字幕表 動画を再生する 英語字幕をプリント 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.