[BLANK_AUDIO] Hi and welcome to Module 30 of Two-Dimensional Dynamics. For today's learning outcomes we're going to define or derive actually, angular momentum about any point in a rigid body, for a rigid body in 2-D planar motion. And we're also going to define something called Moments of Inertia and Products of Inertia. And so, first I want to recall from an earlier module when we talked about angular momentum and develop it for a, a rigid body in 2-D planar motion. And so, if we had a particle or a system of particles, each of those particles has a momentum m times v. And if I take and I cross r, a position vector with that m times v that becomes the moment of momentum, and we call that the angular momentum, and we give it the symbol H. Now if let's extend that to a rigid body and so we're going to take an infinite number of particles and make this into a, a rigid body. Or, and with now we have around some point P a position vector crossed with a differential piece of mass d m with its velocity and we integrate it over the entire body. And that becomes our definition of the angular momentum, about point P. And so here's where we left off. And now let's restrict that so far I haven't restricted the motion at all. It's been three dimensional but I can now restrict it to two dimensional motion. And about some Z axis, and so, here is my body, okay? It's just the generic body. It's going to be rotating around this Z axis, okay? So it's going to only be in a pla, planar motion. So, will look at a point Q in a reference plane. And any point that lies along a axis parallel to the Z axis at that point Q will have the same velocity, if the body is moving in planar, rigid body, two-dimensional motion. And I'll show you that with a demo in a second. But dm will now be any point in the body. And Q is in the plane of motion and it has the corresponding x and y coordinates of the Q point. And so, we call Q a companion point to all of these points that are along the line that is parallel to the, to the Z axis. And so let's go ahead before I go onto the next 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.