Toy models, Tadashi Tokieda — LMS Popular Lectures 2008

[00:00:00] Speaker ?: Gracias. [00:00:30] Speaker 1: Meanwhile, you run into somebody who happens to be carrying around an inclined plane. And he produces, from the recess of his toy box, two jars. One of them is filled with top-quality basmati rice from India, and the other one is filled with top-quality air from Birmingham. And they can be made to roll down this slope. And you can see that they roll down quite fast, not quite exactly the same way, because the moment of inertia is doing something, but they roll down quite fast. Now what I propose to do is to investigate how the speed of descent, how fast it goes down, depends on the amount of grains inside. Here we have a big question mark, only two data points available. We know that when the jar is practically empty, that's represented by zero. It's quite fast. And on the other hand, when the jar is practically full, that's represented by one. It's also going down quite fast. So how does that work? Well, let us experiment immediately. And in order to make the contrast striking, I shall leave one of the jars about two-thirds full, or one-third empty, depending on the degree of optimism, and the other one one-third full, or two-thirds empty. So I shall do that. Is that a good amount? Is that a good amount? [00:02:05] Speaker ?: Maybe a little more? [00:02:05] Speaker 1: Maybe a little more? [00:02:06] Speaker ?: Good. [00:02:07] Speaker 1: Let's begin with the two-thirds full one. Now, as far as I can imagine, there are three possibilities on there. Perhaps it will go down at the same speed as before, between those two points, because if you remember the story and believe the story of Galilei dropping those two heavy balls from the Tower of Pisa, the speed of the sand didn't depend on the weight, and perhaps it's the same with grains. In that case, it will be the same speed no matter what amount of sand, amount of grains, amount of rice inside, so perhaps it will go down at the same speed. On the other hand, maybe it will go down faster, or perhaps it will go down slower. So let's take a vote. Who thinks it will go down at the same speed as before? Who thinks it will go down faster than before? Faster, faster. Who thinks it will go down more slowly than before? Who is too shy to vote? Well, let's do this, OK. So this is two-thirds. Are you ready? It starts going down, and it goes down very, very slowly, very slowly. Not only does it go down slowly, but there is a very interesting phenomenon going on. Along what we may call the free surface, the surface that's exposed between the grains and the air, there is a sort of avalanche landslide going on. The sand and the rice grains are going down, sliding down all the time. And the rest of the rice, on the other hand, the rest of the rice grains are sticking to the receptacle, to the jar. In some sense, therefore, if you have studied something like fluid dynamics, you can say that it's acting like a viscous fluid, sticky fluid. By the way, you can do the same experiment with honey, and if you do it with honey, something interesting happens. We shall come back to that in a moment. OK, so that was the first experiment. Now, let's go to the next one, one-third full. Now, there are four possibilities out there. Perhaps it will go down faster than when it was full or empty. Or perhaps it will go down faster than this, but slower than when it was full or empty. So, halfway between. Or perhaps it will be very sluggish and go down even more slowly than this. Or, those of you who have symmetric minds, like mathematicians, might say, there must be some symmetry between f and 1 minus f, if you see what I mean. So, perhaps it will go down at the same speed as this one, so a little bit slow down. So, who thinks it will be the champion and go down very fast, faster than 0, 1? Those are the risk takers. So, who thinks it will go down faster than this one, but slower than 0, 1? By the way, reasonable people vote for this option, yeah? OK. And, who thinks it will go down even more slowly than what we have seen right now? And, who thinks it will go down more or less at the same speed as the previous experiment? So, symmetry is cool. Those are the mathematically minded people. Congratulations. OK. So, let's try to do this. Yeah? Nothing up this sleeve. Nothing up this sleeve. Are you ready? Not only does it stop completely dead, but it's quite stubborn. I mean, I can push it and then, no, no, no, I don't want to go. It stops completely dead. [00:05:56] Speaker ?: Yeah? [00:05:56] Speaker 1: It stops completely dead. It stops completely dead. Which means that the graph has this shape. Certainly, if you are full or empty, it rolls down. But somewhere in between, there is quite a long range where it stops completely dead. On the other hand, it is quite clear also that if I keep diminishing the amount of grains, the system had better start rolling again, because after, when it's empty, it rolls again. So, let's try to experiment with this amount of grains. It is a very small amount. OK. So, that's a very small amount of grains. And let's watch how it goes down. It goes down very smoothly. And there is something quite strange about this, because you remember how, in the previous experiment, when it was two-thirds, it went down, there was some sort of avalanche going on on the surface of the grains. This time, there is no avalanche. You can see that all the grains are together, and it's just sliding quietly along the jar. It's sliding, and there is no avalanche. There is no wave. And it is acting just like an inviscid fluid, in other words, a fluid that has no viscosity. This is a quite extraordinary phenomenon. And you can, as I pointed out earlier on, do the same kind of experiment with very, very sticky fluid. For example, honey. That's what I referred to as viscous fluid before. But if you do the same experiment with honey, which I encourage you to do at home. By the way, all of these things you should do at home. If you do it with honey, something extraordinary happens. You put it there, and if you put the right amount of honey, it doesn't look like it's moving. And you wait, and it doesn't look like it's moving. And it doesn't move. But if you walk out of the room, and come back 30 minutes later, it has moved a lot. That's because honey is ever so slowly creeping. It's invisibly, but surely it's moving very, very slowly. So we shall now analyze all that kind of thing. But in the meantime, you can see that not only is this experiment quite surprising, but you can make a clock-- design a clock with this. Instead of an hourglass, you can fill a jar with grains and do something like that. OK, now let us see also-- thank you-- a numerical simulation that was created for us by Nicolas Taberlet, one of my collaborators at École Normale Supérieure in Lyon, which will show us what's going on. So if you have a lot of grains, this is what happens. And then eventually, it accelerates, accelerates, accelerates. And then it centrifuges. Everything is thrown outside, all the grains. I think it just keeps going down. Those prickly things that make the jar look like a hedgehog, those are the lines of force that the grains are exerting on the wall of the cylinder. On the other hand, if you don't have much, but the grains are such that it's barely sliding-- it was the last experiment that you saw-- this is what happens. It goes steadily, steadily, steadily. And you can see that there's an avalanche that's going down along the surface. I said it's the last experiment. No, but it was the first experiment, actually. So it's the one with two-thirds. OK, good. So let us now try to investigate why this is happening. In order to do so, we have to look at the basic and little-known physics of grains. When you go to a beach, looking like this, and try to make a sandpile, you experience that you can make a sandpile only so steep. I'm talking about a dry sand. If you wet the sand, of course, you can make castles and all sorts of sculpture. But take dry sand and you try to make a mountain, then you can make it only so steep. Beyond a certain steepness, the sandpile just avalanches and settles to a very gentle slope. And that critical angle is denoted here by alpha, and that is known as the angle of repose for the grains in question. And this is the angle at which the avalanche starts, if it becomes too high, and at which also the avalanche once started, settles and rests there. In fact, those people who work in Grania physics know that those two angles are slightly different starting angle and resting angle, but that difference is not going to matter too much for our purposes, so I shall neglect that fine detail. So that's the angle alpha. Now, what happened when you had a lot of grains inside the cylinder? This is a cross-sectional picture that I have drawn, and you see another angle, beta, which is the angle of the inclination of the inclined plane that we had. You see that when the grains have more or less filled the receptacle, the center of gravity of that object is practically in the middle. In the middle, on the other hand, I have indicated by a red cross the point of contact between the jar and the slope. And you can see that center of gravity is to the left of the point of contact. That means, doesn't it, that the gravity which is pulling us down--I think we are being pulled down in Birmingham, gravity acts downwards--tends to pull the jar down on the downhill side of this point of contact. And therefore, what's happening is that its effective torque--in other words, the tendency to make things rotate--is making it go downhill. Contrast that with the scenario when we didn't have so much grains. The same kind of explanation, except that the point of contact, which is still indicated by red, is there. But now, because there aren't enough grains, the center of gravity of the grains is to the right or uphill side from the point of contact. Which means that the torque exerted by the gravity, which is still pulling down, has the effect of rolling the thing uphill rather than downhill, thereby opposing the descent. That is exactly why this thing did not go down. You can also see from the picture that it is very important for this mechanism to work that alpha, the angle of repose of the grains, should be strictly bigger than beta. You can imagine, for example, when alpha and beta are the same. Alpha is the slope of the grains that you see there, and beta is the slope of that inclined plane. And if they are parallel, you can see that no such thing as the second scenario can occur. Only the first scenario rolling down slowly can occur. And that is the solution of the mystery. This sort of thing, very surprising and yet simple phenomenon that had never been looked at before, is characteristic of all the toys that we shall be studying and investigating and playing with this evening. And they all have to do with a very common and yet little studied mechanics of contact. You must have seen things that involve friction. You must have seen things that involve rubbing. You must have things that flow very, very viscously, very thickly. All these are particular cases of contact. And the fact of the matter is that mathematics and physics of contact are still in the incipient stage of study. People still do not understand what is going on very well. It is really the cutting edge of what you can study. And all these things tend to produce surprising phenomena. And particularly, I'd like to show you, with lots and lots of toys, the phenomenon of symmetry breaking. Symmetry breaking means that there might be something that you notice which looks symmetric. It might be, you know, future and the past look the same. That's a time symmetry. Or it could be translational symmetry. It's the same here, same here, same here. Wherever you move, it's going to be the same thing. Or rotational symmetry. It doesn't matter whether you roll this way or you roll this way. Or reflexive symmetry, you and mirror image in the mirror are indistinguishable. Those symmetry breaks are very characteristic and surprising phenomena that arise from mechanics of contact and which we shall illustrate with lots and lots of different toys. Okay. Here's another way of going downhill. This penguin is very dainty. And you'll see that it has a wonderfully elegant way of stopping at the bottom of the hill. And you see it's going down gradually and unlike-- oh, now it's going to stop. Very elegant. Reminiscent of the way my lectures finish usually. Now, a literally more striking toy is afforded by an experimenting collision. Here, I have brought a number of balls that I align. And you have probably seen this while playing billiards, or you can do it with coins. If I line them up and then shooting a ball from one end, out comes the end ball from the other side, while all the intermediate ones stay put there. This is the phenomenon called "canon" in billiards. And it's a matter of very common experience that if you're shooting a lot of momentum, out of momentum and if you're shooting a tiny amount of momentum, only a tiny amount of momentum comes out. And even more famous example is this. You can see that those things are suspended from a support and if I shoot in tom-tom-tom-tom-tom-tom-tom-tom-tom. But if I input a little amount of momentum, only a small amplitude oscillation occurs. You can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. And you can play with this in a number of ways. So what goes in must come out. And if you input a tiny amount of momentum, a tiny amount of momentum comes out. Right? I'd like to show you that this is all wrong by introducing a tiny, tiny difference. I'm introducing another ball at the end which differs from all the others only in color. So this is a golden ball. And I'm going to input a tiny, tiny amount of momentum. Okay? Wimpish. Negligible. Almost invisible. Okay? Just very, very gentle amount of momentum. Are you ready? Gentle amount of momentum. Ooh. [00:17:23] Speaker ?: Oh. [00:17:24] Speaker 1: What happened? Tiny amount of momentum went in, so a tiny amount of momentum should come out. Well, let's do that again. Tiny amount of momentum goes in. [00:17:39] Speaker ?: Ooh. [00:17:40] Speaker 1: Congratulations, you have all seen that this is magnetic. Well, that's one thing to say that it's magnetic. It's important to notice things, but the next state is science. Magnetic, so what is the question? It's not the magic word, magnetic. I mean magnetic, yes, but the conservation of momentum says that whatever goes in must come out. So a black box, whatever mysterious construction you have, whatever strange mechanism you can set up, magnetic, telepathy, what have you, I mean what goes in should equal what comes out. So what's going on? It is quite extraordinary. So in order to analyze what's going on, let's think about the energy first. This energy is easy to understand, because when I first position the magnetic ball against each other, they are being attracted by this force, magnetic force. In other words, this initial ball, going in ball, is sitting, so to speak, at the top of a very, very steep magnetic field, potential well, and it's rolling down this precipice and then boom, goes into the whole thing with enormous energy. So this potential energy being converted into kinetic energy, that's why the thing has so much energy. But where does the momentum come from? Indeed, in the beginning, nothing is moving. At the end, something is moving. I mean, momentum has to be conserved. In order to understand what's going on, and in particular, how I cheated, let's put a penguin guarding this magnetic ball over here. OK? So that's where the whole system used to be. Now, I'm going to input a tiny amount of momentum. And have you seen where the position has shifted? This used to be in front of the penguin. Now, it's gone back. There was a recoil. And this recoil of the whole system going back is absorbing or counterbalancing the momentum that has gone out. Why didn't we notice the momentum the first time? Because I told you not to notice it in some sense. I was calling your attention to the ball that was going out. And of course, I also put my finger to prevent the recoil. And so, that's where the momentum comes from. The whole momentum is being compensated for by those balls that are coming back. Because there are many more balls that come back than one that goes off, they come back at a slower speed. Because the momentum equals speed times the mass. And mass is larger, so the speed is smaller. And because it's slower, friction stops it much more efficiently. So, again, you see that there's a recoil. And by the way, this ball came back, bouncing back. Okay, so that's how it worked. Now, perhaps I would like to entertain you with a story of how I got hold of those extremely strong magnets. These are made of neodymium. That is one of the most magnetizable elements in the periodic table that appears quite late in the table. The chemical symbol is ND. And I knew that neodymium was extremely magnetizable. And on the other hand, I make it my business to invent a new toy maybe every month or so. And while visiting Japan more than a year ago, that was the time of the month when I had to invent the toy. So, I thought something like this might work. I went to the largest hardware store in Tokyo. The whole thing, a building, is a hardware store. It's an amazing place. And you think it's a department store, but the whole thing is a hardware store. And I asked them whether they had neodymium magnets. They said yes. And they even had spherical ones. So, I carved out this groove and the experiment worked. I was very happy. But then it occurred to me to go back to the store and ask them what they were selling these magnets. I know how I made use of them, but why do people buy them? And the shop assistant couldn't answer it, so he went back. And then the owner of the shop, a very smart gentleman, came out. And he sounded a little miffed when I asked the question. He said, sir, we are the largest hardware store in Japan, probably in the Pacific. So, we are proud to stock everything under the sun. Of course, we have such magnets. And we sell one or two of those every few years. And we have no idea what these are used for. Now, this was the first experiment that I had in mind. But I would like to show you something that has never been seen before. And this was discovered, in some sense, by accident by my playing with these magnets. And probably it was unpredictable to the shop owner as well. So far, we have been shooting the magnet in from the magnet side. This is the strong magnet. All the others are just normal bearings, dime a dozen. And we were shooting the magnet from this end so that they attracted very, very strongly. But now, I'm going to shoot in the magnet from the other side. Okay? What happens? Well, we have seen before, with the example of this, that one goes in, one comes out. By the way, two go in and two come out. The same thing goes out as what came in. So, one goes in and should come out. Besides, there's a symmetry in the problem. We discussed symmetry earlier on. Imagine filming the whole thing. Okay? If you film the whole thing and run the film backwards, what would you see? Well, you would see one thing going in and something happening. Well, if you run the film backwards, something that came out would be going in now. And one thing would be coming out. I mean, because of symmetry, one thing goes in and one thing must come out. Right? With the same momentum. Of course. Let's try this. Are you ready? I mean, one thing goes in and two come out. Two go in, three come out, and three go in, and four come out. Zero, one, two, three, four. This device knows how to count. If you had many more bearings in the middle, the pattern keeps going. But because if it's too long, the magnetic influence at the end is becoming weaker and weaker, there is some irregularity in the pattern. For example, one goes in, two come out. That's okay. Two go in. Three go in, three come out, three go in, four come out, four come in. Oh, it's working very well. It's a magnetic abacus, if you like. This is an extraordinary phenomenon, and at the moment, there is no theory. In other words, have you ever encountered a phenomenon like this? It's an open problem. The cleverest mathematicians and physicists in the world do not know what's going on. I'm working on it, so if you do figure something out, please let me know, and we'll write a joint paper on this. Now, after violating the conservation of momentum, linear momentum, that has to do with translation, I thought it would be very nice to go on to violate other principles of nature. And here is one toy that illustrates a very important class of motion called rotation. You know, rotational phenomena seem to be intuitively hard for most people. For one thing, it took our ancestors about two million years before they realized that we are living on the rotating support. Now, just as linear momentum is conserved, which means that things that are moving have a tendency to keep moving the same way, for rotation, there is something that's conserved. That's called rotational, or more commonly, angular momentum. Things, in other words, have a tendency to keep spinning the same way. And when I say spin in the same way, not only does it spin at the same speed, at the same rate, angular spin, but also the axis of spin tends to stay the same, unless there is some external perturbation. So here is a good toy that I brought from Japan. It's called Bill Bokeh, but the fact that the name is French doesn't mean that it's French. The toy is more than a thousand years old. This particular one looks particularly old. And the challenge is this. You see there are lots of cups around. And you are supposed to flip this ball up at the end of a string, and then catch it on cups. It is extremely difficult, of course, because the cups are so small, and gravity is very strong in Birmingham. And you never do this in public, and you never manage to do this. But if you are really ambitious and shameless, you can try this smaller cup. You see, this is small, this is large. Large you could probably do, but smaller cup, in absolutely no way. But if you are really crazy, you might say, ah, look, there is another cup at the end of the stick. Of course, it's not designed for that purpose, and nobody ever manages to do that, of course. But what is absolutely undemonstrably, you see, it's getting harder and harder, what's absolutely undemonstrably impossible is this. You see, there's a stick, and there's a hole at the end of the thing. So, you might try to flick it, and then plant it on the stick. Now, that's not going to work, because, why is it so difficult? Because, when I flick the ball, you see that the ball starts spinning all over the place, and it's extremely unlikely that the hole will be pointing straight down when it comes into contact with the stick. And unless it's pointing straight down, it's not going to go in. So, that makes me think, the trick, if it is possible to have a trick, is to make sure that the hole is pointing all the way down. Well, conservation angular momentum. If I spin the thing, and lift it, you saw that it lifts straight up. Because it's spinning, things have a tendency to keep spinning the same way, in particular, keep the axis of the spin. It doesn't go haywire. It keeps spinning the same way. So, if you spin this, what is impossible becomes sometimes possible. Thank you. By the way, compare this with how things behave, if I don't spin it at all. So, I don't take advantage of the angular momentum conservation. It's more or less the same. That illustrates for us the angular momentum conservation. And here is the first story that takes advantage of the conservation of angular momentum. You probably remember the story about Columbus, who was challenged after he discovered America. That means that he went there, people had lived there for thousands and thousands of years to begin with. But, anyway, he discovered America. And he came back and people said, oh, you know, it's very easy. You just have to think of it. Well, Columbus got annoyed and said, well, ladies and gentlemen, probably only gentlemen at the time, can you make an egg, both eggs stand on its head? And people tried, people tried, and they couldn't. And Columbus just took an egg and broke the shell a little and then made it stand up. And that's the story. And he said, well, everyone complained. Well, no, that's easy. Well, it's easy once somebody has thought of it. Now, the story is often told, but you and I must agree that this is cheating, of course. When you say, make an egg stand up, I mean an unbroken egg standing up. And Columbus might have been a great person, but he certainly cheated. I brought here an egg. And in case you have any doubts, it says EG on the cover. And it's an artificial egg. But the experiment that we have about witness works with real boiled eggs. So please make sure to boil it at home and then try. Actually, if you don't boil it, you know there's a nice way of distinguishing a boiled egg from a raw egg, which is to spin it. A boiled egg, because it's a solid body, spins very naturally. But a raw egg doesn't want to spin like this because there's some kind of liquid stuff sloshing about inside which opposes rotation. So it's very, very stubborn and it resists rotating. By the way, if you do keep coaxing the raw egg to spin, eventually, after a lot of tries, you'll get it to spin. Once you have started spinning it, it's hard to stop. Indeed, try to keep doing this to a raw egg until it spins and then catch it to stop it. And an instant later, release the egg and you'll see that it resumes spinning. Again, because liquid stuff inside is carrying the spin or the angular momentum which keeps spinning. That's not what I wanted to talk about. What I wanted to show you is what happens when you spin a boiled egg fast enough. Have you seen it? Have you seen it? It stands up. [00:30:40] Speaker ?: It stands up. [00:30:44] Speaker 1: It stands up. It stands up. You don't actually need a mirror. I can make it stand up. Boop! [00:30:52] Speaker ?: Stands up. [00:30:53] Speaker 1: Actually, just about anything, it turns out, which is spinning in frictional contact with the floor has a tendency to raise its center of gravity. Now, why do we find it so surprising? Because it is surprising. We find it so surprising because ever since primary school, we have been learning that nature wants to be lazy and wants to lower its center of gravity as much as possible. And indeed, when you leave an egg like this, that's what's happening. It's just lying flat, its center of gravity is as low as possible, and it doesn't want to be standing because that would raise its center of gravity. On the other hand, when you spin an egg, somehow or other, it raises the center of gravity. It is quite remarkable. And here is another toy, much more traditional, which takes advantage of this phenomenon. This is a top, and it wants to look like an apple. And it doesn't want to look like a mushroom, not upside down, because it's heavy on this side and it's light on the axle side. But can I make it stand upside down? Not really easily. How can I make it stand upside down? Ah, we remember that just about anything, as I said, in spinning motion and in frictional contact with the floor has a tendency to raise its center of gravity. So let's try to spin it and see what happens. [00:32:15] Speaker ?: Poof. [00:32:17] Speaker 1: I can spin it with my left hand, if you like. Poof. You might be wondering what happens if I spin it upside down to begin with. So I'll spin it upside down. And you see that it's very robust. It really wants to keep its head up, upside down, because the center of gravity is higher that way. It's spinning. It's spinning. While it's spinning, it wants to keep its center of gravity as high as possible. But as it starts losing spin, please observe, it goes into a wobble. And then it starts having doubts. I'm not spinning anymore, so I should really go back to my natural state, which is to keep the center of gravity low rather than high. And end up looking like an apple rather than like a mushroom before. It's called a tippy top because it tips over. Now, let us see if we can theoretically understand what's going on and, in particular, predict how much time the tippy top takes to tip over completely. This is a very, very difficult problem. I make it spin and puff. But there is a way to handle this. Right. To this astonishing problem, we shall apply an astonishing method. It's called a dimension analysis. First, let us list all the relevant variables in the problem. There is certainly friction with the flow. It turns out that if you do it on a very smooth surface or practically friction on the surface, the thing doesn't stand upside down. So friction is certainly relevant. And I denote it by mu m g. Also, probably mass is relevant of the top, as well as its radius, denoted by r. And the spin, which is angular velocity, how fast I'm spinning it initially, if you like, which I denoted by the Greek letter omega. Now, dimension analysis tells us that then we have to think about in what units or dimensions, as the physicists say, these things are measured. Friction is a force. And anything in particular force should be expressible in terms of the so-called fundamental dimensions. The fundamental dimensions in which all mechanical phenomena can be measured are mass, length, and time. You can think of them as kilogram, meter, second, if you like. Or I should simply denote them by m for mass, l for length, and t for time. m, l, t. So friction, in particular, should be a combination of m and l and t. And that's easy to figure out because force, friction, is mass times acceleration. And mass is m, l divided by t, divided again by t. It's the second derivative of the distance traveled, so l divided by t squared. So that's m, l, t minus 2. Mass is, of course, m. And the radius, it's length. And spin is how much angle is being swept out per unit time. So angle being dimension, this there is no dimension. It's 1 divided by time. So those are the relevant parameters. And in the friction, I mentioned some thing about the force. Mg, that's a familiar expression on the force. But what's that mu? Mu is the so-called coefficient of friction that has no dimension, which is characterized by the contact between particular material. For example, in this case, you had this kind of glass and wood. And it measures, really, how hard it is for one thing to slide upon the other. And it turns out that you can measure it by measuring the angle at which it starts barely sliding. It's a bit reminiscent of the grains, and that's no accident. And if you look at what's happening, I would say this angle is, I don't know, maybe 30 degrees, 40 degrees, something thereabouts. It's less than 45 degrees. And the tangent of that angle, it turns out, is the coefficient of friction. In this case, because it's 30, 40 degrees, somewhere thereabouts, coefficient of friction, let's say, is about 1/2. All these estimates are rough, but they'll lead us to an order of magnitude estimate. OK, now, on the other hand, what are we trying to analyze? We are trying to analyze how this thing tips over. In other words, I'd like to see how that angle that I denoted by theta, it's the angle of inclination over the top, keeps increasing, increasing, changing. Theta is, again, an angle, so it has no dimension. But what we really want is theta dot, the rate of change in time of theta, and that's one over time, so t minus 1. Dimensional analysis tells us to combine, somehow or other, those variables that we have listed and produce that answer, t minus 1. Whatever else we do, dimensions must match. Otherwise, the solution cannot be correct. So let's try to manufacture t minus 1 by combining all those things. Well, it doesn't actually take too much to figure out that what you can do is to put mu m g, the friction on top, and m r omega at the bottom, because on the top, then, you have, as indicated by the list, m l t minus 2. At the bottom, you get m r omega, and that gives you, in turn, m l and t minus 1. And if you divide one by the other, you are left with 1 over t. So, that combination, and that combination alone, gives you the correct answer. What dimension analysis does not determine is the possible presence of a proportionality constant in front. For example, maybe it's not that formula itself, but half of that formula, or maybe ten times that formula. That kind of thing we cannot tell ahead of time. But we know that at least the form of the answer must be exactly this, because dimensions tell you so. It is an astonishing method. A bit like cheating, but it works. OK. A more fine and careful analysis actually shows that the constant proportionality in front is very close to 1. So, this happens to be the right analysis. OK. Now, let us see if we can start plugging in numbers. How can you calculate tip-over time? We have seen that theta changes at the rate of that awful fraction that we have seen. But in order for the top to tip-over completely, theta must change by 180 degrees. Because when it's standing straight up, its theta is zero, and theta increases, increases. And when it stands upside down completely on its head, theta has changed by 180 degrees. Or if you like, pi in radians. So, the time is given by pi divided by the rate of change, theta dot. And by putting in various things that we have figured out, you get that it is equal to pi times r, radius, times omega. Omega is the angular spin, so it's the number of times that it spins per second, times 2 pi. That's the angle that's swept out per turn. Divided by mu g. M has dropped out already in the previous line. OK. So, can we translate that into some numbers? Let's estimate roughly. What's r? R is the radius of this thing. How long would you say this is? 2 centimeters? 3 centimeters? I took 2 centimeters. G, the gravitational acceleration, is about 1,000 centimeters per second per second. You might remember, if you are over-educated, 980 or something. But it's about 1,000. So, let's be rough. Mu, as I mentioned before, is about 1/2. And what is the number of turns per second? That's the trickiest one. When I launch it, it spins quite fast. How many turns per second would you say? Any takers? [00:40:42] Speaker ?: Huh? [00:40:44] Speaker 1: 5 turns. 5 turns. 1 second is a long time. 1, 1, 1, 1, 1. 5 turns per second would be . That's about 10 turns per second. You think that's 5. I think it's more than 10. For one thing, you cannot see that it's spinning, really. Right? It's beyond visual precision. I think it's more than 10. Do you think it's going to be more than 100? [00:41:24] Speaker ?: Probably not. [00:41:25] Speaker 1: Probably not. If it's more than 100, we will probably be hearing a buzz or something like that. So, it's between 10 and 100. So, let's take the average. People tend to think, ah, average, that means that I add two numbers and divide by two. But that's not the correct average. Because we are multiplying all these numbers, we should take the multiplicative, not the additive average. We shouldn't add and divide. We should multiply. And what's the multiplication equivalent of dividing by two? Take the square root. That's called a geometric mean. And if you take the geometric mean of 10 and 100, it's 10 times square root of 10. And square root of 10 is about pi, isn't it? So, that's 10 pi. So, let's plug those numbers, estimates, into the previous formula. That was pi times r times number times per second times 2 pi over mu times g. Putting all these things in, we get the following. Now, pi times pi, pi squared is about 10. So, you have 10 up there and another 10 there and 1,000 that cancels. And then, one half downstairs, that's from mu, comes upstairs. So, it becomes two and so on. And if you do the arithmetic, it's about 2.4 or something. It's just over two seconds. According to this, astonishingly naive and slightly cheating model. Now, let's see if we can go back to the experiment. OK. One, one, one, one, one. That's the pace of seconds. One, one, one, one. One, one, one. Not bad. Not bad. About two seconds. John provided a stopwatch measurement and it's 2 seconds 68. Good. OK. Now, that was a tippy top. And by the way, I should mention that this I studied with John and Hillary Ockendon of Oxford University. Those were the toys that have been satisfying the conservation of angular momentum. Next, I would like to go on to a toy that violates the conservation of angular momentum, or so it seems. Please meet my little turtles who are mounted on a boat-like object. And I already call your attention to the fact that it's symmetric either way. Completely symmetric. These turtles can be positioned in any way you like. You see they are facing each other. For example, they are in love. And you can spin them one way or the other. And they spin nicely. Or perhaps after a tiff, they are not speaking to each other anymore. They are back to back. You can spin them anti-clockwise positively or clockwise negatively. They still spin nicely. Or perhaps one of them is in love with the other, but alas, affection is not reciprocated. Then you can still spin them clockwise or anti-clockwise either way. So far, no surprise. But now, if you position the turtles like this, in an S shape, thereby changing the pattern of symmetry. Nevertheless, there is some symmetry left. It's no longer a reflexive symmetry, but you see that the center of gravity is still smack in the middle. That didn't change. So the center of gravity is still in the middle. But certainly, the turtles are sticking out their necks in different directions. So the symmetry has changed. Well, you know that turtles like to walk forward. Ideally, you have never seen a turtle go backwards. So if you ask them, politely and gently, because these are sensitive animals, to please go forward. They are willing to oblige us. As before, smoothly. But if you try to force them to go backwards, they say, no, no, no. We don't want to go backwards. And it's not a matter of left or right, because if you position the turtles like this, what used to be the wrong direction, they can now go forward because they are going forward. But what used to be the wrong direction, they still don't want to go backwards and reverse the spin. Everyone in this room, I'm sure, is wondering what happens if I position the turtles like this. Right? Okay, let's do this. As seen from above, I shall spin them in the clockwise direction. So like this. Well, in that case, this turtle will be happy. This turtle will be unhappy because it's going back. And these are very democratic turtles. Happiness and unhappiness vote each other and cancel each other out so they can spin. And if I reverse the direction of spin, the happiness and unhappiness reverse and they can still spin. So it is a very sensitive toy that we have here. The more readily available commercial version is this one. I brought it from the Science Museum in Boston, but you can find it in many, many places. And it has a certain funny shape because of which the same kind of asymmetry is manifested. It wants to spin one way but not the other way. Now, those are examples of types of dynamics called chiral. Chiral is a funny and yet English adjective of Greek origin. The Greek word that gave rise to this word is cheir, which means hand. It means handedness. Right handed, left handed, twisted in different ways. I have just written underneath the Chinese character for hand. The left one and right one look like mirror images of each other. But the chirality is important. The left one is correct, the right one is incorrect. As you have two hands and they look the same in some sense, but they are not superposable in each other. They have handedness. That's chiral. Those dynamics are then examples of chiral dynamics. And I'd like to make a few comments about these creatures. First, there has been a lot of study on these things. Starting about 100 years ago, a meteorologist called G.T. Walker wrote the first paper. People have been studying this, but many people still believe that the chiral behavior is due to the funny shape of the contact surface. And it's a bit too far for you to see, but if you run your finger, there's a sort of mountain ridge that's skewing the surface. It's not completely symmetric. And people even wrote papers saying that, oh, that must be the reason this dynamics occurs. But it has nothing to do with the contact surface. This example shows that it can be completely symmetric, and yet chiral dynamics can occur. So you can say that this is red herring. And what's really important, the only thing that's important, is that the turtles stick their necks in wrong direction, in opposite directions. In other words, the distribution of mass must be skewed. Or to be completely precise, the so-called ellipsoid of inertia must be skewed with respect to the axis of the symmetry. In order to drive this point home, here is one version that I produced in Paris. For all intents and purposes, you can believe me that it's completely symmetric either way. But he has a preferred direction. He wants to spin this way, anti-clockwise and not clockwise. Anti-clockwise, but not clockwise. You agree? Now, I rub it against my shirt, and lo and behold, now he wants to spin clockwise and not anti-clockwise. I can also rub it against my hair. That reverses the chirality spinning, as before, in the first instance, anti-clockwise and not clockwise. If I shake it, it doesn't seem to do anything. It still spins anti-clockwise and clockwise it doesn't want to spin. What's happening? Have you been watching very carefully? So let's watch carefully. It's now spinning anti-clockwise, yeah? And clockwise it doesn't want to spin. So I pick it up, rub it against my shirt. You imagine that that's very important. I rub it against the other arm. And before I put it down, I turn it over and then, of course, it's... What's happening is that there is a metal bar inside, which is skewed. So mass distribution is skewed with respect to the geometry. And that's the only thing that matters. OK. I shall not go into the details of the analysis of this phenomenon, because it's taken more than 100 years to analyze completely. But last year, Keith Moffat of Cambridge University and I wrote a paper and explained the main features of this. But I would like to share with you one feature that turned out to be very crucial and sort of a breakthrough point for the whole analysis. It is this. The tendency of those turtles to go forward, turn, spin forward, is so strong that I don't even have to begin by spinning them. If, from rest, I wake them up by tapping a little bit, they spontaneously go forward. And that's something that you can do at home as well. If you wanted to make something like this, you don't have to go and fish turtles out of ponds. You can take an old-style telephone handset. It has to be curved at the bottom. Not one of your fancy mobile phones with a flat back, but it's curved. And disconnect it from the cord and put it on the table and attach hefty masses like batteries with scotch tape. And if you tap, it will spin one way but not the other way. So try it at home. If you think of this as a boat, then this vibration is in the nautical terminology called pitching. It's a pitching vibration. And pitching makes them go forward. Indeed, if I try to force them to go backwards, you can see that the pitching gets excited followed by the reversal. But if it is a boat, there's another kind of vibration, which is the sideways vibration. That's called rolling. That's the one that makes everyone seasick, by the way. So what happens if I make it roll rather than pitch? It's amazing that people haven't tried this before. So to remind you, pitching makes them go forward. In order to excite rolling now, I'm going to tap here. What happens? They go backwards. They become the reversed turtles. So pitching makes them go forward. Rolling makes them go backward. It's a bit weaker than the forward spin, but nevertheless. So I was cheating a moment ago. Even if I spin them in the forward direction, actually those turtles start nodding their heads. That's rolling. As a result, sometimes if there isn't too much dispersion from the floor, they can reverse even from the forward motion. You saw them reverse. So sometimes if the conditions are right, they can go forward and reverse and then backward and reverse. They can have successive reversals. These particular turtles on a marble top table in a cafe next to the British Museum reversed three times. That was the record. I wanted to buy that marble top table with my research grant, but somehow they didn't. But the world record, as far as I know, is held by the former Cavendish professor, Brian Pippard, who unfortunately passed away last week. And he fashioned a toy like this, but by carving out a wine bottle. And he claims that he reversed five or seven times, up to seven times. And unfortunately, that particular object seems to have been lost somewhere in the recess of the Cavendish. Those were the key to the analysis. And I would now like to show you how that analysis works. In fact, if you write down the simplest possible model that captures all those things that we have been observing, you get the following. First, let's look at the turtles. By P, I denote the pitching motion. By R, the red one, I denote the rolling motion, sideways. And S is the spin. And if you want to write down the simplest set of equations that takes into account all the things that we have been seeing, that turns out to be the equation. The derivative in time, plus or minus t, sign depending on whether the turtle directions match with the right or left handed motion of the spin. You can ignore that for the time being. Equals some vector product of two vectors. One of them is r lambda p zero, and the other one is p r s. Where lambda is the ratio of frequencies squared. Ratio of frequency in p and the ratio of frequency in r. That turns out to be the simplest equation that you can write down, which takes into account all the observed phenomena. And that equation has never been seen before. It's very simple. It's first order. It's something that you can study, but it has never been seen before. A very simple equation that emerges out of the blue. You notice, first of all, that because it's a vector product on the side, the vector product takes two things and produces an answer, which is perpendicular to those two ingredients. This velocity of the motion, p r s, is perpendicular on the one hand to p r s itself, from which you deduce that the sum of the squares of those things is constant. That turns out to be none other than the conservation of the energy. So that's something that we could have predicted before. But the other thing that is very striking is that the velocity vector is also perpendicular to the first vector, p r lambda p zero. And from that we can easily deduce that p times r to the power lambda is constant. This, as far as I know, has no sensible traditional physical interpretation, but nevertheless it is constant. So there are two constants of motion for a three-dimensional problem. All the curves must be, all the solutions must be closed curves. So it's a periodic motion, as we have seen before. Everything is periodic. And if you don't have too much dissipation, the turtles go back and forth, back and forth forever. I have kept, until the end, the best of the toys. So I would like to conclude by showing you a big, big, open problem. I have brought here another tippy top. It is made of plastic, transparent. And let's spin it one way. Sorry. It tries and pops, pops upside down. So it is a very efficient tippy top. It works very well. I will spin it again. It works. Now, you laughed, all of you, when I said I can do it with my left hand too. I will now spin it with my left hand. In other words, I will give it a spin exactly the same in magnitude, but reversed in sign. With my left hand, the initial conditions are all identical, except that it's being spun in the other direction. Now this top tries, tries, tries, but it does not tip over. It tips over one way, but spun the other way, it does not tip over. You have in front of you what was, for a while, the world's first and only chiral tippy top. This object, wonderful object, was produced by accident by David Atchison of Oxford University. And I begged him to lend it to me, and I have been keeping it ever since. Just like the other open problems that I mentioned before, it is open. And the problem, by the way, is real. I had this made in my own laboratory, and now I know how to make such a thing, but there is no theory. And I tried it on all sorts of surfaces, with all sorts of people, left-handed, right-handed people. In every single way, I know how to reproduce the phenomenon, but there is no theory. We don't know how this works. It is an open problem that challenges the mechanics experts. It is probably the greatest open challenge that confronts us today. It is time now to bring this lecture to a conclusion. Before you came to the lecture, you might have wondered what toy in the title could have meant. After all this playing and investigating, I am sure that you understand now what toy is for us. If something is pre-packaged and sold in, say, a toy shop as a toy, it is almost certainly not a good toy for us. On the contrary, a toy is something that we can pick up from daily life, which surprises us precisely because marketing people did not recognize it as a toy. You know, there is a misconception among the public, and even among scientists, that science is something that must be officialized. Something that is practiced by men wearing spectacles, that's true, wearing a white shirt in super-duper laboratories. Or, science is something that teachers make students do in classrooms, and once the bell rings, people file out and science stops. But, nature does not care what we humans pre-packaged, pre-label prejudice. She just keeps doing science every day, every minute, every second, everywhere. So, there is a sense in which we have to look for science, because it is happening non-stop all around us, whether we pay attention or not. And, every time we pay attention, nature repays us with bountiful gifts. There is a story concerning a French mathematician and his daughter, who used to live in Princeton. And, this daughter was caught in an educational stunt called New Math, which pretended to teach abstract set theory at primary school levels. And, one day, she came home and said, "Papa, today we learned set theory in class." "Really? What did you learn?" Well, you know, the teacher asked the class, "Those of you who had a toast for breakfast, please stand up." And, some of us stood up, and the teacher explained that this was the set of people who had a toast for breakfast. And, then the teacher said, "Those of you who had cereal for breakfast, please stand up." And, some of us stood up. And, teacher explained, "This is the set of people who had cereal for breakfast." Okay, and then? And, then the teacher said, "Well, those of you who had both cereal and toast, please stand up." And, a smaller number of people stood up. And, teacher explained that this is the intersection of the set of people who had cereal and who had toast for breakfast. And, this mathematician was getting more and more suspicious and said, "Well, alright, so this dish of carrots." There was a dish of carrots on the table. This dish of carrots, can they be a set? Of course not, Papa. Look, they are not standing up. Science, and more particularly mathematics, do not come already standing up for us. On the contrary, with intelligent work and open-mindedness, it is us who have to pick out phenomena, fresh phenomena, and make them stand up into mathematics and science. Thank you very much for your... [01:00:40] Speaker ?: Thank you very much. [01:00:41] Speaker 2: Thank you very much. [01:00:42] Speaker ?: Thank you very much. Gracias.