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]