All right. So, let's pick up where we left off last time. Um, just a couple I've gotten a couple questions regarding the reaction board and the math. So, the reaction board math is not going to be on your exam, but it is on your worksheet and it is this week's lab just to help you understand the concepts. So, you should know how to do it for those purposes. You're just not going to be tested on the exam. So hopefully that clarifies things. If I was a little bit unclear or confusing with that, I do apologize. Um, but that's where we're at with the reaction board. So where we left off last time, we talked about the conservation of angular momentum. So angular momentum is always conserved. It's the same in all situations. But we could do things like um change our radius of girration, which changes our moment of inertia, which will also change our angular velocity. Now even though whole body angular momentum is going to be the same angular velocity um it can be transferred and what I mean by that is um or this can really be shown in the writing reflex in cats. So if any of you have cats um and maybe that cat has jumped out of your arms or squirmed out of your arms, you'll notice that cats always land on their feet. So, no matter how low or how high they jump from, they will always land on their feet or on their paws. And it it isn't chance, it's actually physics. And so, the cats perform something called the writing reflex that always helps them land on their feet. And it has to do with angular momentum and the transfer of angular momentum and the manipulation of angular velocity. So when dropped, right, and I use, you know, quote unquote dropped, what cats will do is they have a really, really flexible spine. And so what they do is they flex that spine and basically create two segments within their bodies. And each of those bodies has its own axis of rotation. So it has an upper body axis of rotation and a lower body axis of rotation. So it goes from basically one body to two. Now instinctively the priority of the cat is to orient its upper body because that's where its head is. So what it's going to do, it's going to orient the upper body first by tucking its front legs and extending its back legs. By doing this, by tucking the front legs and extending the back legs, the upper body moment of inertia is reduced. There's a lot of mass close to the upper body axis of rotation. And so the angular velocity of the upper body increases. The upper body rotates about 90°. And now the upper body is facing the ground. by the lower body um or by the cat extending the back legs, the lower body moment of inertia goes up. The angular velocity goes down. So the lower body doesn't rotate as much, right? Because we're prioritizing the upper body. Next, what the cat will do is it'll tuck its back legs and extend its front legs. So, the lower body moment of inertia decreases and the angular velocity increases. And now the lower body is oriented towards the ground along with the upper body. And so cats will repeat this process two to three times um so that they always land on their feet. No matter what, angular momentum is always the same. But the moment of inertia and the angular velocity go up or down to orient their upper or lower bodies. Just give me like a thumbs up or just let me know when you're ready for me to move on. Our next variable is impulse. And if we think about linear impulse, impulse is the influence of a force over a given period of time. And so when we talk about the angular version of impulse, what we have to do is we have to get the angular version of force. Time is just time. So angular impulse is torque times time. It's the influence of a given torque and how long that torque is applied for. And it's measured in Newton me seconds. The angular impulse momentum relationship says when angular impulse acts on a system, the result is a change in that object's angular momentum. So for instance, if there's like let's say there's a a ball on the ground and we apply a torque to that ball, right? That ball is going to generate angular momentum, right? So angular momentum would be like that ball spinning. So if we apply an angular impulse to a system, there is a change in that system's angular momentum. So torque times time equals change in angular momentum. Or if we or if we expand on this, force time distance time, which is torque and time equals moment of inertia time angular velocity final minus moment of inertia* angular velocity initial. So basically what I'm saying is angular impulse is required to create angular momentum or to stop angular momentum. So this brings us to math. And so I have good news and bad news regarding math. The bad news is that today is probably one of the it's the most heavy math lecture of of the semester. Um we have about I think five practice problems that we're going to do together. But the good news is that after today there's no more new math. So in other words, after today, all the math for that third exam is going to be covered, right? So there's light at the end of the tunnel. So let's go ahead and go over one of these math examples. Um let's say we're trying to run. And for a runner to start running and initiate leg rotation, they have to achieve a leg angular velocity of 3.5 radians per second. and they have to do it in a time of 0.2 seconds. The leg has a moment of inertia of 7 kilogram meters squared. How much torque must be generated by the hip flexors to create this angular momentum? So what I want you to do is on your own take a minute. I want you to just list your known and unknowns. So make this known unknown table and just list all the variables I gave you and categorize them under known or unknown. We're going to go a little bit faster on this practice problem than the other so that uh we get through the lecture. Um but if you don't get it the first chance or the first try, don't worry about it. So in this case, I've given you some known variables. We know the initial angular velocity is zero because they're not moving yet. But they want to achieve this angular velocity of 3.5 radians/s. The moment of inertia is 7 kg m and the time is 2 seconds. What we don't know is we don't know torque. So, we're going to take these variables and we are going to plug them into the equation. Torque time equals moment of inertia* angular velocity final minus moment of inertia* angular velocity initial. So, go ahead, take another minute. If you want to do the math, you can. Uh, but just fill out the equation the best you can. If you want me to check your answer, just shoot me a private message. So we don't know t. We know time is 2. The moment of inertia is.7 kg m. The final angular velocity is 3.5 radians/s. The moment of inertia is still. But this final angular velocity is zero or sorry this initial angular velocity is zero. So it knocks out this part of the equation. So we get torque *2 seconds equals 2.45 kg m^ 2/s * 3.5 is 2.45. That kilogram m it's just a unit of measure. You don't actually have to square 7. So, let me know in the chat what answer do you get when you isolate torque? Right? I'm going to wait for a couple answers publicly, privately, I don't care. What do you get mathematically? It's 12.25 Newton meters. Right? So we have to produce a torque of 12.25 Newton m to achieve that angular momentum required to run. Now, how much force must these hip flexors apply to initiate leg rotation? And the hip flexors have a moment arm of 0.03 m. Well, torque equals force time distance. Torque is 12.25 Newton m. The distance or the moment arm is 03 m. So if we want to isolate force, we divide both sides by 03 and we get 408.33 newtons as force. The torque is 12.25 Newton m. The force requirement is 48.33 newtons. I just made this number up here. So take 30 seconds, write down anything you missed, and then we're going to do another practice problem. A runner is swinging their leg forward at an angle. angular velocity of 3.5 radians/s. The leg has a moment of inertia of.7 kg m. How much torque must be generated by the hip extensors to stop leg rotation in4 seconds? So go ahead list your known and unknowns and then we will um move forward. So I'll give everyone three minutes and then we'll reconvene. So our known we know our initial angular velocity is 3.5 radians/s and our final angular velocity is going to be zero because we want to stop rotation. that moment of inertia is.7 kilogram meters squared, time is point4 seconds, and we're solving for torque. So if we plug these variables into the equation, right, we get torque*4 seconds equals and then this first part of the equation, it gets knocked out because our final angular velocity is zero. And so we get torque*4 seconds equals -2.45 45 kg m^ 2/s. That negative is important here. We divide both sides by4. We get a torque of -6.13 new m. So that must that's how much torque we have to apply. If I give you the moment arm of the hip extensors and ask you to isolate force, you divide that torque by the moment arm I give you and you'll get a force of -153.25 newton. So take a minute, write that down, and then we'll move forward. All right, so there's our wrap-up. What is the center of mass and how do we use torque knowledge to calculate center of mass location? What is the moment of inertia and how does it relate to gate in baseball? What is the conservation of angular momentum and how does it apply to skaters? How is angular momentum transferred and explain the angular impulse momentum relationship conceptually and mathematically? So in part two we'll do another math practice problem. We'll talk about angular work and power and then we'll discuss Newton's angular laws both mathematically and conceptually. So here's another uh very similar but a little bit more complicated problem. Your arm has a mass of 3.5 kg and a radius of giration of 2 m. The elbow is rotating at 5 radians/s. What is the force not the torque? What is the force your elbow extensors need to produce in.3 seconds to stop elbow rotation? And the moment arm of the elbow is 03 m. So it's the same problem as this. And you need to list your knowns and unknowns. But remember I in the last two problems I just gave you the moment of inertia. Here I did not. I gave you the mass and I gave you the radius of giration. So you have to calculate moment of inertia on your own. So list your known and unknowns and then plug the variables into the equation and then we'll see where you're at. So, we'll reconvene in four minutes and then we'll go over the answer. So, our known and unknowns, we know our initial angular velocity is 5 radians/ second. Our final angular velocity is going to be zero. Our mass is 3.5 kg. Radius of gerration is 2 m. Uh the time is.3 seconds and our moment arm is 03 meters and we don't know force. So we're going to plug these variables in. Torque time equals moment of inertia* angular velocity final minus moment of inertia* angular velocity initial. Right? That's torque* time= mk^2 omega final minus mk^2 omega initial. So if we plug in our known we get force * 03 m *.3 seconds equals this left side is irrelevant because anything multiplied by a zero is a zero minus 3.5 kg *2 m^ 2 * 5. We only square the 2. So we get force times 03 m*.3 seconds is.7 kg m^ 2/s. Divide both sides by.3 to get -2.33 Newton m. And we divide both sides by the moment arm. And the force should be -77.67 67 newtons. So that is our final answer and your exam question will look very very similar to that. So I'm going to move forward, but remember um this full PDF is going to go live um at 9:45. Um so you can just copy it then. Um, I believe on the PDF it says 77.78. Whether it's 78 or whether it's 67, those would both be close enough to the final answer to where Canvas will mark you correct on an exam. So our next concept is work. And when we talk about work in an angular sense, we talk about angular or joint work. It's the same thing. And angular or joint work is used to express the influence of a torque at a given joint. So joint work is torque times angular displacement. What is the magnitude of the joint moment and how much does the joint move? That is angular work. And we can measure it in jewels or newton meters. So joint work can be positive or negative. Positive joint work is very similar to concentric uh muscle actions. Positive joint work occurs when the joint moment and the angular displacement are in the same direction and negative joint moment occurs when angular displacement and the joint moment are in opposite directions. So if we produce a elbow flexor torque and we get elbow flexion, we're performing positive work. If if we produce a flexor moment but we get extension, we are performing negative work. So just think about work as how much torque does that joint produce and how many degrees do our joints rotate. When you combine those two, we get angular or joint work. So these next bullet points, they won't be on your exam. It's just to provide you a little context. But um the distance of how far you shoot a basketball shot from, it influences the joint work requirements. So, when you compare long range shots or three-pointers compared to shots like free throws or layups, uh long range shots require more lower uh lower limb joint work. So, they require more joint work from the ankle, knee, and hip in order for that shot to be successful. So their hips, ankles, and knees, they either have to produce more torque or they have to undergo more angular displacement or both to create more joint work to successfully make that shot. Long range shots also require more elbow joint work than short range shots. So they have to produce more of an elbow extensor torque or get more elbow extension or both in order for that shot to be successful. Our next variable is angular power. So angular or joint power is the product of torque and angular velocity. So how big is that joint moment and how fast does that joint rotate? When you combine those two, we get joint power and our unit of measure is watts. So like joint work, joint power can be positive or negative. Um, and they're very similar. So we get positive joint power when the joint moment and the angular velocity are in the same direction. So if we produce a knee extensor moment and we get knee extension velocity that's positive power. So positive joint power is associated with power generation. Now you don't have to know this example. It's just to help you pro or help provide some context so that um you get a better understanding of joint power. But this is um this is a graphic of a professional or I believe collegiate baseball pitcher. And what I've done is um our program can create a model and it can show us which segments are creating power and which segments are not. So if you look at this graph, if a segment or a joint is red, it means it's generating power. So if you notice during pitching, the pelvis and the back leg are bright red. So during pitching, our trail hip and our leg generates the power needed to throw that ball really fast. Negative joint power occurs when the joint moment and the angular velocity are in the opposite direction. So maybe we get a hip extensor moment, but we're actually moving in hip velocity or hip flexion. Negative joint power is associated with energy absorption. So if you look here, this front leg here, it's not generating any power. It's bright yellow. So that front leg is absorbing the energy of the pitch and absorbing the ground reaction force so that it's really stable so that this back leg can generate the power. You don't have to know the right side of the slide. You do have to know the left, but that baseball example might help provide a little bit more context so that it makes a little bit more sense. to give you a little bit more context. So, right now in baseball, there's an obsession about throwing the ball faster and faster for pitchers. And what research shows is that two of the most important things uh a pitcher can do to uh to throw a pitch fast is to have uh a lot of pelvis rotational power and trunk rotational power. These two contribute most to the velocity of the ball. And so a lot of the times I get questions of like, you know, how are joint power, how is joint power relevant, right? Like I'm not going to be a biomechanist when I grow up. Um, and you wouldn't be wrong. Of the 63 of you that are involved in this class, maybe one of you are going to pursue a master's degree in biomechanics, if that. It's just not a very common field. But if you choose a field like you know strength perform or like athlete performance or or physical therapy and you don't understand this concept then in my opinion you are doing your client or your athlete a disservice. So here's what I mean by that. So let's say you're a strength coach or a performance coach or a physical therapist and you're working with a baseball pitcher and that baseball pitcher comes to you and says, "I want to be able to throw the ball faster. How can I do that? Well, if you don't understand what joint power is, then you don't really know where to train. And so, there's a lot of strength coaches who, you know, they'll maybe emphasize lower body strength and power um for pitchers throwing fast. And sure, that's important, but if you don't understand that trunk and pelvis rotational power matter more, right, then you're doing that pitcher a disservice. If you're a physical therapist or a clinician and you're working with a pitcher who's trying to get back to sports and you don't address their pelvis power or their trunk power, right, then you are doing them a disservice as well. You're they're just going to get hurt again when they go back to pitching. So, understanding how things like rotational power influence um a sport like pitching um it's actually quite important even if you're not going to be a biomechanist. So, just keep that in mind. All right. Lastly, we are going to go over Newton's laws and we are going to translate them into an angular sense. So, if you know Newton's laws, the three laws in a linear sense, all we have to do is we just have to translate them angularly. So, here's what I mean by that. Newton's first law says a body in motion stays in motion. A body at rest stays at rest unless an external force is applied to us. So in an angular sense all that means is a rotating body will constantly rotate. A body that's not rotating will not rotate unless acted upon by an external torque. So instead of linear motion, we're talking about rotation. So an object that's not rotating, it's not going to rotate. And an object that's rotating, it's going to constantly rotate unless we apply an external torque. Not a force, but a torque. This is basically the conservation of angular momentum. The second law is the law of acceleration. It says that a force applied to an object causes an acceleration that's proportional to how big that force was and in the direction of the applied force, which is F= ma. What I want you to do is I want you to try to think of the angular version of that. So take 30 seconds and try to translate that into an angular sense. You don't have to explain it to me. Just think about it. So it says a net torque applied to a body will produce an angular acceleration that is directly proportional to the magnitude and direction of the applied torque. So if we apply a large torque to an object, it's going to angularly accelerate a lot. A smaller torque would cause a slower angular acceleration. If that torque is clockwise, that object is going to accelerate clockwise. So instead of F= MA, we get torque= I alpha or the moment of inertia* angular acceleration. So, some of you might be wondering, well, how is torque equals moment of inertia time angular acceleration? I thought torque was force time distance. And that would be a very good question. So the the torque that we've calculated so far uh torque equals force times distance that's torque in a in a static situation right at a moment in time. Torque= I alpha is a dynamic analysis of torque. It's the torque in a non-static situation. So that's how those two differ. So, make sure you write this formula down so that we could do the next problem. So, for this practice problem, I'm going to walk you through it and then the last one I'm going to have you do on your own. The quadriceps insert on the tibia at an angle of 30° 003 m away from the knee axis of rotation. The lower leg and foot have a mass of 4.5 kg and the radius of girration is 23 m. If you want to achieve a knee joint angular acceleration of 1 radian/s squared, how much force do the quadriceps need to produce? So, here's our diagram. Our quadriceps are producing force at an angle of 30° 03 m away. So, we're going to list your known and unknowns. So, I've given you some known. I gave you the mass. I gave you the radius of geration. I gave you the angle of pole of 30°. I gave you the distance of 03 m. and I gave you angular acceleration. What we don't know is we don't know force. What we're going to do is we're going to take these variables and we are going to insert them into the equation of torque= I alpha. So we plug our variables in. Torque= I alpha which is mass time radius of geration squar time angular acceleration. So torque equals force * 03 m. Um that's the distance we got. mass is 4.5 kg. The radius is 23 m. And our angular acceleration is 1. Remember we only square the 23. So 4.5 *23 2 * 1 will give us.24 kg m^ 2/s squared. It's a bit of a mouthful. We divide both sides by the distance to isolate force. and our force is 8 newtons. However, we have an angle of pull of 30° which means we found this rotary component. To get this resultant quadriceps force, we have to divide by the sign of that angle. And so our final answer is that force equals 16 newtons. Remember, if I give you an angle of pull, you have to make sure you're solving for what I ask. If I don't give you this angle of pull, then this initial force is enough. If I give you that angle of pull, you have to incorporate it to get the right component of force. So, take four minutes, try to do this one on your own, and then we'll go over it together. So here we have our diagram and if we list our known and unknowns here they are mass is 7 kg the radius is8.18 m angle of pole is 50° the distance is 03 m and our angular acceleration is 1.9 radians. per second squared and we are trying to get force. So we're going to plug our variables into the equation. 7 *.18 2 * 1.9 will give us43 kg m/s squared and we're trying to get this force component. We divide both sides by 03 and we get 14.33 newtons. But we have an angle of pull of 50°. So that 14.33 is this rotary component here. So if we divide that by the sign of 50 or the angle of pull, our force requirement is 18.71 newtons. So that's how we solve for Newton's second law of angular motion. Now for Newton's third law, the third the third law in a linear sense says that when a body exerts a force onto another body, the second body exerts a reaction force to the first body that's equal in magnitude but opposite in direction. For every force, there's an equal and opposite reaction force. Well, the angular version of that, for every torque applied by one body on another, there is an equal and opposite reaction torque exerted by the second body to the first. So for every torque, there's an equal and opposite reaction torque. That's Newton's third law in an angular sense. So when we're running, walking or running, um our arms and legs, they swing in opposite directions. The right leg comes forward while the left arm comes forward. Why is this the case, right? Like why don't we go right leg, right arm, left leg, uh left arm? The answer for this has to do with torqus and reaction torqus. So this is a bird's eyee view of someone running. So we're looking down at someone and that black circle represents the axis of rotation. So just like a figure skater rotates around that central axis of rotation. That is what that little black circle represents. So let's say we're walking and our right leg is moving forward. In order to move forward, we push back on the ground. The ground applies a force back to us and that allows us to go forward. Well, that force we apply is at some distance away from our central axis of rotation. So, let me know in the chat, does this force relative to that axis, does it produce a clockwise or counterclockwise torque? counterclockwise. Now if our right arm comes forward while moving with our right leg, that right arm creates a clock a counterclockwise torque as well. So if we move with ipsilateral arm and leg swing, they produce torqus in the same direction which over twist you and it makes walking in a straight line difficult because you're always being rotated. But if we walk with contrlateral arm swing, that left arm swing will create a clockwise torque. And so the torqus go in opposite directions which basically balance you instead of overt twisting you. So, there's another example of this. So, I have a video for you. Um, pay attention to what the uh what the bo the hitter's body looks like. Try not to pay attention to where the ball is going. Pay attention to the person hitting the ball. There's a slow motion at the end. So, if you pay attention to the hitter, right, what you'll hopefully notice is that when he goes up to hit and his upper body moves one way, his lower body rotates the opposite way. So, the upper body is applying a torque to the lower body, the lower body applies a reaction torque to the upper body. When we run, the same thing happens. Your upper body twists one way and your lower body twists the other way. So those are examples of Newton's third law of angular motion. So to wrap up, what is angular work? What is angular power? And how can they be positive or negative? What are the angular versions of Newton's three laws? And how do we solve for torque equals I alpha questions? And remember how to incorporate the angle of pole. What are two examples of Newton's third law of angular motion. So that concludes angular kinetics. We'll do tissue biomechanics next week. If you have questions, stick around. If not, I will see you in lab or I will see you in meetings or I will see you next week. Have a good weekend.