[BLANK_AUDIO]. This is module 17 of three dimensional dynamics, and I'd like to begin with a recap of where we're at in the course. And so we have actually finished the kinematics portion of our study for three dimensional dynamics. Kinematics again are the geometrical aspects of the motion, relating things like position, velocity, acceleration, and time. And so we did angular velocity and angular acceleration of bodies in 3D. We did velocity accelerations, in moving reference frames. We looked at Eulerian angles, and then we finished up with, rotation matrices. And now we're going to move into kinema, kinetics, where we look at the forces acting on a body and relate it to that motion of the body. And so today's m, learning outcome is to review particle kinetics and Newton's laws for particles. And review Euler's first law for motion of the mass center of bodies. And now I've covered these actually in my two dimensional course. And so we're going to take a look back for the remainder of this module at, the development that I did in that 2-D course for particle kinetics and Newton's laws for particles. And then we're going to go on to Euler's first law for the motion of mass centers of bodies. And then we're eventually going to move into and extend this for 3-D motion. So, kinetics of particles and mass centers of particles. So far we've looked at the kinematics of particles and systems of particles, and that was the geometrical aspects of the motion, relating things like position, velocity, acceleration, and time. Now, kinetics relates the forces acting on a particle or a body to the changes in the motion of that particle or body. And so here's, as a re, as a review from my earlier courses, here's Newton's First Law. Remember now, Newton's laws apply to particles. And so the First Law says that if I have a balance of forces on a particle, then the particle is at rest or remains in constant velocity, okay? And so what I want you to do now is to write down, write out in your own words what's Newton's Second Law for particles. What you've should of come up with is that Newton's Second Law says that the time rate of change of the linear momentum or the momentum of a particle is equal to the sum of the forces acting on the particle. So no longer do we have a balance of forces, we don't have some of the forces equal zero. But we have some change in the momentum. And we're going to be talking about in this course, classical dynamics, where there are no relativistic effects, and so we have constant mass. And I can take the mass, which is constant, outside of the, the derivative and I get mass times acceleration. So this is your famous F equals MA for particles. And then finally write down in your own words what Newton's third law is. Okay, so Newton's third law says if I have two particles, the forces between those particles are equal in magnitude, opposite in direction, and along the same line of action, or colinear, so here's my notation. This is the force on particle one, due to particle two, is equal to negative, because it's in the opposite direction. The force on particle two, by particle one. So, equal and opposite forces. So now, let's extend these laws for particles, to Euler's Laws for bodies now. We want to look at bodies as a whole or systems of particles and then onto bodies. And so here I have a, a body composed of capital N particles. Where capital N is a large number. I've only drawn a few of these particles. But I could have millions of these little particles in this body. And so that's a system of particles there is a bunch of external forces acting on this system of particles or body so I'm saying that f sub i is the net external force acting on particle i and here's particle i right here. I've labeled it m sub i because it has a mass m sub i, here's particle j. So I would say there are i through j, i through n particles in the body. So a total of n particles. In addition to the external forces felt by particle i, the i particle also feels these internal forces from all the other particles in the body. And so, now you can write the Newton's Second Law for the ith particle, and so for the ith, try and do it on your own and come back and see how you did. And so what we have here is the external forces acting on particle i, plus the internal forces acting on particle i, or the total forces acting on particle i, equals the mass times the acceleration of particle i by Newton's second law. And then we can, for the entire system of particles or body, we can sum up all of those equations. So we're going to sum N equations over N particles and now I have, I put the summation sign on here in addition to all of these individual particles. We sum them up from i equals 1 to N for the total number of particles in our body. So my next question to you is, let's look at this second term, and can you tell me, can you simplify this second term for me? And after you think about that for awhile, come on back. Okay, so what you should have realized is that by Newton's third law this term is equal to zero because if I go through my indices from I equals one and J equals one, I have F one two, F two one. Remember they cancel out, F one three, F three one, they cancel out. F two three and F three two cancel out and so if I sum up over the entire body, because each of those internal forces are equal and opposite, they end up zeroing out. And so all I'm left with is the total summation of forces, external forces on our system of particles is equal to the total of these, particles mass times acceleration. And so here again is, is, is that same equation. And now I've drawn my my system of particles or body here again. And Ive got, and I'm looking at it from an inertial reference frame where I'll call that point O, and the origin of which I'm looking at it from, the body from. And I've got a position vector R to each of the particles. This is a position vector from O to the Ith particle, and then I also have written the position vector from O to the mass center of the, the body. And so since I've shown the position vector from O to the ith particle, I can sum up over the entire, body again and, and pull the double integral out because we know that for the acceleration, it's the second derivative of the position and so, these are equivalent. Once I've done that let's look at the definition of center of mass. If I sum the mass of each of the particles times their position vector, so I go to this particle, and this particle, and this particle, sum them all up and then I divide by the total mass of the body. That by definition of the mass center will give me a vector from O to C, the mass center of the systems of, of particle or the body. Okay, and I can rearrange that. And so I have m times R OC, instead of a summation sign. If I change this to a continuous body, so I have an infinite number of particles, then it becomes the integral of the position vectors over each little piece of mass or differential piece of mass in the body. Okay. We can now use this definition and put in for the summation of mR. For a continuous body, is equal to r, m-r-o-c, or the interval of, R dm. And, so, this is two forms for a continuous body I can write. And then I come up with what's called Euler's, first law. If I take the second derivative, remember, mass is constant. We're talking about classical dynamics. And so the second derivative of the vector from O to the mass center is just the acceleration of the mass center. So, what this says is for a system of particles or body, the sum of the external forces of the acting on that system of particles or body is equal to the mass times a very particular point, mass times the acceleration of the mass center. And so you can see Euler's first law for bodies is completely analogous to Newton's second law for particles. For particle we don't have to have the subscript of C here, but for the body we have to specify the particular point that we're talking about the acceleration. And so we, this is the equation that describes the motion of the mass center of bodies. So that was a look back at the development of the particle kinetics Newton's Laws for particles and Euler's First Law of motion in the mass center of bodies. And we'll continue on to look at Euler's Second Law for the motion of bodies and review that, in the next module.
