字幕表 動画を再生する 英語字幕をプリント I brought a little tube whose ends have been colored black and orange and I'm going to launch it into spin and it will spin around the middle and because it is spinning around the middle a priori, there is no reason why you would see black better than orange orange better than black. Ok, but now when I launch what do we see? We see black. And every time we seem to be seeing black. That's really funny. We seem to be seeing, be seeing black but you don't like black, I'm going to shake it and launch it and now we'll see orange instead. That's really curious. It's not a matter of launching from right to left because I can launch it with my left hand and it's still orange. After all, you don't like orange then so I'm going to rub it against my black hair and that changes the color again, and it's black. So the first mystery is why sometimes do we see black and sometimes orange? Mystery number two is when I launch this how many black spots do we see? I think we see four. Why four? It's not as if I'm launching it with very much care. Its initial conditions seem to be random. So why do we see four? When it's stabilized, it's four. And I have to shake it around. And... four oranges. See, in order to understand it better, I brought another tube. Similarly marked, black and orange. Now, this time I'm going to show you black. There's a much longer transient, in other words, you have to wait for it but when it settles into steady state we have three. Again. We see three. That's interesting. You might have noticed... I hope you didn't, that when I launch from the black end it shows black. And when I press down the orange end and launch it's orange. So why is this? I'd like to begin by giving you a zeroth order theory, and then we'll do the full theory later on. So the zeroth order theory goes as follows... you see, this motion I'm pressing down, making it spin can be thought of as a combination or superposition of two motions. First... the whole thing is spinning like this around the vertical axis. We shall call it revolution. Yeah. At same time, this thing spins around it's own axle. And we shall call that motion rotation. And the whole thing is a combination of revolution and rotation. Let's say for the time being that we launch it by pressing down the black end, okay. Always. And, please look at the other point here at the orange end. That point will be advancing like this, because of the revolution. But you see... the... rotation that I gave the black end is a back spin like this, which means the orange end is going in top spin. So that's a rotation which is also going forward. [Brady]: Yeah. So the revolution makes it advance like this but the rotation which is like... like this is making go like that also. So, the rotation and revolution ADD to each other, so this end is going fast. In contrast... let's look at the black end that I pinch down. Well again, the revolution... the revolution will make it advance like this. But at this end, the rotation is back spin so the rotational speed is actually pointing backward. So those two speeds subtract from each other so this end's going to move slower. And the fact of the matter is that the end that we see better is the end that is moving slower and the end that's moving faster we don't see so well. So, black end is moving slower and orange end is moving faster and the zeroth order theory is that black end is... the slow end is the one end that we see and the orange end, the fast end, we don't see. So far, so good, but you see, this explanation, zeroth order as I called it is not totally satisfying, because first of all, why is it that we see the slow end rather than the fast end? It sounds plausible, but maybe human eyes have a... you know... preferred frequency that it wants to capture and maybe at some frequency -- even if it is fast -- we see best, and so on. Maybe. And also, this zeroth order explanation doesn't do anything to explain why four, why three. So we'll wade into the now full analysis It so happens that when you... pinch it Initially, everything is sliding because I'm launching it. But soon, this sliding motion goes into ROLLING motion. Rolling, by the way, means the following thing: when two bodies roll against each other it means that at the point of contact, there is no relative velocity In other words, these two bodies are moving in exactly the same way. That's what rolling means. If they have a relative vevlocity if they are not moving the same way at point of contact they slide against each other That makes sense. So they roll against each other. That means that their relative velocity is zero at the point of contact. But if an object is rolling on the fixed floor Well, the floor is not moving, so that means that the point of contact at any instant is instantaneously at rest Has zero velocity... at the point of contact. That's what rolling means. So let's say it starts rolling like this and it goes around, around like... Okay. Now, as you can see... It's tilted a little, but let's ignore this tilt for the moment. It's very, very small and it just keeps going around and around The point of contact then is describing a largish circle on the floor. On that... large circle you can see there's a cross sectional circle of the tube which is... you can imagine a coin rolling on the circle like this going around, around Ok, so there is a small circle rolling on the large circle. Well... How many times does this small circle roll on the large circle? Well it has to do with, of course, the perimeter of the small circle to the perimeter of the large circle. But that's equal to the ratio of the diameter of the small circle to the diameter of the large circle which is approximately the length of this end So, it's really the aspect ratio of how long it is to how wide it is. And that's -- it turns out -- 4 to 1. By the way, it looks a bit curious, I think. Most people looking at this tube would say that it's longer and thinner than 4 to 1. But actually it is, because, for example... These two have the same width and if I measure this in units of the small one that's one... two... three... four So it is really four to one. I don't know why human eyes think of it as longer and thinner than 4 to 1 but it is 4 to 1. And that's 3 to 1. And that is number that we were seeing. 4 points, 4 dots, 3 dots. But there is something even more beautiful going on so let me go into this... We have been so far neglecting this slight tilt but actually, there is a slight tilt When I launch this... it so happens that the part that I push down starts... immediately rises it's head and goes into this motion. Why does it rise it's head? Well... it's easy, but... an awkward explanation. It has to do with the gyroscopic effect. So let me, er, slide over that then. Let's accept that rise... it raises it's head and starts going like this. While this thing is rolling, rolling... with a slight tilt against the floor in steady state In your imagination, please place a transparent ceiling from above So as the touch this tilted tubes from above... this motion is happening between two planes the imaginary ceiling above and then the real floor below Well... you can see that there is an up and down symmetry between the floor and ceiling of this motion, it's just going around and around like... like that So... in particular... if this thing... thing is rolling on the floor... at the bottom, at top its rolling under the ceiling. On the ceiling, or under the ceiling rather, this thing is describing a larger circle. And this... sort of coin like cross-sectional circle is hanging from the ceiling circle going around, around in a circle. Rolling means, as we explained, that in fact instantaneously whatever... point is a point of contact to the... with the ceiling is instantaneously at rest, because it's now touching the ceiling from below. Yeah. So... earlier we said something to the effect that well the black end is the slow end then the, erm... orange end is the fast end. But never mind slow, it's actually instantaneously at rest the velocity becomes ZERO Every time it comes up it goes paff, paff, paff, paff, paff! Stops for us And that's why we see them so clearly and how many times did it stop per circuit? Well, four exactly because that's the ratio... aspect ratio would we talk about [low frequency humming -- slowed down rotation] [Brady]: Stopping four times in every possible position too, though? [Brady]: It's not just stopping with the black part up, [Brady]: it's stop... Ahh... that's right, but you see... the... in other positions we are seeing the white bit stop so between the black bits, in fact you see lots and lots of white bits that are stopping for us and that's why we see in fact the white circle with four black dots Indeed. So every point, whenever it comes upward is stopped. Similarly, every point... down below is stopping whenever it comes into point of contact so if you did this experiment, maybe you would like to do it together... between not just a floor like this, but it's a glass floor and transparent floor and we filmed from, ah... below then we would see the orange bits... boom, boom, boom, boom paff, paff, paff, paff and then all the rest would be white. There is a very beautiful curve called a cycloid. So a cycloid is roughly speaking this curve. If you look at the trajectory of this black thing as the... the cross-sectional circle rolls I'm going to exaggerate the size It... describe a... describes a curve like this and that's called a cycloid. The point about the cycloid... it has cusps. You see whenever it touches the floor the stationary support it... it has a cusps. and this cusp turns out to be locally vertical you know it doesn't make an angle like this... but actually it's like this because the point is curving in and then kind of bouncing vertically, and going out... and going out. (breath) So... this has the following very very interesting consequence as well. If you go back for a moment to this motion... Orange one, which is touching the floor, is then describing a cycloid. If you like those lobes standing on the, er, circle on the floor. Similarly, by the same token, this black dot is describing a sequence of upside down cycloids whose lobes are hanging from the ceiling... like this. And each time the black dot comes up you are at the cusp of this cycloid.... upside down cycloid. But you see this cusp is... vertical cusp and we are looking at the motion from above You see, so that means that whenever the black dot comes into contact with the ceiling and goes into instantaneous rest. Not only does it stop instantaneously, but in that vicinity it's moving only vertically. and we are looking at... from above so we see only the horizontal motion. And for the observer from above vertical motion is invisible, right? I mean it looks like stationary. See, in other words, near this stationary point the cusp of the cycloid when the black dot comes into contact with the ceiling from above it not only comes... becomes stationary at that moment but it's velocity is really exaggeratedly shrunk with respect to the observer from above so not only do you get zero velocity, but around that... moment the... its velocity actually shrunk because of the angle of the observation. That's why we see this effect so clearly so it has the, erm, the effect of exaggerating this stop motion. Now, when I, erm, started thinking about this I didn't know, of course, what was happening and so I went to a hardware store in Tokyo and I bought a long, erm, plastic tube and then cut them in aspect ratios... integers and then started playing with this and figured out everything. By the way, if you want to do this at, er, at home and so on I recommend that you use hollow tubes because, you see a hollow tube means that most of the mass is concentrated on the rim and at the same time because it's hollow, it's not so heavy So making it like... has the advantage that you can launch it easily. but concentrating the mass on the rim and has the advantage that what is called the moment of inertia is large for the same mass so once it gets going, it gets going... erm, it keeps going in a stable fashion. I made those things in... aspect ratios of integers: 4 to 1, 3 to 1, and so on. I also have at home, 5 to 1, but I regret doing this because you see I really would like to know what happens when the aspect ratio is not an integer. And instead of listing the vertices consecutively like this in order what you see is that you see those dots skipping and going every second vertex like this, like a... [metal object spinning on a wooden surface] In the beginning it's a mess There's a... [audio fades]
B1 中級 奇妙な回転管 - Numberphile (Strange Spinning Tubes - Numberphile) 3 0 林宜悉 に公開 2021 年 01 月 14 日 シェア シェア 保存 報告 動画の中の単語