 All right, any questions, department? We've been looking at a couple different coordinate systems. As we look at our two dimensional motion, we get the regular x, y cartesian coordinates which were real good for projectile motion, work real good, I guess for just some general motion, they can apply, they'll be okay. Then we looked at normal and tangential coordinates which are pretty good for the general curvilinear motion we've looked at with some, especially where the path has a changing curvature, the normal tangential coordinates could apply, be fairly useful. And then now we're gonna look at what are called polar coordinates work pretty well for the type of problem where an object that we can use as our origin is fixed with normal tangential coordinate systems. We had these radii of the different curves, but that the radius could change, the center of curvature could change as we went along the path. And so that made the normal tangential coordinates pretty useful because it traveled with the object. If we happen to have a fixed origin as we would for a problem where we might have some kind of tracking satellite dish that's watching a particular object, for example, if I flew overhead in my plane and was being tracked by local radar at the airport or something, the tracking stations not going to be moving itself however the object is tracking is. So we're gonna use polar coordinates that will give us a distance from this origin which we'll easily call OR. And that's a pretty easy thing for something like radar to come up with, certainly distance to an object is no big deal for the technology that we have there. And then we'll also take it as an angle from some reference line for a tracking station that might mean the horizontal local ground level we could measure R. And then that's the coordinates that place the object we would say just like we'd say with X, Y, like we would say this object is at some distance away with some angle on that. Well that's just what we used in physics one as the position vector. So keep it a little bit more generic. Here's some path an object might be taking an object on that path and then we have some idea of where it is at any one time. Our unit vectors to help us with this will be a unit vector in the direction of increasing R. And then since we have always done this before we have no reason to stop this where we have a orthogonal coordinate system at all times we're gonna do the same thing here so that there's no new concern of that. We'll also have a unit vector in the direction of increasing theta that's perpendicular to the unit vector in the radial direction. So again this is sort of like the normal tangential coordinate system in that the coordinate system we're using travels with the object itself. However it's a little bit more regular in that our origin is considered fixed. Not quite the case with the normal tangential coordinate system as it is with this one. So that's our basic setup. First thing we look at then is the position of the object at any one time. It's at this distance R in the direction our unit vector in the R direction. So there's our magnitude. If we had a number in there we'd have units with it and direction and we've got our full idea of a vector as we always have before. Simple as that for the position. The velocity, well things start to get a little bit more involved but it's nothing we can't handle. We know that to be the time rate of change of that position vector if you'd rather, and it'll be a little bit easier here if we do this from now on for these since all we're working with is time for this we'll put a dot on that. Well simple enough, we'll take the time rate of change of the position vector. Let's see that'd be our dot in the R direction. Any velocity it might have in the radial direction. Simply no matter where it happens to be isn't getting farther away from our origin or is it coming closer in some measure. That's all that one in particular looks at. Here's where the complication steps up just a little bit for us. We're not done with that derivative. We have to do the chain rule because both of these things can change with time. The unit vector itself is changing with time. Not its magnitude, but its direction is. So we have to take account for that too. So we'll do the chain rule, that's where we leave the first part alone and take the time derivative of the second part. Now we haven't had something like that before. We haven't had a coordinate system that we need to look at the time derivative of how the coordinate system itself is changing. But it's pretty straightforward to look at that. I'll kind of blow it up here. Here's our original unit vector. Blown up a little bit, remember its magnitude is one. No units on those, just magnitude of one. It just gives us the direction of these things. And some little tiny bit of time later when our object, say, has moved along the path, a little tiny bit causing the unit vector to change a little bit. You don't have change to maybe here. This is sort of exaggerated. We'll call that ER, the unit vector in the R direction with a little prime on it just to show that this is something that's happened. A little tiny moment of time later. Now let's see. So that gives us a little tiny delta theta in there has caused a change in this unit vector. Let's see. Let's see what that delta ER itself would see. That's just the arc length of this little bit of a triangle we've got here, which is the distance times the angle subtended. Right for arc length? Well, the distance, that's a magnitude one. The angle is delta theta. Remember, this doesn't work in degrees. It works in radians. And we're talking about very small angles here. So this is nothing more than the arc length of that little piece there. Is that fair enough to just sit well? Yeah? What is that? That's a one. It's a one? Yeah. I thought it was like similar to a zero point. I don't know. Have you ever seen the number one before? Benduvia? Don't you, have you seen how they do ones in Europe? They don't like that, right? They're very, very unrecognizable to our much more civilized eye. Whoops. That's okay. We're all European Americans here anyway. All right, so that's, so we got to do something with that now. We know what the change in the unit vector is. We don't have the time rate of that change. So we need to do that next. So what, remember what we're looking for is er dot, which is d dt of er. That's the thing we're looking for. That's what we need to put in here. We haven't had that beast before. So let's see, we can take a little bit of step closer to it. We can take the limit as delta t goes to zero. All that delta er we've got over there, right? That, if we take the limit as delta t goes to zero, we will have this then time derivative. So we've got that delta er thing there. So let's put it in. Limit of delta t goes to zero of one delta theta. I'll just leave the one off, especially since it's not recognizable as a one by some people. So we have, we have just this thing. Oh, oh yeah, I lost a piece. I lost, almost lost a piece. This is a vector. We got the little vector symbol over it. This is not a vector. So I have to have some direction on this. That's just the magnitude. The direction on that, this is a, this change in the radial vector, radial unit vector, has that magnitude and it has that direction. Well, that direction is the same as that. So we'll put that direction on it. That's the piece we needed. So the change in the radial unit vector has a magnitude of delta theta. I remember that's because this triangle has a, has a distance of one along there since it's a unit vector. And it changes in the theta direction. So we need that on there. That's delta theta in the theta direction is the way in which the radial vector itself changes. We're almost there because the limit of delta theta, the limit as delta t goes to zero of delta theta over delta t is simply theta dot. Or what we called in physics one, omega. Physics one, we tended to give that. That's the angular velocity in the theta direction. So now we have the velocity. It's r dot er, that's just the thing we had before. But now we have this time rate of change in much simpler form. So we have r theta dot, that's the velocity in polar coordinates. Let's put a pink box on it. It's almost Valentine's Day. That serves as a general reminder that some of you need to do some shopping. Right, doobie doobie, is this your first Valentine's Day? Is it married, yeah? Yeah, I know what's in your first ever. First one there, cool. All right, let's see what that looks like. Let me take this down. We're gonna put something off very similar to it. It's just gonna have the velocity in it. So here's our path, I don't know. Whatever it is that's going along there does make that noise. Our object is right here. Of course, has a tangential velocity only in that direction. Remember, in this curvilinear motion, velocity is always tangential only. It's at some position we can describe with the r position vector, with theta measured increasing counterclockwise from the horizontal. And so our velocity is r dot in the er direction. So that's the er direction perpendicular to that. So that right there is r dot in the radial direction. And the other component in the theta direction, that's r to the two components in the polar coordinate system. For some object moving along a path. Let's see, just so this doesn't look terribly unfriendly, I know it's new in some regards because we've certainly never taken the time rate of change of the unit vector before. But just to remind ourselves what we've got here, the velocity vector r dot er r theta dot, just rewrote it over here, very same thing. Just to help it look a little bit friendlier, let's imagine this path to be circular and our origin is at the center of the path, which would certainly make sense. The path, how does this change in any way? If an object's going in a circular path with the origin at the center, what then happens to any of these terms? For example, r dot, since r is a constant for a circular path, r dot is, so for a circular path, that term disappears. And we have then the velocity is r theta dot theta direction, but you knew that from what we looked at in Physics 1's circular motion. Anyway, an object going in a circular path, a coordinate system at the center, has only on it to it. Remember when we looked at this in Physics 1? My students have had me in Physics 1. The ones are smart enough to get here from the beginning. They've come in after the fact, which is half the size. But we did that in Physics 1, didn't we? That very simple. In fact, any time you have a velocity that's always perpendicular to the radius, you have circular motion. That's one of the defining features of circular motion. So for the special case, this isn't something too terribly new. We just have the possibility that we're not on a circular path. We can handle a more general situation. Yeah. For a problem in polar coordinates, can we use omega's and alphas, or do you want us to stick to like beta dots and stuff like that? Yeah, you can use them if you want, I guess. It's, the book doesn't. But I do, you know, I recognize that that's how we started looking at things. Did you guys do that in Physics 1, using omega and alpha before? Okay. Yep. As long, you can use any system you want for communication, technical communication. As long as you know your reader or your listener understands it. You wouldn't want to write a paper to a German colleague in Chinese, if that's not what he understands. It's not gonna go anywhere. I do understand omega. As you can guess, we're gonna get to alpha in a second because we're gonna do acceleration now. But I'm gonna have to clean the board first, so is this okay? Everybody comfortable with this? Everybody, this was a little bit different than anything we've done before. So that rest okay? The trick, not the trick, but the thing to remember here is the fact that the magnitude of this vector is never changing, this unit vector. It's only changing in direction. And the direction of the change is already in one of our coordinate directions as well. Helps make that sit a little bit better because we're gonna need that idea, that kind of idea again. But now we're gonna do acceleration. So it's dv dt or v dot here. Well, we've got v dot, or we've got v here, so we'll take the time derivative of that. Let's see, we'll take, well, I wanna be a little more complete, so bear with me while I write it out a little bit before we actually do it, just to make sure we've got it. We're gonna take the time derivative of the first piece. Remember when you're taking the derivative of things added together, you take the derivative of each and then add those together, same thing, that's all right now here. So this is a d dt of r dr, so that's nothing more than the time derivative of the velocity where that's the velocity, oops, hang on. Nope, I want the part in the pink box, r theta dot, sorry. The time derivative of the velocity over there in the pink box. Get yourself another power drink, Frank, you're ready. Eight o'clock's too early for those, but 11.50's okay. All right, there we go. So let's see, you gotta take the time derivative of this, and again we're gonna need the chain rule. So it's r double dot, time derivative of the first part times the second part left alone. That's how the chain rule works. At least the last I taught calculus, that's how it worked. Times the first part and the time derivative of the second. So that's this first piece here, because we've already looked at it. We already know what it is, we already know what this piece is right there. Time derivative, let's see. Now we'll do the time derivative of the second piece. This is gonna be a little bit bigger. We've got three parts in here, I'll change it with time. First one, r dot, leave the second two alone. Dot the second one, leave the third one alone. The dot of the last one. That sounds right, because you have grown to know it and love it. Yeah, product rule. Oh yeah, I guess chain rule means something a little bit different. So when you have a function inside a function, that's your name. The math teacher will worry about the names, we know what we're doing. Does that look okay? Everybody, it's easy to drop a dot or put a dot on it. I did, yeah, I did that. I made a dot there. Is that it? That's no trouble, because we already did it. That came out to be theta dot e theta. We'll fix that in a second. The thing we need to double check on is this piece, this piece right there. So we need to double check. We've got something like that. Time later, it's moved over to here. So I'll call that e theta hat prime. Just to indicate to us that a little bit, as seen in a change in the angle has gone by. That gives us an idea of how this vector is changing. So like before, the time rate of change of the vector is related to the time rate of change of the angle itself, since it was both happening at once. Just, I'm gonna move that out of the way a little bit. So the magnitude in the unit vector in the theta direction, the magnitude of that is just like it was before. It's the arc length up there. I know a friend is coming back for me. Del theta, direction of it, vector form of it. Well, I won't put the one down. I'll just put the del theta. The direction of it is, well, it's parallel to the r. It's parallel to the radial direction, but opposite to, because it's going in as the theta increases with time, which was our convention there. So we need a minus sign in front, and it's in the r direction, the theta dot. So we'll take the limit as delta t goes to zero. The unit vector put in what we now know it to be, minus del theta. Of course, that'll come out, the minus will come out, and then the del theta, del t, as delta t goes to zero, is theta dot. That's the time rate of change of the theta direction unit vector. All back together, we'll collect some like terms. So we have l, sorry, r double dot. The rate at which the position is, the distance is changing, distance from the origin. Other terms that have E, r on them. Well, well, this new one does. So it's gonna be a minus r theta dot squared, because there's two of them, minus r theta dot squared. Miss any dots, miss any minus signs? Well, we have that one minus sign. There was already a theta dot here. I picked up another one, and it's now in the r direction, so I collected those r direction changes together. It is in the theta direction. Well, that's this one. Remember, this is r dot, and then this was theta dot E theta, right? Things are following. It's a real life lesson here. Theta dot. Now I got dual. This is E r dot, which is a theta dot E theta itself. But had to raise, that's what we came up with when we had the velocity. Let's, let's re-convince here a little bit. Position is just simply in the r direction. Angle, remember, is inherent in here, because the direction of this depends upon the angle. So we don't need an extra component for the theta direction. Then we add the velocity. The velocity is r dot E r. Let's double check this like we did before. There are the circle, so r is a constant. Let's see, if r is a constant, this term disappears. Because if r is a constant, our dot is zero, our double dot is zero. So we have minus r dot is zero, because r is a constant. So we have just r omega squared. Also might know that at uniform circular motion, talking about circular motion, what do you mean if I throw in the word uniform circular motion? What's uniform circular motion? Remember, for uniform circular motion, circular path, we've already guaranteed that. Velocity in any instance is always tangential. What's the situation? What's the meaning of uniform circular motion? Already got circular. The speed is constant. Velocity remembers a vector. Velocity vector is always changing, but its magnitude never is. Formed circular motion, that's all we have. Because r theta double dot, which you might also know as alpha, is the tangential acceleration. Formed circular motion, there is none. Human motion, we have centripetal component, always directed towards the center. That's what the minus sign does, because the unit vector in the radio direction is always away from the origin by convention. We might have some to uniform circular motion, looks even more uniform circular motion in physics one, so we got that piece of it there. All right, I review. I look at that and say, man, that stuff's easy. That stuff's easy. Give me something to do. So here, we're gonna do a problem. Tracking radar, got a bead on my plane there. Quick design, you don't want to just jettison it and start all over again. So yeah, my car and my plane do look a lot alike. Some of you guys, you're just now seeing the car I drive. Right up there in the parking lot. All right, so under heavy surveillance from the Department of Homeland Security, for some reason they feel they need to keep an eye on me. They have to do with the death threats against students. At a particular instance, here's some of the stuff that's known. And this is all very easy stuff to read right off of radar readings as they're going by. Distance from the installation's 6,400 feet. No trouble for a radar to know that kind of thing. R dot, the rate at which that number is increasing is 312, increasing, let's see. Yeah, that makes sense. As shown, I'm getting farther and farther away from the radar's installation. And this number at this instant, again a fairly easy one for output from a radar reading 9.751. At this instant, we'll say it's 40 degrees. Angle is decreasing with the direction I'm flying there. We're gonna go to less than 40 degrees as time goes by here. Minus 0.039 radians per second. Angle or acceleration, theta double dot, or alpha if you wish, 0.0807. And the velocity and the acceleration of the plane. Forward is pretty much this chugging plug. We've got a bunch of r's and r dots, theta dots and stuff and we got them all there. So it's a matter of putting them together and then double checking to see what they mean. So I'll let you do the putting them together part and to see what they mean part together. So if you make little jet noises while you're doing this, that's okay. Do be, understand that genetically incapable of making really cool engine noises. That's okay. And double check to see what we got. Watch your units, make sure you get the right things in the right place. There's a lots of dots and double dots and r's and theta's and everything here. So get the pieces right. Let's see, 300 on the line connecting the plane and the station. And then minus 250 in the theta direction. Plus theta direction is counterclockwise. So this component is clockwise per second in that direction. Well by golly, look at that, doesn't just give us a nice horizontal velocity like we expected. In fact, if you check the angle between those two, you should get 40 degrees. So that makes a perfect sense, just what we're looking for anyway. And then the acceleration and the yard damage. Workout, check with somebody. What are you talking about, you did it. It was a fraud really doing it to social being. The pieces we need here for the lot we've got numbers there, just got to watch your units. Just make sure you don't mix anything up. Watch your minus signs, your acceleration. Someone there, those are much bigger numbers than I got. Stone feet per second, I mean two feet per second square. Yeah, so just double check your man. Oh, you were doing it on your cell phone and you didn't bring a cop later. Don't they, doesn't TI have a cell phone app? Yeah, I don't know. Yeah, me too. Don't forget, we got squares in here, we got minus signs in here. So we didn't have on the other one. So we've got to make sure you got each little piece of it just right on our component and a theta component. Sounds like we're all getting the same things I got. 0.0166 feet per second squared for that component and then 288, the plane there at that second. So we have a little bit of acceleration in the tangential direction or in the radial direction because this is positive and the radial direction is always away from the origin. And then what does this one look like? It's not on the same order magnitude of size, but in what direction? In the radial direction, but actually a little bit more spontaneous acceleration at that time would be something like that. That remember, acceleration's not nearly as intuitive as velocity is, so you have to trust what you've come up with. Remember, that's the acceleration with respect to that, that single origin back there. All right, everybody pretty much get those numbers, couple, got your mistake there, do we? Okay, third problem, looks like maybe this should be a get out of class problem if we can get Bob jump-started. It'll warm a couple of you up for technical free instant. All right, so here's our coordinate system we're gonna use, three-dimensional coordinate system just to lay things out. What we have is a crank arm of some kind. At the moment, it is laying in that one direction. It turns only if we did have a coordinate system just for viewing this. That crank arm can only turn in the XZ plane. The shaft part of it is in the Y direction. So that's what we have, that's our setup there. Right here, we have, we have on your Sunday clothes a pink collar. And here's what we know with it. Reference theta will measure from there. It's worked just right. We'll take it as that, plus it's milliliters. Along the velocity, the velocity, we can get a picture of it, but yeah. Theta dots, R double dots, theta double dots. One second. This number's correct and it'll be fine. It's all, it's all we're looking for. Down a velocity vector and then only wrote down the speed. When you sketch those, at least the velocity on there, the magnitudes of two vectors are perpendicular. Cooperation time, Bob, if you can jump on it. The other thing that we don't use on the tone, but we do need R dot and theta dot. All of those we can get is probably just a simple math. You can't shout the answer out to anybody else. They can go too. The easiest thing to do is just say that's not a vector. That's magnitude. This one, velocity vector. I think everybody's got okay. Component in that direction, there's milliliters for the component in this direction. Let's sketch out the acceleration in the same way. Minus 700 and 1800. Minus 700 and plus 1800 milliliters per second squared.