http://scienceblogs.com/builtonfacts/ An exploration of physics and the quest to understand our world. en Copyright 2009 Fri, 20 Nov 2009 13:36:06 -0500 http://www.sixapart.com/movabletype/?v=4.261 http://blogs.law.harvard.edu/tech/rss BuiltOnFactshttp://feedburner.google.com <p>Around ScienceBlogs, people who don't accept global warming as a real phenomena tend to get called denialists. In the interests of full disclosure, I should admit that I'm not a denialist but rather a global warming <i>defeatist</i>. Doesn't matter how bad or not CO2 is, ain't nothin' gonna stop it. People will not give up electricity and transportation in the developed world (nor should they), and people in the developing world will not be give up the quest for developed-world living conditions (nor should they). As such it's either massive and immediate worldwide switches to nuclear power and electric transportation, or we'll just have to live with whatever happens, wherever it is on the spectrum from "nothing" to "Day After Tomorrow". CFLs and conservation and even the climate change bill currently stalled (and honestly pretty much admitted dead even by its supporters) in congress will only make the barest scratch in US emissions. The US is increasingly a side issue anyway - the 2,400,000,000 people in China and India will matter much more than the 304,000,000 in the US in terms of CO2 release.</p> <p>And since such massive rollouts of nuclear power and electric transit don't seem to be forthcoming, I'm a defeatist.</p> <p>But that's just disclosure. I know it's a minority opinion around here, so you should at least know that I hold it.</p> <p>All this is just preliminary because ScienceBlogs needs an open thread on <a href="http://noconsensus.wordpress.com/2009/11/19/leaked-foia-files-62-mb-of-gold/">these leaked climate change documents</a>. They're all over the web at this point, so you may have seen them mentioned already. (<a href="http://wattsupwiththat.com/2009/11/19/breaking-news-story-hadley-cru-has-apparently-been-hacked-hundreds-of-files-released/">Here</a>, for instance.) Allegedly there's all kinds of nasty misconduct possibly constituting scientific fraud.</p> <p>I make ZERO claims about the credibility of any site discussing this. Since I generally don't follow the subject (on defeatist grounds), for all I know the sites discussing this might like to rail about the Illuminati and sell alien urine to pay their bandwidth bills. But it has been confirmed that the leaked emails are real by the Climate Research Unit, the organization that got hacked.</p> <p>If the whole thing is overblown, I want to hear about it. If not, tell me about that too. You as a group of readers are pretty astute and I'm interested to hear your analysis of this mess. Thus: open thread.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/leaked_climate_change_document.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/5cug-AJm8hY" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/5cug-AJm8hY/leaked_climate_change_document.php http://scienceblogs.com/builtonfacts/2009/11/leaked_climate_change_document.php Fri, 20 Nov 2009 13:36:06 -0500 http://scienceblogs.com/builtonfacts/2009/11/leaked_climate_change_document.php <p>Ok, see counselor Troi firing her phaser?</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="troi.png" src="http://scienceblogs.com/builtonfacts/2009/11/19/troi.png" width="400" height="300" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>You see this kind of thing all the time on film in scifi. Whether it's Star Trek, Star Wars, or pretty much anything else, energy beams fired from future weapons are visible. Usually someone will point out that in fact laser beams are not visible in this manner. To see light, it has to reach your eyes. This is clearly not possible when all the light is actually traveling down the beam path. You can see this in action with laser pointers - only the spot where the light hits and diffusely reflects is visible. The path is not.</p> <p>Writers of TV shows usually explain this by saying that the beam is not strictly light, but some stream of particles that slightly emits to the sides along its main path. While this has its own problems, at least it acknowledges the issue.</p> <p>But what's even more interesting is that in fact there are already automatically particles present along the beam path in the atmosphere. Some of them are sizable particles like dust, others are individual atoms and molecules. Generally they don't scatter much light, but if the light is intense enough then the small amount they do scatter is enough to see. And so you have a visible laser beam. Here's one in my lab:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="laserbeam.png" src="http://scienceblogs.com/builtonfacts/2009/11/19/laserbeam.png" width="400" height="300" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>The beam scatters off the air and you can actually see it as a straight line. Apologies for the terrible camera phone picture, I really need to get a classy camera that can take nice pictures. This is not actually a laser I'm working on, so honestly I'm not sure which variant of frequency-doubled Nd:something laser this is. Probably Nd:YLF.</p> <p>This is used to pump an infrared ultrashort-pulse laser with a repetition rate of 1 kHz. This can itself be focused to a point in the air, which becomes visible as a little stationary spark as the intense beam ionizes the air. This produces a 1 kHz buzz which can easily be heard by the unassisted ear.</p> <p>I have to say it's a nice job perk that I can see old science fiction tropes come to life pretty much every day. :)</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/seeing_laser_beams.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/YXsH-dmxVCA" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/YXsH-dmxVCA/seeing_laser_beams.php http://scienceblogs.com/builtonfacts/2009/11/seeing_laser_beams.php Thu, 19 Nov 2009 16:15:34 -0500 http://scienceblogs.com/builtonfacts/2009/11/seeing_laser_beams.php <p>There's a little bit of buzz burbling around over Al Gore's scientific goof during a Conan O'Brien interview. <a href="http://www.youtube.com/watch?v=Ns_4pzfOSTc">Discussing geothermal energy</a>, he said the following:</p> <blockquote>It definitely is, and it's a relatively new one. People think about geothermal energy -- when they think about it at all -- in terms of the hot water bubbling up in some places, but two kilometers or so down in most places there are these incredibly hot rocks, 'cause the interior of the earth is extremely hot, several million degrees, and the crust of the earth is hot ...</blockquote> <p>Of course the interior of the earth is extremely hot, but not that hot. It's several thousand degrees rather than several million. If the earth were several million degrees it would be a rapidly diffusing cloud of metallic vapor. Even the center of the sun is only perhaps 13 million degrees C.</p> <p>But I'll let him slide; pretty much everyone blanks out from time to time. And it gives us a chance to do a little thinking about just how much thermal energy is in the earth.</p> <p>First of all, just because something is hot doesn't mean you can squeeze energy out of it. You can only squeeze energy out of temperature gradients - you need something hot and something cold. This is why we can't just set up a temperature-to-energy machine in the desert and have free energy. In your car, for instance, you need both the heat of the burning gasoline and the much cooler ambient temperature from the outside air via your radiator to turn the hot gasoline vapors into forward progress. Power plants frequently have large cooling towers for that very reason. It's not the energy of the hot substance, it's the process of moving that heat to a cooler place that creates useful work. Think of it in the same way as water flowing downhill can turn a paddlewheel - it won't work unless the water starts off high and ends up low.</p> <p>But that's not a problem here. The interior of the earth is hot and the exterior is much colder. The difference in temperature is such that the efficiency of heat-to-work conversion could be near 100% in theory, though in practice it would be much lower. And we're not likely to run out of geothermal heat any time soon. As a slightly wild Fermi calculation, assume that the earth is uniformly iron at 3000 C. The specific heat of liquid iron is about 611 J/kg K, so cooling the earth to room temperature this yields about 1.8 million joules of energy per kilogram. Multiply by the mass of the earth, and the total energy content might be in the neighborhood of 10^31 joules. The total energy consumption of the world's human population is in the vicinity of 5e20 joules per year.</p> <p>Divide out, the earth's geothermal energy could support that consumption rate for about 21 billion years. We're not likely to use it up.</p> <p>So why isn't it in widespread use? After all, every nation has domestic access to it - all you have to do is drill straight down. The main problem is that the temperature really doesn't start getting ramped up until dozens of miles down. Drilling a hole that deep and pumping water (or whatever) down and up is technically unfeasible. Geothermal is at its best at those places which are close to geological activity that brings the heat closer to the surface. Volcanic and other geologically active locations often do very well with geothermal power. Iceland in particular produces vast quantities of usable energy from the internal heat of the earth. Most other places are much farther from the hot regions of the earth's interior and geothermal is correspondingly much more difficult to get.</p> <p>Sadly Al Gore's hopes for geothermal as a major clean energy technology are probably futile until deep drilling develops into a much more mature form. It would be nice if that happened; the energy to be tapped is pretty close to inexhaustible.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/al_gore_and_geothermal.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/WQ6FQvLr2hc" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/WQ6FQvLr2hc/al_gore_and_geothermal.php http://scienceblogs.com/builtonfacts/2009/11/al_gore_and_geothermal.php Wed, 18 Nov 2009 16:33:56 -0500 http://scienceblogs.com/builtonfacts/2009/11/al_gore_and_geothermal.php <p>We're doing two functions today. If I'm not mistaken we've done each of them separately, but there's a famous and interesting relationship between the two that's always interesting to look at. Like very many interesting mathematical facts, it has to do with the prime numbers.</p> <p>As such the first function is the log integral Li(x), usually defined in the following way:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="1.png" src="http://scienceblogs.com/builtonfacts/2009/11/15/1.png" width="128" height="42" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>We'll plot it in a minute, but if you're interested in a rough idea of it's behavior it so happens that Li(x) ~ ln(x)/x. That is, those two functions have a smaller and smaller percentage difference as x becomes larger.</p> <p>Now the second function is &pi;(x), which is the prime counting function. The notation is traditional; the pi has nothing to do with the number 3.14159..., rather the p in pi is supposed to suggest the word "prime". &pi;(x) is defined as the number of prime numbers less than or equal to x. For instance, &pi;(10) = 4, because there's 4 primes less than 10 (1 is not a prime, if you're curious.).</p> <p>Now if we plot these two functions on the same graph we may get the accurate impression that the two are related.</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="graph1.jpg" src="http://scienceblogs.com/builtonfacts/2009/11/15/graph1.jpg" width="361" height="229" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>The log integral and the prime counting function get closer and closer to each other in percentage terms as x gets large. This is the prime number theorem, and it is tremendously important in understanding the properties of the prime numbers.</p> <p>You may also notice that while the log integral is a very good approximation, it is an approximation and over the interval of the graph it's always a little higher than the actual number of primes. This is true if you plot the first million primes or the first billion primes or the first trillion primes. For a while it was thought that this property of Li(x) > &pi;(x) held for all x. There was some theoretical support for the idea that this was universal, but in math theoretical support isn't worth a whole lot. You want a rigorous mathematical proof. And it turns out that you wouldn't be able to prove that property because it's not true. In 1914 the great mathematician John Littlewood proved that in fact the property was not universal. At some large x, the prime counting function would pass up the log integral function (though of course their percentage difference would continue to decrease). Then it would in turn be passed up, and they would go on trading off an infinite number of times. But his proof of this fact was not a so-called "constructive" proof, which means that though he proved that some x existed where one passed the other, his proof did not in fact tell you what that x might be. Could be relatively small, could be unimaginably huge.</p> <p>In 1933 another mathematician named Stanley Skewes was able to shed some light on the issue. He wasn't able to find the specific number for the first crossover, but he was able to show that whatever it was, it was mathematically certain to be less than a number that's usually now called <a href="http://en.wikipedia.org/wiki/Skewes%27_number">Skewes' number</a>. That number was really <em>unimaginably</em> huge. If you packed the universe shoulder to shoulder with tiny print zeros you wouldn't even be able to write the number, much less the quantity that number represents. You have to use a tower of exponents to write the number. The number is approximately equal to 10^10^10^34.</p> <p>Isaac Asimov tacked the subject of explaining such a huge number in simple terms in his classic essay "Skewered!" I don't think it's online but it's in his excellent essay collection "<a href="http://www.amazon.com/gp/product/B002R2WPPW?ie=UTF8&tag=buionfac-20&linkCode=as2&camp=1789&creative=9325&creativeASIN=B002R2WPPW">Of Matters Great and Small</a><img src="http://www.assoc-amazon.com/e/ir?t=buionfac-20&l=as2&o=1&a=B002R2WPPW" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" />".</p> <p>For a while it was the largest number to naturally appear in a mathematical proof. It's since been passed up by others which are much larger. On the other hand, since Skewes' number is an upper bound subsequent work has been able to show that the first crossing actually occurs at a vastly smaller number. The number is still huge, but not nearly so huge as Skewes' number - around 10^316, to be specific.</p> <p>By physics standards this is a gargantuan number. But by mathematics standards it's not so much. After all, there's an infinite number that are larger.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/sunday_function_53.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/_hbAzIu4B_Q" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/_hbAzIu4B_Q/sunday_function_53.php http://scienceblogs.com/builtonfacts/2009/11/sunday_function_53.php Sunday Function Sun, 15 Nov 2009 17:49:03 -0500 http://scienceblogs.com/builtonfacts/2009/11/sunday_function_53.php <p>Whew! Interesting day around here yesterday, no? There's more controversial topics out there: abortion, health care, gay marriage, Iraq, and a few others. But not many. It's good for sparking discussion, but I also know that some large (probably majority!) fraction of you would prefer to hear about physics. Which is good because generally I prefer writing physics, and I know that I can get irritated when my otherwise favorite nonpolitical blogs go off on political crusades for causes I dislike. So as always I'll try to continue to keep the partisan politics to relatively rare and easily skipped posts.</p> <p>How about some of the <em>mathematics</em> of politics instead? I promise it's not partisan!</p> <p>As you know, the various states are divided into congressional districts which each have their own representative in congress. These districts are drawn in very arcane and shady ways, with the majority trying to carve out districts in a way to give them the most votes, the minority trying to do the opposite, and politicians of any party pushing those considerations to the side to give <i>their particular district</i> the greatest possible incumbency advantage for themselves. Gerrymandering is nothing new, but it's especially bad these days.</p> <p>There have been a number of mathematical proposals to fix this and take out any possibility of district manipulation. One of my favorite is <a href="http://rangevoting.org/GerryExamples.html">the splitline algorithm</a>, which works in the following way:</p> <blockquote>1. Start with the boundary outline of the state. <p>2. Let N=A+B where A and B are as nearly equal whole numbers as possible. <br /> (For example, 7=4+3. More precisely, A = ⌈N/2⌉, B=⌊N/2⌋.) </p> <p>3. Among all possible dividing lines that split the state into two parts with population ratio A:B, choose the shortest. (Notes: since the Earth is round, when we say "line" we more precisely mean "great circle." If there is an exact length-tie for "shortest" then break that tie by using the line closest to North-South orientation, and if it's still a tie, then use the Westernmost of the tied dividing lines. "Length" means distance between the two furthest-apart points on the line, that both lie within the district being split.) </p> <p>4.We now have two hemi-states, each to contain a specified number (namely A and B) of districts. Handle them recursively via the same splitting procedure. </blockquote></p> <p>My own state is a stellar example of gerrymandering:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="texas.png" src="http://scienceblogs.com/builtonfacts/2009/11/13/texas.png" width="345" height="315" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Using the splitline algorithm, the districts would be much more fairly and sensibly placed:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="texas2.png" src="http://scienceblogs.com/builtonfacts/2009/11/13/texas2.png" width="450" height="430" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>The algorithm only takes into account the geometry of the state and the density of the population, and produces a unique result. It does so in a transparent way, and prevents any tweaking by elected officials. Certainly it would immediately result in true contests for many formerly entrenched officeholders. I doubt we'll ever see anything like this implemented, but it's an interesting application of mathematics to the process of fair governance even if it remains purely theoretical.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/the_physics_of_gerrymandering.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/-HgALqkOA6E" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/-HgALqkOA6E/the_physics_of_gerrymandering.php http://scienceblogs.com/builtonfacts/2009/11/the_physics_of_gerrymandering.php Fri, 13 Nov 2009 12:25:11 -0500 http://scienceblogs.com/builtonfacts/2009/11/the_physics_of_gerrymandering.php <p>God help me, I resisted mightily. If my fellow SB friend Greg wants to spin the Ft. Hoot shooting as a cause for gun control then frankly there's pretty much nothing further to say. You'd think a @#$% major in the @#$% army on a @#$% army base just might not have been terribly inconvenienced in procuring weaponry even if every civilian gun in the hemisphere vanished in a puff of sunshine and wishful thinking. But I was going to leave it alone, assuming that that particular point makes itself. To each his own.</p> <p>But he wrote a <a href="http://scienceblogs.com/gregladen/2009/11/nidal_hasans_weapons.php">follow-up post</a> asserting a few points of fact, pretty much all of which rather wildly miss the mark. As a physicist, semi-pro educator (this blog!), and enthusiastic firearms owner and advocate, I simply can't help setting the record straight as to the points of fact. The political gun-control points I will grit my teeth and let slide. Let's begin:</p> <blockquote>He apparently carried two pistols, and both are designed to be effective killing weapons. The more newly designed Five-sevN that he had purchased under the noses of the FBI who was busy investigating him is specifically designed to be very effective at killing large numbers of people in close quarters, to have more controlled "follow-up shots" and to pierce body armor.</blockquote> <p>The Five-Seven (weird trademark capitalization is goofy even when Apple does it) is not designed to kill large numbers of people in close quarters, except insofar as any pistol is most effective relatively close. It's a pistol like any other, and does the same thing. With the exception of the last three words, you could replace Five-Seven with pretty much any centerfire pistol except the niche wilderness big-bore revolvers and have a statement that works just as well.</p> <p>But the Five-Seven was designed to fire a rather unusual 5.7x28mm round which is itself designed to pierce body armor. That much is entirely true. But what's <em>not</em> true is that the armor-piercing handgun ammunition is available in the US. You cannot buy it, it is a violation of federal law. To be clear: if Hasan bought ammo for this pistol in the civilian market he bought ordinary, standard 5.7x28mm ammo. And the 5.7x28mm round sucks at pistol velocities for the purposes of incapacitating or killing. Though Greg sarcastically speculates it might be good for hunting moose, in fact in many states it would be banned for hunting even small critters on the grounds that such a small round would be cruel to the animal by virtue of the injury being too small to reliably kill quickly. For that matter the armor-piercing ammo wouldn't have been much better - the ability to penetrate armor is more or less precisely the inverse of what's needed to damage tissue. This is why police and self-defense ammo is almost exclusively hollow-point, which is good for quick incapacitation but terrible at armor penetration.</p> <p>In short, the very fact that the pistol was designed for military use against armor-wearing opponents makes the pistol poor for anything else. Even if he had the armor-piercing bullets, which he didn't.</p> <blockquote>The other gun was a magnum, a.k.a., miniature cannon. </blockquote> <p>Magnum doesn't mean that. I suppose you might think so if you didn't know anything about guns, but that wouldn't make you correct. Magnum means all kinds of things depending on context. The .357 magnum Hasan possessed but apparently didn't use is a solid but not particularly unusual round. The .22 magnum is a very small bullet. The .44 magnum is ginormous, but no one uses it for actual combat because it's unwieldy. Magnum shotgun shells have more pellets but they're slower-moving. Some of the largest pistol bullets (.50 AE, .454 Casull, etc) aren't labeled magnum at all.</p> <p>None of this is to say the weapons Hasan chose weren't dangerous and lethal. Certainly they were, and for his actions he deserves nothing but a short drop and a sudden stop. It <em>is</em> to say that a matter of empirical fact the weapons he carried were relatively ordinary, and that the weapon he actually used was in fact among the least effective weapons he could possibly have picked - certainly orders of magnitude less dangerous than a standard combat rifle. (The M-16, by the way, actually shoots a slightly narrower 5.56mm projectile, but the bullet is about twice as heavy and moving around 50% faster. This makes it much more effective, and yet it's still the target of continual controversy among military circles for its not-always-impressive terminal performance. In my state it's illegal for hunting deer for that very reason - too high risk of an escaped, injured deer rather than a quick kill. UPDATE: The previous sentence is not quite right - the Texas requirement is just that the round be centerfire, so effectively the 5.56 mm is the smallest legal hunting round. Other states vary, some do enforce a larger cutoff.)</p> <p>You can make up your own mind about the politics of gun control (on an army base!) - as I said I'm not going to argue it here now - but at least now we can be clear as to what actually happened.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/re_nidal_hasans_weapons.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/U2qcnk407lQ" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/U2qcnk407lQ/re_nidal_hasans_weapons.php http://scienceblogs.com/builtonfacts/2009/11/re_nidal_hasans_weapons.php Wed, 11 Nov 2009 23:56:19 -0500 http://scienceblogs.com/builtonfacts/2009/11/re_nidal_hasans_weapons.php <p>On Veterans Day we commemorate the living veterans of the American armed forces. On Memorial day we commemorate those who lost their lives. We should also spend a moment to remember those who helped make sure more soldiers fit into the first category. Though I've made the point before on this blog, I'd like to commemorate two men in particular who between them likely saved the lives of tens or hundreds of thousands of Allied soldiers during the Second World War, simply by doing brilliant science. Their names are Alan Turing and Robert Watson-Watt.</p> <p>Turing was a mathematician and computer scientist who lead the British code-breaking efforts which broke the Nazi Enigma code and various other forms of wartime message encryption. Though it's impossible to accurately judge counterfactual history, I have read more than one historian speculate that the Allied codebreaking successes may have shortened the war in Europe by a year or more.</p> <p>Watson-Watt was one of the early pioneers of radar, and the first person to develop it into a practical means of finding range and direction of enemy aircraft. The Battle of Britain might have been a very different story if his invention had not allowed the vastly outmanned and outgunned Royal Air Force to hold its ground against the Luftwaffe. (In a cute coda, many years later he was cited for speeding in Canada by a policeman with a radar gun)</p> <p>We owe these two men a lot more credit than they get. May their names never be forgotten.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/veterans_day_and_the_scientist.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/Hus5bD_-VxU" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/Hus5bD_-VxU/veterans_day_and_the_scientist.php http://scienceblogs.com/builtonfacts/2009/11/veterans_day_and_the_scientist.php Wed, 11 Nov 2009 13:07:36 -0500 http://scienceblogs.com/builtonfacts/2009/11/veterans_day_and_the_scientist.php <p>Again I have to apologize for the sparseness of posting lately, but I've got two research projects going full blast and time has not been something I have a lot of. I'll still be writing at least a few times a week, and you can't beat the price. ;) In any case once things cool down just a little I should be back to a more regular schedule.</p> <p>Today's function isn't interesting because of the function itself, the interest comes from what we'll do with it. Let's say we have a function like this:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="1.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/1.png" width="116" height="23" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>If we want to see where the function is equal to zero, it's clear that 0 = x^2 - 2 is solved by x equal to the square root of two, either positive or negative. That's the snappy analytic solution, but let's say we want to use this function to compute a decimal approximation to whatever accuracy we feel like. The brute force method is just to try various decimal numbers and see what gets us closest, refining our guess each time. But this is ugly, slow, and requires a lot of continual work. We'd prefer a faster method where we can just turn the crank easily and get a good answer. To do this, let's plot the function and (for reasons I'll explain in the next paragraph) its tangent line at the point x = 4.</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="graph.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/graph.png" width="369" height="234" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Let's say we picked x = 4 for our initial guess as the value of the square root of two. It's an awful guess, obviously, but that's ok since we're looking for a procedure that can turn any terrible guess gradually into better and better approximations of the actual answer. So we pick our initial point and draw the tangent line. We notice that it cuts the x-axis pretty close to the square root of two (which is exactly where the parabola cuts the x-axis, since the square root of two is by definition the number that makes the function equal to zero). So why don't we take the location where the straight line cuts the x-axis, and use that as our second guess, and draw the tangent line there:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="graph2.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/graph2.png" width="369" height="234" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>This line cuts the axis even closer to the point x = the square root of 2. If we take that point as our next guess and repeat the process, we should be closer still. So first we need to actually work this procedure into mathematical language so we can actually get some numbers out of it. The equation of a line is:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="2.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/2.png" width="98" height="18" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Where m is the slope and b is the y-intercept. We know the slope of a tangent line is just the derivative of the function at that point, and y is just the value of the function at that point. That means we can solve for b:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="3.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/3.png" width="172" height="22" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Where the prime denotes differentiation, and we're subscripting the x so we know it's the particular value of the guess we're using. Our new x for the next iteration of the guess will be the value that makes our line equation y = mx + b equal zero, i.e., 0 = m x + b. Substituting all the previous stuff into this equation and solve for x. After a little simplification, we get:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="4.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/4.png" width="134" height="48" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Now this is a very general expression that works to find the zeros of an arbitrary function f, whatever it happens to be. Our particular function can be plugged in (for us, f'(x) = 2x by a little bit of calculus), which gives us the complete procedure:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="5.png" src="http://scienceblogs.com/builtonfacts/2009/11/09/5.png" width="136" height="46" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Let's give this a try. Plug in our initial guess of 4 and the procedure tells us our next guess is 2.25. Plug that in and the procedure gives us 1.56944. Plug that in and we get 1.42189. Repeat again and get 1.41423. So on and so forth closing in on the rounded-off real value of 1.41421, and if we didn't round off at 5 decimal places as I'm doing eventually we'd get as many digits of the square root as we wanted to arbitrary accuracy.</p> <p>This procedure is called <a href="http://mathworld.wolfram.com/NewtonsMethod.html">Newton's Method</a>, and it's a fine way of calculating the zeros of a function if you know its derivative. The method is not quite perfect - if a function has more than one zero the one you get will depend on your initial guess in a not-always-predictable way. And while the method is quite general, there are certain functions that don't fulfill the relatively generous convergence conditions. Still, it's a great method with a long and continuing history of use. It's pretty likely that your pocket calculator uses a very similar method when you hit the square root button. And now if you ever find yourself without such a button, you can do it yourself if you're patient.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/sunday_function_52.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/8LR5J55I02Y" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/8LR5J55I02Y/sunday_function_52.php http://scienceblogs.com/builtonfacts/2009/11/sunday_function_52.php Sunday Function Mon, 09 Nov 2009 12:13:27 -0500 http://scienceblogs.com/builtonfacts/2009/11/sunday_function_52.php <p>Of late president Obama has taken a little bit of heat for his frequent (and mostly male) golf outings. Before him, president Bush took the same sort of heat for his golf and vacations. If you were willing to dig a bit through the news archives, I'd bet you could find similar tut-tutting about previous presidents taking time off. It's a common theme for criticism of just about any important federal officeholder - it's no coincidence that so many congressional "fact-finding" missions are to tropical paradises or European vacation spots. In that case it's a criticism I vigorously share, as the taxpayer dime ought not be spent ferrying already rich congressweasels around the globe. What they do on their own dime I don't worry too much about.</p> <p>But the president comes in for special criticism no matter who pays for his downtime, in view of the fact that his job is so critical and demanding. Critical I'll grant, but contra the received opinion I'd like to argue that in fact it's easily possible to be an perfectly effective president while spending shockingly little time behind the Resolute desk. In fact for much of early American history presidents did just that - ie, very little. The presidency and the country have both changed, but in my opinion even today the president simply doesn't have to do much to do perform his job with great competence. Now I don't expect that any modern president will actually take as little time as I'm going to suggest; the very type of person who is attracted to the job and can campaign effectively is necessarily the kind of person who's willing, able, and eager to manage as much as possible. But he doesn't have to be. Let's go down the list of his constitutional responsibilities:</p> <p><strong>Sign or Veto Laws</strong><br /> How many of these does congress generate each week? I don't know, but given the glacial pace at which anything of consequence gets done in congress it can't be many. I believe it's a few hundred per year. The president has ten days to sign or veto each bill, so there's nothing wrong with just knocking out the previous week's bills over a Monday morning. No need to read them in their entirety - God knows congress doesn't bother. Working with official summaries, your various adviser's opinions, and the opinion of your constituents ought to be enough to decide the fate of a piece of legislation.</p> <p><strong>Take Oath of Office</strong><br /> Takes about a minute.</p> <p><strong>Be Military Commander-in-Chief</strong><br /> A huge responsibility, to be sure. But the president's job here is to set overall policy and strategic objectives at the highest levels. Getting involved in the details is both inefficient and actively a bad idea (Most famously in history, Hitler's terrible mis/micromanagement of his armed forces was a huge boon to the Allies). Given the fact that the president has a full-time civilian SecDef (with a huge staff) whose only job is to manage the military, the National Security Council, the Joint Chiefs, and a huge stack of generals and other military officials to translate strategic objectives into concrete plans, being CinC should be something the president could do each Monday after bill signings and before lunch.</p> <p><strong>Appoint Government Officials</strong><br /> The president has to appoint the cabinet and various other officials. This is also an important responsibility, but it's something that's pretty much done after the first week or two in office at least for the cabinet. The scores of lower officials that continually have to be appointed might be a pain, but that can easily (and in practice is) delegated to the cabinet and other advisers and then signed off on.</p> <p><strong>Make Treaties</strong><br /> Obviously the president doesn't actually write these. If he needs a treaty he tells the Secretary of State what he wants, and then subject to senate approval it happens. Treaties don't happen much anyway, so if there happens to be one in the pipeline give it an hour after lunch for progress reports and making any needed changes.</p> <p><strong>State of the Union</strong><br /> We think of it as a speech, but it is not thus mandated and in fact plenty of presidents just wrote up a SotU letter and mailed it to congress. So if you happen to have one coming up and want to actually do the speech, pencil in another after-lunch hour for practice and revisions with the speechwriters.</p> <p><strong>Ancillary Executive Duties</strong><br /> These amount to a few procedural matters with respect to congress, executive appointments as already dealt with above, and most importantly "receive Ambassadors and other public Ministers; he shall take Care that the Laws be faithfully executed". The ambassador stuff is easy: the State Department does all that. Faithfully executing the laws is obviously highly critical, but as chief executive this boils down to making sure your subordinates are doing their jobs with competence and fairness. So we'll say from 2-5 pm Monday grill your cabinet officials and lower agency officials (and heck, even random lower employees and the general public they affect) about the way their agencies are doing their jobs. Bring down the hammer when poor governance is spotted.</p> <p>And that's that. Assuming your staff is even marginally committed to their jobs you can do your job perfectly well working one day a week, maybe less. Now sure you're neglecting the statecraft presidents like to do: visiting dignitaries, traveling the country giving speeches, campaigning for your party, wheeling and dealing with recalcitrant congresscritters, and all that. But will the wheels actually come off the country if those things don't happen? I seriously doubt it.</p> <p>So go ahead Mr. Obama, golf to your heart's content.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/being_an_absentee_president_is.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/Q25cM_i-Ifs" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/Q25cM_i-Ifs/being_an_absentee_president_is.php http://scienceblogs.com/builtonfacts/2009/11/being_an_absentee_president_is.php Sat, 07 Nov 2009 10:30:07 -0500 http://scienceblogs.com/builtonfacts/2009/11/being_an_absentee_president_is.php <p>Here's a question which pretty much everyone gets wrong. But the readers of this blog aren't just a random sample, so I bet most of you will get it right:</p> <blockquote>Two identical vehicles A and B, both traveling at speed V directly toward the other vehicle, collide exactly head-on. At another test track, car C collides with an indestructible concrete barrier at speed V'. All other things being equal, the crash-test dummy occupants of vehicles A, B, and C will experience the same forces (and therefore injuries) if... <p>1. V' = V/2<br /> 2. V' = V<br /> 3. V' = 2V<br /> 4. Something else (explain)</p> <p>How might the answer change if the A and B do not have equal masses?</blockquote></p> <p>I'll let my astute readers answer, but I'll give a place to start for those who want to make sure they're thinking from a physics perspective. Momentum is mass times velocity. Force is the time rate of change of momentum. Go from there. Also, in this case I have to suggest you not try to find the answer experimentally!</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/cars_and_crashing.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/d_DmksVyPJk" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/d_DmksVyPJk/cars_and_crashing.php http://scienceblogs.com/builtonfacts/2009/11/cars_and_crashing.php Fri, 06 Nov 2009 14:02:44 -0500 http://scienceblogs.com/builtonfacts/2009/11/cars_and_crashing.php <p>For most of human history, our ability to perceive and understand very fast optical events has been limited by the temporal resolution of the human eye. Things that happened too fast were a blur, and all we had to go on was guesswork. Take, for instance, this 1821 painting of a horse race:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="horse.png" src="http://scienceblogs.com/builtonfacts/2009/11/04/horse.png" width="350" height="246" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>If it looks a little bizarre, it's because it guesses wrong on the character of one of those very short events - in this case, the position of a horse's feet during a gallop. Not until the invention of high-speed photography later on in the 19th century could the gallop be better resolved and understood. As it turns out, there is a point in the stride where the horse is entirely out of contact with the ground, but it's the point where all the hooves are brought inward. During the extension of the hooves, one is always in contact with the ground:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="Muybridge_race_horse_animated.gif" src="http://scienceblogs.com/builtonfacts/2009/11/04/Muybridge_race_horse_animated.gif" width="300" height="200" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>(Both images from the Wikipedia <a href="http://en.wikipedia.org/wiki/Horse_gait">horse gait</a> article)</p> <p>Time went on and science discovered and used faster and faster events, and the technology has had to keep pace. The overall procedure is to find an even faster event than the one you're trying to measure, and use it as part of your examination. In the case of the horse, the faster event is the flash and shutter speed of the camera. This sort of thing can get you down to events as small as a fraction of a millisecond with standard cameras, and specifically designed high-speed cameras can do a bit better. Much below the millisecond range though, and shutters simply can't move fast enough to provide that faster event. Photodiodes can respond much more quickly than film and shutters, and using those along with careful and sensitive measurement it's possible to measure flashes of light with nanosecond resolution. Using streak cameras which trade off spatial resolution for temporal resolution can improve this to resolution of a couple of picoseconds. (The one in our lab can get to about the 1ps range. State-of-the-art can improve on this by perhaps a factor of 10, though at the cost of massive expense for not so much experimental value).</p> <p>A picosecond is a trillionth of a second, by the way. It's very fast - light only moves around a third of a millimeter in a picosecond - but lots of atomic processes happen much faster. We can both use and study these processes with pulsed lasers, which today can easily reach pulse lengths of a few femtoseconds. A femtosecond is a thousandth of a picosecond, so it's pretty much an unbelievably small amount of time. A sort of typical way of thinking about the orders of magnitude is to note that the speed of light is 300 nanometers per femtosecond, and an average atom might be a few tenths of a nanometer in diameter. These femtosecond pulses are close to the shortest optical events we can currently produce. We'd like to measure them, but remember how we need that shorter (or at least comparably short) event? We just haven't got any laying around, and if we did we'd be studying them which would put us right back at square one.</p> <p>But we can cheat. What if we could use the pulse itself as that short event? What we do is split the pulse into two copies of itself with a beam splitter. Then we make one pulse travel a slightly longer path to the detector, and by varying that path length the crests and troughs of the two pulses will interfere with each other in a way that depends on the path length difference. This is all detected on a photodiode or photomultiplier or equivalent with only nanosecond resolution and so all we see is a total integrated intensity as a function of path length difference, but we can still extract information from that:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="400px-Optical-interferometric-autocorrelation-setup.png" src="http://scienceblogs.com/builtonfacts/2009/11/04/400px-Optical-interferometric-autocorrelation-setup.png" width="400" height="216" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>This image is also from Wikipedia. M1 is the movable mirror, and SHG is a nonlinear crystal that responds to the square of the intensity, which allows us a bit more information.</p> <p>In a way it's like examining the rainfall in a region by looking at a rain gauge once per month. You have no idea if it's been sprinkling all month or if you just had a short but intense cloudburst. In meteorology you'd not be able to say all that much about the day-by-day rainfall by looking at the bucket, but in optics our job is easier because the variable path length gives us a way to tease more information out of the intensity.</p> <p>What does an autocorrelation look like? Here's one fresh from our lab. It's just a calibration test with standard equipment, so I'm afraid you won't be able to use it to scoop us on our hopefully interesting upcoming results. ;)</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="femtosecond.png" src="http://scienceblogs.com/builtonfacts/2009/11/04/femtosecond.png" width="279" height="198" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Y-axis intensity (au), x-axis femtoseconds. From it, we can say the pulse is about 9 fs long and a little bit messy - the wings indicate the pulse isn't quite a perfect sech^2 shape as we might like. But it's pretty neat that we can so easily measure such a short event with a procedure that's relatively simple and fits in a box not much bigger than one you get a Big Mac in.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/horses_lasers_and_interferomet.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/-O-mAceg_hI" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/-O-mAceg_hI/horses_lasers_and_interferomet.php http://scienceblogs.com/builtonfacts/2009/11/horses_lasers_and_interferomet.php Wed, 04 Nov 2009 12:31:33 -0500 http://scienceblogs.com/builtonfacts/2009/11/horses_lasers_and_interferomet.php <p>These days pretty much everyone knows that mass and energy are two sides of the same coin, as discovered by Einstein. In fact this is so well known that the average man on the street - knowing nothing at all of physics - would still recognize the expression even if he didn't know what it meant or how to use it:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="1.png" src="http://scienceblogs.com/builtonfacts/2009/11/02/1.png" width="74" height="18" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>E is energy, m is mass, c is the speed of light. As with all such formulas, you need to have properly matching units. You can't just use mass in pounds, e in British thermal units, and speed in furlongs per fortnight. But any properly consistent set of units will work; for macroscopic purposes we usually use joules, kilograms, and meters per second.</p> <p>Since the speed of light c = 299,792,458 m/s, the speed of light squared is about 8.99x10¹⁶ m^2/s^2. As such one kilogram of mass is the equivalent of a ridiculously huge amount of energy, as we see in nuclear reactors and weapons. It's a little less commonly understood that this isn't just true for nuclear reactions, it's true for everything. Burn some logs in your fireplace and the total mass of the wood and oxygen will have been just microscopically higher than the final mass of the combustion products. The remainder has gone into the energy released to warm your house. It's the very same equation on a much smaller scale.</p> <p>To get a specific number for this, we need to know how much energy is released by wood. Some googling seems to indicate a figure around 15 million joules per kilogram, so if we assume the burning of 10 kilograms of wood, we get 150 million joules total. Divide that by the speed of light squared and you'll get a mass of about 1.7 micrograms. Very tiny, much smaller than your average sand grain.</p> <p>In fact the total solar power output absorbed by the earth is about 1.8 ¹⁷ watts. Over the course of a year, this comes out to a staggering 5.68²⁴ joules. It's a huge amount of energy, which you might expect since it's responsible for warming the earth, driving the weather, and powering essentially the entire biosphere. Dividing that by c^2 and you'll see that a total of about 63 million kilograms of solar mass is converted into sunlight each year in order to illuminate the earth. It sounds like a lot, but the same mass of water would fit in a cube 40 meters on a side.</p> <p>The sun is quite a bit bigger than that, and though it doesn't come close to achieving perfect conversion efficiency, it probably won't be running out of nuclear fuel any time soon.</p> <a href="http://scienceblogs.com/builtonfacts/2009/11/mass_energy_and_the_sun.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/R9_Lq0DtDHg" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/R9_Lq0DtDHg/mass_energy_and_the_sun.php http://scienceblogs.com/builtonfacts/2009/11/mass_energy_and_the_sun.php Mon, 02 Nov 2009 12:35:57 -0500 http://scienceblogs.com/builtonfacts/2009/11/mass_energy_and_the_sun.php <p>There's an <a href="http://www.newscientist.com/article/dn18041-seven-questions-that-keep-physicists-up-at-night.html">interesting article in New Scientist</a> that purports to describe "seven questions that keep physicists up at night". The list is very heavy on the "deep questions" that tend to percolate around the more esoteric quarters of the high-energy physics world, and not so much on the vast bulk of physicists who (ad Chad and I like to harp on) do physics that's much more directly connected to the real (i.e., observable) world. For instance, it's the question of high-temperature superconductivity probably dominates the dreams of lots of solid state physicists, but it's not on the list. So let's look at the questions and rate them on Matt's AMO Physicist Insomnia Index:</p> <p><strong>Why this universe?</strong><br /> In other words, "why <em>these</em> laws of physics and not others?". I can't say I worry about this much. The laws are what they are. I hold out zero hope for the idea that only one set of laws will turn out to be mathematically possible.</p> <p><strong>What is everything made of?</strong><br /> Electrons and photons - wait, you mean there's more? Seriously, it would be nice to know what's what in the world of fundamental particles and fields. But it doesn't keep AMO physicists up at night, since as far as we're concerned atoms are pretty much as small as we need to worry about.</p> <p><strong>How does complexity happen?</strong><br /> Ah, now this is an interesting question and has lots of bearing everywhere in physics. The trivial answer is "because the mathematics puts it there", but nonlinearity and all the rest of that difficult math is not completely understood despite its vast importance. This one actually registers at a modest level on the insomnia scale for me. (Not literally of course.)</p> <p><strong>Will string theory ever be proved correct?</strong><br /> Nope. But I wouldn't be at all shocked if it's proved incorrect. In any case I worry about it exactly none, as it affects experimental physics exactly none.</p> <p><strong>What is the singularity?</strong><br /> What is <em>a</em> singularity, not what is The Singularity. I don't really worry about this, but it's an interesting and important question. We have great mathematical descriptions of gravity and quantum mechanics, but they don't mesh well in the domains where both effects are of similar scale. How to resolve this? Beats me, but I'd say this one at least registers at a small level on the insomnia scale.</p> <p><strong>What is reality <em>really</em>?</strong><br /> Ok, this is just ridiculous.</p> <p><strong>How far can physics take us?</strong><br /> The article is a little vague and airily philosophical about this, but I think it's asking how close physics is to being "finished". This isn't a purely hypothetical question. As in chemistry, large pieces of the subject are - if not finished - then at least only in need of some work around the periphery. But I'd say even if the Theory of Everything were found tomorrow, we'd have many decades of work left both exploring its consequences and continuing work in non-fundamental physics. In any case, not worth worrying about.</p> <p>So of the seven questions, I think two are actually relevant to most physics. Well, it's a start.</p> <a href="http://scienceblogs.com/builtonfacts/2009/10/physics_and_insomina.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/dv70O5sNlFQ" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/dv70O5sNlFQ/physics_and_insomina.php http://scienceblogs.com/builtonfacts/2009/10/physics_and_insomina.php Fri, 30 Oct 2009 13:41:23 -0500 http://scienceblogs.com/builtonfacts/2009/10/physics_and_insomina.php <p>Grab a book, or an empty DVD case, or anything else that's a uniform rectangular solid. If it's a book, you might want to secure the book closed with tape or a very lightweight clip, because we'll be throwing it in the air. We want to test a theory.</p> <p>In classical mechanics we know that each solid object has three special rotational axes, called "principal axes". Intuitively, imagine the object is shaped from styrofoam. The principal axes are the axes along which you can spear the object with a wooden dowel, and the object doesn't try to "wobble" when you rotate the dowel. Aligning a car tire is simply the process of making sure the axle and the principal axis of the tire are collinear. There's a somewhat involved mathematical procedure you can use to calculate the principal axes (there's always three of them*), but frequently we can find them <em>a priori</em> without any math just by looking for axes of symmetry. Pulling a picture from Wikipedia, here's one principal axis of our book:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="Recplane.JPG" src="http://scienceblogs.com/builtonfacts/2009/10/28/Recplane.JPG" width="153" height="176" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>There are two more principal axes, and we can find them in the same way. One runs through the center, parallel to the spine. One runs through the center, parallel to the lines of text. If we throw the book in the air such that it's rotating exactly about one of those axes, it won't wobble.</p> <p>But it's not possible to be perfect. We will inevitably be a tiny bit off. In that case, what happens? To answer, we look at Euler's equations which describe this sort of scenario:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="1.png" src="http://scienceblogs.com/builtonfacts/2009/10/28/1.png" width="230" height="69" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>The various "I" represent the moment of inertia about each principal axis, and the little omega represent the angular velocity about those axes, which is just the rate of spin. Notice the equations are coupled: the rates of spin about one axis affect the change of the rate of spin in the others. However, we see that if the rotation is initially about just one axis (say, &omega;1 ) the rotation about the other two will stay zero because their initial rotation rates are zero.</p> <p>But if the initial rotation isn't perfectly aligned with a principal axis, all three omegas may be nonzero. What happens then? Well, we can solve these equations approximately and we'll see that in fact the rotation stays nearly aligned with the principal axis, with a slight precession with a frequency given by &Omega;. I'll skip the math. It isn't hard, but it's not very instructive either. In any case, the frequency of precession is:</p> <p><span class="mt-enclosure mt-enclosure-image" style="display: inline;"><img alt="2.png" src="http://scienceblogs.com/builtonfacts/2009/10/28/2.png" width="217" height="47" class="mt-image-center" style="text-align: center; display: block; margin: 0 auto 20px;" /></span></p> <p>Here we've numbered the axes such that 3 is the one about which we were mostly initially rotating modulo the very slight rotations about the other two. &omega; with the 0 subscripted is just the initial rotational speed.</p> <p>Now here's the tricky part. &Omega; is square, so it must be a positive number. This will be true if I3 is the largest moment of inertia, as both terms in the denominator will be positive. This will also be true if I3 is the smallest moment of inertia, as both terms will be negative, and negative times negative is positive. But if I3 is the middle-sized moment of inertia, clearly we get a negative, and a squared real number can't be negative. Our assumption that the tiny imperfection in our throw doesn't matter must therefore fail if we're initially rotating about that middle-inertia axis.</p> <p>So throw that book such that it's spinning about the axis in the picture. It'll work great. Throw it such that it's spinning about the axis parallel to the spine. It'll work great. Throw it so that it's spinning about the axis parallel to the text. It will wobble crazily, no matter how hard you work on the careful precision of your throw. Bet your friend a dollar that he can't do it, and physics will make you a dollar richer.</p> <p>*With especially symmetrical objects like spheres and car tires the axes are not necessarily uniquely specified. There are still three of them, just with sort of a "phase freedom" in their orientation.</p> <a href="http://scienceblogs.com/builtonfacts/2009/10/book-throwing_and_physics.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/oU6n9Nx0DOQ" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/oU6n9Nx0DOQ/book-throwing_and_physics.php http://scienceblogs.com/builtonfacts/2009/10/book-throwing_and_physics.php Wed, 28 Oct 2009 12:57:35 -0500 http://scienceblogs.com/builtonfacts/2009/10/book-throwing_and_physics.php <p>There's a stereotype that it's declasse for us intellectual aesthetes to enjoy football, but I don't care. I enjoy it anyway. Whether you spent any time this weekend watching football or not, I'd like to pose a quick and (maybe!) easy vaguely football-related problem to exercise your brain to make up for all the boob tube time:</p> <blockquote>You're standing on the goal line of a standard football field and walk the 100 yards to the opposite goal line at a uniform speed of 2 miles per hour. Upon reaching the other side, you immediately turn around and jog back. How fast do you have to jog so that your average speed over the whole round trip is 4 miles per hour?</blockquote> <p>Remember: average speed is total distance divided by total time.</p> <p>Bonus: Rework the problem by developing a nonstandard definition of "average speed" that better fits this particular question.</p> <a href="http://scienceblogs.com/builtonfacts/2009/10/football_and_trick_questions.php#commentsArea">Read the comments on this post...</a><img src="http://feeds.feedburner.com/~r/BuiltOnFacts/~4/tdTvgJA5Ibo" height="1" width="1"/> http://feedproxy.google.com/~r/BuiltOnFacts/~3/tdTvgJA5Ibo/football_and_trick_questions.php http://scienceblogs.com/builtonfacts/2009/10/football_and_trick_questions.php Mon, 26 Oct 2009 11:33:25 -0500 http://scienceblogs.com/builtonfacts/2009/10/football_and_trick_questions.php