Built on Facts

I would never have thought that so recently after starting this site that I’d be invited to do something as cool as be a guest on SETI radio. But I was, and it was awesome. This week they’re doing a fascinating show about the science of Indiana Jones and the Kingdom of the Crystal Skull. Tom Rogers of Insultingly Stupid Movie Physics (a favorite site of mine long before I started writing here) and I discuss the physics of the film, where it goes wrong, and the couple of places it goes right. Ian Freestone of the University of Wales talks about the actual crystal skulls which are known to archaeology (they’re human shaped, and almost certainly fake). Phil Plait of Bad Astronomy talks about Mayan archaeology and the infamous 2012 deadline for the doom of the earth. Finally, Jim Underdown of Center for Inquiry West in Los Angeles discusses how critical thinking might have been put to some good use in the film.

It’s a really fun show, and I hope you enjoy listening as much as I enjoyed doing the interview!

SETI Radio

A few days ago I saw the movie Rudy for the first time.  It’s a story of a young guy in the 70s who wanted to play for the Notre Dame football team.  He was a small guy, and after tremendous adversity finally managed to get on the practice team to function essentially as a training dummy for the actual team.  His dream was just to sit on the bench as a backup player for just one game.  Not my cup of tea, but good for him.  On the last game of his senor year, he’s not only allowed to play, he’s actually put in at the very end of the game as a defensive player.  He participates in the last two plays of the game - on the last one he sacks the opposing quarterback.

“Ok,” thought I, “I can understand a feel-good story but this is so implausible it defies all logic.  Who writes this stuff?”

Turns out it’s a true story.  Oops!

Turns out there’s also a lot of physics in football.  The most obvious example is projectile motion in the throwing or kicking of the ball.  There’s the added bonus that the aerodynamic shape of the ball makes air resistance less of a factor than in baseball and similar sports with spherical or very lightweight balls.  One of the cleanest particular uses of the football as a projectile occurs during punts and kickoffs.  You might want to pick the angle that gives the maximum possible range so as to best establish field position - but your team might not get to the punt returner before he can start running.  If he’s a good runner this longest-distance strategy might not be the best.  Another alternative is to kick the ball at a higher angle to maximize the time the ball spends in the air.  It won’t go as far but the extra hang time allows your team to get into place to best tackle the returner once he gets the ball.  There’s also the wacky on-ground squib kick, but that’s not really projectile motion.

Taking the y-direction as vertical and the x-direction as downfield, the ball’s motion is governed by

and

Where v₀ is the initial speed of the ball.  We can solve the first equation to find t at y = 0, and plug that into the second equation to find the range.  A little algebra and a trig identity gives a range which I’ll call R.

Now we know that sin has a maximum of 1 where its argument equals 90 degrees.  This means we need 2θ to be 90, which means θ is 45 degrees as expected.  That gives us the maximum range.  but the time of flight T (obtained by solving the y equation for t at y = 0) is

This is a maximum where the angle is 90 degrees.  But 90 degrees is a kick straight up, which common sense and the range equation tells us is not very useful since the ball lands where it was kicked.  The time of flight for a full-range 45 degree kick is only about 5/7 of the same kick angled straight upward.  Maybe some angle in between might fit the bill for a particular situation.  It’s a tradeoff, one considered many, many time on any given Sunday.

So next time you see a football game, don’t be prejudiced and think it’s just something for jocks to bump chests about.  The kickers and quarterbacks are doing real physics without even having to think about it.

Is the universe made of math? That’s a question going around internet science-fan circles of late, and it’s a pretty difficult question. Roughly, a cosmologist named Max Tegmark believes in a very concrete form of mathematical Platonism - the idea that math is “real” in some sense. Now I and many other people agree that math isn’t just something we humans made up out of thin air (the formalist position) because math simply works too well. The richness of mathematics and mathematical physics is not something that was planned out from the start, which leads me to think mathematics is built in to the universe independently of anyone in the universe to write it down.

Now that’s a much weaker statement than “the universe is made of math”. Take Tegmark’s view:

I said there could be a whole universe that is nothing more than a dodecahedron, a 12-sided figure the Greeks described 2,500 years ago. Of course, I was just fooling around, but later, when I thought more about it, I got excited about the idea that the universe is really nothing more than a mathematical object. That got me thinking that every mathematical object is, in a sense, its own universe.

Tic-tac-toe is formally and completely described by math (as is the machine code of the Barbie Fashion Show computer game, for that matter), but the idea that somehow there’s an actual tic-tac-toe universe out there seems… far-fetched. At the very minimum there’s a lack of corroborating evidence, and not much hope of getting any.

He has a factual error as well:

But if space goes on forever, then there must be other regions like ours—in fact, an infinite number of them. No matter how unlikely it is to have another planet just like Earth, we know that in an infinite universe it is bound to happen again.

Not quite.  There’s an infinite number of prime numbers, none of them are the number 15.  Infinite trials do not necessarily lead to every result.  There could be an infinite number of earths, but there doesn’t have to be even if the universe is infinite in extent.  I notice John Derbyshire on (of all places) the National Review website has also pointed this out.

It’s not a bad attempt at an answer to Hawking’s “How come existence?” question, but for the moment I think it’s better to take the wet blanket stance on this particular idea.  There aren’t any experimental ways to check it, and so it’s speculative philosophy but not yet anything approaching science.

Here’s a new feature for this website that I’m going to try out.  One of the daunting things about studying physics is the profusion of textbooks.  Some are terrible, most are mediocre, and a shining few are really excellent.  I’m going to keep a list of those books which I’ve used in my classes and found to be at least pretty good.  There will also be short comments about what I thought of them.  So far this list is not complete but it’s got a pretty broad base to start with.  I have a few more to add in the coming weeks, and as I take more graduate classes I’ll be able to evaluate the textbooks I’m assigned and I’ll add the good ones to the list.  You can leave reviews or comments at that page as well.  So here we go:

Recommended Books

As an added bonus for me, books purchased from Amazon from the links here will give a small percentage of the profit to this site, at no added cost to you at all.  Even if you just get to Amazon from the links but buy something else, the site still gets a small bit of the purchase price.  So if you want, you can support this site for free when you buy books!

Famously there are these exercises called Fermi problems, which are basically questions designed to be solved only by reference to approximation and dimensional analysis.  They aren’t even necessarily physics problems - the canonical example is “How many piano tuners are there in Chicago?”.  The reason why they’re interesting to try to solve is because it develops your intuition for what right answers look like.  If you know what the rough order of magnitude of the answer ought to be, it’s a good first check of your actual formal calculation.  Let me make one up:

I have a pretty standard ceiling fan above my head right now.  If I installed my own electrical system for it, how fast could I spin the blades before they tore off due to the centripetal force?

There’s so little information here it’s ridiculous.  You might think the problem is impossible, and in some sense it is - but if we want a rough estimate we might be able to make some progress.  After all, experience proves that the number is greater than the few-hundred RPM it spins under normal operation and it’s certainly below… I dunno, a few billion RPM.  So we’ve already eliminated two ranges of possibilities, one infinitely large.  It’s a start.  Let’s try to do better.

How much force will it take to break a fan blade?  While there’s no way to get a good answer short of destructive testing I think it’s reasonable to guess that if you mounted a blade vertically instead of the usual horizontally it might be able to support maybe a hundred pounds hanging from it.  Maybe not exactly, but I think it’s reasonable to peg the figure between 50 and 200 pounds.  In Fermi problems rough estimates are all we’ve got.  So 100 pounds it is.  Now the force required for a point mass m in circular motion is F = mv²/r, where v is the velocity and r is the radius of the circle. Now the velocity is rω, where ω is the angular velocity. We can convert that to RPM when we’re done. All that means the force is F = mrω². A fan blade is not a point mass, but we can assume the mass is all at the center of mass. As far as egregious approximations go, it’s not terrible.

I don’t know what the mass or the distance to the center of mass is, but those can both be estimated. Distance to the center of mass is about 18 inches, by my rough guess. The blade itself is pretty much particle board, a quick Google search shows a typical density of 200 kg/m³. The volume is probably about 36in*6in*1/8in. Multiplying and converting units gives a mass of 88 grams.

Solve for ω. We get ω = (F/(mr))^1/2. Plugging all our estimates in gives ω = 105 s^-1. Divide by 2π to get frequency, multiply by 60 to get RPM, you get a final breaking speed of: 1002 RPM. We’ll call it a thousand.

There’s an answer to a Fermi problem. That process applied to more cutting-edge physical situations is often the first step for developing new theories. The next step is to refine the idea and make the wild guesses into something more rigorous. In our example we might calculate the forces throughout the entire blade and find out something much more precise about the strength of particle board. And finally we’d test our theory by actually spinning up the blades and seeing just when they broke.

Still, you might not want to be in the room when you run this test.

You’re standing around minding your own business when suddenly a mile or two away, a rocket launches into the air. It curves upward in a lazy arc and begins its descent toward you. This rocket has several pounds of high explosive in its tip and so you’d prefer to find a way to improve your outlook. The old method was to get lucky and hope you didn’t get hit. Most people in target zones don’t generally prefer this method. The new way is to install radar to give you a couple dozen seconds warning to seek cover before the rocket reaches your position. It’s a definite improvement, but it’s not going to do much to protect anything that can’t move out of the way.

Close-in weapons systems are a possibility. Shoot enough bullets at the incoming missile and you’re bound to hit the thing eventually. The problems with this idea are that it’s very difficult to hit a missile with a bullet, the thousands of bullets that don’t hit are a great danger to the surrounding area, and the systems are expensive, bulky, need constant resupply.

A laser might be an improvement. Targeting is easy since the beam travels in an extremely precise straight line. Once installed, there’s no supplies required except electricity. There’s almost no possibility for collateral damage since a miss with a laser beam will just continue up into the air. It would be an ideal system if only a laser could destroy an incoming missile. That’s the current sticking point. Practical lasers just don’t have the power necessary to heat a missile to the point of destruction in the required very short timeframe.

There is a somewhat impractical laser that can do it. Boeing has developed an airborne laser that is powerful enough, but it’s huge. It requires almost the entire volume of a 747 and is tremendously expensive. Worse, it’s chemically pumped. Instead of electricity driving the lasing process, it uses a chemical reaction. The chemicals involved are toxic, bulky, and expensive. The entire system can only make a few full-power shots before running out of chemical fuel. These are acceptable problems for defending a large area against a small number of ballistic missile launches, but generally not for defending small areas against numerous cheap tactical missiles. It has been done with ground-based systems, but it’s not ideal.

This is the current state of the art, but progress keeps moving. The lowly laser pointer produces laser light in a solid-state medium. This laser light is generally at a quite low power, but solid-state lasers can in fact generate quite high power. They do so without the fuel or cost problems of chemical lasers, just requiring electricity. Beam intensity is a bit of an issue, with the most powerful solid-state lasers being on the order of 10kW of power. A practical system needs to increase that by another 1 or 2 orders of magnitude. Heat dissipation becomes a problem as well. Raytheon is making progress, and Northrop is pushing the 100kW solid-state mark as we speak.

It will probably be a few years yet before these are ready for deployment. But when that day comes, both soldiers and civilians in areas of rocket attack will have one less danger to worry about.

My own primary research interest is laser physics, so I find this practical application doubly interesting.  Now the lasers in my group’s lab have a vastly lower average power and one of our main goals is to get the pulses as short as possible - attoseconds, ideally.  They will not do anyone’s military much good as far as I can tell, but of course that doesn’t mean the more mundane military lasers are any less remarkable.

Next semester I’m taking the graduate classical mechanics class, and so I’m trying to spend a little time this summer brushing up on my old undergraduate mechanics. It’s been about three years since I’ve had to do much of it, and I’ve dug out my old books and am working on some problems. My undergrad class text was Thornton and Marion. Opinions varied among the class as to how good it was, but I really liked it. This problem is from Chapter 7, and is another problem using the Lagrangian formulation of dynamics. I think a sufficiently clever person might be able to do this problem using forces, but I’d hate to be the one who had to try it.

A particle of mass m rests on a smooth plane. The plane is raised to an inclination angle θ at a constant rate ω (θ = 0 at t = 0), causing the particle to move down the plane. Determine the motion of the particle.

The position of the particle is uniquely determined by two coordinates: the distance up the ramp r and the angle of the ramp θ. And θ(t) is fully determined by the fact that the particle is on the ramp* and the ramp’s tilting is not affected by the particle. So really we just have to find r(t). To construct the Lagrangian we need the potential and kinetic energy of the particle. The potential energy is

The kinetic energy is just K = (1/2)mv², but v has components in the r and θ directions. If you’ve followed this far you probably know how to write the kinetic energy in polar form, but if not here’s a brief summary. First we need the velocity in polar coordinates. This is done in considerable detail in Marion and Thornton, and in a good if dense form here. The result is that

Where the dots denote differentiation with respect to time as usual, and the hat symbols denote unit vectors. Kinetic energy is proportional to v² so we need to square the above expression:

And the dot products for the unit vectors are just 1 if the unit vectors are identical and 0 otherwise because the vectors are orthogonal. This means we finally have

Ok. Subtract the potential energy from the kinetic energy and we have the Lagrangian:

Now we need to find the partial derivatives with respect to r and r-dot so we have something to put in the differential equation. It’s pretty straightforward:

and

Which we can plug straight into the Euler-Lagrange equation for r.

leading to

So after taking the derivitive of r-dot, canceling the m’s, moving the rightmost term over, and casting θ in terms of ω as instructed in the problem statement, we have

Which is a fairly straightforward differential equation. You can find the homogeneous solutions and then get the particular solution using Green’s functions. The final result with an initial position r₀ and initial velocity 0 is:

A lot more complicated than the stationary inclined plane, no? But it’s the right answer, and it was found by a reasonably straightforward procedure. Attempting this problem with forces would have been nightmarish. Just for fun, let’s graph this with g = 9.8, r = 10, and ω = 0.2 in the appropriate units.

Looks about like we expect. The particle starts off slow and picks up speed as the ramp angle is increased. Problem done!

*If this ceases to be true our assumptions about θ(t) are no longer valid and the solution method fails. This corresponds to the ball falling off as the ramp angle goes about 90 degrees. This need not happen if the angular rotation is fast enough, and we could use undetermined multipliers to actually find the minimum required angular velocity, the constraint forces (Lagrange’s method can be used to find the forces, just as the forces can be integrated through distance to find energy), etc. This is a somewhat more difficult problem.

For a little lagniappe to the regular daily post, here’s a few things things I thought were really cool.

The world’s largest air vortex cannon. How far away can you blow out a candle? Unless you have a jet engine handy, probably not this far.

Hrm. Carl over at Mass is working on some heuristic modeling of fundamental particles. It’s a little out of by depth, but reading stuff like this makes my depth a little greater.

Via Dr. Pion, the science(?) of Donald Trump’s hair. Come on, Mr. Trump. Don’t you know that thanks to those enterprising biologists there’s an actual full-blown cure for baldness that’s already in Phase II trials? Surely your millions can get it now if you want.

Firefox 3 is out of beta and available for download.

Anyway, never let it be said you don’t get variety at this site!

The Boston Globe has a beautiful retrospective on some of the most visually stunning photographs the Cassini probe has taken from the vicinity of Saturn. All of them are beautiful and I highly recommend the link so you can see all the images. This image is of Dione, Saturn’s 4th largest moon. It’s absolutely surreal.

This particular photograph was taken from just a few hundred kilometers above the moon’s surface. That human beings can send such an exquisite piece of instrumentation to fly just above another world a billion miles away is mind blowing. In some sense we can comprehend what earthly places are like even if we haven’t been there. “Dione, moon of Saturn” seems a little unreal, a small blurry dot in a book. A picture like this dispels those vague impressions and makes it seem like a solidly real place our children or their children might one day stand and run their gloved fingers through the Dioneian dust in silent contemplation.

I wouldn’t personally want to be an astronomer for a living. I like my experiments to fit comfortably on a tabletop. But I’m very glad there are astronomers so I can see the beautiful end results of their missions.

Which isn’t to say there’s not a lot of physics on these planetary science missions. There’s a reason many universities have a combined Department of Physics and Astronomy. Even the design of the spacecraft sometimes has both the physics and the PR problems of physics to deal with. For instance: Saturn is roughly 1.4 billion kilometers from the sun. The earth is about 149 million kilometers from the sun. Take the ratio, square it, and you’ll find that there’s only about 1/93 of the solar power per square meter at the distance of Saturn as there is at the distance of the earth. Cassini needs a lot of power, and there’s not a lot of solar power to be had way out there. The solution developed was the use of radioisotope thermoelectric generators, or RTGs. This is essentially a very small scale nuclear reactor. The radioactive material produces heat, outer space provides the cold, and a thermoelectric material turns this temperature difference into electrical energy. It’s reliable and stores lots of power. In the case of Cassini, it’s hundreds of watts for the entire decades-long duration of the mission.

The problem is of course that it’s radioactive, and so the plan generated some protest. If the launch vehicle were to blow up or reenter the atmosphere during a gravitational assist the radioactive material could be dispersed into populated areas. While a nuclear explosion is totally impossible due to the nature of the material, the radiation by itself is not insignificant. Fortunately, exhaustive analysis indicated that the risk was extremely minimal. Basically there weren’t any plausible ways for the material to dangerously disperse even if the rocket exploded, and even if so the Pu-238 fuel is an alpha emitter which is the easiest form of radiation to shield against. A t-shirt will block it, though it’s still dangerous upon direct skin contact, inhalation, or ingestion.

Not perfectly safe, but as a matter of probabilities it’s a lot more safe than the radiation your local hospital uses to treat cancer. Hopefully the continued progress in solar panel efficiency, cost, and weight will make it unnecessary in the future. Personally I hope that the safety and success of Cassini might be one more thing in the public consciousness to help improve the acceptance of modern and safe nuclear power as one pillar of the fight to rely less on fossil fuels. It’s not as though coal power isn’t radioactive, after all.

Cassini wasn’t exactly cheap either, at a cost to the US of around $2.6 billion. But somehow I can’t get really worked up about this, as the federal government spends that much roughly every 8 hours. The Cassini mission will produce about 6 years of exploration of the Saturn system while Congress has spent $17 billion on pork projects in FY 2008 alone. I know which I’d rather see more of my money supporting.

For the last few hundred years, one of the most fundamental principles in physics has been that energy is conserved. You simply can’t get it from nowhere. Every experimentally tested physical theory from Newton’s laws to Maxwell’s equations and beyond is completely consistent with energy conservation. Noether’s theorem makes the possibility of an error in our understanding of energy conservation even more unlikely.

None of this has deterred people from trying to invent perpetual motion or other free energy / “over unity” devices. “What if the scientists are wrong? They’ve been wrong before. They should at least give serious examination and consideration to my idea.” say the hopeful inventors. I can understand their perspective. It’s frustrating and offputting to be met with an eye-roll and a “You’re wrong” with no friendliness or consideration. It’s also frustrating and offputting for the scientists to keep being approached by people who are wrong (and often belligerent), and the cycle feeds on itself. Even worse are the outright fraudulent companies preying on unfortunate investors, and the media which is often so spectacularly ignorant that you can’t tell if they’ve been fooled by fraud or just screwed up a real research story so badly that it seems to be claiming something it’s not. Swans on Tea recently noted an instance of one of these, though it’s hard to tell which one.

To avoid this, I have a suggestion for those people who are trying to build a free energy machine and want to be taken seriously. There’s two issues to resolve. You have to make sure you’re not fooling yourself about what your device can do, and you have to make sure that it’s easy for a scientist to see that the device can do what you say. Here therefore is the Built on Facts Protocol for testing perpetual motion or free energy. It is a literal instantiation of a black box test.

1. Put the machine in a completely closed box. Metal and airtight is preferred, but not required. Absolutely nothing can be connecting the outside of the box to the inside of the box under any circumstances.
2. Attach a standard light bulb socket to the outside of the box, with the wiring running into the inside of the box to the power output of your free energy machine. If your device does not produce 120VAC at 60Hz, attach a small generator, inverter, or whatever else is required to provide the appropriate power. If this is too onerous in terms of total wattage, a small 12V bulb or even a LED is acceptable. Try to use as high an output power as possible, as the difficulty of step 3 is inversely proportional to the power output. When your device is running, the bulb must remain unambiguously lit.
3. The device must run and the light must shine continuously for the amount of time T in seconds given below without outside intervention of any kind.

T = (34,600,000,000)(V/P)

Where V is the volume of the box in cubic meters and P is the power of the bulb in watts. So if your bulb is a 120W light and the box is 1 cubic meter, you’ll need the device to power the bulb for 9 years. Now most people don’t have 9 years to wait. You can take this figure down quite a bit by ramping up the power and shrinking the size of the box. If you can fit your invention into one cubic foot and power a 1000W halogen you’ll only have to operate for about 11 days. I understand this is quite a difficult requirement but there is a concrete reason for it. It could be worse - perpetual means forever, after all.

What is the reason? Why 34,600,000,000 for the constant? That’s the energy density of gasoline in joules per cubic meter. The box ensures that you’re probably not inadvertently connecting to some outside source of energy. The large time constant ensures that you’re probably not using some internal energy storage. If you can run for the required time with no contact or input from the outside, you pass the test and the device is worth a more careful examination by a qualified scientist.

Best of luck.