Hi, this is module four of two dimensional dynamics, our learning outcomes for today are to describe a rectangular Cartesian coordinate system, a cylindrical coordinate system and to describe the kinematic relationships of position and velocity in a tangential and normal coordinate system, so the one you are probably most familiar with for studying curvilinear motion, or curvilinear motion of a particle. P is a rectangular cartesian coordinate system of x, y, and z, or i, j, k for unit vectors. Since our coordinate system is fixed, i, j, and k do not change and only the magnitudes. Of the x, y, and z components change. And so this is what the velocity and acceleration look like. Another common coordinate system that we use for curvilinear motion is cylindrical coordinates. In this case, we use a polar coordinates are in theta to describe the projection of the motion of point P in the XY plane, so we have a distance r, radial distance r. The distance r is in what we define as the e sub r direction. And the, direction in the theta direction is called e sub theta. And ,we can actually relate those to rectangular and, and Cartesian coordinates, through the following relationships. We see that here if I want to go out a distance x, and over a distance r The x distance, we go out. This is r theta. So that's r cosign theta. And this is opposite side, so this is r sign theta. So we see that ,x is the same as r cosign theta. Y is the same as r sign theta for the projection in the xy plain. And then, we have the same z coordinate which is common to both systems. And so again, position would be, in this case going from O to P would be in the r direction, east of r direction. A distance r, okay, plus, oops, plus Z distance in a z direction. And then you can look at any standard dynamics text to see the development of velocity and acceleration in the cylindrical coordinates system. So both cart, rectangular Cartesian and curvilinear Motion in cylindrical coordinates are called exter, external, excuse me, extrinsic coordinate systems because they do not depend on the path of motion of the particle. The next coordinate system I want to look at is tangent. Tangential and normal coordinates, which in fact, are dependent upon a path, and so we call this and intrinsic coordinate system. So here's our particles, starting at some arc length s equal zero, moving along some Curvilinear path. Here's our fixed frame f, and so our position vector would be from a. O to point P, and we see now that position is a function of where we are at along the the path of the particle, or what we call the arc length, s. So r, OP, the position vector, is a function of the arc length. So, when I go to find the velocity I need to look at how that position changes over some small period of time. And so we're going to take a small instant of time. We're going to go from r, original rOP to rOP S plus some delta S, since the position is a function of the arc length. And so the change In the during that small incremental period of time, it has a vector change from this point to this point, or delta rOP, or an arc length change of delta S, and so we shrink that time frame down to become a derivative and we get the velocity of P is equal to the derivative rOP Which is the derivative with respect to time. And, since the position is a function of arc length, I'm going to break that derivative into two parts. I'm going to say dr op ds, and ds dt. And so, it's going to have a motion quantity. Which is associated with the change of the arc length itself with respect to time, and a geometric quantity which is the change of the direction of this vector from p to p prime, as it moves along the path. And so here again, we have our chain rule and so, in the limit, what we see is that as I shrink this time period smaller and smaller, the magnitude of this vector delta rOP is the same as the magnitude of delta s. And so, the direction, as I shrink it down. The direction of rOP is going for a very, very small change in time period, is going to be ,along the tangent to the path. And so ,the direction becomes tangent to the path, therefore the direction of this vector rOP is tangent to the path. We'll call that e tangential. And so the velocity is always tangential to plath and so I get. The velocity of p is equal to. Now I've got ds dt, which I'll call s dot, and dr OP ds, is that shrinks down. We said that the change of the direction of rOP with respect to the arc length is e sub t, so we have a magnitude and a direction. Magnitude being how the arc length changes with the respect to time the direction being tangent to the path at any point along the, the curvilenear motion. And so, this is actually called, let me circle this because this is very important .This is called the speed, it's the magnitude of the velocity. And so it's equal to s dot, or the magnitude of the velocity of p, and that's our motion quantity and our geometric quantity is dr op ds. So it's for today's module. We'll come back next time and continue on with looking at tangential and normal coordinates.