คำบรรยาย (622)
0:00This little mechanism shrinks when you
0:02try to stretch it. You try to pull it
0:04apart and all of a sudden it pulls back
0:06on you.
That's so weird, right?
Here it
0:10is under controlled conditions. There's
0:11a cup hanging from the mechanism. But
0:14now look what happens as you add water
0:16to it. All of a sudden, the cup shoots
0:18up. The amount it shoots up is tiny. But
0:21the physics behind it is so
0:22counterintuitive, nobody thought it was
0:25possible.
It feels like it violates
0:26physics. That's why it's fun. The
0:29paradox that controls this mechanism
0:30governs everything from mechanical
0:32systems to food chains, from traffic
0:35jams to power grids. And to understand
0:37it, you just need to ask a simple
0:39question. What will happen to this
0:42weight if you cut the green rope?
Where
0:44is this going to end up? Is it going to
0:45go up? Is it going to going to go down?
0:47Or is it going to stay the same?
0:50Can I touch? Yeah. Yeah. Yeah, you can
0:52try it.
Nothing.
You don't think
0:53anything's going to happen?
In the same
0:54place. Uh, weight is going to be right
0:57over there.
Like it's going to fly off
0:59or what?
No, not not too much, but
1:01probably it's going to go to this way.
1:04If you cut the green rope, this is going
1:05to come down.
It will go down.
It will
1:07go down.
It drops. It'll fall down.
The
1:11first thing that occurs to me is as soon
1:13as you cut that, the weight is going to
1:14drop. I imagine the weight ends up lower
1:16than it started.
Here's a closer look at
1:18the setup. You have a spring hanging
1:20from a hookup here. And then via this
1:22green rope, it's connected to another
1:24spring that's carrying this weight below
1:26them. There are two extra ropes here as
1:29well. So the red one and the black one
1:31are slack. They're not under any tension
1:34whatsoever. So they're not actually
1:36carrying any weight. What's going to
1:38happen if you cut the green rope?
You
1:40can pause the video here and try to
1:42figure it out for yourself.
1:48Wait, what?
Here it is in slow motion.
1:52So, even though the ropes on the side
1:54were slack and we cut the only rope in
1:56tension, the weight somehow went up.
2:00Okay, so if you're unconvinced that
2:02cutting the green rope actually makes
2:04the weight go up, here's a huge version
2:05of the experiment. So, the black and red
2:07ropes are still very much slack and it's
2:09just held together by this tiny piece of
2:12green rope here. So, let's see what
2:14happens now. Okay, you ready?
2:18Okay. Three, two, one.
2:24That was actually insane.
That was
2:26pretty pretty good, right? I
still don't
2:27believe it looking at it.
Okay, so why
2:30does this actually happen?
Cuz the
2:32springs are contracting back and just
2:34pulling it together.
Maybe the tension
2:36in the, you know, this part is changed.
2:39It releases the tension in the springs
2:41and only goes to the length of the
2:47Look at what happens when you remove the
2:48slack side ropes from the initial setup.
2:51You're left with this a mass hanging
2:53from a spring hanging from another
2:55spring. So these springs are connected
2:57in series. Obviously when you hang a
3:00weight from one spring, it extends just
3:02like you'd expect. And the amount it
3:04extends by, call it X, is proportional
3:07to the force exerted by the weight.
3:08That's Hook's law. But if you add
3:11another spring in between in series, now
3:13both springs extend roughly the same
3:15amount, X, because both springs feel the
3:18same force of the weight pulling from
3:19below. So in the case of ideal massless
3:22springs, you would end up with exactly
3:252x of displacement. Now there's another
3:28way to connect these two springs to the
3:29weight, and that is in parallel. This
3:32way, both springs are independently
3:33connected to the hook above and to the
3:36weight. So, each spring is only carrying
3:38half the weight of the mass below, which
3:41is why both springs extend only half as
3:43far or x over two. If you look at the
3:46setup right after the green rope is cut,
3:48you'll notice that this is actually
3:50exactly how the springs are laid out.
3:53So, the red rope is connecting the
3:54bottom spring directly to the hook
3:56above, and the black rope is connecting
3:58the top spring to the weight. So these
4:00springs are in parallel.
4:03So by cutting the green rope, you're
4:05actually forcing the springs to go from
4:06a series to a parallel. And that change
4:09is what causes the contraction to
4:10happen. When you cut the rope, each
4:12spring only extends by about half as far
4:15as before, which is why you can add so
4:17much slack on these black and red ropes
4:19to give the impression that the weight
4:21is going to fall down.
When you cut the
4:22rope, you go from series to parallel and
4:24that pushes you up. The slack ropes,
4:26that's where I get to cheat. And that
4:27that's that's the misleading bit, you
4:31the key to getting this paradox right
4:33comes down to the length of the slack
4:34ropes. Each one has to be longer than
4:36the length of one of the springs in
4:38series plus the green rope. That's what
4:40adds the slack. But they also can't be
4:43much longer than that because too much
4:44slack will nullify the contraction you
4:46get between series and parallel and the
4:49weight will still fall.
4:51Now, you might think that this paradox
4:52only really works with the springs in
4:54this demo, but the first time it was
4:56discovered was actually because of its
5:00In April of 1990, New York was getting
5:02ready for its 20th annual Earth Day. It
5:05was going to be Manhattan's biggest
5:07celebration of environmentalism to date.
5:09Stop the war against the Earth. On the
5:12day, Central Park was turned into a
5:14massive festival ground with almost a
5:16million people pouring in to see a
5:18stacked lineup of performers, including
5:20Hallen Oats and the B-52s.
5:24But the boldest stunt of the day was to
5:26ban traffic on some of New York's most
5:28important streets, including 42nd
5:31Street, one of the busiest streets in
5:32Manhattan. It stretches from river to
5:35river connecting Time Square to Grand
5:37Central Station and it's almost always
5:40jammed with slowmoving traffic. The only
5:43thing that's an hour from 42nd Street is
5:4543rd Street.
And maybe not surprisingly,
5:48people were really against this idea,
5:50insisting that just a 6-hour closure of
5:5242nd Street would mean doomsday. As the
5:55commissioner of New York's Department of
5:57Transportation put it, you didn't need
5:58to be a rocket scientist or have a
6:00sophisticated computer queuing model to
6:02see that this could have been a major
6:04problem. But the city went ahead with it
6:07anyway, and no cars were allowed on 42nd
6:12Now, to everyone's surprise, the traffic
6:15in the surrounding area actually got
6:17better. The number of cars was reduced
6:20by 20% with bystanders claiming the
6:23whole area was a ghost town compared to
6:25the way it normally is. But one man
6:28wasn't surprised by this result. In
6:29fact, he predicted it over 20 years
6:34His name was Dietrich Braze, a German
6:36mathematician. And back in 1968, Braze
6:40was studying road networks. As part of
6:42his research, he imagined a scenario
6:44where drivers from one side of a
6:46fictional town were trying to get to the
6:48other. But there were only two possible
6:50routes the drivers could take. Route one
6:52starts with a wide highway that takes
6:54you halfway across town. The road is so
6:57wide that regardless of how many cars
6:59are on it, this part of the trip always
7:01takes 25 minutes. The second half of
7:03this route turns into a narrow city
7:05street, and the time to drive through
7:07this street depends on how many cars are
7:09on it. For every 100 cars on the street,
7:12the time to pass through it takes an
7:14additional minute. So 100 cars will take
7:161 minute, 200 cars will take 2 minutes,
7:19and so on. The second route through town
7:22starts with a similar narrow city street
7:24that depends on the number of cars. It
7:26takes you halfway across and then turns
7:28into another 25-minute highway stretch
7:30where the transit time doesn't depend on
7:32traffic. So which route would you take
7:34to get across town? Well, you can see
7:36that the routes are identical but
7:38flipped. So, it doesn't really matter.
7:41Since this is just a mathematical model,
7:43both will get you there at the same
7:44time. So, say there were 2,000 drivers
7:47trying to get across the city. Half of
7:49the cars would end up on the first route
7:51and half on the second route. And since
7:53there are now 1,000 cars going down each
7:55narrow city street, the travel time on
7:57these segments increases to 10 minutes.
8:00So, the total time on both routes is 10
8:02minutes for the narrow street plus 25
8:04minutes for the highway. A total of 35
8:06minutes regardless of route. But now,
8:09say the city decides to connect these
8:11two routes at the halfway point with a
8:13small piece of highway to give drivers
8:15more options. This piece only takes a
8:18minute to travel across. So, which roads
8:21would you use now to get across town?
8:23Well, as an individual driver, you
8:25should just go straight down. It will
8:27take you 10 minutes to get through the
8:28first city street, 1 minute on the new
8:30connecting road, and another 10 minutes
8:32for the second street. So your total
8:34journey time would now be only 21
8:36minutes compared to the 35 minutes for
8:38everyone else on routes 1 and two. Okay,
8:41great. So you minimize your own time and
8:43that's that, right? Well, not really.
8:46See, drivers like you are selfish and
8:48everyone wants the shortest possible
8:49travel time, which means everyone starts
8:52flooding the narrow city streets.
As
8:54drivers switch to this new shortcut, the
8:56narrow streets become more and more
8:58congested, making the route slower and
9:00slower. But this makes the original
9:03routes worse, too. Because the time to
9:05get through the street segments keeps
9:07increasing for every driver. So,
9:09everyone decides to switch to the
9:10shortcut. And now all 2,000 cars are
9:14driving down the city streets. Now, the
9:16time to traverse each city street jumps
9:18to 20 minutes. So, the total journey
9:21time for everyone increases to a
9:23whopping 41 minutes compared to the 35
9:26minutes we had before the new road was
9:28constructed. So traffic actually got
9:33To fix it, the drivers could simply go
9:36back to their original routes, right?
9:38Well, who's going to be the first to
9:40switch back? If any one driver goes back
9:42to Route 1 or two, their journey time
9:44will be the 25 minutes on the highway
9:46plus 20 minutes on the now congested
9:49city streets, or 45 minutes in total,
9:52which is even worse than the now
9:53congested streets. So, no single driver
9:56would ever want to go back to the
9:58original route. And because humans are
10:00humans, it's not like we could all just
10:02agree to ignore this new road. So, even
10:05though every driver was making a
10:07rational decision to try to minimize
10:09their own travel time by just using the
10:11city streets, collectively, this made
10:14the situation worse for everyone, and
10:16there's no way out. But if the city were
10:19to destroy this new connecting road,
10:22everyone's journey time would drop from
10:2441 minutes back to the original 35
10:26minutes on route 1 and two. So removing
10:29the road would actually make the traffic
10:32better. It's just like cutting the green
10:34rope from before. That's because both of
10:37these are examples of the same paradox.
10:40The springs are like the narrow city
10:42roads. The more weight or cars you add,
10:45the longer they get. And the ropes are
10:48like the highways. It doesn't matter how
10:49much weight is on them, they don't
10:51change. That is unless you don't know
10:53how to tie them properly.
10:56My god.
This is the paradox Dietrich
10:59Braze discovered in 1968. It's now known
11:03as Brazy's paradox, and it is the reason
11:06why New York traffic got better after
11:0842nd Street was closed on Earth Day.
11:10Now, sure, you'd be right to argue that
11:12the reason the traffic decreased on
11:14Earth Day in New York was simply because
11:16people decided to walk or cycle more
11:18that day, but it turns out
11:19mathematicians actually modeled the
11:21whole city in 2008. And they found 12
11:24roads that were redundant and could be
11:25cut to actually reduce traffic. And it's
11:28not only New York. The paradox showed up
11:31in Boston, London, Seoul. In fact, if
11:34you were to randomly add a new road to
11:36just about any city, you'd have an equal
11:39chance of making the traffic better as
11:42But there's nothing special about the
11:44flow of cars. Say instead you want to
11:46send electricity from one station to
11:48another. Well, now you're looking at the
11:50flow of electrons in a power grid. And
11:53just as before, you could try to improve
11:54the grid by increasing the capacity of
11:56existing lines or by adding new lines.
12:00But it turns out that this can actually
12:02destabilize the grid or even cause a
12:04blackout. And virtually any other
12:06network, anytime you're sending things
12:08from one place to another, it can fall
12:10prey to Brazy's paradox. Be it a food
12:13chain, blockchain, or even the internet.
12:16Adding elements to the network can make
12:19So less can actually be more. And that
12:21kind of got me thinking. It's the same
12:23with your data on the internet. The less
12:25of your private info is on the web, the
12:27better. See, I got an email from someone
12:29a couple of months ago suggesting that
12:30they can create tailored solutions to
12:32accelerate growth. Okay, I thought it
12:34was spam, so I didn't reply. Uh, and
12:36then I got a couple of emails and I felt
12:37bad and I thought, okay, I'd be nice and
12:39actually respond. I drafted up a little
12:41email, but once I hit send, suddenly my
12:44inbox flooded with spam emails.
12:50Grace and Jenny offering price lists for
12:53well, I don't really know what. Oh, and
12:55the best email actually suggested we
12:56should turn Veritassium's existing
12:58content into high performing YouTube
12:59videos to increase our audience and
13:02extend the brand's reach. That's a
13:04genius idea. But unfortunately, most of
13:06these emails are spam and I'd very much
13:07like them to stop. And I can with the
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13:21requests have been completed, saving me
13:2324 hours of time. Precious time I can
13:26spend tying springs to strings and
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14:15internet. Go get it off. What are you
14:18doing here?
Braz's paradox doesn't occur
14:21every time you modify a network. You
14:23need a very specific set of conditions
14:25for it to occur. But if you can make it
14:27work consistently, you get this.
So,
14:30we're here at the Amolf Institute where
14:32they actually figured out how to make
14:34something shrink when you pull it. So,
14:36let's go check it out. Is it here?
Yeah,
14:38I made all these samples.
14:45Yeah,
right. I guess you never expect
14:48things to yank back on you so
14:49unexpectedly cuz a rubber band, you're
14:52stretching it and you feel it wants to
14:54pull back on you more and more, but here
14:55it just you're never ready for
freezing
14:57suddenly, which it's a weird feeling.
14:59There's something almost like human
15:01about it where it starts tugging back on
15:03you when it gets bad.
What's special
15:05about this mechanism is that everything
15:07else around us works in the complete
15:09opposite way. Try to press one of your
15:12keyboard buttons slowly so that it
15:14steadily goes down into place. You can't
15:18do it. No matter how slowly you go,
15:20there is some point at which it just
15:22gives way and clicks through. The same
15:25happens if you try to stretch a bendy
15:27straw. You can pull on it as slowly as
15:29you like, but at some point the
15:31individual straw joints are going to
15:33expand suddenly. light switches,
15:36eyeglasses, grasshopper legs, these all
15:39have a failure point beyond which they
15:41give way and quickly snap into a
15:43different position. And this is called
15:45well snapping. Here it is mapped to a
15:48force displacement graph. As you'd
15:50expect, the more force you apply, the
15:52more the material bends or displaces.
15:55But eventually you reach a tipping point
15:57and beyond it, the force required to
15:59bend the material further actually
16:01drops. So if you apply a force higher
16:04than that peak, the displacement has to
16:06rush to the next corresponding value to
16:08match the force which is all the way on
16:10the other side of this dip. And as a
16:12result, you get a huge amount of
16:13displacement for that tiny increase in
16:16force. That's what creates the sudden
16:18snap.
It's very intuitive. I mean,
16:20you've all experienced it, at least if
16:22you're in the Netherlands, with your
16:23umbrella. There's a gust of wind
16:25underneath your umbrella. It pops to the
16:27other side. So you sort of go over a
16:29peak in energy or in force and it
16:32suddenly snaps and typically it becomes
16:34softer.
And this is the way all things
16:37snap. Everything used to fail in the
16:39direction of the applied force until
16:42this mechanism came about doing the
16:45exact opposite of snapping. Call it
16:47counter snapping. Imagine that gust of
16:50wind blows under your umbrella but
16:52instead of flipping out your umbrella
16:54suddenly closes in. Or you try to pull
16:57apart a straw and the joints suddenly
16:59contract.
The wind is pushing on my
17:01umbrella, but instead of it folding out,
17:03it
would push itself against the wind to
17:06close itself, right?
But it feels like
17:07it violates physics. It It's just so so
17:10counterintuitive that the displacement
17:12is in
in the other direction to the
17:14force.
That's why it's fun.
So, how does
17:17this thing work? Well, the mechanism
17:19itself is built out of three different
17:21components, and on their own, they all
17:23stretch normally when you try to pull
17:25them apart.
Individually, they behave
17:27like springs in the sense that they
17:29extend when you pull on them,
but then
17:30you combine them together, and then
17:32suddenly they shrink.
Exactly.
If you
17:34draw the system as a set of springs, it
17:37looks something like this. The long and
17:39lanky components represent the two
17:41springs on the sides, but they actually
17:43don't feel like springs at all. You can
17:46easily pull them apart until they
17:48suddenly get very stiff. Meanwhile,
17:50these pieces represent the top and
17:52bottom springs, and they feel a lot more
17:54springy. So, the more you pull them, the
17:56more they pull back. And finally, the
17:59central piece. It looks very similar to
18:01the previous one, but pulling it apart
18:03feels very snappy. In fact, you can even
18:09Put them all together and you get the
18:10mechanism. If you stretch it slowly,
18:13you'll see how tension builds up in the
18:15three middle pieces with the sides
18:16staying mostly relaxed. But if you keep
18:19stretching, the centerpiece will
18:21suddenly snap out and transfer most of
18:23its tension to the side springs. This
18:25causes the system to stiffen and shrink.
18:28If you now let go, the system resets, so
18:31the mechanism can flip between a set of
18:33springs in series to one in parallel.
18:37It's a reversible case of Brazy's
18:39paradox. The network is basically the
18:41same as the brush paradox.
So the the
18:44way they connected, the topology of the
18:45connection is exactly the same.
The
18:47force displacement graph you get for the
18:49mechanism is one that loops in on itself
18:52with two distinct curves. One for the
18:55system in series and the other for the
18:57system in parallel. And this leads to
18:59some pretty remarkable properties. If
19:02you slowly control the stretching force
19:04by adding water to a cup below the
19:05mechanism, the mechanism first slightly
19:08sags like you'd expect. But when you
19:11reach the tipping point at the end of
19:12this curve, the displacement has to
19:14quickly reduce to keep following the
19:16force along the graph. And this jump
19:19back is why the mechanism shrinks. Now
19:22you can also control displacement
19:23instead of force and measure how much
19:25force it takes to stretch the mechanism
19:27at any point. This time when you reach
19:30the tipping point, it's the force that
19:32has to follow displacement along the
19:34graph. So you get a sudden jump up in
19:36force showing that the material has
19:38stiffened.
Oh,
this little force jump is
19:41enough to make it slip out of your
19:42hands. If example,
I mean it even though
19:45you said it is a small jump, it's like
19:46this is the only thing that does this
19:49anywhere probably on Earth.
Uh yeah, as
19:53far as we know, the force jumping like
19:54this, it was not reported.
It is pretty
19:57insane. So what is counter snapping
20:00actually useful for? Well, notice that
20:02there is a force at which the series and
20:05parallel curves of the system overlap.
20:07Which means that at this force, the
20:09mechanism will actually be the same
20:11length in both states. So if you exert
20:14that exact force on the structure by for
20:16example hanging a weight from it, you
20:18can flip between the two states by
20:19giving the mechanism a little tug. And
20:22although that will change whether the
20:23springs are in series or parallel, it
20:26won't change how long the system
20:27actually is. So you can change the
20:30stiffness without changing the length.
20:33Now look at what happens if you give the
20:34mechanism a nudge in its original series
20:37state.
If you poke it, you it's
20:39basically a way to measure the natural
20:40frequency.
Oh yeah.
When the mechanism
20:42is in series, the natural frequency is
20:443.7 hertz. But if you switch to a
20:47parallel setup, the natural frequency
20:49increases to 6.4. 4 hertz.
We switch it
20:52and we look at the natural frequency.
20:54Now we can see it's much higher.
20:57What's unique here is that you're able
20:59to almost double the natural frequency
21:01of the material without changing its
21:03length.
I'm going to move the robot
21:05robotic arm very slightly just up and
21:08down.
Yeah. Okay.
And I put a frequency
21:10of 3.5 hertz. So it's close to the
21:13natural frequency of this system.
This
21:15is going to drive the structure into
21:17resonance. But once the vibrations get
21:19big enough, the mechanism is actually
21:21going to switch states on its own,
21:23change its natural frequency, and
21:25thereby reduce the vibrations.
Like it
21:28gets stronger and stronger, and then it
21:30just locks it out.
Yeah.
Yeah. That's
21:31cool.
The same happens in reverse. If
21:34you vibrate the robot hand at 6.4 hertz,
21:36the mechanism is quickly going to switch
21:38back to its original state and minimize
21:42It's interesting uh the way that you say
21:44it. We're moving the point at which
21:45resonance happens. And that's what stops
21:48excessive vibrations.
Yeah.
Interesting.
21:50Yeah.
Other snapping structures could
21:52also switch upon resonance. But the
21:54problem that once they switch, they're
21:56much more elongated, much more
21:57contracted. So they don't, let's say,
21:59they wouldn't provide the same function.
22:01You could use this effect to keep
22:02structures from vibrating or reaching
22:04resonance.
Could it be easier to install
22:07a system like this where you're actually
22:08moving the resonance instead of like a
22:10whole tune mass damper or or something
22:13like that?
Yeah, I think this solution
22:15is still very complex. It's very
22:17complicated design, but I think the
22:18principle could be used.
I know it's
22:20still super early, but I'm really
22:22excited to see like it pop up somewhere,
22:24you know, in a couple of years or
22:26decades.
It's more about the concept and
22:28showing what it can do. We're going to
22:30try to see if we can maybe make contra
22:32snapping also maybe with different type
22:34of variables.
It's going to be like a
22:36balloon that deflates when you inflate
22:39You increase the pressure and the volume
22:41would decrease.
Wait, really? That's
22:43crazy. That would be the equivalent, but
22:45we it's not there yet, but
22:48we'll see.
That's so cool.
In principle,
22:51it should be possible.
Yeah.
