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## Torque, moments, and angular momentum

# Constant angular momentum when no net torque

## Video transcript

- [Voiceover] 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 doesn't look like a point, it looks 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. Where functionally a mass of string and 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 wire that is holding it, or I guess you can say perpendicular to 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. Well, the mass times its velocity you could do that as the
translational momentum row 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 can also think about it in terms of angular velocity. Now 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 can describe or you could predict the type of behavior, explain the behavior that you might see. The figure skating competition where if someone pulls their
arms in while they're spinning and they're not, you know,
applying anymore torque to spin, and if they pull their arms in, well 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. When 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 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 this is going to be mass times the acceleration in this direction which we could view as, which we could view as the change in this velocity over time, and we're talking about magnitudes. I guess you could say, it's through the magnitude of velocity
in that direction. And then of course we have times r. Times r. Now if we multiplied both
sides of this times delta t, we get and actually we do
tau in a different color. We do 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 might remember from kind of the translational world. The translational world
you have this notion, if you take your force and you multiply it times how long you're applying the force, should do this in a different color. So we multiply it by how long
you are applying the force. So this quantity we often call is impulse. Impulse. That's going to be equal to your change in translational momentum. Your change in translational momentum. And if you have no force then you're not gonna have
any change in momentum or you're gonna have your
conversation of momentum, or it's not gonna change, it's just going to be conserved. And then you can do all sorts of neat, you know, predicting
where your ball might go or whatever else. We have the same analogy here. The analog for force
in the rotational world is torque, it's obviously this force times the kind of radial, times
that radial distance. But 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 constant. So if you don't apply any torque, like once this 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, they're angular
momentum's going to be constant but they can change their rate of spinning by changing r, by changing how far in or out r is, or how far enough the masses are. 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 rope. You could view a figure skater as kind of being modeled
by a bunch of point masses but hopefully this gives you the idea.