The hosts walk through how Newton's laws generate solvable equations of motion in simple systems: a mass on a spring obeying Hooke's law (F = -kx) produces oscillatory motion, adding a damping term produces the damped harmonic oscillator, and planetary orbits follow an inverse-square force law that can also be solved. They note that an equation of motion can be written down even when it can't be solved, as with the three-body problem. This sets up the core difficulty with fluids: a fluid contains an enormous number of self-interacting particles, so tracking Newton's laws molecule by molecule is intractable, and a new mathematical language (fields rather than individual trajectories) is needed to describe fluid motion.
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Critical damping, the specific damping value that kills oscillation fastest, is cited as the physics behind car suspension design.
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The three-body problem is offered as an example where the equation of motion can be set up from Newton's laws but has no known solution.
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A dust grain dropped into a moving river is used to show that individual fluid particles still obey Newton's laws even though the whole fluid cannot be tracked that way, with a cup of water estimated at around 10^23 molecules.
2,237 words · auto-generated from the episode video
10:20So these are the Navier Stokes equations. There's two of them. And our goal is to understand these equations. These equations are an equation of motion that describes fluids. How fluids behave. Um what are the forces that they exert? If I know what the fluid is doing right now, can I predict what the fluid will be doing at some later time? These are the equations that allow that to happen. And when I say they're an equation of motion, that's what I mean. They're like they're telling you how the the equation how the fluid is going to move around in some space. Now in classical systems the starting point for all equations of motion is always Newton's laws. Okay,
11:02Newton's first law is an object at rest stays at rest and an object in motion will keep going unless it's acted on by another force. So you know if you're in empty space and you just have a satellite that's moving in one direction, it's going to keep moving. What happens when there's a force? Well, when there's a force, there is an acceleration. It's not a velocity. It's not like the you need a force to keep it going. If it's going, it'll just keep going. If there's a force, it'll change how it's moving. So, that's called an acceleration. It's a change in velocity. And the relationship between the amount of force that I put on an object and how fast it's going to accelerate is
11:42dependent on the mass. It's a mass is a kind of inertia. It's like how much how much resistance to um a force do I have is literally what mass is. And there there's a pretty common demonstration that you see in physics classrooms in high school where you've got two rubber bands. They're the same rubber band, but one of them is attached to a block of wood and the the other one is attached to two of the same blocks of wood. And you can see that the the single block of wood accelerates a lot faster than the one with two blocks of wood. That's because the the two blocks of wood has twice the mass. So, it's going to have half the acceleration. There's Newton's second law. And Newton's third law is something about conservation of momentum, which is like, you know, if I'm pushing on an object, then that
12:23object pushes back on me. So those are the three laws. And if we want to understand how something moves around, all we have to do is figure out what is the nature of the forces on this object. And if we're good enough at the math, we can figure out how this object is going to move forever. This is partly what enables us, for example, to land on the moon, everything or at Mars, all of these kinds of obviously much more surface area than that, but this is why we can make predictions over such long periods of time. >> Yeah. Really, in order to get to to the moon, you just need Newton's laws. I guess you need electricity to like work the capsule or whatever, but like you know, if there was some analog version where like you had fluids that did
13:05computation or something, you could land on the moon with just Newton's laws. So let's take for example the mass on a spring. I want to go through a singular example of a very simple problem where I can have a new a mo um an equation of motion derived from Newton's laws and I can then figure out what that thing is going to do forever. Okay, a common example is a mass on a spring. Okay, a spring is a substance that obeys Hook's law. Meaning if I pull on the thing, it's going to resist and it's going to pull back. And how much it's going to pull back is dependent on how much I'm stretching it. If I stretch it twice as far, the pull back the force backwards is twice as much. That's called Hook's law. And you usually see it as um the
13:47force is equal to kx. K is how stiff the spring is. X is how far I've pulled. And so, you know, the the larger I pull, the more it pulls back. And the negative is because it's pulling in the opposite direction of how I'm distorting the spring. Okay. >> So, that's Hook's law. F= Kx. Now force according to Newton's second law is mass times acceleration. Yes. Right. And the acceleration is really a second derivative of the position of the x. Right? The acceleration is a change in velocity and the ch velocity is a change in position with respect to time. So it's really a double change which is why I have mass times a second derivative in
14:28position is equal to kx. Notice this is an equation of motion. I've got a single variable that I need to solve for which is X, the position. And the key thing is the force is dependent on the position, >> right? But the force is related to how fast the the position accelerates. So now I've got an equation that is in a single variable X and it's a differential equation with respect to time. This is something I can solve. Okay? And that's what most people do. You can solve this thing. It's a differential equation and if you solve it, you get a very nice solution. You get a very nice solution that is an
15:08oscillation. And I think we've got a little video of that. This is what happens. You take your equation of motion and you say, what is something where if I take a second derivative, I take a derivative once and I take the derivative again, I get the negative of the thing back. >> And that's cosine and s. The derivative of cosine is negative sign. The derivative of s is cosine. So I'd go from cosine to negative sign to negative cosine. Well, that's just the negative of what I started with. And there I have an oscilly motion. So this is the paradigm >> of how we do physics in general. We have Newton's laws. We analyze our system and we figure out what is the differential equation that the system obeys. And then
15:48we solve this. And if we're lucky enough, the differential equation is simple enough that the mathematicians have figured out how to solve this differential equation. And the physicist can then borrow that intuition and boom, you've got um a solution that tells you for any time I can plug in the time and I can tell you where the mass is going to be. >> And it it's a way in which we can now simulate and create replication such that we can predict how a system >> will act under certain circumstances. >> Yes. Exactly. And you can think you can make things even more complicated. For example, you can take the same spring and you can add friction. In this case, there's like some kind of damping force. And what that means is that in your
16:30differential equation, you've got a term that's related to velocity because like the faster I move through some like let's let's say I dunk this thing in honey, right? The faster I move through this thing, the more the honey is going to resist my motion. So that is dependent on the first derivative of of position, just the velocity. And if I add that term, then I get a new solution, which is the damped harmonic oscillator. And that's the red that you see over there. So in the undamped case, you've got an oscillation that just bounces over and over and over again. And then in the damped case, you've got an oscillation that dies out very quickly. For those of you who are listening to us in the car, um you can thank damped harmonic motion for your
17:10suspension because there's a particular value of the damping that makes all these oscillations go to zero very fast. It's called critical damping. And a lot of times car manufacturers want their springs, the suspension in their springs to have critically damped so that you know you get bounced on a on a what are those called? The holes, the potholes. You get you bounce on a pothole but you don't like bounce up and down forever. You just immediately bounce and then you go right back and there's like no bouncing whatsoever. And that's because the spring is critically damped. That has to do with how how much damping is happening. My point is you can make the equations of motion a bit more complicated and still be able to solve stuff. >> This is the visual I think of is seeing
17:51those meme videos of those giant trucks with giant suspension that's visible going over, you know, this humped dirt, you know, dirt track and they're chilling. >> Yeah. Yeah. Yeah. There's all these like Chinese propaganda videos, too, where they show like the luxury cars of China versus the luxury cars of Germany. And it's like the luxury cars of China are just like it's they're like on a cloud and then the ones like the BMWs like it's like okay well is that a new BMW or like you know you never know with those propaganda videos or is it just straight up AI. Anyways for every case the logic has been pretty simple. Okay the point is we start from Newton's laws. We understand what the forces are and then we try to set up the equations of motion. This works for springs. This
18:33works for planets. >> For a planet the force is proportional to 1 / r 2. how far away my thing is and it's pointing directly at the central the the central body and again that's a force that is a function of position and now I can solve for what the position of the planet is because I can set up a differential equation that does so okay I can't I can set up a differential equation for a three-body problem but crucially I cannot solve that so I can set up the equation of motion but I don't have a solution for it so there's caveats right okay so now let's try to understand fluids I see the setup we're going with here because we're saying we can understand the world around us and make predictions
19:14using math. >> Yeah. And Newton's laws. >> And Newton's laws for certain physical systems. >> Yes. >> And we just we just defined examples of those physical systems in which we can do so. >> Yes. >> Now I think where we're going is fluids maybe have some unique aspects. >> Yes. >> That make that different. >> Yes. It makes it different and difficult. >> Okay. >> It's different in the it's so so first of all it's same same a little bit because each part of the fluid still obeys Newton's laws. If I were to look at a single particle of of fluid, let's say I drop a dust grain into a moving river, right? That dust grain is going to experience forces because water
19:56molecules are pushing on the thing >> and those forces are going to make it accelerate. >> Okay? So Newton's law is still a work >> 100%. >> Right? The problem though is that Newton's laws are tenable and they're like tractable if it's a single thing that's like moving around in a fluid. For example, even in this cup, right, of of water, I have 10 the three molec No, no, no, it's not 10 the 3. That's only a thousand 10 the 23 molecules of water moving around. Okay. So, what do you expect me to apply Newton's laws to every single molecule that's in this water? It doesn't make any sense. That's sure that's one way of
20:36doing it. If I had infinite compute, which I don't, >> I'd like to I'd like to understand this thing at a level of granularity where I can make predictions, but maybe I don't have to worry about every single molecule and what it is doing. >> Well, you don't want to have to simulate everything because the system is so large as compared to what we were doing before. >> Yeah. And the system is large and it's self interacting is the other point, right? the water molecules are interacting with other water molecules and like it's just it's just a whole mess, right? So, it's not a new kind of physics. It's the same physics, but it's being applied everywhere in this system. >> Everywhere. All at once. >> And all at once. Exactly. And so, we need a new language. Okay.
21:17>> And perhaps here's the point. What do we want to actually describe about this fluid? We don't want to describe the individual molecules. As I just said, it would be crazy to try and describe the position of every single molecule as a function of time like we could with that mass on a spring. Right. Right. Um that