Gresham College Lectures

The Maths of Gyroscopes and Boomerangs

June 13, 2022 Gresham College
Gresham College Lectures
The Maths of Gyroscopes and Boomerangs
Show Notes Transcript

Spinning things are strange. Why does a spinning top stand up? Why doesn't a rolling wheel fall over? How does a falling cat always manage to land on its feet? How can the Hubble Space Telescope turn around in space? How do ice-skaters spin so fast? 

Taking a look at gyroscopes, this lecture explores the common threads that link all spinning things. The law of Conservation of Angular Momentum is far more subtle than we may think and there are many counter-intuitive observations.


A lecture by Professor Hugh Hunt

The transcript and downloadable versions of the lecture are available from the Gresham College website:
https://www.gresham.ac.uk/watch-now/gyroscopes-boomerangs

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- It's my very great pleasure this evening to introduce the annual Gresham College London Mathematical Society lecture. This is a longstanding partnership between the society and the college, and we have fantastic speakers every year, and this year is no exception, with a wonderful speaker, Professor Hugh Hunt, who I'm about to introduce. Hugh is professor of engineering dynamics and vibration at Cambridge University. His research specializes in climate change and how we could potentially refreeze the Arctic. Some of his other research interests include noise and vibration from underground trains, pendulum clocks, and spinning things that fly, you're going to hear something about those tonight. Hugh is a regular presenter on television documentaries,

including "Dambusters:

Building the Bouncing Bomb," "Attack of the Zeppelins," sounds fantastic,

and "Guy Martin:

Wall of Death," which I think may be one of the pictures there. He's also keeper of the clock at Trinity College, a clock which is the most accurate tower clock in the world. He has an impressive collection of boomerangs, which he uses to inspire students in the study of dynamics and mechanics, and tonight he'll be speaking to us on the maths of gyroscopes and boomerangs. Professor Hugh Hunt. (audience applauds) - We're talking about things that spin, and so what I want to do is to start off with a demonstration, and the demonstration involves a bouncy ball. Now, if I take a bouncy ball, you kind of expect a ball bouncing between two tables would behave like a ray of light. If that was a light, photons, they're particles, bouncing between mirrors, you'd expect it to bounce kind of like that. So let's now do a demonstration. So I've got a table here, I can move my cat, the cat will come in later, I have a table here, so there's two tables, this is the top table and this is the bottom table, and I've got a bouncy ball. Here we go, a bouncy ball. And I'm going to bounce the ball. Are you good at catching? I'm going to bounce the ball sort of this way. You'd expect the ball to go here, here, here, and out there. And it's a pretty reliable experiment, so I can make myself a bit of a target, and the ball will come back towards me. Now the question is why does that happen? I'll get rid of these glasses because they're just in the way. If I try that again, I can do this way. Why does the ball come back to me? Now, the thing about this is anybody can do this experiment, we've probably never thought about it. You can do it, you can throw the ball under a table, the ball comes back the way it came. There's three bounces, which is kind of what you'd expect, you can hear that, (ball bangs) one, two, three, but why does it come back? So what we've got to do is to think about each bounce as it happens. The first bounce, we get some spin happening. You kind of expect that because if you imagine a plane landing on a runway, the plane comes in like this, you'd expect the wheel to start spinning. If it didn't, you'd have thought the brakes were on, or something. So why shouldn't that happen with a ball? So I've got my ball here, the ball starts spinning after the first bounce. So straightaway, this is a problem about spin. Then what happens? The thing is, we know that if I put backspin on a ball, that the ball will come back towards me, so is this backspin or topspin? All right, let's do a show hands, who thinks this is topspin? Lots of sporty people. Who thinks it's backspin? It looks like topspin, but that's because we're looking at it from the perspective of the ground, let's tip it upside down. Now it looks like backspin because it's spinning. It is backspin, so the backspinners got that right. So if we think about what happens, with backspin, the ball comes back the way it came. And then, also, you've got to think, what happens to the direction of spin? If you look closely, when you take a ball, I don't know whether anybody can see what's happening, but that ball comes back towards me, but also changes the spin direction. Try that for yourself, and that's quite interesting.

So now let's recap:

the ball comes in, starts to spin, comes back the way it came, spin direction reverses, now we've got some topspin, and out it goes. You kind of think, well that's not maths, but the first thing about solving problems is to know what problem it is you've got to solve. And you look at this and you think, hmm, I think I can see what's happening here, so now we solve the problem. The maths we want to do, we're going to assume that the ball is really good at conserving energy, 'cause it's a nice, bouncy ball, so conservation of energy on each bounce, conservation of angular momentum, I'll talk a bit more about that this evening, and if we just put the maths in for those things, conservation of energy, conservation of angular momentum,

then this is what happened:

ball comes in, no spin, conservation of energy, conservation of angular momentum there, it starts to spin, the equations there, it starts to spin, the same equations here, it starts to spin. What's really neat is that the equations are just multiplying, adding, square root, nothing very complicated, so it's really easy to code up, so if anyone's interested in coding up this sort of thing. Then if you do a high-speed movie, which you can all do with your phones these days, that's what it does. In it comes, up it goes, spin direction reverses. So it's pretty amazing that this is what's going on, and a little bit of maths helps us do it, but the first thing you have to do is to observe the thing in the first place. Why is this important? Well if we think of ball games here, so this ball hits the crossbar, and that's fine. (audience laughs) So what's happening there is the ball hits the crossbar, it bounces back, but it's spinning like anything, we know that now, and so it's just quite fun, now that you know about spin. I was very pleased in Cambridge, when I first arrived there, to find that this chap, Ken Johnson, was just in an office just down the corridor from me, and he'd published this paper called "The Bounce of the Superball." And in it, at the very beginning, he works out angular velocities of each bounce, "As we will now proceed to show, "this behavior cannot be accounted for "by classical rigid body theory of impact." The thing is that all I've just been showing you now is the classic rigid body theory of impact, so I was a bit worried about this. So I spoke to Ken, and he was saying, no, it doesn't work. And then I showed him my videos, I think it does work. He wrote a lovely paper on the contact of elastic solids, and he came up with this nice idea that you take a rigid body, a rigid ball, like a solid steel ball, and for the impact, you just put a tiny, little elastic particle at the point of impact, and when you do that, it all comes out very nicely. So this is pretty much the only maths I'm going to do, I thought I'd better put some maths in for those who've come here for maths, but this is, as some people in the audience would call, dirty maths, applied maths, that's all I do. But what's interesting is, if you look at the kinetic energy for a spinning ball, then it's a formula that looks kind of like this, where v is velocity, and omega is the angular velocity, and I've got the kinetic energy before collision, and the kinetic energy afterwards must be the same, and then I do the same thing for angular momentum about the contact point. And then you can solve those equations, that's fine, you do that. But then, the neat thing is, you do the next collision, and then you get the third velocity, depending on the first one, and this is approaching the bottom table again, and you think, hmm, well now I've got, every time it approaches this bottom table, I know what the velocity is depending on what it was the last time it approached the table. So what you have is, it's a difference equation, so the n+1-th velocity and angular velocity is related by this ridiculous matrix, and we want it to be some kind of proportionality, and that leads to an eigenvalue problem, and the problem here is you end up with complex numbers, you think, oh God, why do I, there's no complex numbers in bouncing balls, it's a real thing. But you solve it all, you go through the maths, and you end up with that the velocity and the angular velocity are sinusoids, and you think, okay? So you plot it all out, and you get these sinusoids, but it turns out that you sample the sinusoids at particular points. So what it means is that if I start here, the first collision is there, and that's the first little circle on the curve, and then the next one's there, which is the next little circle, then the next one's here, so that, I've shown with the big thick lines, is the first three bounces, which is what we've observed. And what's really nice is when you look at it, it all makes sense, more or less, it's not perfect, but it's pretty good. And you kind of imagine if I could make the table long enough and get rid of energy loss, this ball would go bouncing backwards and forwards forever, and I just quite like that sort of stuff. But now some of you will remember a more important goal, do you remember this one? So this was Frank Lampard's disallowed goal, and it bounced over the goal line, and then back up. Look at him, he's a bit older now, isn't he. So the thing about that goal is that if you do the maths on it, it turns out that it's impossible for a ball to hit the crossbar twice without going over the goal line, and you can do the angular momentum and energy calculations, but the newspapers the next day didn't mention any of that. (audience laughs) But it's just quite fun that you can, with very simple maths, look at that sort of thing and enjoy what's going on. And ultimately, it comes down to Isaac Newton's F equals ma, force is mass times acceleration. Before we get onto gyroscopes, I want to tell you about this wall of death thing, because what was really good, this chap, Guy Martin, you might have seen him on the telly, he's a bit of a nutter, but he wanted to ride the wall of death and set the world speed record for the wall of death, he wanted to go 100 miles an hour on a wall of death. So what I did was to write down some equations for him, but let's have a look at a wall of death first. This is what you might find in a fairground, you've got a slopy bit at the bottom, you've got a motorbike, and you get these motorbike riders that will go round and round. Typically, the diameter of this thing is four meters, and they'll go at 25 miles an hour. And all I said to to Guy Martin was if you want to go 100 miles an hour, well we're going to have to do some sums on this because the thing that is difficult for the rider on this wall of death is the forces on their body, because you get the g-forces, in other words, your eyeballs sink, and your bladder sinks, you can die just from the g-forces. So there they are, the g-forces are holding the bike onto the wall, this is the wall of death, but otherwise, as far as I know, nobody's ever died doing a wall of death, so it's a bit of a misnomer. So, circular motion, what are the forces involved with circular motion? Well let's just have a look at this. if I take a tennis ball, it's a regular tennis ball, and I've got here, it's full of water, so this is two kilograms, two liters, two kilograms, and I'm going to tie a string to this two liters of water. Normally, a tennis ball is 50 grams, or something. In a game of water bottle versus tennis ball, the water bottle wins, but it doesn't take much to lift the two kilograms. So you get an idea that this centrifugal force, this force of circular motion, is very big. So when Guy Martin is going round on his bike, he wanted to know what forces can I put up with. The force in that string is dependent on how fast you're going, squared, divided by the radius of the circle. As you do for television, you spray-paint the formulae on a wall, and so that's what we did. The small wall, 25 miles an hour at 4 meters, if he wants to do 100 miles an hour, we need a 64 meter diameter wall of death. So what are we going to do, what do we have to do? Let's just think a little bit more about the force. If you imagine your bike sitting on the ground, you might have noticed on the wall of death, that the bike was at an angle, because as you're going around on a circle, you need a horizontal force to make you go around in a circle, that's this mv-squared over r thing, but you still have to hold your weight up. If you're up on the wall, then this resultant force is always along the line of the slope of the bike. So when you're going around on a corner, you lean your bike in, that's what's happening. So you can see, more or less, what the g-forces are by the slope of the bike. And by the way, this theta has become a q here, nevermind, the coefficient of friction, mu, is 10-theta. What's really nice is that we can link the coefficient of friction, what coefficient of friction do you need to hold the bike up, it's a nice bit of trigonometry, which is good. What it means is that you've got to have a dry surface, no oil on it, no water on it, to hold yourself up. Is this video going to run? - [Commentator] 70 miles an hour, five-g, that's five times his body weight pushing him down to the ground. - [Hugh] So you see how big the wall has ended up being. - [Commentator] Pushing him towards seven-g. - [Hugh] So we made a wall of about 40 meters diameter, 40 meters diameter, an lo and behold, he gets to about 80 miles an hour, 'cause if you do the sums. So there we go, 80 miles an hour. So we did the sums, and it all worked, it's great. Why do you build these big things? I don't know, but it was good fun, and it's good to know that the maths works, that's what I really like to see about this. The other thing where the maths works as well is in a loop-the-loop, and this was another thing that we tried to do, which was to build a loop-the-loop, and to drive a regular car, a Toyota Aygo, around the loop. So here we go, this car heading towards the loop. I told the driver you had to go in at 37 miles an hour, I'd worked it out, to be able to go around the loop, and he just said, you know what, I have to trust you. When you're doing these things, you've got to do some sums, and make it work, exactly the same way as when astronauts go to the Moon, or wherever they go, you do sums, it's got to work, and it's just great that it works. But now, let's think about it, why did we build a circle? Because let's start looking at other loop-the-loops. If you look closely at them, they're not circles. Why are they not circles? Humans can't withstand very high g-forces, but as you go around your loop-the-loop, you slow down, so you can have the circle tightening up. So this shape, it's got a funny name, it's called a clothoid, if ever you want to find a word that nobody knows about, but these are not circles. And if you look at places where there's no track, if you've got a plane doing a loop-the-loop, they naturally go on these shapes, they're not circles. And you kind of think, well yeah. So why do we think, when we have our toy Mattel, whatever it is, Hot Wheels thing, we do circular loops, when we really ought to be doing loops that look a bit like this. So next time you go to a theme park, just have a look, and you'll see that the circles are not circles. We've looked at spin from a point of view of circular motion, not really getting at the gyroscopic effect, now I want to do gyroscopic effect. And when we have a spinning top, like this one here, we know that a spinning top doesn't stay up on its own, and it doesn't stay up unless it's spinning fast enough. And when it's spinning fast enough, what you see, it starts to move around, this kind of motion. And you think, I'd like to know, why does it stay up? It's one of these things that, at school, we don't learn about spinning tops anymore, even if we ever really did, because it's reckoned to be too complicated, but yet we do learn about genetics, and double helices, and all that sort of thing, because, somehow, biology's allowed to be complicated, but physics and maths, it's got to be stuff that you can explain from one step to the next. But I think that's a bit of a shame, really, because there's so much fun to be had with spinning tops. The gyroscopic effect, if you want to Google this and try and figure out what's going on, you need words like gyroscopic effect and gyroscopic precession, and once you've got those words, it's much easier, 'cause if you just Google spinning top, you'll find out that you can buy one from Argos for 7 pounds 99, and it won't tell you anything about how they work, so you need these words. But what I quite like is that it's not just a spinning top that behaves like a spinning top, and if you get a regular bike wheel, and you can go to a bike shop and get a stunt peg, this is a stunt peg, and you can screw the stunt peg on like that, that gives you a handle, and now, with this thing, I can do exactly what the spinning top does, it's exactly the same, it's a spinning thing, and you see it's precessing around. But what's nice is that I can make it precess around at whatever angle I like if I've got a piece of string on the end of it there. And the thing here is that it kind of looks like that this is some kind of anti-gravity device. There was a famous Royal Institution Christmas lecture in 1976, I think, a chap called Eric Laithwaite, and he demonstrated, basically, that gyroscopes were anti-gravity devices, because if they weren't, then how could you do this. All you have to do is let go of the string and you realize that it's being held up by the string. And if you put a pair of scales, a balance on here, you'll see that the mass of this thing hasn't changed at all as it precesses around. So what's going on? And this is where you can do it with maths if you like, but I quite like doing it with visualizing it. So what we've got are two forces, I like to call them a couple, some people will call it a moment, but if you've got two forces that are not in line with each other, one force is the force in the string and the other force is the weight of the wheel, those two forces create a couple, which, if the wheel is not spinning, will cause the wheel to twist around and develop some angular momentum, but if the wheel is already spinning, then there's another way for this wheel to generate some angular momentum. Which direction do I want to generate the angular momentum? I want to generate the angular momentum that way, because that's the direction of the couple. But if it's already spinning, then I can generate that angular momentum by turning the wheel around to there, and now it's spinning in exactly the direction I was wanting it to do. So this gyroscopic precession is continually moving the direction of spin, it's spinning in this direction, put a couple that way, ah look, it's spinning that way. That is what's going on physically. Now, the maths you can do, and you get at with the equations, which is fine. We need to understand though what's meant by angular momentum, and for that, I'm going to do a demo here. It's quite handy to have a volunteer, anyone want to? Ah, we've got a volunteer, there we go, that's good, we've got a volunteer here. Right, yeah, come up there, that's good. Hi, what's your name? - Olly. - Olly, right. I have a swivelly stool here, which you're going to stand on, are you happy to do that? - Yeah, that's fine. - Are you good at keeping balance? - Yeah. - Right, there we go. So if you stand on that, there you go, happy? (audience laughs) Yeah? - Yeah, that's fine. - Good, that's good. So now what I'm going to do is I'm going to give you a couple of, this one's two kilograms, hold that one, this one's two kilograms. Now what I want to do is, I want you to hold those in front of you like that. If I try to spin you around like that, you have a certain angular mass, I can feel that mass, but now if you put your arms at full stretch, I know you haven't got heavier, but that feels, you can feel it too, can't you, in your arms? So it matters how far out you put your arms. So this is what you might call angular mass, or we might call it moment of inertia if we want to use its proper name. But if we think of momentum, like a car is moving along a road, its momentum is its mass times its velocity. Angular momentum is its angular mass times its angular velocity. Nice frictionless stool here, so we know that if a car is going on a frictionless road, its speed stays constant because mass stays constant, momentum stay constant, but you can change your angular mass by bringing your arms in and out. So if you've got constant angular momentum, which is angular mass times angular velocity, then your spin speed's going to change. Right, so you're going to do this as a demo, arms out. What I want you to do, without any spin, at this sort of rate, bring your arms right to, there we go. Now we do the same thing spinning. Arms out. Wait, wait, wait, wait. Okay, nice and gently, all the way in, all the way, all the way keep going. And slowly out again. (audience laughs) Okay, we'll go backwards, we'll unwind. You get a bit dizzy doing this, don't you. Are you okay? Arms out. In you go. All the way in, how far can you get? Very good, fantastic, Olly, brilliant, a big round of applause. You all right there? Thanks very much. (audience applauds) So that is an example of conservation of angular momentum, and we can see that with ice skaters. This ice skater here, with not much friction at the contact with the ice, she can move her arms in and out to change her angular velocity. She's given herself a kick here, she's still got her bum stuck out like that, and now she stands up thin and spins around really fast, conservation of angular momentum, so you don't need extra weights to be holding. So if I try to do this, I can stand on here, with my bum stuck out like she had, and then. (audience laughs) So all I have to do is stand up straight, and I start to spin around faster. So now this thing, conservation of angular momentum, you might kind of think it's only to do with ice skaters, and ballet dancers, and stuff, but it's a really useful thing out in space. So let's suppose I'm out in space, and I want to point a telescope. All right, let's imagine I've got a telescope here, and I would like, there's a star over there, but there's a better looking star over there. How do I? Conservation of angular momentum, I can't point my telescope over there. But if I've got a wheel, then I can do some fun things. So the first thing I can do is, if I start my wheel spinning, then if the wheel is turning clockwise, then I have to turn anticlockwise, and I can stop wherever I want, and that is conservation of angular momentum. If I start and stop the wheel, that's fine, but what if I start the wheel spinning clockwise, and what if I were to tip the wheel like this? Because now the wheel is spinning anticlockwise, it's now spinning in the opposite direction. So what would you expect to happen? So I'll get on my stool here, so I'll start the wheel spinning clockwise, and now turn it around, and I'll spin it anticlockwise, and now I'll go back up there. So what it means is that I can use this as a steering wheel when I'm out in space. Anyone want to come up and try, another volunteer from anywhere? Ooh, we've got one up there, come on. So this is just to prove that I'm not cheating here, so here we go, very good. Your name is? - Premjo. - Cranjo? - Premjo. - Premjo, great. So what I want you to do is stand on my swivelly stool, there we go. Are you happy with that? - Yeah. - Good with that? So what I'm going to do is I'm going to spin this up, and what I'd you to do is to grab that with both hands, and now you can tip it up, and you can tip it down a bit, now point it downwards, keep pointing downwards. Yeah, that's it. There we go, and back up again. It feels pretty strange, doesn't it? - [Premjo] Yeah. - The thing is, it feels stranger than it looks. So there we go, conservation of angular momentum. Very good, Premjo, thank you very much. (audience applauds) So they key thing here is that because we're on a swivelly stool, as you change the angle of a rotor in your spaceship, you can change the direction that you're spinning, conservation of angular momentum. Conservation of angular momentum, again, happens in places you least expect it. If I take this tennis racket, it's painted red facing you there, and it's white on the other side. Normally, you'd kind of expect, if I toss the tennis racket up in the air, you would expect the white is facing you all the time, that's what conservation of angular momentum would do, you wouldn't expect it to tumble around. Well unfortunately, that's exactly what does happen. If I take my tennis racket and spin it this way, red facing you, toss it up, now it's white facing you. White facing you, toss it up, and now it's red, toss it up again. And this is just one of these things that happens when you do the maths of a spinning body, it turns out that, it doesn't have to be a tennis racket, it can be a book, I've got a book here, and a rubber band on it to stop the pages from opening. If I spin the book like this, it's pointing upwards, toss it up, catch it, and it's now upside down, toss it up, it's now back the right way up. In fact, if I do it high enough, I might be able to get a double flip. So it's the right way up, it's back the right way up. Why is this happening? I can show you a bit of the maths of that later on if you want, but what's really interesting is the maths isn't too difficult, and it's why we study maths, because it's only by getting into the maths of things that we start to understand crazy things like that. There's even a crazier one, which, again, is conservation of angular momentum, which is why my cat, if dropped upside-down, will land on its feet. Unfortunately, this one is a defective cat, (audience laughs) but this one is a non-defective cat, this is a real cat. I'll show this NASA video, 'cause if anyone's got any complaints, they should contact NASA. But here is the cat, that was reversed, by the way. (audience laughs)

So let's see what's happening:

the cat is being dropped upside-down, and the back end of the cat, the tail and the hind legs, are going, say, clockwise, and the front end is a cat is going anticlockwise, and then the cat does a bit of a maneuver to get the legs in the right place, and lands on its feet. And of course, cats have worked out how to do this over generations. This is a picture taken about 120 years ago, a series of flash photographs, but the problem was that the cat had to be dropped a few hundred times to get this sequence of photographs, it's good that we've got video cameras now, cats are in a happier place. But I think I can do what the cat is doing, sort of, let me just demonstrate. If I put my arms out at a distance, I get large angular mass, and my body moves quite a lot, and my arms don't move that much, whereas if I put my arms in, small angular mass, my body doesn't move at all. So that means I can go around one way with a large angular mass, bring my arms in, go back out. So that is what the cat is doing, it's kind of swimming around in a circle. And again, it's really nice to be able to see how that works. And maybe some of you are trampolinists, or maybe some of you have done any kind of acrobatics, you can tumble around. You watch this with diving in the Olympics, you think, how is it that this diver has managed to start off this way, but lands in the pool pointing the other way? Well there you go, that's all this conservation of angular momentum stuff. I'm going to wind up now on my favorite bit of gyroscopic stuff, on boomerangs, and how they work. We have to accept that wings work, airflow over a wing generates lift. If ever anybody asks you why is there lift, if you start to talk about Bernoulli, that's the complicated way to think about it, because there's nothing at all, by looking at that diagram, to say that the air moves over one side of the wing faster or slower than the other, but the easy thing to think about is that the wing deflects air downwards, and every action has an equal and opposite reaction, so if I'm deflecting air downwards, there's got to be a lift force on the wing. And we know that the lift force on the wing is bigger the faster the air is moving. So how does that relate to a boomerang? The first thing is, the boomerang I've shown there is, if you like, two of these regular boomerangs joined together to make a kind of cross shape. It turns out they're easier to analyze, they're easier to make, they're easier to think about generally. And what we do is we have some spin, just like our spinning wheel, and we're going to throw the boomerang in a certain direction. And what you'll notice is that because we've got spin and moving forwards at the same time, our boomerang is, well, if the middle is going at a certain speed, and it's spinning, then the top is moving forwards faster than the middle, and the bottom is going backwards a bit, so it's moving forwards slower than the middle. So when moving through the air, this bit of wing, because the boomerang's got wing-shaped surfaces, this bit of wing, the faster one has got more lift, the slower bit has got less lift. That means we've got an unbalanced set of forces, which was just like on the bicycle wheel, we have this couple, more force at the top, less at the bottom. And when we've got a couple acting on a spinning object, well here's our couple acting on a spinning object, we get this gyroscopic precession, it wants to go around to change its direction. Does that happen with our boomerang? And the answer is yes, it does. There we go, it's gone round on a curved path because of this gyroscopic effect. You might notice that it levels out a bit because there's a bit of drag, it slows down, various effects, but actually, it's very handy if it does level out a bit because when it comes back to me, it makes it nice and easy to catch. I've got other ones here, this one's a left, oh no, where's my left-handed? This is a left-handed boomerang, mirror image, it should go round the other way. It goes round a bit fast. And then this one, these are quite fun. And you can make these yourself, they're nice, easy things to do. If you do the maths on them, it turns out that the radius of the flight of a boomerang, it doesn't depend on how hard you throw it, the radius of the flight of the boomerang, that's big R, depends on our angular mass, J, the moment of inertia, density of air, the lift coefficient of the wing, pi, which is three, and how big the boomerang is. The radius doesn't depend on how hard I throw it, so if I throw it gently, it kind of comes back to me, and it goes two or three rows back. If I throw it hard, it comes back on the same path radius. Surely, throwing it hard makes a difference? Well you need this flick of the wrist, the faster you throw it, the more you need to spin it, and that's the thing about learning how to throw a boomerang. So I've given you a bit of a whistle-stop tour of gyroscopes and boomerangs. The maths? Well it's all buried in underneath, but I hope you enjoyed seeing some of these demos, thank you very much. (audience applauds) - Professor Hunt, that was fascinating, entertaining, it looked like miracles and magic from where I was sitting. - But no maths. - No, there was maths as well, and we have some questions. We have some questions from the online audience, and I'm sure we'll have some questions from the room as well, and I'm going to start off with one of the online questions,

which is this:

"With gyroscopes pointing telescopes, "why do they eventually run out of capacity "and have to realign the spin with thrusters?" - When you've got a gyroscope. - Should I get out of the way so that you can? - So imagine, if you've got a spinning wheel, the effect you want to get is best when the spin is at right angles to the motion you want to change. So on a spacecraft, you're going to have three of these things, one in this direction, one in this direction, and a third one in that direction, and hopefully then, with any luck, you can use them to do fully three-dimensional aligning of your spacecraft. But what happens if, there I am, I'm doing fantastic alignment of my spacecraft, but what happens if, for one reason or another, this one has ended up pointing up this way, in exactly the same direction as the other one? So now I've got one direction which I can't control. So what I've got to do, turn off my gyro, get it aligned back to where it is, use my thrusters to get everything lined up properly, and start up again. There was a famous bit in "Apollo 13," the movie, where they turned off their inertial navigation system. That was a scary moment, you're out in space, the one thing you don't want to lose is your inertial navigation system, but they had to turn it off, and then you turn it on again, and then, yes, they had to use their thrusters to line up and switch everything back on again. - [Questioner] Thank you very much, Professor Hunt, for a most interesting lecture. I didn't realize a boomerang would actually have four arms of equal length. I tried, actually throwing a boomerang when I was in Australia, I was completely hopeless at it. - Well that's because it's in the Southern Hemisphere, it would have been fine if you were here. (audience laughs) - [Questioner] With a traditional Aboriginal boomerang, does that involve far more skill than the boomerang you've used? - If you buy a boomerang, or get a boomerang from a tourist shop, you're probably going to find that it's not going to work, or if it does work, it's going to take a bit of practice. The best boomerangs to get are online. If you go to, believe it or not, the best boomerang society is the British Boomerang Society, there you go, and if you go to the British Boomerang Society, they've got a fantastic list of where to get a boomerang, and you can get little ones, tiny, weeny ones, big ones, left-handed ones, right-handed ones, lead-weighted ones that go for miles, there are sport boomerangs where they go really quickly so you can do a record of how many throws can you do in a minute, those are easy to throw. Probably, you just had one that was hard to throw, it ought not to be difficult. So what's the key? Firstly, make sure you know whether it's a left-handed or a right-handed boomerang. If it's a right-handed boomerang, it's going to go around that way, if it's a left-handed boomerang, it's go around that way. You can throw a left-handed boomerang right-handed, but it's going to go around the other way. Hold it in the vertical plane, and lots of spin. You don't want to hold it in the horizontal plane because then it's going to lift up and go straight up. Vertical plane, lots of spin, you've got to get the spin going. Have you still got the boomerang? - [Questioner] I haven't. What about an Aboriginal? - They're probably quite good at throwing it. Anyway, there you go, it ought to be possible. - [Questioner] If you don't mind, this is not on your lecture, it's just a question that I'd like to ask, but it does involve you. The "Today" program, which, I'm sure you know, comes on between six and nine in the morning, up until about two years ago, it used to have a bit that was called Puzzle for the Day. You used to set questions for that, - Yeah, I did. - [Questioner] and they were always the most challenging, I enjoyed them. Why did those questions stop? - It might be perhaps equally interesting as to why they started, and I think that it was thought that it would be a good idea to promote maths, and to have these puzzles for the day. I'm not sure that they quite thought through how it was going to work, having a new puzzle every day, it was quite a challenge, and I think the quality was variable, and it got more and more variable as time went on, and it got to the point where there weren't that many. What I think was hoped was there'd be a much bigger pool of people coming in to set these puzzles, but they decided to stop it. I think it was probably the right thing. One thing that I found slightly frustrating was that there was an answer online, so if you wanted to figure out what the answer was, you would look online the next day, but I think not many people did, and I would rather have had something like, where you have puzzle for the day, and then half an hour later, you say, right, have you figured it out? Here is the explanation, and actually have it read out by the person who set it. So I think it was difficult, some of the presenters didn't really, sometimes, they say things about maths which they really ought not to say on a radio program. I'm sad it stopped, but it was fun while it lasted. - Okay, we've got some more questions flooding in online.

Here's a question about motorbikes:

"To initiate a left turn with a motorcycle, "you need to turn the handlebars to the right. "Could you please explain this." - This is an extraordinary thing, and it's not just on a motorbike, it's on a regular bike as well. It's called the counter-steer, and it's counterintuitive. Imagine you are riding in a straight line on a bike, and then imagine you want to get from that straight line to going around on a corner. If you want to be going around on a corner, you want to be tilted over, but actually, you don't need to steer, being tilted over is really all you need to go around this corner, that's just what's going to happen, you hardly need steer. How are you going to get from this point to this point? And the easiest way to do that, if you've got a spinning wheel, is if I want to tilt the wheel that way, the easiest way is to, I'm actually trying to push the wheel to the right. I'm trying to do this, but because it's a spinning wheel, and the gyroscopic effect, I'm trying to turn it to the right, and look what it does, it tilts over, which is what I wanted to do. So the quickest way to get onto that corner, turning to the left, is to do a quick turn to the right, and it's really weird. And it's what we all do, you might not have noticed it, but it's what we all do. If you suddenly need to turn left, you do a quick turn to the right, it's amazing stuff. - [Questioner] In the formula for the radius of the flight, you had inertia over the size of the boomerang. Would that not cancel out because the size of the arms of the boomerang would cancel, what is meant by the size of the boomerang? - The size in my formula there was the radius. There's a nice thing you can do, which is dimensional analysis. These are the only parameters that can be put together to create a formula for the radius of the boomerang. Let's assume that the speed of light doesn't matter, and let's say that viscosity doesn't matter, and let's suppose that temperature doesn't matter, various things don't matter, these things all matter, and it turns out that these two formulae are the only way that you can put them together, so you're going to need a size parameter. When you say things cancel out, well they can't cancel out because otherwise you won't get dimensional consistency. But I think what you have to bear in mind is that the gyroscopic effect if one thing that's happening, the other thing that's happening is circular motion, mv-squared over r, and I think if you go through the maths on it, you'll find that this does work out. But I'm glad you're thinking about the maths, that's exactly what we want in this room. - Your next-door neighbor had a question as well. - [Questioner] I had a question about the wall of death, you said that because of the g-forces, the driver wouldn't survive if the radius was too small, so what would happen if you put a robot, or something on there that could survive the g-force, would it be able to go at 100 miles an hour around a radius of 20? - Yeah, absolutely. If you want to do a wall of death, if you want to get a slot car thing, if you imagine a velodrome, and you're on a bike, and you're going around, and it slopes up, you can imagine getting to the point where it's pretty much vertical, and if you got a driverless car to do that, you can go as fast as you like, and the g-forces can be as fast as you like. That tennis ball was experiencing, because 2 kilograms, a tennis ball is 50 grams, so that tennis ball was experiencing g-forces of 80-g, and because it's not a person, it's okay. So yeah, you're right, if you could do it with a robot vehicle, that'd be good. Guy Martin did try to go faster, but he just found that his vision got blurred, and that's one of the classic things with fighter pilots, that they have to train how to deal with the g-forces, you really have to train for it. Good question. - There's a related question here, a couple of questions about fluids, and this one is about fluids and the capability of the human body, which is a bit what you're saying. The question is, "What happens to the fluid within an ice skater's ears," the cochlear fluid, "when they spin, "and does this affect their ability to continue a spin?" - I'm told that ice skaters and ballet dancers who do a lot of these spins, they become immune to it, they don't get dizzy, whereas most of us, we do. I don't know whether Olly is still feeling a bit, (Simon laughs) but you notice it for quite a few minutes afterwards. Is that true, did you notice it? For those of use who don't do it all the time, but if you do it all the time, I'm told it's something you just get used to. The cochlea is a very multi-talented organ in our body, it's used for hearing, but it's got fluid in it which, when you spin around, the pressures in the cochlea can tell you that you are spinning, and that's where we get dizzy from. - [Questioner] You described the bicycle wheel spinning on the end of a string as an anti-gravity device. Is that similar to how astronauts simulate zero gravity by spinning around a plane? - When they simulate zero gravity, what's the film, it's one of those films? "Interstellar," or "Gravity," or "The Martian," where they've got a spin. - [Audience Member] "2001." - That one too, but this audience doesn't know, yes, but it's true. But they've got this spinning spacecraft so that you can have gravity out in space. But if you want to get rid of gravity when you're on Earth, that's really difficult. There's two ways of doing it, one is to go up into a plane, go up as high as you can so you've got plenty of time for the experiment, and then the plane falls down, projectile motion, Stephen Hawking went up in one of these things, you can feel weightlessness for not very long, because you've only got a certain amount of time before you get to the Earth. But then what they did a lot of in the Apollo program, and in the shuttle program was to practice being weightless by being in water tanks, so where your buoyancy is equal and opposite to gravity, but it's still not the same as being out in space. I don't know whether that answers your question, but the spin is to create gravity when you're out in space rather than to get rid of gravity when you're on Earth. - Fantastic, thank you very much for answering those questions. We are now going to move to a vote of thanks. - I hope I've turned that on, and I can hear myself I think, so that's good. First of all, I'm Kevin Houston, I'm the education secretary of the London Mathematical Society, and normally, what I do at these events is say the London Mathematical Society is a very old institution, and goes back other 1865, but I always feel that when I come to Gresham, that that no longer cuts is, as Gresham is at least twice as old. But the talks that we've had over the years go back maybe 15 years or so with this connection between the two institutions, has been fantastic, and we've had another fantastic talk here today. I was glad to see, it was very dangerous, there were all sorts of things which could have ended in disaster, so it's good to see that, because coming back to being in person, (audience laughs) online, you don't really get that feeling of danger, so it's great, indeed, it's great to see so many people coming out, and I'm assured that there are people online, so hello to the people online, I hope everyone has enjoyed the talk today. I would like you to join me in thanking Professor Hunt for such a fantastic talk. - Thank you very much. (audience applauds)