[BLANK_AUDIO]. Hi and welcome to module 18 of Three Dimensional Dynamics. Today's learning outcomes are to review the definition of angular momentum, and to review Euler's 2nd Law for the motion of bodies. And I actually did these two topics in my two dimensional course again. And so we're going to take a look back at that earlier that earlier course, and then we'll come back and move onto three dimensional applications of these laws. So, today's lesson is quite a little bit of theory, but it's important theory that we're going to use throughout the course. And so I, I hope you can stay with me and follow each of the steps. You should be able to. And so let's look at Euler's 2nd Law. First of all, let's recall Newton's 2nd Law for an inertia reference frame. Newton's 2nd Law says that the sum of the forces acting on a particle is equal to the time derivative of the linear momentum of that particle, or the momentum of the particle, since a particle can only have linear momentum. Then if I take a system of particles, I have a bunch of particles here I can then cross. I can look at the system particles from some inertial frame F. Maybe the Earth or some fixed frame, inertial reference frame. I can look at it and I can draw a position vector from the origin of that fixed frame to each of the particles. I've done that there. I've crossed that position vector on both sides of my equation for Newton's 2nd Law. And I know from way back in my earlier course, Introduction to Engineering Mechanics, that R cross F is defined as a moment. So I've got the moment about O is equal to the time derivative of this stuff on the right hand side. And this stuff on the right hand side, it's the time derivative of what we define as the angular momentum. And we give it the symbol capital H. And so it's, it's like the moment of momentum, it's called the moment equation, so we have R cross mv, mv is the momentum. So we have the moment of momentum, or angular momentum. That's for a fixed reference frame, let's look for any arbitrary point P. For an arbitrary point P, the angular momentum about point P is the angular momentum about the mass center plus a position vector from this arbitrary point P to the mass center crossed with the linear momentum of the entire body. And so, H sub P again, angular momentum about point P. H sub C angular momentum about mass center C. And this is the moment of linear momentum about point P. So, You can actually see that development for an arbitrary point P in any standard dynamics textbook. So let's go back now and look at the system of particles as referenced from a point fixed in an inertial reference frame, and this is the what we had come up with. So, I have the sum of the moments about O, and I can use the product rule here to take the derivative. That's times the time derivative of the position vector crossed with mv, plus the position vector crossed with the time derivative of mv, which is ma. But, you'll note that the position, the derivative of the position vector, Ri, is the same as Vi and so this gives us Vi crossed with Vi and we know that Vi crossed with Vi or Ri. Dot crossed with Vi is then going to be equal to zero. And so what I'm left with is, Euler's 2nd Law for a point fixed in an inertial reference frame is the sum of the moments about point o, is equal to the summation of R cross m a. Or, the sum of the moments about point o is equal to the time derivative of the angular momentum. Of point o, for the definition of angular momentum. So that's one result for Euler's 2nd Law, or the moment equation for a point fixed in an inertial reference frame. The sum of the moments equals time derivative of the angular momentum about point o. Similarly, and again you can look at this in any standard reference texts for the development, you can find that Euler's 2nd Law about the mass center is equal to the time derivative of the angular momentum about the mass center. So here are the results. A moment equation about a point fixed in an inertial reference frame, about the mass center. However, in general, you need to note that this form of some of the moments equals the time derivative of the angular momentum, only works for a point fixed in an inertial reference frame or the mass center. It does not work for an arbitrary point P. And so in general, the sum of the moments about P does not necessarily equal the timed derivative of the angle of momentum about P. To show what that is, let's look at, or recall our knowledge of equivalent force systems from way back in my first course Introduction to Engineering Mechanics. We saw by a balance of moments that the sum of the moments about P balances with the sum of the moments about c plus r cross F. So that's just a balance of equivalent force systems shown from this picture. Now in this case, the dynamics we've developed that the sum of the moments about c is equal to the time derivative of the angular momentum about c. We know that force, the sum of the force is the time derivative of the linear momentum, or d/dt (mvc). And so we get the relationship for the sum of the moments about P is equal to the time derivative of the angular momentum about c, plus rPC cross maC. And so it's not in this form, as shown up here, but for an arbitrary point it's in this form. So here's a summary. Some of the moments about o or c is equal to time derivative about, of the, angular momentum about o or c, respectively. That form is not true for an arbitrary point. For an arbitrary point this is, this is true. And we'll use this, these equations, quite a bit in the remainder of the course and so it's, important to have a, a good understanding of those. And I did want to make one other point. If you recall, lets look at, we had this sum of the forces equals the time derivative of the linear momentum for a body. And so what you'll see is, this is completely analogous. [BLANK_AUDIO]. So this'll maybe give you a little bit better feel. Sum of the forces is equal to time derivative of the linear momentums. Sum of the moments is equal to the time derivative of the angular momentum. And so you see the, the connection there. Okay, so that was a look back at my two dimensional course. We've reviewed angular momentum. We've reviewed Euler's 2nd Law for Motion of Bodies, and when we come back next module we'll continue on with extending this to three dimensional motion.