>> Hi and welcome to module 23 of Two Dimensional Dynamics. So far we've looked at the velocity and acceleration between two points on the same body. So it's all one body. Today, we're gonna look at situations with the velocity can be looked at of the same point expressed to two different reference frames or two different bodies in planar motion. And, then later we're gonna actually do accelerations as well. We're also gonna derive something which is called the derivative formula which is a very important tool that we'll be using for the remainder of the course. For taking the derivative of an arbitrary vector expressed in a moving frame with the respect to another frame. So, here is my generic situation of the velocities at the same point relative to two different frames or bodies. So let's take a Point P out here. P, and I have a Frame F, and I have a Frame B. And we can look at the motion of P, with respect to the Frame or Body B or the Frame or the Body F. And so I will often times talk about F as being what I'm gonna call the Fixed Frame and B as being the Moving Frame. But when I say Fixed Frame it doesn't necessarily have to be fixed in an inertial reference frame. It only would have to be fixed in an inertial reference frame if we were talking about kinetics. Where we're going to apply Newton and Euler's laws. Right now we're only dealing with kinematics, and so we're only looking at the geometry of the motion and so when I referred to F and B. Those are just two different frames. In fact, If I had several bodies connected together, I could link them back together two frames at a time. And keep moving the vectors in different reference frames. And so, there are several situations that we're going to encounter where you have multiple bodies and you wanna look at the kinematics of similar points in two different frames or bodies. Here's one example. As you know, I'm a Georgia Tech and our mascot is the Ramblin' Wreck, are actually the Yellow Jacket. And so he is singing our song the Ramblin' Wreck from Georgia Tech here. But we have a little Yellow Jacket and moving out on a vinyl record. And so it has the motion and we could, in this situation look at the motion or Point P of the Yellow Jacket with respect to the vinyl record or we could look at it with respect to the Frame F which I would have fixed to the earth. Here's another example of a mechanical device and we're actually gonna solve this problem later on. But I can look at this motion, this Point P is attached to Bar R and then there's a slider mechanism that can slide up and down on Bar B. And so we could call in this case, our Frame F the ground. And we could call the moving Frame or Body B, this vertical bar. Because P is going to move relative to that vertical bar as well. One more example, this is labeled Body C, but I'm going to call it Body B because it's going to be my moving frame. I've got this cylinder. Here's an example of what I call a Fixed Frame but it's not fixed in an inertial frame, it's able to rotate. So here's my Frame F, here's my Frame B my moving frame. And we can look at from the perspectives of both of those bodies or frames this common Point P and relate the velocities in that manner and so we'll go ahead and do so now. Let's go ahead and develop the theory here. I've drawn my coordinate system for the Frame F, Capital i, Capital j. And I've drawn the coordinate system for the Frame B, which I'll call Little i and Little j. And we're going to relate the two. So I've got Little i is equal to. For Little i. I'm gonna go out a distance. Cosine theta and the Big i direction. You should be really familiar with this. We've done it several times before. And then we're gonna go a distant sine theta in the j direction. [SOUND]. The Big j direction. And then for Little j. We're gonna go, minus sine theta in the Big i direction. [SOUND]. And we're gonna go positive cosine theta in the Big j direction. And, now let's differentiate those vectors. Unit vectors, in the F Frame. Because i and j are gonna move with respect to the Big i, Big j Frame. And so, if I differentiate in the Big Frame, we get [SOUND] the derivative of i. Respective time for the Frame F. Is equal to i dot. Which is equal to now derivative of cosine theta will be minus sine theta, theta dot [SOUND] in the Big i direction. And plus sine derivative of sine is cosine theta, theta dot in the Big j direction. And that equals, well let's look here. It's minus sine theta Big i plus cosine theta Big j, is the same as Little j and so this becomes theta dot j. So the derivative of i with respect t and the F Frame is equal to theta dot j. We'll do the same thing for j. d j, d t in the F Frame is equal to j dot which is equal to okay. The derivative of minus sign theta is minus cosine, theta. Big i times theta dot. And the derivative of cosine is minus sine theta, so this is minus sine theta Big j. We can see that the information in parentheses is negative, i. And so, this becomes minus theta dot i, okay? So, we have the derivative of i and j, with respect to the Frame F. So, the derivative of the unit vectors in the moving frame with respect to the Fixed Frame. And so, here are those results up here. So now let's look at an arbitrary vector expressed in my moving Frame B. And we'll call that vector A. When I say arbitrary vector it can be any vector. And we know vector quantities for kinematics it could be position, velocity, acceleration. Any of those vectors could be represented by A. And so let's go ahead and express that vector A in the moving frame, in the Little i,j components. And so I've got A is equal to the X component of A in the Little i direction plus the Y component of A in the Little j direction. And now let's differentiate, we'll be careful now. And differentiate that vector that's expressed in the moving frame with respect to the Fixed Frame. And so I've got A differentiated in the F Frame is equal to. Okay, we've got A X dot i, plus A Y dot j. And then we've got plus. I also have to by the product rule, take the derivatives of Little i and Little j because they those unit vectors change with respect to time, when looking at them from the perspective of the F Frame. So I've got plus A sub x times the derivative of i, expected time. Taken by the F Frame plus A sub Y times d sub j, d t taken with respect to the F Frame. So A sub, or A with respect to the F Frame equals. Let's look specifically at these two terms. This says A dot xi plus A dot yj. So, from the perspective of the little or of the moving Frame B. If I was looking at the vector A from this moving Frame B. So imagine yourself shrinking down onto that Body B, looking at the vector A. You can see that in that reference frame, Little i and Little j do not change directions. The only thing that changes could be the magnitude of A in both the x and y directions or components. And so this becomes the derivative of A with respect to the B Frame, or my moving frame. Plus, I can substitute now the derivatives of the unit vectors i and j in the moving frame that I found from the last slide and so I get plus theta dot times Axj plus or excuse me minus Ayi. That's just substituting these into here. Let's see. We'll substitute the id into here and dj, dt into here, and that's the result I get. And then I can just make a mathematical manipulation which says that this is equal to A dot in the B Frame plus I can write this as theta dot k crossed with Axi plus Ayj. And you can see that these are equivalent. K cross i is j, so this ax theta dot j, this term and k cross j is minus i. So that becomes Ay theta dot times i. And, we can also realize that now Axi plus Ayj is our original vector A. Expressed in the moving reference frame, and so our final result. Is A dot in the F Frame is equal to A dot in the B Frame plus theta dot k crossed with A. And that's a really important, formula which I'm going to call the derivative formula. And we'll use it a lot in the remainder of the course. What it says is, if I take the derivative of a vector in a moving frame, expressed in a moving frame with respect to another frame. The derivative of the vector in the F Frame is equal to the derivative of the vector in the moving frame plus theta dot k which is, the angular velocity of the moving frame crossed with the vector expressed in the moving frame. So a really important result. And we'll continue on and use this theory in future models to solve some real-world problems.
