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Kine 3030 Lecture 22

welcome back to can 30 30 online today we're going to look at momentum mostly angular momentum although we will talk about what linear momentum is we're going to define linear and angular we're going to look at the momentum of a single and a multi-segment body and [Applause] talk about conservation of angular momentum and transfer of angular momentum so let's say you are working with a figure skater and you want to work with them on their jumps and you want to get more revolutions in the air how do you do that we'll come back to that be an example of a problem that uses angular momentum so momentum in general is a quantity of motion that's what it is it's a quantity of motion in the linear sense momentum and i know it's another capital m same as i sometimes use for moment which is a torque there are just so many uh letters in the alphabet so in this case though it's momentum and you know that because it's mass times velocity so the quantity of motion you have can be defined in a linear sense of mass times linear velocity in an angular sense the quantity of motion you have we're looking at very similar parallel measures so instead of mass it's going to be moment of inertia remember that mass times radius of gyration squared so it includes both the mass and the distribution of the mass times the angular velocity so in angular momentum um you can see that there's going to be three factors that affect so there were two factors mass and velocity in the linear three factors with angular you've got the mass you've got that radius of gyration squared and you have the angular velocity so that distribution is really important because of course it's squared um and and the um so increasing mass or velocity will also angular velocity will also increase your angular momentum if you have a set amount of angular momentum and you increase the velocity the well sorry increase the mass really can't increase the mass so if you increase the moment of inertia by increasing the radius of gyration then the velocity will drop we'll see that when you have a constant angular momentum but just understand that those three factors you know if they go up then your angular momentum is going to go up recognizing too that your body is multi-segmented and the moment of inertia or sorry the angular momentum h for multi-segmented body is calculated as the sum of the moments of inertia of individual body segments and therefore it's you know your angular momentum is going to be include that sum of the individual body segments so each segment angular momentum is the sum of two components the local term which is the angular momentum around its own the segment's own center of gravity and then that segments angular momentum around the center of gravity of the whole body so of the total body you end up then with these two terms the local term which is the segment moment of inertia around its own center of gravity and it's the segments angular velocity that you're looking at there so if just that one body segment is rotating in time that's the angular velocity you're going to look at for the segment and then you're going to have the remote term which is the segment mass the r in this case that you're going to square is the distance from the center of gravity of the whole body to the center of gravity of the segment itself and the angular velocity of that center of gravity of the segment center of gravity around the whole body center gravity which is where your axis of rotation is so you can see you've got the local and the remote term for each segment and this shows you that whole piece that you've got the calf and foot rotating around the center of gravity of the whole body that's the remote term and you also have the segment rotating around its own center of gravity which is its local term and then you would add up that angular momentum for all the different segments to get the whole bunch so you would look at the local remote term for each segment and then you would add up all the segments to get the total angular momentum total amount of motion angular motion for the whole body a very important factor in angular momentum is conservation of angular momentum the reality is that once we're in the air the total angular momentum of the system when we are in the air when when there's no other torques external torques acting on the system um stays constant so if you start as a diver and you take off on your dive and you have a certain amount of angular momentum um that's what you have through the whole thing well what does that mean think about what angular momentum the equation for it is so angular momentum equals i times angular velocity does that mean that if h is fixed if that angular momentum is fixed that the angular velocity is fixed no um h is fixed but we have this wonderful ability to switch around between the angular velocity and the moment of inertia so a diver can modify their angular velocity by changing their moment of inertia how do they change their moment of inertia well obviously they're not going to change their mass in the middle of a dive not happening but if they talk you can see them go faster if they go into a layout position they're gonna go slower because they they have a certain amount of angular momentum and uh when they increase their moment of inertia by increasing their radius of gyration which means getting more distributed which means going into that layout type position um they will slow down and vice versa if they tuck they're reducing the radius of gyration which is reducing their moment of inertia which therefore allows them to increase their angular velocity the diver um as i said can change their their velocity by changing that radius of gyration and around the principal axis of rotation which is around the center of gravity of your body however doesn't mean because you have this certain amount of angular momentum it doesn't mean you have to keep it all under the same around the same axis and around the same direction you can change your angular momentum that might be in a somersaults uh rotation into a twisting rotation for example how do they do that they do that by doing something that's that actually while they're in the air if you did this that actually requires angular momentum because you've got two arms going around and it's not symmetrical that actually is going to take from because you only have a certain amount of angular momentum what you took off with is what you have so if you do something that requires angular momentum it's going to take it from the other um the angular momentum you have and it's actually going to change that angular momentum to different axes you can do it by doing things around your hip too so that's when you're heir that's the conservation of angular momentum the oh i will also say with the conservation of angular momentum um that is the part with the cat and i there's a description in your textbook and i'm sure you can find videos of it on youtube that if a cat ever got dropped or fell upside down they always land on their feet and they do it by they don't have any angular momentum to begin with but they create angular momentum so if i go if i'm in the air or if i'm on something that is that will twist like a chair you can do this if you have a chair that has uh um that rotates if you go like this what you'll find is the bottom half of your body rotates and that's what cats do around several axes so that when they land they are on their paws that's for the angular momentum when you're in the air it's in the air you have conservation of angular momentum there is no external torque acting on you so how do you get that angular momentum in the first place i mean if you want to change the angular momentum that you have you have to apply an external torque what exactly is that we often talk about the idea of an eccentric thrust which is really just a force so if i have a force ground reaction force that is not going through my center of gravity that's going outside of my center grout if the the actual total ground reaction force the resultant ground reaction force goes ahead of my center of gravity then i'm going to um it's it's giving me a backwards rotation um backwards torque that's really a torque an eccentric thrust is just another term for a torque that isn't going through around your center of gravity and so if as i took off just to take off my force is my ground reaction force before i leave the ground once i'm in the air there's nothing i can do but before i leave the ground um if that ground reaction force goes in front of me um then i would be doing a back flip and if at as i'm taking off it went behind my center of gravity then i would be doing a front flip so the idea here is that that ground reaction force the resultant ground reaction force is not going through your center of gravity and there as you take off and therefore you are going to get a a quantity of angular momentum before you take off um so that application of torque is not an instantaneous thing it you have a torque over time and this is impulse this is angular one but there's also a linear version which would be force times time and it turns out that if you have a you know a large torque over a short period of time or a small torque over an extended period of time you could give the same amount of angular momentum so basically torque times time or the area under a torque time curve is impulse and that is giving you angular momentum so whatever angular impulse you give the system will result in angular momentum and do i would highly recommend that you take the time to go through torque times time or angular moment of inertia times angular velocity one being the angular impulse one being the uh the angular momentum and you'll see that the units work out to be the same so angular change in angular momentum equals that torque times time or angular impulse as i said so we come back to our diver we're diver we is going to jump in a layout uh sorry we we have an example of a diver they jump in a layout position where their radius of gyration is 0.5 meters at an angular velocity of 4 radians per second and again it has to be in radians per second not degrees per second or revolutions per second in these kinds of problems has to be in radians what is the angular velocity one assumes a tuck position or a radius of gyration of 0.25 so this is somebody in the air there's no torques on them while they're in the air and therefore the um we're in a case of conservation of angular momentum so however much angular momentum they have just after they take off in the layout position is going to be the same amount of angular momentum when they go into the tuck position so um we're going to initially figure out what that moment of inertia is or mass times radius of gyration squared before uh while they're in the layout position so that's going to be 60 times 0.5 squared or 15 kilogram meters squared your moment of your angular momentum is then going to be that i times the angular velocity so that 15 kilogram meter squared times 4 radians per second or 60 kilogram meters squared per second and those are the units for angular momentum they're in the air so we're going to figure out their new moment of inertia after they tuck same mass 60 kilograms but now a radius of gyration of 0.25 which will square so we're ending up with 3.75 kilogram meters squared which you know as we've talked about before if we in this case we have the radius of gyration which means we basically divided that moment of inertia by four um we are rearranged and solved for our new angular velocity because the angular momentum is constant they're in the air it's not going to change we have conservation of angular momentum in for this diver so 60 kilogram meters squared per second divided by 3.75 kilogram meter squared which is the new um moment of inertia and that leaves us with an angular velocity of 16 radians per second and that shows you why you have to use those radians because those radians are radians disappear on the left hand side the radians disappeared on the right hand side the radians reappear as because it's a unitless unit so we know that it it represents a change in angle over time because that's what angular velocity is so we just put in that radians back in so we're going to watch now a a video we're going to come back to our original problem which had to do with a um how do you get more revolutions out of a figure skater i'm going to show you this youtube video [Music] every four years we watch the stakes for olympic figure skaters get higher as they try to increase rotation in the air with their triple axels and quadruple toe loops how do they do that it's a scientific principle that we asked olympic hopeful rachel flatt and deborah king a sports scientist funded by the national science foundation to help explain figure skaters make it look so easy leaping off the ice rotating through the air and landing in a graceful arm but make no mistake about it figure skating is one of the most demanding of all the events at the winter olympics for 17 year old rachel flatt the demands of training for the olympics have to compete with other demands i basically head to the rink at around six o'clock i skate from 6 30 to 7 15 and then um i go to school from 7 30 until about 12 30 and then um from there i go straight back to the ring when she's on the ice this ap physics student might want to consider the science that goes into her every jump to see the science in detail rachel agreed to train in front of a special high-speed camera called the phantom cam it has the astonishing ability to capture her jumps at rates of up to 1500 frames per second it's very cool watching myself on the phantom camera you get to see every phase of the jump and it's pretty incredible just to be able to see every aspect of it you know where exactly the placement of your arm is and where my head is everything is very cool we brought the footage to deb king a professor of sports science at ithaca college and an advisor to united states figure skating a finger screen jump is a really complicated skill that combines a lot of different motions in it they need to optimize a lot of different conditions in terms of speed force vertical velocities generating momentum and put it all together in a package with just the right timing to execute the skill deb watch the phantom cam footage to explain what rachel needs to get height and speed in one of her jumps the first factor is angular momentum in figure skating angular momentum determines how fast you're going to be able to rotate in a jump in the air so when you do a spin if you generate more angular momentum you have the potential to spin faster going into her jump rachel generates angular momentum by pushing off the ice with her foot [Music] pushing off the ice also generates vertical velocity which will help rachel get high enough to do her spins the vertical velocity comes from producing forces from their jump during takeoff this is sort of where action reaction comes into play as they contract their muscles and really powerfully extend their leg they're pushing down against the ice the ice will create a force up on them which gives them vertical velocity and it's pretty much the laws of projectile motion that the more velocity you have it take off this is vertical velocity the more she can keep going fast straight up the highest jump when rachel spins on the ice she exploits a law of physics to rotate faster and faster almost as if by magic how does she increase her speed while she's spinning the answer lies in her arms when rachel first starts to spin with her arms extended she rotates slowly but as she pulls her arms in closer and closer she starts to rotate faster and faster rachel's following an important law of physics the law of conservation of angular momentum you can't go to jail for breaking this law in fact you can't break it at all as you get smaller body position your speed goes up if you get a bigger body position your speed those down so they react in opposite directions back in her office deb king spins on an office chair to make the same point what i'm going to do is when i'm spinning i'm going to go from a really open position to a tight position you'll see my speed change so let's give that a try so this is pretty fast slower fast slow fast and i'm going to keep going the only way to stop is going to put my foot down and grab the table i'm really dizzy right now if rachel can keep her body straight and hold her limbs in close she'll achieve a higher rate of speed but it's not as easy as it looks it's hard to stay as straight as possible with every course you know you're basically being pulled out everywhere um so it's easier to stay in when you're crossed with your hands and your legs it just makes the jump more efficient but no matter how much attention she pays for the science of her jump rachel's road to the olympics will depend on her making skating look effortless you never know what's going to happen the unexpected is you know it's amazing so coming back to that problem after you've seen that video now you have several elements that all go together the reality is when this person is in the air they're a projectile so you need to have the highest vertical component of the velocity as you possibly can because the higher the vertical component of the velocity is at takeoff the longer they're going to stay in the air and the higher they're going to go the longer they're in the air the more opportunities they're going to have for revolution so that first part they go as fast as they can and then they dig in in order to to convert that horizontal velocity that they have while they're skating along the ice up to a vertical velocity up in the air is to create a make them into a projectile and to take that uh horizon horizontal um momentum that they have mass times velocity into a vertical momentum and a vertical takeoff velocity component of their velocity so that gets them up in the air as high as they can go as long as they can stay up there to get the twists they have to have that eccentric thrust around the spin axis so as they're taking off they have to actively spin as they're taking off so they can still transmit that torque to their to their body and um which is the angular impulse so at torque times time they so that by doing that the skates will actually transmit a torque to the skater and so that they have a certain quantity of angular momentum as high as possible when they take off once they take off once they're in the air they're going to and they're spinning around that axis they're going to want to come in as close as possible and as tight as possible you'll often see them cross their legs too so that their legs don't spread out and in here so that that is the smallest radius of gyration possible that you can get around your the spin axis and that will allow that will increase their angular velocity once that angular velocity here once they're about to come back down they spread out because if they landed while going that velocity they would fall so they actually do have to um at the not really talk about it there but you you have to do the opposite thing just before landing you have to spread out so that when you land you're not spinning too as fast as you could be so those are that's angular momentum and recognizing that something like a spin problem like this is actually a combination of projectile motion way back from um the first part of the course and angular matrix together which is going to give you that um in combination the most revolutions possible so we've talked about uh linear and angular momentum and how angular momentum is created around a multi-segment body um and that conservation of angular momentum and how important that is particularly in things in the air we'll see you next class

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