[BLANK_AUDIO]. Hello and welcome to module 11 of Three Dimensional Dynamics. Today's learning outcome is rather interesting. We're going to develop the equations of motion for a particle close to the earth, where we're now going to treat the earth as a rotating reference frame. We often treat it as inertial but it's not truly inertial. And then, we're going to do a real interesting problem next time, once we have an example once we have developed those equations of motion. So here's our situation, we have the Earth as our, as, as a moving reference frame, now certainly this is is not the scale, the earth is rotating with a constant angular velocity. Omega, the radius from the center of the earth out to a moving reference frame, is the distance R and then we have little R out to a point, or a particle, close to the earth's surface. And so, we're going to fix our inertial reference frame at the center of the Earth and it's not going to turn with the Earth, but we are going to embed or kind of weld a moving frame, turning with the Earth. And so let's look at this with an example. Okay, so here's a model of my Earth, the Earth is rotating with a constant angular velocity Omega. And then the fixed frame is going to be at the center of the Earth, with its origin at the center and it's not going to rotate at all. The moving frame is going to be embedded oops, embedded with the Earth and turning with the Earth. And so in that case we have our little y axis in this direction, our little z axis in this direction, and the x axis is pointed back towards me as it rotates with the Earth. Okay. So this a very similar situation to the generic situation we developed with the hang glider. In this case, the inertial frame F is actually the center of the, of the Earth. Okay? And frame B, which was the car before the moving frame is now embedded to the Earth, out here at this point. And then point P is our particle looked at from both frames. And we came up with the accelerations expressed in those moving frames of reference. And this was the equation that we came up with. And so, now if I know the acceleration of point P, just as a review from my two dynamics course, ho, how can I relate that acceleration to the forces acting on the particle? And what you should say is, we just used Newton's second law, the external forces acting on the particle is equal to the mass times the acceleration of the particle, so I can substitute now the acceleration of P in here, and this is the result I get. As far as the external forces are concerned, I could have a vector of f, forces of, of, of vector force, with x, y and z components. I have the minus n, g, z or minus n g and the k direction is the gravity force directed towards the Earth's center, and then mass times the acceleration expression. And so the first thing I want to look at in this equation is this term R double dot. Now R is the, the position vector from the center of the inertial frame, out to the origin of the moving frame. And so we want to find out what the acceleration is of of Capital, o, o, of O prime. Or R, capital R, double dot. So our next step is to find the acceleration of the origin out here, O prime. And to do that, we're actually going to set up another set of two frames. This time I'm going to have my inertial frame again at the center, but I'm going to put another frame, frame C, a moving frame, which is also at the center, but turning with the earth. And so let's look at that situation over here. So in this case, I want to find the acceleration of the O prime point out here. Which is R, capital R, double prime. And so the inertial frame again is fixed at the center, not turning, the rotating frame in this case, frame C is at the center, but it's rotating with the Earth. [BLANK_AUDIO]. Okay. So in that case I can write the equation of the acceleration of O prime with respect to F using frame C is the, is the rotating reference frame or moving reference frame. So my first question to you is, given that situation, what is the acceleration of point O with respect to F? And you should say 0 because O and the moving frame, and C the moving, excuse me. the origin of the moving frame C and the inertial frame are right on top of each other so there's 0 acceleration. What about the acceleration relative to the moving frame of point O prime? What is that? And again, you should say that that's 0. Because if I'm in the center of the Earth and I'm rotating with the Earth and I look at this point O prime out here, when I'm rotating around, from my reference, O prime is not moving. It doesn't have any velocity and it doesn't have any acceleration relative to me. And so that means a rel is equal to 0. Since the earth is rotating with a constant angular velocity, alpha is equal to 0 and we also already said that VRL is equal to 0 and so this equation reduces down to what's show here. Now, this is back to our original situation, where I had my inertial frame at the center and frame B embedded at the Earth. What I want to do next, is I want to compare the angular velocity of B with respect to F and C with respect to F. What's that relationship equal to? And what you should say is that the angular velocity of both of those moving reference frames are the same. And so, let's go ahead and look over here again. [BLANK_AUDIO]. So if I have a frame out here, embedded out here, and it turns with the Earth, that's going to be the same as I can't get this frame inside the ball, but if it was inside the ball and turning with the Earth, they would both rotate with the same angular velocity. And so I can therefore, write this equation just by substituting Omega B with respect to F for Omega C with respect to F. 'Kay, and so here, here's where we're at as a recap. I've got the acceleration of P with respect to my inertial frame. I know Newton's second law. I substituted it in. I looked at this first term, which was the acceleration of R double pri, prime. Capital R double prime, I found it to be equal to this. And so I can go ahead and substitute that in. And I may go a little quick here on my equations, but you can always stop the video and make sure that you can see how each step takes place, because it's fairly straight forward algebra. Okay, so here's where we're at. Now I know that R from my little reference frame is just some distance to the X, some distance to the Y, and some distance in the Z direction. V rail from my moving reference frame B is going it, it, it can have a relative of, of velocity and it can have a relative acceleration point P with respect to my moving frame here. This particle next to the Earth. So this is the expression for V REL. This is the expression for A REL. And finally I, again, I know that the Earth is rotating with an angular acceleration that's 0, so this, it's got a con, a constant angular velocity. And so here is the result that I get, again if you substitute these in, all right? And so, here again is that same result. Now let's make some important assumptions, I said that this was not to scale at all because this, this particle is very close to the Earth, the radius of the Earth is obviously very large and so little R is much much less than big R, which means I can cancel out little R or neglect it. I know that my force external force acting on my particle can have X, Y and Z components and so I can substitute that in here, I also need to have Omega B with respect to F in my little reference frame. Well, my moving reference frame. I see that Omega B with respect to F that's the rotation that the Earth is omega in the big J direction, and so what I want you to do now is change that to the little ijk coordinate system. And you've done this several times before, hopefully it'll be straight forward to come on back and see how you did. And so this is the results you should have arrived at, Omega j, the big J component is cosign data and the little J direction and Sin data in the little Z direction or K direction. Okay so now I can substitute all of this in, again this, this will go in here, here and here and the result I get is Fxi plus Fyj plus Fzk minus MGK equals now, m times, okay Omega B with respect to F we just found to be Omega Times. Cosine lambda j, plus sin lambda k, that's crossed with omega again, omega being with respect to f which is omega. Cosign Lambda j, plus sign lambda k and that is crossed with R, big R, big R in the little K direction. [BLANK_AUDIO]. And then I've got plus 2 times omega. Omega again is omega times cosine, lambda J plus sin lambda K. And that's crossed with my relative velocity, which is x.i plus y.j, plus z.k and then I've, all I'm left with is my relative acceleration, which is plus x double dot i, plus y double dot J plus Z double dot K. And that's the expression and I'll write that out. Let's, let's clean that up here and this is the result you'd get. Now I'm going to go through again a couple of steps mathematically. It's fairly straight forward algebra. You can stop and watch as I go through these two steps to reduce this. The result I get is here, and now my equations of motion for the particle can be found by equating the I, J and K components. So, if you equate the I components first you find the acceleration, equation of motion for X double dot. If you equate the J components, this is your next equation of motion and if you equate the Z or the K components, this is your final equation of motion. So, these are the three equations of motion for motion to a particle close to the Earth's surface. And we'll use those, this theory, within a really interesting example in the next module.
