[BLANK_AUDIO] Hi, this is Module 19 of Two Dimensional Dynamics. Previously we looked at the velocities for two dimensional motion of bodies. We looked at the velocity between, the linear velocity between two points on a body as related to the, the angular velocity of the body. Now we're going to extend that for accelerations of a body in, in, in planar 2D motion. And so, here was the, the theory that we did before. And, so, we're going to go ahead and apply this acceleration equation to this problem which we did for velocities. And so we have omega one is ten, clockwise. Omega two is 2.42 radians per second, clockwise. We found that from the velocity problem. And we have the angular acceleration is five radians per second counter-clockwise, on body B1. And we want to find again the linear translation of, of piston B. So I'm going to go ahead and apply the, the the acceleration equation we came up with. And so I've got the acceleration of the unknown point A is equal to the acceleration of the known point, O, which is zero, because it's a pin, plus, we're working on body O B, or O A. We're going from knowns to unknowns here. And so I'm going to have theta double dot B 1. And that's in the k direction. Crossed with the position vector from the known to the unknown, or from O to A. And then I've got minus theta dot B1 squared r from O to A. And so let's go ahead and put those values in, so I've got the acceleration of A is equal to, well, we do know that the acceleration at, of O is zero, because it's a pin. And then I have plus, theta double dot K, for body B1 is given. It's alpha one, it's five radians per second. Counter-clockwise is going to be in the positive K direction, so this is going to be 5K crossed with r from O to A, is, I go minus 3 in the X direction plus 4 in the Y direction. So minus 3i plus 4j, minus, now, theta dot B1 squared. The magnitude of the angular velocity of bar B 1 is 10, so this is going to be 10 squared times r from O to A, that's again minus 3i, plus 4j. If you do that multiplication, I'll let you do that on your own, you'll get the acceleration of A now, is equal to 280 i minus 415 j inches per second squared. So let's continue on from there. Here's our acceleration of A, which we just found. So I've gone from this known to this unknown, which is now known. And now I want to go down to the acceleration of B which is unknown, and, and what we're trying to solve for in the problem. So, I've got, the acceleration of B now, is going to be equal to the acceleration of A plus, theta double dot B2 k, crossed with r, now from our known to our unknown, r A to B minus theta dot B2 squared times r from A to B. Okay, do I know anything special about the acceleration of B to reduce the unknowns? Again B is moving in rectilinear translation, so it can only have one component of acceleration, and that's going to be in the idirection, so I can reduce A as a vector to A sub B in only the i direction, equals A sub A, which I found, which is 280i minus 415j, plus, now I don't know what theta double dot B2 is, so that's going to be an unknown. So I've got theta double dot B 2 in the k direction, crossed with, now r from A to B. We're going to have to use a little bit of geometry here. We know this distance is 4 inches, because this is 13. And this is 13, this is a 4 on 3 triangle with 5 inch hypotenuse, so that's 4 inches. So if you do your, your Pythagorean's theorem you'll find that this side is 12.4 inches. And so in going from A to B, I go 12.4 in the i direction and minus 4 in the j direction, so this is 12.4i minus 4j. And that takes care of that term. I still have my, my radial or normal acceleration term. So that's going to be minus the, the magnitude of B2's velocity is 2.42. So I'm going to get 2.42 squared and that's going to be times r again, so that's 12.4i minus 4j. And again I'm going to let you do that tho-, that mathematics, multiply that all out. This is the result you should come up with. AB in the i direction equals 207 plus 4 theta double dot B2 in the i direction, equals minus 392 plus 12.4 theta double dot B2 in the j direction. And so you should be able to do that, that, that mathematics. 'Kay, so here's that result. And my question is, okay, we have one equation, a vector equation, and we have two unknowns. How can I solve for those two unknowns? And you should be used to this now. We're going to go ahead and equate components. Let's start by equating the j components. [SOUND] And for the j components, I get theta double dot B2 equals 392 over 12.4 equals 31.6 radians per second squared. And then I'll do the i components. Why don't you go ahead and do the i components on your own and come on back and see if you got the answer correct. And for the i components, I've got AB is equal to 207 plus 4 times theta double dot B2. But I just found theta double dot B2 was equal to 31.6, and so this equals 333, or is a vector A of B is equal to 333 all in the i direction, inches per second squared. And we've solved our problem. [SOUND] As I've said all along, practice is going to make you better and better and these engineering problems. So I have an angular velocity and angular acceleration problem for you to complete now. You have to find, at the instance shown, the angular acceleration of B2. I always like to put these problems in terms of, of real world systems and you recall back well, no you may not recall back, but you hopefully you've been to amusement park rides and you may have seen this amusement park ride. Which is of, like a pirate ship going back and forth. And you can see that that's somewhat related. It's not exact, but here's somewhat of the similar configuration here. If we think of this as being the passenger compartment for the, the, pirate ship ride. This would be a fixed vertical poll. In the, in the case that I have here, it's not fixed. But, you can see that it's a similar type configuration. And, you know why is that important, why, why would we even care about accelerations on amusement park rides? And so what, what you should say is that you know, the human body can only sustain so many G's of acceleration before you black out or have problems. And so you know when, if you were a designer or a, an engineer working on these amusement park rides, you'd want to make sure that you stay within realistic standards. And so these, these have real world applications, these problems and, I think it's important to remember that, and we'll see you next time.