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  • I made two bizzare objects, they were both made from two cups of, uh, esspresso

  • one of them glued like this and the other one glued like this

  • so this one I can balance very carefully like this

  • on a support

  • whereas the other one I have really no chance of balancing

  • it's very unstable

  • so you might have the impression that this one

  • is the stable one and this one is the unstable one

  • because this bulges out and

  • and hugs whatever supported standing on

  • with this is well, going to tip.

  • Let's do the dynamic

  • version of the experiment so this is an

  • inclined plane and we'll try to let this roll down the slope.

  • here we go. And it rolls down

  • and rolls all the way down very happily.

  • next, this one. let's try to make it roll down the slope and

  • when I do that, oops, it went off the rails

  • oops, however hard I tried to center it, it really

  • doesn't want to go all the way down it's

  • very very difficult

  • almost made it, but not quite it doesn't

  • why is this such a good roller all the way down

  • and why is this such an unstable roller?

  • the difference is very interesting and

  • it has to do with one of the most

  • fundamental if simplest ideas in science

  • and engineering which really is at the

  • heart of today's technology called

  • the stability and instability. let's first look at

  • this one.

  • well as long as I can center this thing

  • exactly in the middle

  • well thats fine goes straight down, rolls straight down, there

  • is no problem there. But, you know in nature

  • you always make mistakes so there is

  • some error initial error. let's see if it has

  • erred slightly to the left like this so

  • it's now deviated slightly to the left.

  • in that case you see

  • this thing is going to be supported on

  • the left the rail here and on the right

  • rail here that is on the left

  • it's rolling on effectively a wheel that

  • has a large radius over here it's

  • effectively rolling on the wheel that has

  • small radius so you have a large wheel

  • on the side and small wheel on this side

  • and you know what they're going to do

  • going to start deviating rightward in

  • other words in such a way as to cancel

  • the initial error to the left so this is

  • a restoring mechanism or an automatic

  • correction mechanism whenever there is a

  • slight deviation to the left, the system

  • automatically corrects itself by going

  • right and if it deviates a little bit to the right

  • into the same thing the other way around

  • you start automatically going to the

  • left so there's an automatic correction

  • mechanism and that's why it's so stable just wrong

  • (Brady) Why, professor? Why can't the big one roll along slowly

  • and what's the physical thing that happens?

  • Aha, because if you have a

  • small wheel on the side a large wheel on

  • the side and if they roll both at the

  • same time you see the per same amount of

  • angular roll the smaller one will advance a small

  • distance and large one advance a large

  • distance so you see this one goes a

  • little but it this one goes large so it starts curving in this

  • direction and if you have a small one on

  • this side and large one on this side, it's the other way and starts curving

  • on this side, so whichever side is smaller that side is

  • the side toward which the whole thing

  • starts reeling

  • (Brady) it's because they're connected, isn't it

  • Yeah they are connected. That's right.

  • ok so that's why this thing is such a

  • stable roller because whenever there is

  • an initial error which is inevitable

  • automatically is correcting its course

  • so why it's wobbling around the center

  • it can still go down in contrast this

  • is a disaster because again you know if

  • you can if you can at all center it in the

  • middle there is no problem to roll straight down but

  • that's only the ideal scenario in

  • practice there's always initial error let's

  • as before assume that there is a slight error to

  • the left but then you see the left part

  • of this object is supported here on the

  • rail so it's rolling on a small wheel

  • whereas the right part is supported here

  • which is rolling on a large wheel so

  • whenever it's deviating slightly to the

  • left you have a small wheel on the left

  • and a large wheel on the right and you know what those

  • wheels will do they'll start moving

  • towards left so if there is an initial left error

  • there will be a tendency to start going even

  • further left so the error grows and grows

  • and goes out of control

  • and eventually of course the whole thing

  • moves off the rails so there is a mechanism

  • which is opposite to what we had before

  • that is, whenever there is a small initial error which is

  • inevitable there is some mechanism

  • a devilish mechanism that makes the error grow and

  • grow and that is called an instability

  • and in general in abstract terms

  • whenever there is a system that we want

  • to control or something that nature

  • gives and you know there are some

  • initial errors in the positioning well if

  • the error has a tendency to diminish

  • there's an automatic correction mechanism

  • that system is stable and if the error

  • tends to grow and goes out of control

  • that's the unstable mechanism and that is why this can

  • hardly go down the slope whereas this is very very

  • stable thing that keeps going down

  • (Brady) Professor, anyone watching is immediately right now

  • is going to start thinking about trains maybe. How does a train wheel work?

  • So, if a train is going straight there is no problem

  • but suppose it wants to round the corner

  • and it has the wheel that say on this

  • side and the outside wheel on this side

  • and they are connected. As it goes around the corner

  • the distance that this outer wheel has to travel is

  • actually longer than the distance this inner wheel

  • has to travel so there's a

  • problem because they are rotating at the

  • same rate

  • one of them must skid and that's a very

  • bad the situation so they cannot both roll

  • because the distances are different

  • so how do you correct for this effect as

  • long as the wheels are connected it

  • seems to be completely insurmountable as a

  • difficulty but today is a really nice

  • mechanism here is what you should do: effectively

  • you should put this kind of design on

  • the pair of rails you see so when it wants

  • to go around the corner the centrifugal

  • force make this go slightly outside but

  • that's very good news because as long as

  • it's on the outside that part of the wheel

  • becomes effectively larger and this part effectively smaller so the

  • distances that they travel are ultimately

  • adjusted and indeed the outer wheel does travel

  • the longer distance and the inner wheel a short distance and

  • because the wheels radii are different

  • and of course in real train

  • wheels it doesn't look

  • like that but actually the designs kind

  • of look like this cross-section and this

  • is sitting on the rail like this and

  • this is sitting on the rail like this

  • so as the wobbles left and right the

  • effective wheel size varies and that

  • automatically adjust for the unstable

  • rounding of corners its radial on the

  • full thing. And by the way if you designed

  • the wheels in this fashion that would be

  • really a disaster train that

  • immediately come off the rails just as this

  • couldn't go down the slope.

  • I wonder if I can actually make it

  • go down all the way. AHAHA -- the first time.

  • [laughing]

  • You've just undone

  • all that math there

  • one way to do it, is from my point of view, the right hand and pull it

  • over and then do this which produces

  • a perpendicular knot.

I made two bizzare objects, they were both made from two cups of, uh, esspresso

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安定したローラー - Numberphile (Stable Rollers - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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