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Main content
Current time:0:00Total duration:11:19
AP.PHYS:
CHA‑4.D (EU)
,
CHA‑4.D.2 (EK)
,
CHA‑4.D.2.1 (LO)
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CON‑5.E (EU)
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CON‑5.E.1 (EK)
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CON‑5.E.1.1 (LO)

Video transcript

if we have some mass m and it is moving with some velocity let's say the magnitude of that velocity we say is V we know that this object right over here has momentum translational momentum and that momentum and we use the Greek letter Rho to represent momentum translational momentum is defined as being equal to the mass times the mass times the velocity this is all review we have other videos where we talk about translational momentum and one way to think about it is well how hard is it to to to stop this thing I mean you literally in everyday language you think well how much momentum does something have you know the more momentum is has the harder it's going to the harder it is to just to stop it in some way and so we know if we want to get a little bit more mathematical that if we want to change momentum we have to apply force for some amount of time and so the magnitude of our force the magnitude of our force times the duration of the time that we apply it for Force Times time and this is called impulse and this is once again I'll review this is equal to change in momentum so this is equal to change we do this in that yellow color that is equal to change in momentum so if you don't have any impulse especially if you don't have any net force acting on an object its momentum is going to be constant you have a conservation of momentum and we use that idea in all sorts of interesting physics applications in the world and especially a lot of cases using billiard balls and whatever else so now let's try to take a similar idea but go into the rotational world so let's imagine you have a mass for the sake of this we're going to assume it's a point mass so you have a mass there and let's just say it's attached by essentially a massless wire a massless wire to to you know it's just nailed down right over here and so this right over here would be its center of rotation and so you could imagine if someone applied a torque to this mass this mass could start rotating in a circle and you could just assume that it's maybe it's sitting on a you know the screen this this is a kind of a frictionless surface there's no air resistance and so then it will if you apply a torque here it will start rotating and so you could think about well there might be an idea just as momentum is this idea of well how hard is it to to stop something you might say well how and this is stop translating something something from moving you might think well maybe there's a similar idea of how hard is it for something to or how hard is it to to stop rotating something and you could imagine that that idea has been defined and it has been defined as angular momentum so let me make this clear this right over here is momentum mo momentum and over here we'll talk about angular momentum angular angular momentum angular momentum and actually both momentum and angular momentum are vector quantities so here I just wrote the kind of the magnitudes of velocity and momentum but momentum is a vector and it could be defined the momentum vector could be defined as equal to the mass which is a scalar quantity times the velocity times the velocity vector now the same thing is true for angular momentum but I'm going to stay focused on the the magnitude of angular momentum angular momentum can have direction as you can imagine you can rotate in two different ways but that gets a little bit more complicated when you start thinking about taking the products of vectors because as as you may already know or you may see in the future there's different ways of taking products of vectors but just to get the intuition of angular momentum I'll focus on the magnitudes so angular momentum is defined and the letter used is L I did a lot of research to try to figure out why it is called L and I could not find a good reason so in the in the message board below if anyone has a good reason I would like to know it's why angular my angular momentum is called a lot of the best arguments I saw is that almost everything else was all the other letters were used up for other ideas in physics but anyway angular momentum is defined and it's defined very similarly just as kind of torque is the thing that can change how something rotates and force is the way that something changes how something translates it and torque is Force Times distance from the center of rotation angular moment end of the rotational world is is is defined in a similar way you kind of take the analog in the translational world and you multiply it times the distance from your centre of rotation so angular momentum is defined as angular momentum is defined as mass times velocity times velocity times distance to from the centre of rotation so let's call this distance right over here R R for radius because you could imagine if this is traveling in a circle that would be the radius of the circle MV R and actually let me be a little bit more careful here it's the magnitude of the velocity that is perpendicular to the radius sometimes it might be called a tangential velocity so this symbol right over here this is the magnitude of the velocity that is perpendicular to the radius so it would be this it would be that magnitude right over here this is what we define as angular as angular momentum and what I will tell you here is just as in the absence of a net force momentum is constant we know and I haven't shown it to you I haven't proven it to you yet mathematically but in the absence of torque so if torque is equal to zero we do torque in pink if torque is equal to zero if there's no net torque going on here if the magnitude of torque is equal to zero then then we will have then we will have no change no no change in angular momentum and we will look at that mathematically in a few seconds but just from this there's a very interesting thing that arises and something that you might have observed that even the Olympics or another thing and this is the idea that you can by changing your radius you could actually change your tangential velocity and as we've seen in previous videos tangential velocity is closely related to angular 2 angular velocity so let's explore that a little bit so when we write it in the world where well actually you see it straight out of this if L is constant if R went down so let me write this down so should we rewrite it over here so L whoops so let me write it L is equal to mass times tangential velocity or actually well tangential velocity or the velocity that's perpendicular to the radius times the radius now what happens if we assume that this is constant if we assume that this is let me write this down if we assume that there's no torque so we're in this world so this over here is going to be constant if this is going to be constant so what happens if we were to reduce R somehow you know this this wire started to reel in a little bit or started to wrap around here and actually that's actually a reasonable thing you could imagine as it rotates it starts to wrap around this thing so the wire gets shorter so if R goes down if R goes down and this is constant the mass isn't going to change well if L is constant mass isn't changing R is going down tangential velocity or the velocity is perpendicular to the radius is going to go up velocity that is perpendicular to the radius is going to go up and if we want to think about it and we can think about it in terms of angular velocity we know we know that angular velocity which we would measure in radians per second we would use a symbol Omega Omega is defined and we go into much more depth in this in other videos as tangential velocity tangential velocity the velocity the magnitude of the velocity that is perpendicular to the radius divided by the radius divided by the radius or if you solve for tangential velocity you get V is equal to is equal to Omega R is equal to Omega Omega Omega R and so if you substitute back into this really this definition for angular momentum you get you get angular momentum is equal to mass times this times R so mass times I'm just substituting for velocity here times Omega R Omega R times R which of course is just Omega Omega R squared so once again we do the same exercise if our radius goes down what happens to our angular velocity remember we can measure this in angles per second or radians per second well if this goes if this is constant remember we're assuming that there's no net torque being applied to the system so we're still in we're still in this world right over here if we assume that this thing is it's changing but the radius were to change what's going to happen to Omega well Omega is going to go Omega is going to go up now likewise if the radius got longer if the radius got longer so the radius got longer what's going to happen to Omega Omega is going to go Omega is going to go down so if you reduce the radius you're going to start spinning faster if you increase that radius you're going to start spinning slower and you have seen this where I think there's a high likelihood that you have seen this probably at the Olympics when you have seen figure skaters where they might start spinning and they have their arms out so when their arms are out you could say that their radius is further out and obviously a figure skater is a much more complex system than a than a point mass you can imagine a figure skaters a bunch of a point masses how much detail well you could just model a figure skaters a huge number of point masses at different radii and you would want to sum up their angular momentum but the the essence of what happens is happening is is when her arm is out the the average radius for when you're calculating all of the point masses in her arms and all the rest the average radius is higher and then when she pulls them in when she pulls them in that radius goes down and her angular velocity goes up and you see that they start spinning and then without applying any torque when they pull their arms in they start spinning faster and then if they push their arms out without once again without applying any torque they start spinning slower