[BLANK_AUDIO] Hi, this is module nine of two-dimensional dynamics. I want to, first of all, look back at, where we're at in the course because we're moving right along. We've been looking at particles and systems of particles, and we've looked at the kinematics, the geometrical aspects of the motion, and now we're starting to look at kinetics. And so far we've looked at the Newton's Laws or, and Euler's Laws for assembling the equations of motions for a body. And today, we're going to move into the work energy principle. And so, we're moving right along. The second hardest, half the course we'll extend it to bodies in rigid planar motion to include rotation. Right now, we've only worked with translation and so, today's learning outcomes are to develop the work and kinetic energy principle for particles, systems of particles. And to calculate the work of a linear spring. So let's start by doing a, a, development of the work energy principle. I'm going to start with Euler's first law, which we developed when we looked at the, Newton Euler's law. Sum of the forces equals mass times the acceleration of the mass center of the body, or a system of particles. We're going to do a standard integration of that. But first I'm just going to dot each side by the velocity of the mass center. And that's going to equal this. Now that may not be all that obvious to you and so over here I've, I've put the details of the math that allows me to go from this step to this step. So you can go through there and, and, and, and, see how I, I, I, I did that. Once I do that, I'm going to integrate both sides from t1 to t2, over the period of motion. And, when I, integrate that, I've got on the left hand side force times velocity dt, but I'm going to put in for the velocity dr dt or the, the derivative of position. On the right hand side, you see I have m over two times the 2nd velocity or the velocity at time 2 squared minus the velocity at time 1 squared. And, so, on the left-hand side, then if I integrate again, I get F dotted with, the change in r. On the right-hand side I get one half m times v squared, minus, for t2 minus v squared for t1. And so, if I look at this, this what I've, what I've come up with. On the left-hand side, I have force dotted with distance which is essentially work done by the external forces acting on the body. And on the right-hand side, I have one half mv squared. And you should know from your, you basic physics, prerequisite for the course that, that is equal to the change in translational kinetic energy from t2, going from t1 to t2. And so, the work energy principle, write simply as the work done by external forces on a body is equal to the change in the translational kinetic energy. Or, or particles or systems of particles. And so, here it is written again. Work done equals change in kinetic energy. Let's talk about the work terms. Forces do work, okay. It's a scalar quantity, because I've got a vector dotted with a vector here. And there's a scalar quantity on the right hand side as well. The sign convention that I'm going to use that is that if the force, is aiding the motion of the particle or body, then it's going to be positive work. If the force is hindering the motion, then it's going to be negative work. As far as energy is concerned, one half mv squared bodies have energy. It's a scalar quantity as well. And, the sign convention is that the body is always going to have positive energy. So let's first of all look at the different ways that work can be done on a body. I'm going to look at a linear spring to start. So I've got this linear spring, it's got an unstretched length, L0. And then I'm going to stretch it so I'm out delta one so the total distance is L1 and that's position one. And then I, I've got my mass, it's going to stretch out to position two where delta two is the stretch from the un-stretched length of the spring. And so, this force due to a liner spring is conservative. And I want you to think for a second what I meant, mean by conservative and do a little research on that and then come on back and I'll explain to you what I, what it means. And so what conservative means is that it is path independent. Okay no matter how, if I, if I start at one point no matter how I go back or forth and I end up at the, the same point, the net work done is zero and there's a conservation of energy. There's no change in energy for the system. And so, for that work of the spring, I've got again, I've written my my derivation of what work is, the time integral of the sum of the forces dr. In this case, the force of spring is only in one direction, so I'm just going to work with scalars. The force in the spring, by Hooke's Law, is k times the stretch or, k times delta. Since the spring force is hindering the motion and going from position one to two, it's negative, and according to my sign convention. And so F is minus k delta, and we're saying dr is d delta in this case. And if I integrate that, I get work done by a linear spring is minus K over 2 times the deflection squared at time two minus the deflection squared at time one. And so, that's the work of a spring. The initial work energy principle for particles, systems of particles, and we'll come back next time and continue to look at more work terms and, and finish up.