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  • 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]

I brought a little tube

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奇妙な回転管 - Numberphile (Strange Spinning Tubes - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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