- Look at this arrangement
of springs and rope. These two springs are joined together by this short bit of blue rope and there's a weight on the bottom. There's also a red rope and a green rope. The red and green ropes are slack. They're not taking any of the weight. So the question is what will happen when I cut the blue rope? I mean, look, you could think
about the forces and energy and try to work out if the
weight will go up or down. Or you could just consider the fact that I'm making a video about it. So it's probably the
counter-intuitive answer. Like, this blue rope is in tension. So when I cut it, I will
release that tension. So surely the weight will go down. Let's see what happens. This video is sponsored by Fasthosts. Viewers in the UK have a
chance to win the ultimate work from home bundle worth 5,000 pounds, if you can answer my techy test question. The link is in the description. More about that at the end of the video. (weight thumps) Isn't that crazy? The weight goes up. Why does it do that? Well, when you add a weight to
a spring, the spring extends. And of course, if the
spring were twice as long, it would extend twice as far like this. But actually if you
consider the initial setup before the blue rope was cut, that's exactly what we have. The blue and green ropes are slack. They're not doing anything. They might as well not be there. And so we have a spring
that is twice as long, plus this little bit. So these two springs together
will extend twice as far as one spring would on its own. You could say that these
springs are in series, but what would happen if you
put the springs in parallel? Okay, so start again with
a mass on a single spring and then add a second spring like this. Now the weight is shared
between the two springs. This spring takes half the weight. This spring takes half the weight. If you half the weight on a spring, it should extend half as far. And look, that's what we see. Actually, it's not exactly what we see because my springs aren't ideal springs. They don't follow Hooke's Law. But anyway what's interesting
is springs in parallel is what we have after
the blue rope is cut. Look, two springs in parallel. It's just that they've both
got this extension rope on them as well. So the reason the weight moves
up when you cut the blue rope is because you're switching
from two springs in series to two springs in parallel. But then I was like, "Hold on, hold on. If I were to slowly extend the blue rope, instead of cutting it, surely
the weight would go down." Like my brain wasn't happy, still. So look, here's the set up here. I can slowly release the blue rope and watch what happens as I do that. Actually the weight does go down. But it only goes down until the red and green ropes become taut. And then as I release more
and more of the blue rope, the weight goes up again. It's at that point when the red
and green ropes become taut, that you begin to transition
from one regime to the other, from springs in series
to springs in parallel. What's really cool is
that this arrangement of springs and rope is actually an analogy for a traffic-based paradox
called Braess's Paradox. Imagine we've got two towns, A and B, with a network of roads between them. The red and green roads here
are really wide and quite long. The silver roads are really
narrow and quite short. The blue road is really
wide and really short. The red, green and blue roads are so wide, they never get congested. It doesn't matter how many
cars you put on those roads, it always takes 11 minutes
to travel down the green road and 11 minutes to travel down the red road and one minute to travel
down the blue road. The silver roads are so narrow, the travel time depends on how
many cars are on the roads, because they get congested. For simplicity, let's say that the silver roads take
one minute to travel down for every 100 cars that
are on those roads. Let's imagine there are 800
cars trying to get from A to B and they all take this route. So it's eight minutes
here, one minute here and eight minutes here. That's 17 minutes in total. But is that what drivers
would actually do? Would they all take this same route? Or would some drivers
switch to a different route? Well, a driver could switch
to this route or this route. But both those routes take 19 minutes, which is more than 17 minutes. So no individual driver
would take that route. What we have is an equilibrium. All drivers taking this
same route is an equilibrium because it doesn't make sense
for any individual driver to change route. How does this relate to
the springs and ropes? Well, the weight is like the
number of cars on the road and how far down the weight
goes is like the travel time. The green, blue, and red roads are like the green, blue and red ropes. It doesn't matter how much
weight you put on them. They don't get any longer. Just like it doesn't matter
how many cars you put on those roads, the travel time doesn't
go up on those roads. The silver roads are like the springs. The more weight you
balance to the springs, the lower the weight goes. Just like the more cars you
add to the silver roads, the longer the travel
time gets on those roads. In this setup, the green
and red ropes are slack. They're not being used in the same way that the green and red
roads aren't being used. So what happens when
you cut the blue road? Well, there are now two
possible routes between A and B. And assuming that drivers can have up-to-date
information on congestion, maybe because they've got
Google Maps or whatever, then a new equilibrium will be reached, where half the cars take one route and half the cars take the other route. That means there are now 400 cars on each of those silver roads. So the total travel time for any one car is now four plus 11 equals 15 minutes. And that's less than the 17 minutes we had before the blue road was cut. And that's Braess's Paradox. By removing a short fast
road from a network, you might actually decrease travel time. It's almost like a multi-player version of the prisoner's dilemma. And this isn't just an academic curiosity. It's happened in real life. We know that in Seoul and
Stuttgart it's happened where the removal of a road has
actually caused travel times to go down without the number of cars on the road going down. But surely if you added the
blue road back in at this point, drivers would choose to
continue using the better, faster outer routes. But with everyone using
those outer routes, look how attractive the blue route is. Four minutes plus one
minute plus four minutes equals nine minutes, compared to the 15 minutes
of the outer routes. So it makes sense for
any one individual driver to defect to the blue route. And unfortunately,
every driver that defect makes the blue route a little bit worse, but it also makes the outer
routes a little bit worse as well. In fact, the blue route
is always preferable. So all drivers will
switch to the blue route and it will be worse. You know, if all the drivers could agree to not use the blue road
as if it wasn't there, then we could keep it
for people who want to go from here to here. Wouldn't that be nice? Unfortunately, humans
don't form hive minds, so it can't happen. Except maybe one day, we
will solve the problem with a hive mind. Like if self-driving cars take
over the majority of driving and these self-driving
cars can act cooperatively, then maybe one day we can
experience this utopia of optimization. It's never going to happen, is it? Like even if all the car
companies could agree to write software that acts
cooperatively with each other, there's always going to
be a human in the car, ready to override the software and say, "Do you know what? Take me the fastest route this time." Anyway, thanks to George Maveron Nicholas for the idea for this video. I appreciate it. This video is sponsored by web
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