have to do it on our own. On the other hand if we had thought of it ourselves it would have been a free trip to Sweden...

* Yes, I'm thinking the same thing you are, so it's officially on the to-do list for Sunday Function: Taylor expansion of the Lorentz factor, showing equivalence to classical kinetic energy

Posted by Matt Springer at 10:00 AM • 11 Comments • 0 TrackBacks