All right. So, let's start um angular kinetics. So, angular kinetics, it might take us two or three days uh just depending on how far we get with um the exam review. Um so, let's go ahead and get started. So, for angular kinetics, uh we're going to understand the center of mass, grav center of mass and center of gravity. momentum. We'll understand the angular momentum impulse relationship, understand angular work and power, and then identify the angular analoges of Newton's laws of motion. So, we've used the uh the term center of mass pretty liberally um throughout this semester and it's it's probably because we all have a general idea of what the center of mass is. But by definition, the center of mass or coom is the location on a body or on an object where mass is evenly or perfectly distributed. So mass is equal in all directions relative to the center of mass and it really just depends on the density of the object or how the mass in the object is distributed. So if the object has uniform density, the center of mass is exactly in the middle of the object. So what I mean by that is if we look at this wooden board, there's just as much mass no matter where you look on the board. It's not like one part of the board is really really heavy while the other part of the board is really light. The density and the mass is perfectly distributed. So if this is the situation, the center of mass is exactly in the middle of the object. But if we look at an object like a baseball bat and mass is not evenly distributed and it's not of uniform density, meaning there's more mass on one end than the other. If you pick up a baseball bat, the barrel of the bat is a lot heavier than the handle of the bat. So, the center of mass will shift in the direction to where there's more mass. On an exam, I'm not going to put a picture of an object and ask you to point out the center of mass. I'm not going to ask you to to determine whether an object has uniform density or not. This just helps you visualize where the center of mass of an object can be. Now, importantly, the body weight or segment weight vector begins at the center of mass location. So this next slide looks like there's a lot of math on it, but uh it's just a copy and paste from previous lectures. So you don't actually have to write anything down. In one example, I said that the weight of the leg is 57 newtons and it acts 32 m away from the axis of rotation. Well, this 32 m, it's actually the center of mass distance. So we're saying that the center of mass of the leg is 32 m away from the knee joint and that's where the vector of the weight starts. Same thing here. The weight of the forearm is 13.35 newtons and it acts.15 m from the axis of rotation. So the center of mass of the lower arm is.15 m away from the axis of rotation. So the vector of weight starts at the center of mass. To be honest, we're it's not even important to to include what the center of gravity is. Um I never put it on an exam anyway. Um so all you have to know for the center of gravity is that it's at the same point on a body. They are different concepts but they are at the same point. They have the same location. So just highlight that part and then you don't have to really know anything else. Am I going too fast or am I okay? All right, I'm just going to assume I'm okay. So, anatomically, our body is not of uniform density, right? So, think about it. Um, most of us are right-handed and so right-handed and right-footed. So, most of us probably have a little bit more muscle mass on the right side of our body than than the left side. um we have more mass in our lower body than we do our upper body. And if we were to divide our body in half in the frontal plane, the front half of our body probably has less mass than the back half of our body. So because our body is not of uniform density, our center of mass isn't just exactly in the middle. We have to find the point where our mass is perfectly balanced in the upper and lower body, left half and right half and the front and back. And honestly, this is a really difficult and impractical process. So what we do is we estimate it and in anatomical position our center of mass is right around the intersection of all three planes of motion. So where the frontal plane, sagittal plane, and transverse plane all meet, which just happens to be right at your navl or belly button. So if you were to estimate your center of mass, it's somewhere around your belly button. Now our center of mass is not a static point. It's dependent on mass. So if we shift our mass, our center of mass is going to move. So if I raise my arms up, I am putting more mass in my upper extremity than my lower extremity. So if I were to raise my arms, my center of mass would shift vertically a little bit. If I were to move my arm to the right, um, my center of mass would shift to the right as well. So, what we're actually going to do is in past semesters online, I've actually gone over the method on how to calculate the center of mass. Um, but this semester, I'm trying something new. I'm not going to test you on this mathematically. So, what we're going to do, um, we are just going to skip a couple slides. So this slide we're going to go over, but then the math you're not responsible for for center of mass. So the center of mass can be quantified two ways. One is as a position in meters and how far from the bottom of of our foot. So we could say from our the bottom of our feet, our center of mass is 1.5 m away. Right? That could be one of the ways we express our center of mass. We can also express our center of mass as a percentage of our total height. So from the bottom of our foot, our center of mass is located at 55% of our total height. So that's how we can express our center of mass. Now, it's really difficult to say as position how far from the bottom of the foot people's center of mass are because people have very very different heights. So, it's a little bit more common to express it as a percentage of total height and typically it'll be right around 50 to 60% of height. Now, usually on average the center of mass percentage is different in biological men and biological women. Biological women on average tend to have a lower center of mass location at about 54% of height and men typically have a higher center of mass location which is about 57% of their height. Why do you think there's some differences in center of mass percentage? Let me know in the chat. What do you think is driving this difference? I'm seeing some answers um that that are are worth mentioning. So one of the answers was height, right? But if we think about it, we are talking about the percentage of height. So yes, on average, men are taller than women. But we're expressing center of mass location as a percentage of their height, not as an absolute position. So in this case, we're assuming that this man and this woman are equal height. Women have a lower center of mass and men have a higher center of mass. Someone said muscle mass and someone else privately said the pelvis. So on average women have wider pelvises for childbearing uh compared to men. So this is a male pelvis and this is a female pelvis. So this wider pelvis shifts the center of mass location down a little bit. The second answer is not just muscle mass but it's the distribution of muscle mass. So research shows that um in equally trained men and women, muscle mass in the lower extremity is actually pretty similar. But in the upper extremity, biological men have uh more muscle mass um when when training uh experience is normalized for. So if there's more uh mass in their upper extremity, the center of mass is going to shift up a little bit. So it's again the the pelvis anatomy and not the amount of muscle mass but the distribution of muscle mass. So make sure you include that distribution part. Now there are two techniques to estimate the center of mass on a person. They are called the segmentation method and the reaction board. So, we're not really going to worry about this segmentation method. We're going to talk a little bit about the reaction board. So, a couple minutes ago, um I might have misspoke. So, we are going to go over the math for the reaction board because your lab is uh we're actually going to do the reaction board in lab this week. Um and it is important to understand how that works. But I'm not going to test you mathematically on the exam for the reaction board. So if I misspoke, I apologize. We're going to go over the math, but I'm not going to test you mathematically. So what would probably be a good idea is just to listen to how it works. If you want the notes, um you can get it from my full version that's going to be available to you right after class. But don't feel like you have to scramble to write down all the words. So try to understand how it works and what I'm doing rather than getting the right uh answer exactly. So the reaction board technique requires two force plates and a rigid board like a wooden board with some supports. It's really good for a static analysis of the center of mass and it gives us the distance between the axis of rotation and the center of mass location. So it'll give us this as a position. How far from the bottom of the foot is the center of mass. This is what the reaction board will give us. And what it does is it relies on the formula of sum of all torqus equals 0. So here here is our reaction board, right? This gray horizontal bar is our wooden board and these two vertical boxes are the supports that the uh the box will lie or the board will lie on. And so we put it on two force plates. So these black uh rectangles are force platforms. And we have a person just lay down on the reaction board. So in lab today, this is what it's going to look like. You're going to have your group members just lie on a board that's sitting on two force plates. In this situation, we have four forces. There's two force plates. So, we have reaction force one and reaction force two. We have body weight of the person and we have the weight of the board. So if we're relying on the formula of the sum of all torqus equals zero and we have four forces, we're going to have four distances as well. So each of these four forces is going to have a corresponding moment arm. So we have moment arm of reaction force one, moment arm of reaction force 2. Then we have the center of mass of the person. Remember the um the body ve or the uh vector for body weight starts at someone's center of mass. So the distance between the axis of the bottom of their feet and their body weight, that's the distance of their center of mass from the bottom of their foot. This is what we're looking for. And then our fourth distance is the center of mass of the board, right? How far is the center of mass of the board from the axis of rotation? So these distances are all from the axis of rotation. Now, what you might be wondering is, okay, well, if these distances are from the axis of rotation, where the heck is our axis of rotation going to be? And what we do is we set the axis of rotation at the bottom of our feet. It's right here. So, at the bottom of our feet, it is right there. So, when you lay down on the reaction board today, make sure your feet are at the very bottom of the board. We set our axis of rotation at the bottom of our feet for two reasons. One is so that we know how far from the bottom of our feet our center of mass is. It would be really weird to say, "Oh, let's set our axis of rotation at our knees." And our center of mass is x amount of meters from our knees, right? That would be a little bit strange. from the bottom of our feet makes a lot more sense conceptually. Second, reaction force one goes right through the axis of rotation. Therefore, there's no moment arm for reaction force one. So, one of the torqus that we have to deal with this torque here, reaction force 1 time moment arm of reaction force one, it gets eliminated because the moment arm is zero. So it makes our math a little bit easier. So we have four torqus or really three torqus that will add up to zero. Reaction force 1 * moment arm 1. Reaction force 2 * moment arm of reaction force 2 which is just the length of the board. Reaction force two for relative to our axis of rotation is just the entire length of the board. Then we have body weight time center of mass of the person and then weight of the board times the center of mass distance of the board. These four torqus will add up to zero. So, I just want to gauge where we're at. Even though I'm not going to test you mathematically, I do want you to understand it conceptually. On a scale of 1 through 10, one being very hard, 10 being very easy, how are we feeling about center of mass so far? Just let me know in the chat. I'll wait for about 10 answers. So sevens, sixes, eights, right? That's that's pretty solid, right? So let's continue. Now the board, it's a wooden board and so it's of uniform density. So the center of mass of the board is half the length of the board. So I'll always give you the length of the board, but again, you won't have to worry about this because I'm not going to ask you mathematically. But for the lab, you'll have to know. So if the wooden board is let's say 5 m long, the center of mass of the board is 2.5 m, it will always be half the length of the board. So let's add some numbers, right? The mass is 73.45 kg. Mass of the board is 44 kg. Length of the board is 2 m. Reaction force one is 1450 newtons. Reaction force 2 is 650 newtons. There's a problem here though. Mass is not a force. So, we have to convert mass to force. So, we multiply by gravity and we have these values. I'm going to pause just in case you want to follow along with the math. So, you could write these numbers down. Um, but you don't have to. So, I'll wait about 30 seconds to a minute uh for everyone to catch up and then we're going to move on. So the sum of all torqus are going to add up to zero. So reaction force 1 time the moment arm of reaction force 1 which is going to be zero plus reaction force 2 * moment arm of reaction force 2 times oh plus body weight time center of mass plus weight of the board time center of mass of the board is going to equal zero. So here are the corresponding numbers. We do all the math and we get 1.21 m. So, what we're saying is from the bottom of this person's foot, the center of mass is 1.21 m away. Again, if you want to do this math on your own to maybe help you understand the concept a little bit better, you're more than welcome to. We're just not going to spend too much time in class. This would be something really good to refer to uh during today's lab. Now this is the location of the center of mass relative from the axis of rotation or our foot. But if I give you the person's height, we can find the center of mass as a percentage of their height. So if they're 1.95 m, and I just made this up, and we want the percentage, we just divide 1.21 m by 1.95. And now we know that this person's center of mass is 62.05% of their height. So it seems plausible. It's between it's 50 to 60%ish. Let's take about a two-minut break and then we'll talk about our moment of inertia. Um, if you have any questions, please let me know in the chat or you can ask. But let's take about a two-minute break to kind of regroup. Um, also don't forget that some of you signed up for those meetings with me uh to discuss your project. They are in person in my office in BHS 341. Uh, so just don't forget to come by if you reserved a time slot. Okay. So if we think back to linear kinetics, we learned about something called inertia. And inertia is the resistance to change in acceleration or the state of motion. It's proportional to mass. So large mass, large inertia. Small mass, less inertia. So if something has a lot of inertia, it's harder to accelerate or to stop moving. And if something has a little bit of inertia, it's easier for them to accelerate and to stop moving in a linear sense. I think in your version, I made a little bit of a typo and this arrow is going up. Just make sure this arrow is pointing down in your notes. Right? This was the example of like in football when a running back gets matched up against a bigger person, they could just go right around them because it's easier for them to start and stop moving. That's what inertia is. Inertia exists in an angular sense as well and in an angular version uh inertia becomes the moment of inertia. You're going to see it as m oi or i. So the moment of inertia is the resistance to change in angular acceleration. It's the resistance to change in rotation or spinning. Now we when we're looking at the moment of inertia, mass it still matters, right? Mass is definitely one of the contributors to the moment of inertia, but we add a a second dimension or a second variable. And what we look at is how is that mass distributed relative to the axis of rotation. If there's a lot of mass far away from the axis, the moment of inertia is greater. This could be a little hard for you to visualize at the moment. I have a couple examples that will hopefully um cement your understanding. So just try to just try to um stick around for that. What I really want you to understand from this slide, the take-home point from this slide is what does the moment of inertia mean conceptually? A greater moment of inertia means it's harder to rotate or to stop rotating. So rotating could be like swinging a baseball bat, swinging a golf club, um doing a bicep curl, a lateral raise, anything to do with rotation. So if something has a large moment of inertia, it's harder to rotate and it's harder to stop rotating. If something has a smaller moment of inertia, it's easier to rotate and it's easier to stop the rotation. That is what moment of inertia is. So mathematically the moment of inertia is equivalent to MK^2. I = MK^2 where M is mass of the object and K is the radius of girration of the object. And before you write this down, look at the formula. The K is squared. So mathematically, the radius of girration has the biggest influence on the moment of inertia. And the unit of measure is kilogram me. So give you a couple seconds to go ahead and write that down. Now, this K, the radius of girration, um, if you've taken physics, you have an understanding of what it is. Um, it's quite a, um, I don't like using the word complicated because I think we're all intelligent enough of understanding what it is, but I think it's one of those things that for the sake of our class, the technical definition just isn't very important. So, I'm going to give you this technical definition and then I'm going to give you the definition that we need for this class. So, if you're an overinker like I am, this might make a little bit of sense. So if we look at an object like a baseball bat, right? This baseball bat is one object. But within an object, multiple particles will make it up, right? Like these red circles represent one particle. And this bat could have thousands or millions of little particles that make up the entire bat, right? Like we have a bunch of cells that make up the human body. There's a lot of particles that make up a bat. Each of these tiny particles has its own mass and a distance between that mass and the axis of rotation. So if we take all those distances and we average it, that is the radius of gation. It's a bit of a complicated definition. So in this class when you think when you hear radius of giration just think of it as the distance between the axis of rotation and the center of mass of an object or the center of mass of a of a body segment. Right? If you tell your physics professor this definition, he or she might have an aneurysm. Right? It's not the right definition, but it's the right definition for the context of this class. It's the distance between the axis of rotation and the center of mass of an object. So, let's apply the moment of inertia to a couple sports. So, if you've played baseball or softball, or if you watch baseball, you might have noticed, but um if we look at the player that that's not necessarily hitting, but the one that's next up to hit, um they're taking practice swings and what they do is they add this little cylinder looking object to the bat. It's called a donut. And so, they'll add a donut to this bat and it makes the bat heavier. So they'll take practice swings with a really heavy bat and then when it's their turn to hit, they'll take the weight off so that the bat feels really light and so it feels easier to swing when they're actually up to hit. I don't know if baseball players still do this. I have seen it in professional baseball lately. Um I don't know if it's still done at the high school or college level. It's been a while since I've played. Um but it is still relevant. So we have two uh bats here. Bat A and bat B and donut A and donut B. So bat A, bat B and these two donuts have identical masses. So if we put bat A and donut A on a scale, it would have the exact same mass as bat B and donut B. But what do you notice about the location of donut on bat A versus B? Let me know in the chat. A or B? Which uh which bat has the donut further away from the axis of rotation? A or B, right? It's a. So here the donut is this far away from the axis of rotation. And bat b that donut is a lot closer. So the center of mass of bat A is further from the axis of rotation than the center of mass of bat B. So K is bigger in A than B. So, bat A has the greater moment of inertia compared to bat B. Even though they have identical masses because there's more mass away from the axis of rotation in bat A, bat A has the greater moment of inertia. So, let me know in the chat which bat is easier to swing or rotate, bat A or bat B. it would be bat B. Bat B has a smaller moment of inertia, so it's easier to rotate and swing and it's also easier to stop swinging. On exams, I have put a picture of bat A and bat B on an exam and have asked a similar lo uh question as this bullet point. So, just make sure you mark this on your notes to pay a little extra attention because it more than likely will be an exam question for uh exam three. Now within baseball or softball, there's a way we can reduce the moment of inertia. So if we look here, this baseball player, he's holding the bat, not at the handle like most of the time. He's holding it away from the axis of rotation. Right? Uh in baseball or softball we call this choking up. So originally the axis of rotation would be here. But when we choke up and grip further down the bat, our axis of rotation will move to there. So now we've brought our axis of rotation closer to the center of mass of the bat. So the radius of girration goes down. The moment of inertia goes down and now it's easier for this person to swing the baseball bat. It feels lighter. So it's easier for him to swing. You'll notice that when you compare, you know, someone who's played a lot of baseball or softball to someone who's never even swung a baseball bat, if you give the experienced person a bat, they'll hold it right at the handle. But if you give it to someone who's not as experienced, they'll automatically grip it away from the axis of rotation. It's because they want that bat to feel lighter, so it's easier for them to swing. It'll happen in tennis. It'll happen in golf. Um, it's a very common thing that people do when they're not familiar with holding an object. It's a big hint for your project. If you chose like swinging or or like tennis or anything like that, you might notice this. Give everyone about 30 seconds to catch up and then we're going to go on to another example. Now, before we talk about gate, um I I thought of just another example um at at this time. Um I'm assuming that we've all done a a a bicep curl, right? A bicep curl with one arm. Now, let's say you want a bicep curl 10 pounds and in front of you there's a 10 lb dumbbell and there's a 10 lbs kettle bell in the let me know in the chat which one are you going to pick up to do a bicep curl for the most part a kettle bell or a dumbbell we're going to pick a dumbbell every time if you think about the mass of a dumbbell versus a kettle bell I'm going to try to be an artist um just you know bear with Okay, a dumbbell kind of looks like this. Hold on. I'm just going to use my hand because this app is horrible. One second. It won't work. Okay, so let's just try to visualize it. A dumbbell has its mass perfectly distributed. There's just as much mass on the left side of the dumbbell as the right side. So, as long as you grip it in the middle, the mass is perfectly distributed. But a kettle bell, most of the mass is in that that sphere. And so, when you grab it by the handle, there's a ton of mass away from the axis of rotation. So, the moment of inertia is greater. So, that's why doing dumbbell curls with a kettle bell feels so awkward. The mass is distributed very differently. That's why it's harder than with a dumbbell. Not because it's heavier. It's because the moment of inertia is greater during running. Um I don't know if many of you are runners, but you'll notice that certain phases feel easier because the leg feels lighter in in certain phases of running, but the mass of your leg remains totally constant. So let's think about the swing leg. The swing leg is the leg that's actively rotating. So when the swing leg is flexed relative to the hip axis of rotation, there's a lot of mass really close to the hip, right? The knee is bent. So the foot and lower leg are really close to the hip. So if the formula for the moment of inertia is mk^ squ, mass remains the same. Your leg isn't getting any heavier or lighter, but the radius of girration is uh shorter, right? There's mass closer to the axis of rotation. So, the moment of inertia is decreased. So it feels lighter and easier to rotate. Now, when this rotating leg is extended, right, relative to the hip axis of rotation, there's more mass further away. The leg and the foot are now far away from the hip. mass is still the same, but the radius of girration is longer. So, the moment of inertia is increased. So, your leg feels hard uh heavy and it's harder to rotate. There have been exams in the past online and in person where I put this picture on on the question and I ask you to basically explain it. So you might want to put a big star by this slide as a potential free response question on your exam. It will be on your exam in in some capac in some capacity. Just let me know when I'm good to move on. Okay. So, let's talk a little bit about the moment of inertia and how that influences how fast someone can rotate, otherwise known as angular velocity. So, I don't expect you to remember, but in linear kinematics, I showed a video of Conor McGregor coming to um Cal State Fullerton when I was there. Um, and we showed how motion capture worked. And I made a comment that said, "There's something on here that's not very accurate, but we'll talk about it later. I don't expect you to remember, but we did have that conversation. So, let's go ahead and let's rewatch this video." thing about Connor is his special movement. He talks a lot about how he moves differently than other people in his division. So, we wanted to actually take a look at that. We attached retrflective markers all over Conor's body and we put him in a room 360° of high def slow motion cameras. We'll look at his hip, his knee, his feet, and figure out where all this power is coming from. What we notice is the hips and the core are probably the most important factor to generating force. Golfers typically have the highest angular velocity in their hip. Fact, Rory Mroy has the highest angular velocity at about 720 degrees per second. So, we looked at that in Conor during that spinning back. His angular velocity was about 800 degrees per second. Okay, so the claim is that Conor McGregor has elite hip angular velocity. That's measured at about 800 degrees per second. But let's dive a little bit deeper into this claim. You don't have to uh this won't be on the exam. It's just something to help you understand. So the moment of inertia is mass times radius of generation squared. Now I was there I was one of the people that calculated that velocity of 800° per second. We calculate his angular velocity when his uh hip angular velocity is at its fastest. Right? like we don't want to know what's his slowest angular velocity. We want to know what's his fastest angular velocity. And it basically happens at this instant when the knee is tucked really really close to the hip. Well, when your mass is so close to your hip, the moment of inertia is very low, right? All the mass is really close to the hip axis of rotation. So the moment of inertia is relatively reduced. So it's really easy to rotate the hip. If m is the same but k goes down, moment of inertia goes down, which means it's pretty easy to rotate. Let's look at a golf swing. The golf club head, that thing you hit the ball with, it's really far from the hip, right? The heaviest part of the club is really far from the hip. So the moment of inertia is increased around the hip. The length of the golf club also plays a significant factor. Right? This golf club, it's about 4t long. And so that also increases the moment of inertia around the hip. So the moment of inertia is relatively high. And it's really hard to rotate the hip. If the moment of inertia is high, it's harder to rotate. So, it's not necessarily the best comparison. So, when we were there in post-prouction, a lot of the producers for the UFC um asked us to compare Conor McGregor's, you know, like mechanics with something that the average person could really relate to. Um and so when we measured how much force he produces with a kick, they were like, "Well, I don't know what x amount of newtons is like." And so we had to do a literal Google search and we were like, okay, it's similar to dynamite. Obviously, it's not the same. And so really that angular velocity of of 800 degrees per second, it's not that elite. It's it's pretty average, right? So the whole point of that is is just to show you that context matters. It's just a really good example of how the moment of inertia can influence angular velocity and how easy it is for someone to rotate. That presents a really good segue into angular momentum. So linear momentum is the quantity of motion. It's mass times velocity. The angular version of momentum, it's called angular momentum. Don't ask why it's an H. Some physicists who, you know, discovered angular momentum called it H. It's the quantity of angular motion. So mathematically, angular momentum equals the moment of inertia time angular velocity, which is the same thing as MK^ 2 * angular velocity. The unit of measure is kilogram m/s. Now keep in mind an object can have linear velocity but if it doesn't have any angular velocity it doesn't have angular momentum. So for instance, if there is an object moving in a straight line and it's not rotating at all, it has linear velocity. So it has momentum. But because it's not rotating, it has no angular velocity. It can have which means it has no angular momentum. So you could have momentum but no angular momentum. If we look at the formula for angular momentum, this radius of gration, this K is also the driving factor. It has the biggest influence. It's squared. So the next slide has a ton of math. Um, don't worry about writing the numbers. I'm just using the math to uh to kind of cement a point. So if an object has a mass of 10 kg, the radius of dation is 2 and the angular velocity is 3 radians/ second. What is angular momentum? And then what if we double the mass or double the radius or double angular velocity? How does that influence angular momentum? Well, I did the math for you and angular momentum is 1.2 2 kg m/s. If we double the rad uh the mass right we go from 10 to 20 our angular momentum doubles. If we double the radius from 2 to 0 4 our angular momentum quadruples and if we double our angular velocity our angular momentum doubles. So this is just mathematical uh evidence that the radius of girration has the biggest influence on angular momentum. That takes us to the last concept of today the conservation of angular momentum. The conservation of angular momentum states that uh once an object has angular momentum, it stays constant. It doesn't change unless we apply an external torque. Gravity alone cannot influence angular momentum because gravity acts through our center of mass. So if angular momentum is moment of inertia time angular velocity, if moment of inertia goes up, angular velocity goes down. If moment of inertia goes down, angular velocity goes up because angular momentum remains constant. One goes up, the other goes down. The practical application of this is way more important than the math. So take about 30 seconds, catch up, and then we're going to go over a really, really good example of the conservation of angular momentum. Now the best example of this conservation of angular momentum is in figure skating. So watch what I believe uh if I remember correctly this figure skater is a female. Watch what she does when she's uh wants to spin faster and look at what she does when she slows down. Okay. So, when a figure skater wants to spin faster, they will tuck their arms and legs. They'll either do this pose or that pose, or they'll just bring their arms and legs really close to their body. This makes their mass. they have more mass closer to the axis of rotation which reduces the radius of girration. This reduces the skater's moment of inertia which will increase the angular velocity and they spin faster. When a figure when a figure skater wants to slow down, they extend their arms and legs. So they're that same mass is far away from the axis. So their radius of girration goes up. This increases the skater's moment of inertia, which will decrease their angular velocity and they slow down. Either way, angular momentum is conserved. It's the same in both situations. I promise this will be on an exam. You're going to have to know why they slow down or sorry, why they spin faster when they bring their arms closer. Why they spin slower when their mass is far away. You're going to have to know that the radius of girration influences the moment of inertia, which influences angular velocity. So, just put a big star by this. Okay. So, if you notice um if you're looking at my version of the notes while taking notes, which again I always recommend, you're going to notice that we have a couple slides left. In fact, we have about 10 12 slides left. Instead of doing that today, we're going to delay that until next or until next time, which is going to delay um the schedule on the syllabus a little bit, which is totally okay because I've I've accounted for that. Um so, what I would like you to do between now and Wednesday, you don't have to read these notes uh verbatim, but just compare the blank notes with my full version and then just see if you can follow along. And then on Wednesday, we'll go over it together. Um, the more you look at the notes beforehand, the more sense it'll make. Um, but this is where we're going to stop today. The reason we're going to stop today is we're going to talk about the exam. So, before I end the recording, are there any questions that you have regarding um, Angular kinetics today? If you don't care to go over the exam, please feel free to leave the meeting. I'm not taking attendance today. Um, so if you want to leave, go for it. If there's any questions about Angular kinetics, I will wait about a minute before I end the recording. You probably do want to stick around though because I'll go over the answer. I'm going to open up the exam for you to look. Um, so I I do recommend sticking around. You just don't have to.