- [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.