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# Constant angular momentum when no net torque

## Video transcript

so right over here like I've done in previous videos we have a diagram of a mass we really should conceptualize it as a point mass although it look like a point it looks like a like a circle but imagine a point mass here and it's tethered to something it's kind of it's tied to a massless string a theoretical massless string right over here or functionally massless string it's kind of nailed down and let's say it's on a frictionless surface and let's say it has some velocity and right here we have the magnitude of its velocity in the direction that is perpendicular to the the wire that is that is holding it or I guess you say perpendicular to this the radial the radial direction now based on that we've had a definition for angular momentum the magnitude of angular momentum is going to be equal to the mass times this velocity times the radius and you could also view that and this is kind of always kind of tying the connections between you know translational notions and rotational notions we can see that angular momentum could be the same thing as well the mass times its velocity you could do that as the translational momentum Rho in the magnitude of translational momentum in this direction times R so once again we took the translational idea multiplied it by R and we're getting the rotational idea the angular momentum versus just the translational one and we could also think about it in terms of angular velocity and this comes straight out of the idea that this V is going to be equal to Omega R so you do that substitution you get this right over here now in previous videos we said okay like based on this and based on the idea that if torque is held constant then this does not change you you can you can describe or you could predict the type of behavior explain the behavior that you might see it a figure Eight's figure skating competition where if someone pulls their arms in while they're spinning and they're not you know applying any more torque to spin if they pull their arms in well if this thing is going to be constant because there's no torque being applied well their mass isn't going to change so they'll just spin faster and you do the opposite the opposite would be happening but you might have been left a little bit unsatisfied when we first talked about it because I just told you that I said hey look if torque is if there's no net torque then angular momentum is constant and then you have this thing happening but let's let's dig a little bit deeper and look at the math of it so you feel good that that is actually the case so let's go back let's go to the definition of torque and so the magnitude of torque I'll focus on magnitudes in this video the magnitude of torque is going to be equal to it's going to be equal to the magnitude of the force that is in this perpendicular direction times R times R now what is this this force well this is just going to be equal to the mass force F equals MA so it's going to be mass times the acceleration in this direction which we could view as which we could view as the change in in this velocity over time and we're talking about magnitude so I guess you could say the whole lot the magnitude of the velocity in that direction and then of course we have times R times R now if we multiply both sides of this times delta T we get we get and actually we do tau in a different color the torque in green we get torque times delta T torque times delta T is equal to is equal to mass times Delta V Delta V in that perpendicular Direction times R well what's this thing going to be what's this well that's just change in angular momentum so this is just going to be change in angular momentum and there's a complete analogy to what you you might remember from kind of the translational world the translational world you have this notate notion if you take your force and you multiply it times how long you're applying the force so let me do this in a different color so you multiply it by how long you are applying the force so this can't this quantity we often call as impulse impulse that's going to be equal to in translational momentum your change in translational momentum and if you have no force then you're not going to have any change in momentum or you have your conservation of momentum or it's not going to change it's just going to be conserved and then you can do all sorts of neat you know predicting where a billiard ball might go or or whatever else but you have the same analogy here the the analog for force and the rotational world is torque it's obviously its force times the kind of radial times that radial distance but it's if you take torque times how long you're applying that torque that's going to be your change in angular momentum so if you're not applying any torque if your torque is zero if your torque is equal to zero well that means that your Delta L is zero your angular momentum is not changing or you could say your angular momentum is is constant so if you don't apply any torque like once the figure skater is already spinning and he or she she's not pushing to get more to spin so she's not applying or he's not applying more torque well then your angular momentum is going to be constant but they can change they can change the rate of spinning by changing R by changing how far in or out R is or how far not the masses and obviously I said in the last video a figure skater is a much more complicated system than a point mass tethered to you know tethered to a tethered to a rope you could view a figure skater is going to be modeled by a bunch of point masses but hopefully this gives you the idea