Check it out, so I've got this spout. It's like a sprinkler. It's got these tubes, and then water shoots out the end. So I want you to think about what direction the water is going to go when it comes out. OK, let's turn it on. Woo. Ah.

(Describer) Water comes out three thin pipes.

OK, so as we suspected, the water comes straight out. But now I'm going to spin the pipes, and I want you to think about what the path of the water is going to do. Is it going to stay going straight? Or is it going to curve and trail behind, or maybe curve and go ahead? Check it out.

(Describer) She rotates the sprinkler and streams of water curve behind.

So if you guessed that the water curves and trails behind, you were right. But now I'm going to replace these pipes with something a little bit different. OK, so these pipes curved back inward, so when we turn on the water, it's going to squirt toward the middle. Ah. Ah!

(Describer) They curl like candy canes.

So now the water's shooting straight back inward. And I'm going to spin it again clockwise. And I want you to think about this time which direction the water is going to curve. When I spin it, is the water going to trail behind the tube? Or is it going to curve ahead? Or is it going to this time keep going straight? Let's check it out.

(Describer) The water curves ahead.

Woo hoo hoo. That is so weird. It is so unintuitive. And if you guessed it would jump ahead, I don't know how you knew. But hopefully, by the end of this lesson, we're going to understand why this happens. Hello, I'm Dianna Cowern, and welcome to lesson 15 of Dianna's intro physics class, also known as AP Physics 1 review, also known as Physics by Dianna. Today's lesson is about the property that you feel inside of your head every time you spin around for a while, and then you get dizzy and fall down. Why do you get dizzy? Because the fluid in your head keeps spinning even after you stop spinning. Basically, objects that are spinning will just keep spinning. This is a property of objects. It's like the momentum of a fast car, but it's for spinning momentum. It's angular momentum. And that is today's lesson. Our theme for today is: "The Apocalypse," because if the Earth stopped spinning, that would be the end of our days.

(Describer) A skeleton wears a long wig.

[screaming]

Here's a fun question. If everyone on Earth gathered at the equator and then started running in the same direction, could we change how fast the Earth is spinning? In today's video, you're going to learn the tools to answer that question. OK, we are done being "humerus."

(Describer) She sets the skeleton aside.

[laughs]

As you know from the last lesson, the key to understanding rotation is to link it to linear motion concepts that we've used before. So, for example, a linear force makes this spherical cow accelerate, and the rotational analog is that a torque makes it change its rotational speed. You also remember the definition of torque, which is it's a force times the distance between where the force is applied and the point about which it will spin, which could be either a fixed axle, like a door hinge, or the center of mass of something, like the center of a Frisbee when you spin it. So we did all of that in the last lesson, but we didn't talk about the feeling of spinning or spinning something. It feels much easier to spin a top

(Describer) She does.

than it is to spin a bowling ball.

(Describer) She does.

And it would be even harder to spin a real cow. And the moon? Well, I personally don't have the strength to spin the moon. Some things are just hard to spin. So why is that? It turns out you already know.

(Describer) She pulls out paper and catches a marker.

What we're about to do right now is find a concept like F equals ma, but for rotation. So an equation with torque instead of force, angular acceleration instead of acceleration. And then we're going to figure out what this new term is. That's the rotational analog of our linear mass, and it's called the moment of inertia. But we're first going to derive this equation, then understand what it means, and then understand what that moment of inertia term is. And then from last time, rotational acceleration alpha is related to linear acceleration like this. Acceleration equals alpha linear acceleration rotational acceleration times r. And then we have torque equals force times r. Now I'm going to combine all these equations and simplify. I'm going to put ma into here. So my torque equals mar. And then put my a here with the alpha and r. So torque equals m alpha r times r. So torque equals mr squared times alpha. And now I have a new model. Torque equals mr squared alpha. And if you're just joining these lessons, a reminder that one of the main things that we do in physics is to come up with mathematical models to describe reality as closely as we can. So this looks a lot like F equals ma, especially if I put parentheses around this part.

(Describer) MR-squared.

And it is. This is the rotational F equals ma. Force, torque, acceleration, angular acceleration. But this part here, well, we had mass in F equals ma. And so over here, we've got mr squared. What is that? That is the moment of inertia. Yeah, I said it because that's what it's called. And remember, mass tells us how much force we need to apply to accelerate something linearly. So similarly, moment of inertia tells us how much torque we need to apply to get something to spin, to get something to accelerate angularly. And we're going to give it a new label. We're going to call this I, for a moment of inertia. So I is that mr squared in this case. And then check it out. Torque equals I alpha. Now we've got F equals ma and tau equals I alpha. But hold up. If I spin, say, a record, what is the m in this equation-- I equals mr squared-- and what is the r? And what I mean by that question is, if I swung an egg on a string, it's way more obvious. The m is the mass of the egg, and the r is the length of a string. And it turns out that this I we just found-- I equals mr squared-- is exactly the right moment of inertia for a particle or an object that has all of its mass in one spot. So like the egg on the string. But what about a spinning record? The mass is all spread out. And the record is really a collection of a gazillion tiny little pieces, little record molecules. So when I spin the record, all the tiny pieces will have the same angular velocity. And this confused me at first because the pieces on the outside are going faster linearly than the pieces in the middle. But recall angular velocity refers to revolutions per time, and all the pieces make one round trip, or 2 pi radians, around the circle in the same amount of time. But since the outer pieces have a larger r, a larger radius, then they have to travel farther to make the revolution, which means their acceleration is what? Bigger? Smaller? The same? It's bigger. So the outside pieces need more torque to spin up to angularly accelerate. So to get the moment of inertia for a specific object like this, we actually have to add up all the little moments of inertia of all the little pieces using their mass times the radius away from the center squared. And it's, unfortunately, much more complicated than just the total mass of this record times its radius because think about if the record were some weird shape. What would you use for radius? But, fortunately, people have done these calculations to find moment of inertia for common shapes like spheres, cylinders, donut-like rings. And you can, too, if you take calculus-based physics. Here's some moments of inertia for common shapes, and some intuition for you first. What shape do you think gives you the biggest moment of inertia? Meaning that it's the hardest to spin. It is a hoop because all of the mass is as far from the axis

(Describer) She catches something on the second try.

as you can get. That is, if you're spinning the hoop like this. And the moment of inertia for this is mr squared.

(Describer) She draws a roll of tape.

Now what's the smallest moment of inertia you can get? It's a sphere. Oh, can you pass me Jupiter? Thank you. That's why planets are all spherical, because nature is minimizing the energy to spin-- or pretty close to spherical. So the moment of inertia for a solid sphere of uniform density is 2/5 mr squared. The moment of inertia for a solid cylinder is 1/2 mr squared. So it's half of this one.

(Describer) The hoop.

But that is if you spin the cylinder about the middle. I'm going to grab another cylinder--nice. The moment of inertia if you spin the cylinder instead of about the middle about this middle is different.

(Describer) Lengthwise.

That moment of inertia becomes 1/4 mr squared plus 1/12 mL squared, where L is the length of the cylinder. Woo, confusing. All right.

(Describer) She drops a cardboard roll.

So many rotational analogs to our linear motion. Why stop now? So we did forces. We did acceleration. We did F equals ma. Let's do energy. We know the energy of a cow ball moving linearly.

(Describer) Her toy cow.

That's just 1/2 mv squared. What about a cow ball rotating? Oh, this is so easy using analogs. The rotational world equivalent of mass is moment of inertia. And equivalent of linear velocity is angular velocity omega. So the energy of an object rotating, the rotational energy, is 1/2 I omega squared. Done. I know. It's ridiculous how easy that was. But don't worry. It's going to get complicated again right now. I just gave you all the tools to answer this next question, which is a very tricky quiz question. So imagine that I have two cows-- one spherical cow, one box cow. Now I'm going to slide the box cow down a ramp. And for simplicity, let's imagine that we turned off friction. So imagine this is sliding down with no friction, and it just continues to speed up. And then with friction, the spherical cow is going to roll down the ramp.

(Describer) It rolls.

Now which cow is going to reach the bottom first? The rolling spherical cow, or the box cow sliding with no friction? Now a lot of people think that the rolling cow reaches the bottom first because rolling things get going really fast, right? But let's dig into it. And we're going to think about it using energy. If this spherical cow and the box cow have the same mass and they're starting at the same height, they're going to both have the same amount of gravitational potential energy at the top of the hill before they start sliding or rolling down. So they're both going to start out with the same potential energy-- mgh. Now when the box cow reaches the bottom of the hill, it is just sliding, it's not rotating; so it has converted all of its potential energy into linear kinetic energy. But when the spherical cow reaches the bottom of the hill, it is moving linearly and it is rotating. And we just saw rotation requires energy by 1/2 I omega squared. So some of the spherical cow's initial potential energy has to go into that rotational energy, and some of it will go into the linear motion of a spherical cow. So my energy here is going to be 1/2 I omega squared plus 1/2 mv squared. Box and sphere-- for box cow and spherical cow. So some of this spherical cow's initial potential energy is going to go into rotational energy. So at the bottom of the hill, the spherical cow will have less linear kinetic energy than the box cow. So the box cow reaches the bottom of the hill first simply because it doesn't have to put energy into spinning. This is one of the unintuitive quirks of physics. So now let's actually do some math because on a test, you would get this problem with some numbers. So we're going to assume that the spherical cow has a mass m

(Describer) She moves the paper.

and they're both at a height h up on the hill. Now the potential energy of the ball cow is mgh. We already wrote that down. And at the base of the hill, I'm going to assume all their energy gets converted into this type of energy. So this is their total. 1/2 I omega squared plus 1/2 mv squared. Now I'm going to plug in more stuff that I know. The moment of inertia of a solid sphere, we have it. It's 2/5 mr squared. So I'm going to plug that in here.

(Describer) ...for I.

And we know that omega, we can write that in terms of this velocity. So omega, the angular velocity, is always linear velocity. And I'm going to write vs because it's the spherical cow over r. So this is going to be vr squared. Now if we look, we've got an m in every single one of these terms. So I can cancel those m's, and we can rewrite. Now I've got gh equals 1/2 times 2/5 r squared times vs squared over r squared plus 1/2 vs squared. Now this r squared cancels with this r squared. 2 cancels with 2. And I'm getting 1/5 vs squared plus 1/2 vs squared. And I can simplify all of that to gh equals 7/10 vs squared. Now if I wanted to, I could solve for vs. This seems like it's good enough. But if I wanted to solve for vs, I could. I would get v equals the square root of the 7/10 would flip to 10/7 gh. Ta-dah! So we get that the velocity at the bottom of the hill only depends on the gravitational acceleration of Earth and the height of the hill. So the mass and radius of the ball are totally irrelevant to the situation so long as the radius is not so big that it compares to the height of the hill. But that means that any solid sphere, as long as it has a uniform density, will reach the bottom of the hill at the same time as any other sphere. This surprised me the first time I heard about it. But if you think about it, it's really the rotational equivalent of Galileo's big idea about the feather and the hammer reaching the ground at the same time. The mass and the size don't matter here, but the shape does. Now seem like a good moment as any to introduce the big idea for today. Think about that spherical cow rolling down the hill. When it gets to the bottom of the hill, it's going to just keep rolling. If it fell off a cliff, it would keep spinning in the air. If it fell into a vacuum, it would keep spinning there. Things that are spinning will keep spinning with the same angular momentum until something exerts a torque on it. The way that we would say that in physics terms is the conservation of angular momentum. And as always, angular momentum has a linear analog, which is just momentum. So linear momentum is mass times velocity. And remember that equation p equals mv? So angular momentum is moment of inertia. We use L for angular momentum. I don't know why. I, moment of inertia, times angular velocity omega. And I'm actually going to write this out. I'm going to use mr squared for I so that we can think about this a sec. This is Thea Ulrich. She's an aerialist and a maker. But most importantly for us, she's a fantastic demonstrator of physics.

(describer) She hangs upside-down from a hoop,

(Describer) (no standard option) She hangs upside-down from a hoop which hangs from a bar.

which hangs from a bar.

(Dianna) Thea let me pass her some weights and then give her a bit of a spin, and then showed what happens when she brings the weights in toward her body. She speeds up a lot-- enough to make even an aerialist dizzy. And this is the same concept as our crazy little water demo. You've got these pipes spinning around, all going the same angular velocity, the same number of revolutions per second. But the parts further away from the center where the water comes out have a higher linear speed sideways than the parts of the pipe in the middle, which are going slower linearly, because they're going around a smaller circle. And then just as the water droplets leave the end of the pipe where they were moving with that higher sideways linear velocity, they're now sprayed toward the part of the pipe that has the smaller radius-- the part moving with the lower linear velocity. So the fast water is going to pass those parts closer into the center. And you can see now exactly why you have to spin faster when you decrease your radius in order to conserve angular momentum. When you move inward, your initial linear velocity closer into the center corresponds to a higher angular velocity. So, actually, I don't see this demo as necessarily being explained by conservation of angular momentum. I kind of think it explains conservation of angular momentum-- that's cool. So Thea, the aerialist spinning in the air doing amazing tricks, is a great example of conservation of angular momentum, because there's no motor to make her spin faster or slower. She is controlling how fast she spins by changing the position of her arms and legs. So if you watch closely for a while, you'll notice that even without the weights, when her limbs are closer into her body, she spins faster. When they're further out, she spins slower. That is conservation of angular momentum because when she brings in her arms, her r goes down, so her moment of inertia goes down. Now let's tackle the question that we asked at the beginning of the video. If we got everyone on the planet to start running in the same direction along the equator, could we change how fast the Earth is spinning? We're going to answer this question using conservation of angular momentum. Now the Earth already has some angular momentum because it's rotating. But we want to figure out how much angular momentum we can transfer to the Earth by all of us running together along the equator, and how big of a change in Earth's angular velocity that would translate to. So, ultimately, we want to find change in angular velocity, or also change in velocity. So first, let's estimate the angular momentum of everyone running in a big old circle around Earth's equator. So angular momentum is moment of inertia times omega times angular velocity. So what's the amount of inertia here? Angular momentum is moment of inertia times angular velocity. So I have to figure out the moment of inertia of all those people running. Well, if we treat every single person running on the equator as a point particle, then the total moment of inertia of all of the people running around will be the total mass of people on Earth times the radius of Earth squared, because it's kind of like a ring of people moving around. So we're going to use mr squared. And, actually, this moment of inertia is the same for a ring, which is kind of what. We have this ring of people going around. Let's pick an estimated average mass. Let's say 65 kilograms. So my total moment of inertia is going to be the sum of all my mass times r squared, which is going to be the radius of Earth. So we're going to say this is 65 kilograms times the number of people on Earth, which is about 7.8 billion, or 7.8 times 10 to the 9 people. So that's going to be our mass. And then we need the radius of Earth, which is roughly 6,400 kilometers. So we want that in meters. That's 6.4 times n to this 6 meters. That's the radius we're going to use. And now what's the angular velocity of all those people running in a circle? Well, that's their linear velocity divided by the radius of the Earth, because we know that equation omega equals v over r. We'll say that everyone's running at 5 meters per second. Look, this whole thing is far-fetched. 5 meters per second is about 10 miles an hour. 5 meters per second around the radius of Earth was 6.4 times 10 to the 6 meters. Plug those numbers in to L-- this equation-- and solve for L. So L equals my I, here we go, 65 kilograms times 7.8 times 10 to the 9. All of my people, all their mass times 6.4 times 10 to the 6 meters squared. Then we're going to multiply it by this omega. So times 5 meters per second. And then divide all of that 6.4 times 10 to the 6 meters. That actually cancels with one of these. So I'm going to cancel this squared. And then if we do all this math out, I don't know, in your head, in paper, on the calculator, whatever. L, our angular momentum, equals 1.6 times 10 to the 19. And then our units are kilograms times we've got a meter squared here per second. Ta-dah! So that's how much angular momentum we would be able to transfer to Earth. But would we be able to change how fast Earth spins? Yes, but by how much? We can use this equation for angular momentum, L equals I omega, and use Earth's moment of inertia, I, to see how much of a change in angular momentum this would correspond to. So L equals I omega. I'll say this is our change equals L Earth. Something like that. So that equals 1.6 times 10 to the 19 kilogram meters squared per second equals. And then I need to know the moment of inertia of Earth, which I looked up, because it's not easy to calculate it since Earth is not a sphere of uniform density. It's more dense in the middle. So I looked it up. Earth's moment of inertia is 8 times 10 to the 37 kilogram meters squared. And then we've got omega, because what we're doing is angular momentum for Earth, so that was Earth's moment of inertia times omega. And I can solve for omega. And when I do that, when I solve for omega, I get omega equals 2 times 10 to the minus 19 radians per second. So that's a small number. So our answer is, yes, we can change how fast the Earth is spinning, but only by an amount less than 0.0000000000003%, or minus 2.7 times 10 to the minus 15 of Earth's rotation speed. So not much. Fun fact. Humans running may not be able to change how fast Earth spins by much, but big earthquakes actually can. That magnitude 9.0 earthquake that hit Japan in 2011 made Earth spin faster, and therefore shortened the day by almost two millionths of a second, according to measurements by NASA scientists. So that's our lesson on angular momentum for today.

(Describer) She moves the paper aside and tosses the marker.

When they ask what you learned on YouTube today, here are your two key takeaways. One, the rotational equivalent of mass is the moment of inertia. And two, rotational energy and momentum act just like their linear friends. They are conserved. And as always, don't forget to go solve the problems yourself that we did. So here are all the problems from today's lesson if you want to review, which you definitely should because that's the only way to really learn physics.

(Describer) (no standard option) Problem 1: A box and a sphere are sliding and rolling, respectively, down a ramp. If they start from the same height, and we ignore friction on the box, which will reach the ground first? What if it were a full soda can and an empty soda can? Problem 2: A sphere of mass 4 kilograms rolls down a ramp, starting with a height of 5 meters from the ground. How fast is the sphere rolling when it gets to the ground? Problem 3: If every person on Earth ran around the equator in the same direction, what would their total angular momentum be? Use 65 kilogram as the average mass of the 7.8 billion people on Earth, and let them run at 5 meters per second. Problem 4: Given 8 times 10 to the 37 kilogram meters-squared as the Moment of Inertia of Earth, by how much would all those people running change the angular speed of the Earth?

(describer) Problem one: A box and a sphere are sliding and rolling, respectively, down a ramp. If they start from the same height and we ignore friction on the box, which will reach the ground first? What if it were a full soda can and an empty soda can? Problem two: A sphere of mass 4 kilograms rolls down a ramp, starting with a height of 5 meters from the ground. How fast is the sphere rolling when it gets to the ground? Problem three: If every person on Earth ran around the equator in the same direction, what would their total angular momentum be? Use 65 kilograms as the average mass of the 7.8 billion people on Earth, and let them run at 5 meters per second. Problem four: Given 8 x 10 to the 37 kilogram meters squared, as the Moment of Inertia of Earth, by how much would all those people running change the angular speed of the Earth? And almost lastly, here is an out-of-this-world example of angular momentum. So when some massive stars explode as supernova, supernovae, supernovai-- their cores collapse into super dense, compact objects called neutron stars that are only a few kilometers across. Usually, the massive star was spinning before it collapsed, so it had some angular momentum. And angular momentum has to be conserved. So the much smaller neutron star that's left after the stars collapse typically ends up spinning super duper fast to make up for its now much tinier size. Sometimes, they spin hundreds of times per second. And also, some of these neutron stars shoot out intense beams of light that sweep around like a lighthouse as they spin. These are called pulsars, because from our point of view on Earth, we see these far away cosmic lighthouses as regularly timed pulses of light. These pulses are so strangely regular that astronomers nicknamed the first ever pulsar observed LGM1 for little green men because they thought it might be a message from aliens. And if you keep studying physics, you might get to learn more about pulsars and other really cool examples of angular momentum in a class on classical mechanics or on astrophysics. And I do hope you continue on with physics. And now let's hear a message from a special guest. Hi there, everybody. My name is Kari Byron, former MythBuster, current Crash Test Girl, and big fan of the study of physics. Maybe it was all my time blowing things up and dropping things off cliffs, but I love physics. You get to uncover the secrets of the universe. Well, I hope you stick with this course, and I hope you love everything that you're doing. And good luck, and find yourself in a world of science. Accessibility provided by the U.S. Department of Education.

(Describer) She blows a kiss. Title: Physics Girl Accessibility provided by the US Department of Education.