Laws & Causes

[Music] Hey, Vsauce. Michael here. Do you want to see the most illegal thing I own? It's a penny from 2027. That's right, it is a piece of counterfeit US currency. Or is it? There are no 2027 pennies today, which means that this is a counterfeit of an original that doesn't exist yet.

I mean, sure, if you didn't look at the date, this could pass as real, but will it not truly be counterfeit until 2027? Well, what we do know is that its novelty has a lifespan. It's really cool to show people today. It's a penny from the future! But in 2027, it will become indistinguishable from a common penny and it will cease to be as interesting.

I cannot tell you where I got this, but the point is I can break the law. I am free—perhaps not free from punishment—but nonetheless, here we are. I cannot break the laws of physics; nothing can. Why? Today, I want to look at the lawful behavior of spinning.

Let's start with a physics classic. As a spinning ice skater pulls their body in closer, they spin faster and faster. It's really cool, but what is accelerating their rotation? What is pushing them around faster and faster? You can try this at home with a chair that spins and some really heavy books in both hands.

Now to get started spinning, I'm gonna have Jake give me a push. I don't need Jake; no, no, watch this. With him gone, I can nonetheless speed myself up by just pulling the books in. If I move the books out, I slow down. Somehow pulling the books in speeds me up.

Now, if you look up why this happens, you'll probably be told that it is because of the conservation of angular momentum. Oh, the conservation of angular momentum! I love demonstrations of it, and this is one of my favorites. This is Hoberman's sphere, a toy that collapses and expands. Doesn't it sort of remind you of me with the books far from my body and then the books close to my body? It behaves in much the same way.

I need to give it some angular momentum to start, but once I do, if I bring all of that mass towards the axis of rotation, it speeds up, and then it slows down, and then it speeds up, and then it slows down. Speeds up, slows down. I love this! Thank you, conservation of angular momentum!

But what are you? Well, if you look it up, this is what you will find for a particle in circular motion around some center of rotation. See, the angular momentum of the particle is defined as the product of the particle's mass, the particle's instantaneous velocity ( V ), and the particle's distance from the center of rotation ( R ): ( L = M V R ). This is angular momentum.

If you divided me and the chair and the books I'm holding into a bunch of tiny little particles and found the ( MV R ) for every one of those particles and summed them all up, you would have the angular momentum of the entire system of me, the chair, and the books. But notice that this is just a mathematical expression combining three different measures.

It's not a physical substance you could pull out of a particle and hold in your hand or put in a jar and study. It's kind of like, oh, I don't know, taking my weight and multiplying it by the number of countries on Earth and then multiplying it by the time.

However, unlike something like that, this concept is really useful because we have found that in our universe, over time, it is conserved. If the particle is pulled towards the center of rotation, its ( R ) value will go down. But angular momentum is conserved, so one of these other variables must go up.

Well, at low speeds, mass is essentially constant, so the only option is for the velocity of the particle to increase, for the particle to spin around faster than it was before. If the ( R ) value gets smaller, the velocity must get larger or else a law has been broken.

But how do atoms and molecules know to follow this law? I mean, is there some kind of physics police force in the universe bullying everything into compliance? How do all the atoms I pull in know to speed up? And why do they always obey our laws like embodied presences guiding matter around?

No, I have with me right now three things: a lamp, a nail, and a shadow. Gotcha! I actually have four things with me. The fourth is Ink's astonishing ruler, designed by Vsauce. If you're subscribed to the Curiosity Box, you already have one, or it's on its way to you.

And if you're not subscribed to the Curiosity Box, well, you're missing out on the fruits of my mind. I have always wanted a ruler like this, so I made one. And now I and you can have one. It is exactly one light nanosecond long, which means that this is the distance light travels in a vacuum during one billionth of a second.

If I hold my hand just that far away from my eye, I am seeing my hand as it was a billionth of a second in the past. Pretty cool! Now, this ruler allows you to measure lengths of things in light picoseconds, sound microseconds, Micro Everest's—that's one millionth of the height of Everest, beard fortnights, decimal inches, and of course, good old centimeters.

Now, if you asked me, "Hey Mike, why is the shadow of the nail four and a half centimeters long?" Well, I would say, "Who's Mike?" But that I would be helpful, and I would answer that the length of the shadow is caused by the height of the nail and the position of the light source.

And that's a pretty good explanation because if the nail were taller and/or if the light source were lower, the shadow would be longer. Or if the nail were shorter and/or the light source was higher, the shadow would be smaller. Now, these three measures—the nail's height, the light's position, and the shadow's length—are all related mathematically such that if you gave me only two of them, I could figure out the third. Always.

So I could declare this relationship to be a law. But that does not mean that the law causes them to all have the measure that they do. For example, if you asked me, "Hey, why is the nail six centimeters tall?" what causes the nail to have that height? Well, if I said, "Well, the nail's height is caused by the shadow's length and the light's position," you'd be like, "As if!" Alright?

I mean sure, we can figure out the nail's height by knowing about the shadow and the light, but that does not mean that they are causing the nail to have the height that it does. To know why the nail is six centimeters tall, we'd have to ask whoever manufactured it or whoever sets nail standards or whatever.

The point here is that we should not confuse relationships, laws, with causes. Explanations that involve the causal reasons for things are often better. But what is an explanation? Well, that's easy to answer. Explanations help us understand things.

But what does it mean to understand? Does it mean to not stand up fully, or does it mean to stand underneath and to look at from below? If I can understand something, can I also overstand something?

Well, as it turns out, the "under" in "understand" does not mean beneath or below. Instead, it means inside—to stand surrounded by, to be a part of. Now, this sense of "under" is quite common. For example, when you say, "Ah, well under those circumstances," you don't mean, "Well, when those circumstances are overhead and I'm under them." Instead, you mean, "If I find myself in those circumstances," and that is how "under" is used in "understand."

To understand something is to stand in the midst of it, to be part of it, and to be within it. And that is helpful to keep in mind. I cannot walk into something that's crumpled up and closed off. To truly understand something, it needs to be opened up for me, unfolded. And that is what an explanation should do.

The word "explain" literally means to flatten—"plain out." Now, if you'll excuse the analogy, if this is a story I want to read and it's all crumpled up, a physical law basically just tells me a summary of the story. It'll tell me how the story ends, what language and grammatical rules were used to write it. But an explanation will flatten the page out and allow me to see the details, an actual word, and the word that follows it, and so on, all the way through.

So let's unfold the phenomenon of me speeding up my rotation when I pull the books in and find a physical mechanical cause for that increased rotation. I have here two different circular motion paths around the center of rotation, which we'll call C.

Now, when a particle is traveling in this outer ring and is suddenly pulled in, it doesn't stop moving around and suddenly just go right in. Instead, it takes a bit of a curved path like this. Now, this is really interesting because in order for a particle to be in circular motion, it must have two things: it must have put the particle there, it must have a velocity that is tangential to the path, but it must also have a centripetal force—a center-seeking force that always points towards the center of rotation.

Now notice that these two arrows are at a right angle to each other. So, the centripetal force is not speeding up or slowing down the particle; it's just changing its direction constantly. Tada! Circular motion!

However, when the particle is pulled in, and it follows this curved path to this inner orbit, its instantaneous velocity is no longer perpendicular to the centripetal force. Its tangential velocity, tangent to that curve, is going to look something like this. But notice that now the centripetal force is no longer at a right angle.

The force that is changing that particle's direction is no longer at a right angle to its velocity. Instead, if this is the normal to its velocity, the centripetal force is pulling the particle forward, ahead in its rotation, speeding up its rotational velocity. So as you can see, I speed up when I pull the books in because when I accelerate the books towards myself, I'm not just accelerating them towards myself; I'm also accelerating them around.

Now likewise, if a particle were to be pushed out from an orbit like this into a larger orbit, it's not gonna, you know, it's not gonna do this. It's not gonna go, "Oh, time to go here!" in one straight line. No, it's gonna go like this. "Okay, I'll go to the larger orbit." But look now on this line: the particle's instantaneous velocity is something like this, and the centripetal force is not perpendicular to that line.

Let me give that a little ( V ). Oh yeah, nice labeling. Alright, so oh wait, now you can see it. Whoops! Alright, so this is the one we're talking about, though—this is the one that matters. Alright, so the normal to that velocity might be like this. So now notice that the centripetal force is actually working against the velocity of the particle, decelerating it. So, of course, it slows down when I move the books further out. My rotation slows down because moving them out is also decelerating them.

Now this is why it is harder to pull your arms in when you're spinning than when you're not, because you're not just moving your arms into yourself; you're also accelerating them in the direction of their rotation. Not only is the magnitude of angular momentum conserved, so is the direction.

But what is the direction of something's spin? What seems easy enough, right? Look at this. What's the direction of its spin? Clockwise? Or is it someone standing on the other side would say that it was traveling counterclockwise?

Oh, interestingly, this means that if you were to ask the face of a clock which way its hands spun, it would say counterclockwise. Of course, in order to unambiguously differentiate the direction of rotation, we need a tool that has no axis of symmetry in any of the three spatial dimensions of our universe. And luckily, I've got a tool just like that in here.

I've been keeping it fresh. And oh, because a sworn... oh there it is! It's my right hand! A human hand is asymmetrical in the X, Y, & Z directions. We've got thumb, no thumb, wrist, fingertips, palm, knuckles.

Now, because of this property, if we orient two pairs of opposite sides in a particular way, the one remaining pair will be locked. Now, my fingers can only really curl in the palm direction on the palm side. So if I have my fingertips in the direction of a wheel spin, curled such that an object on that spinning object travels from my wrist around to my fingertips, well then my thumb will only point in one direction no matter which side I'm on.

From behind, this is now going clockwise, but the right-hand rule still puts my thumb backwards. That's very cool! We call the direction that the right-hand thumb is pointing the direction of the wheel's angular momentum. But we could also use the left hand. We'd get opposite directions than the right hand gives us, but so long as we all agree to use the same hand, we will always be on the same page.

But as it turns out, of course, the right hand was chosen. And so determining the direction of angular momentum and velocity is called the use of the right-hand rule.

Okay, so now that we know how to find the direction of angular momentum, we could explore how that part of it is also conserved. I have here a very special wheel. Oh yeah, it's pretty crazy looking! That's because I wanted it to be really massive, so I wrap—I’m not sure it's really that safe, but oh man, is the effect good!

Alright, now let's say that I gotta, you have my right hand free. Let's say that I start spinning the wheel like this. The direction of its angular momentum can be found using the right-hand rule. I curl my fingers in the direction of rotation, so something on the wheel is going from my wrist to my fingertips, and I look at which way my thumb is pointing.

It was pointing up. Let's call up the positive direction. I don't know the magnitude of the angular momentum of the wheel right now; it's some number, but let's just call it ( L ) and since it's up, we'll call it positive. I could call it negative; all that matters is that I can differentiate one direction from the opposite.

Alright, wonderful! So notice that if I spun the wheel this way, I'd have to turn my right hand over to curl my fingers properly. Now my thumb is pointing down, so this would be negative. Let's call the magnitude of this wheel's angular momentum ( L ) positive.

If I got the wheel spinning so that it had an angular momentum of positive ( L ), if I then turn it upside down, its angular momentum will have changed direction. It would have swapped from being positive ( L ) to negative ( L ). However, if nothing else is interfering, that can't happen without angular momentum being conserved. So what? Plus negative ( L ) is positive ( L ).

Our original—well, positive ( 2L ) and that is what we will see happen if I spin this wheel like this such that it has an angular momentum of positive ( L ) and then I turn it upside down. Something else will have to have an angular momentum of positive ( 2L ). So like imagine my thumb being twice as long. What is that something else? Well, it'll be me in the chair. Is this very cool?

So let's make sure we know what's about to happen. I'm gonna spin the wheel this way, then I'm going to turn it upside down, and I should go that way. Whoa! Angular momentum is conserved! That's pretty cool.

But the conservation of angular momentum, either its magnitude or its direction, is not why turning the wheel upside down spun me this way. What caused me to move? Well, if you do this yourself, you will feel the handles of the wheel literally pushing you, putting a torque on your body.

It has everything to do with that concept I have covered in my spinning video and my video ending about Euler's disk. If the wheel isn't spinning and I push it down right here, it'll tilt like that. Right? Not very surprising, that's the kind of tilt we would expect.

But when the wheel is spinning, all the pieces of matter out here, well, they all have some velocity before I hit them. For example, when a piece of matter on the wheel is out here, it's moving backwards, and when I push it down, it doesn't suddenly stop moving backwards. The wheel doesn't stop!

Instead, the pieces I'm pushing are now going to go backwards as they already were and down. So they go backwards and down, and then they come back up, and the entire wheel is found to tilt this way. Look towards me, 90 degrees ahead from where I actually pushed it down.

A push down here causes the wheel to move like this. A push down here would cause the wheel to tilt like this. A push up here would cause the wheel to tilt like, let's see, 90 degrees ahead in the rotation like this, and this, and this is exactly what I'm doing when I turn the wheel upside down.

So when I put a torque on the wheel to turn it while it's spinning like this, the wheel actually turns this way. But I have two hands holding it. My left hand lines up getting pushed backwards and my right hand pulled forwards. That puts a torque on my body, which causes me being pushed back here and pulled forward here. It causes me to be turned, and so I turn this way—the way that we saw in the demo.

And that is the physical origin of the phenomenon we just witnessed. The key point here is that nothing you have seen happen today happened because of the conservation of angular momentum. Physical laws don't cause anything to happen, but plenty of things are demonstrations of their truth.

Now don't get me wrong, I love physical laws. They truly do represent a conquering of mystery when we find a way to project definite numbers onto reality and make predictions. That's amazing! But not coupled with causal mechanical physical explanations; their generality can come across like magic.

Now, magic is fun for sure, but congruences between mathematics and reality—oh, that's a sweet nectar. In mathematics, we get to make the rules, pick the axioms. We can even decide what kinds of reasoning are allowed or not, and after doing so, we get to play around and see what happens.

Mathematics is the plane of games; science is finding out what game you're playing. And we still don't know what game we have found ourselves in, but we should keep wondering and never be happy until we get answers that satisfy us.

Stay curious, and as always, thanks for watching! [Music]

I have some pretty exciting news to share with you. As of today, for the rest of the year, every single episode of Mind Field—all three seasons, all 24 episodes—will be free to view all around the world. I am thrilled! So go and check those out.

There's a very special—well, actually two very special things coming on this channel this month. Can't wait for you to see them! In the meantime, if you need more of me—and I always need more of me—check out Ding! I've done about 15 videos this year over there; it's a blast! We learn a lot over there, and things often get a little bit weird.

You know what? I guess maybe I don't know. Check it out and find out! And as always, thanks for watching!