Hi. It’s Mr. Andersen. And this is AP Physics essentials video 96. It is on the conservation
of angular momentum. Now how is angular momentum different from linear momentum? Remember angular
momentum is due to rotation. And you might think to yourself, why does the earth continue
to rotate? It is due to the conservation of angular momentum. It still maintains some
of that angular momentum from the formation of the solar system. It is gradually slowing
down and it will eventually stop, but it is not something that we have to worry about.
So conservation of angular momentum will always be maintained as long as we have no net external
torque acting on an object or acting on the system itself. Now unlike linear momentum
we can calculate angular momentum in one of two ways. If it is a point object in rotation
then our angular momentum is equal to the radius times the linear momentum of the object.
So how fast it is going in this direction times its mass. If we have an extended object
we use a different equation. If it were an extended object the angular momentum is equal
to the moment of inertia, so that is how much material we have and how far it is spread
out from the center times the angular velocity of the object. But in both cases, as long
as there is no net external torque acting on it, the angular momentum before equals
the angular momentum we have after. Now again there are analogs for linear momentum. Remember
in linear momentum, during a collision like this, the momentum is simply transferred between
the two. Now in linear momentum, linear momentum equals mass times the velocity of the object.
But we have analogs when we are talking about angular momentum. So if we have a rotating
disc like this, how do we calculate its angular momentum? Well our equation is L = I times
omega. In other words the angular momentum equals the moment of inertia times the angular
velocity. Let’s say we have a point object like this in rotation. Then we are going to
use a different equation. Now L equals r p where r equals the radius, the distance from
the center. And then this is going to be the linear momentum. It is the linear momentum
we were talking about just a second ago. So that is going to be a product of the mass
times the velocity. Now let’s look at this qualitatively. Let’s say we have a tether
ball. So it is a ball attached to a string and it is rotating around a pole like this.
And then we were to quickly shorten up the radius. What would happen to the speed? It
is going to go much faster. Okay. Now that does kind of make sense but why does that
happen? Well this was our angular momentum before. It was a product of the radius times
the linear momentum. Now we have decreased r. But our L is going to stay the same. So
we are decreasing r, what do you think happens to the mass? The mass has to stay the same.
And so what happens to the velocity of the sphere? It is going to go up. So it is going
to go way faster. We could apply the same thing to an extended object. Let’s say we
are looking at a figure skater and they are pulling there arms in. So let’s say they
are rotating. So it is an extended object. So they are rotating fairly slowly. And then
what happens when they pull in their arms? Well we are pulling more of that mass towards
the center. And so what is happening in this case is we are decreasing the moment of inertia.
So what is going to happen to our angular velocity? It is going to speed up. And so
the angular momentum is conserved. It is just that we are increasing the angular velocity
because we decreased that moment of inertia. And so we could do this quantitatively as
well. Let’s say we have an object. Now we will do a little bit of math. So this is the
radius from the center. This the linear velocity, how fast it is going in that direction. And
then this is going to be the mass. And so first of all let’s calculate the angular
momentum before we change the system. And so I am going to plug in, here is my radius.
Here is my mass. And then here is going to be my linear velocity. So I could calculate
my angular momentum. And I am going to leave all of my significant digits in it. Because
we are going to see what happens if we take the radius and we cut the radius in half.
So we are going to decrease that radius. So now it is 0.16 meters. And so I know this,
I know my initial angular momentum. And now we are going to solve for my angular momentum
after we have reduced the radius. And so I could plug in these values. It is now 0.16
here. The mass stays the same. But now we have a new final velocity. And so we could
solve for that final velocity and it is going to be twice the speed that it was before,
before it was 3.0 meters per second. We could do the same thing if we have an extended object
like this. So let’s say we have a rotating disc. It has an angular velocity of 9.2 radians
per second. Let’s say it has a moment of inertia of 13 kilogram meters squared. And
so let’s say we take an identical disc as it is spinning and we drop it on it from above.
What is going to happen to the angular velocity? Well try to solve this one and I will put
an answer down in the video description below. But did you learn to make qualitative predictions
about what happens to an extended object or point object as we change it due to conservation
of angular momentum? And then finally could you make calculations related to angular momentum
when there is no net external torque acting on it? I hope so. And I hope that was helpful.