point and go over to a demo here. So here is my generic

body, I've taken it, deflated basketball so

it's just some blob and we're rotating it now around

the Z axis. And when I rotate it around the Z axis in

the plane there is a point Q. And every point that's

parallel to the Z axis at the point will have the same

linear velocity as it goes around the Z axis.

So, back to my theory here. I've got

the angular momentum about P and I put in for R now,

the coordinates from P to any point. Okay?

Even though that point will have the same angular velocity here or here.

And so I cross that with my velocity.

Instead of crossing it with the velocity here, it's going

to have the same velocity as the velocity at Q, okay?

All right, so I've got the angular momentum defined

as R cross vQ, where R is the position vector

to any point on the body and it's integrated over the entire body.

And now we're going to use relative velocity

kinematics to find out what that velocity is

for this point or any point along the

line, including point Q, the comp, companion point.

And so we've got R crossed with okay, v of

Q, we're going between two points on the same body.

And so v of Q will be v of P plus omega.

How the body is, is, the angular velocity of the body

about the K axis crossed with however far we go in

the X axis and however far we go in the Y

axis and that'll give us our, the velocity of point Q.

Again, the same as the velocity of this point

out here, and every point in, on, on the body

that's parallel to that Z axis and we do that

over and over and over again, for the entire body.

Okay, so if I do those cross products, I'm going

to pull this term out and move it over here.

Okay?

So I've got the integral of R dm.

I'm taking the dm with it and I'm breaking it into a second integral.

The integral of R crossed with this velocity of Q, excuse me not the

velocity of Q but a omega cross x plus y. All right.

Now this R, the integral of Rdm is the mass times the position vector

from the point P to the mass center by the definition of the mass center.

And we end up with this.

And now if you this right hand side if you take

everything that's in the integral. And we can use the

vector identity that we've used before to rearrange that.

And so this is this term B is going to be the omega

k and x plus y or xi plus yj is going to be C.

I can go ahead and do that mathematics and this is the result I end up with.

And so if I substitute that result back into my

expression for the angular momentum, this is what I get.

And, now we're going to define these integrals.

This integral over the body of x squared plus y squared dm is going

to be called the mass moment of intertia about the Z axis through point P.

This minus the integral over the body of

xz dm, is called the product of inertia with respect to the X and Z axis.

Through point P and this is also a product of inertia term

with respect to the Y and the Z axis through point P.

And so I'll use that nomenclature these mass moments of inertia and

products of inertia to simplify the equation, and it looks like this.

And so this is the angular momentum. Well we, this is one of our objectives

for today's lesson.

The angular momentum about any point for a body in 2-D planar motion.

If we take a special point, if we say that P is either the mass center, if P is

the mass center, the distance from R to P to C will cause this term to go to zero.

Or if P had zero velocity, that makes this velocity turns goes to zero.

In either of those two cases, the angular momentum

simplifies down to this.

And in the future modules we'll talk physically about what the

meaning is of these mass moments of inertia and products inertia.

But you can see here that there is a little

bit of an analogy remember what the linear momentum was.

Linear momentum is mass times velocity and the angular

momentum, although it's a little bit more complicated here.

The angular momentum is given the symbol H instead of L.

The velocity now is at an angular velocity instead of a linear velocity.

And instead of using M we're going to have

these mass moments of inertia and products of inertia.

Which are going to include information about the

mass and the geometry of the body.

And we'll look at those in more depth in future modules.