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