Hi and welcome to module 23 of an introduction to engineering mechanics. For today, we're going to learn, or we're going to recall actually the 2 dimensional and 3 dimensional static equilibrium equations that we learned earlier. And then we're going to employ rules for drawing a free body diagram, which we've done before, but we're going to do again. And we're going to sketch a free body diagram for a body. So first of all on the static equilibrium equations as a review. This is from lesson, or module 13. And so if you don't remember this, go ahead and review module 13 again. But we said that we would have a static equilibrium condition if there was a balance of forces acting on a body and a balance of moments acting about any point on the body and that's vectorially. In scalar form we, we had the two dimensional and the three dimensional equations for static equilibrium. And again, if you don't recall those I'd like you go back to module 13 and, and, and review that one more time. So, we're going to learn about a free body diagram, more about a free body diagram. We've used it earlier in the course. We're going to. This is a really important tool. This is a tool that engineers will use throughout their career. It's a sketch which represents a body isolated from its surrounding, and it shows all of the forces and moments on that body. And I refer to it as an equilibrium tool. It's a very, very important graphical tool that you must learn to be a successful engineer. So, here the, the free body diagram procedures I've, I've listed them here but the easiest way to learn how to do the, the procedures are to go through an example problem. So, here is an example, this is a great part of the course because now we get to start looking at. Some real world engineering examples. And so, this is a beam, typical beam. Might be in a building, might be in a mechanical structure, might be in an aerospace structure. Beams are everywhere but, this beam has a couple of constraints on it. On the right-hand side. It has what's called a pin reaction. Here, six inches from the left-hand side it has what's called a roller reaction. And on the right hand, or on the left-hand side here we have a five pound external force that's acting down. And so I have a, a model of this. Engineering system over here. So here, here's my beam. I've got a pin reaction here. And we'll talk more about that pin reaction. I've got a roller reaction here. And then I've got a weight acting down on the, on the left-hand side. Now in this case it's not five pounds. But it's a scale model, so you get the idea. So, that's, that's the physical model. So, we want to go ahead now and draw the free body diagram. And so, I'm going to, first of all identify the body of interest. And in this case that's going to be the beam, okay? And then, I'm going to sketch that body free of constraints or separated from the rest of the world. And so, here's my beam. [inaudible] completed step two. Now I want to apply any external fo-, forces or moments acting on the body. In this case I don't have any external moments acting on the body, but I do have an external force over here on the left-hand side which is five pounds. So I can replace that with a force acting down on the left hand side which is 5 pounds. The other external force which is the body force for the beam would be its weight and that would act right in the center here. I'm going to push that down and I'll just call it weight, it would be known, it would be given and so we've completed step 3. The next step is to replace the constraints with forced reactions and moment reactions. And we have two constraints on this body. We have this pen reaction and, or pen constraint and we have this roller constraint. So, let's look at those on my scale model. So, first of all I'm, I'm taking off the five pound weight. I've replaced that with a force and I'm going to look at the pin reaction now. This pin reaction alone, it prevents motion left or right, it promotes, prevents motion up or down, but it doesn't prevent rotational motion. So there's no, what we call moment reaction there, it only constrains motion in two orthogonal directions. See, if I remove the pin I can now move left or right. I can move up and down. In fact, I can move in, in any direction linearly. And so, that pin reaction can be replaced on my free body diagram with two, pin constraint can be replaced on my free body diagram with two force reactions orthogonal to each other. And so let's go ahead and label these points. I'm going to call this point a, and this point b. And, on my diagram here, this'll be point a, and this will be point b. And so on the pin reaction, when I pin restraint when I remove that constraint I replace it with 2 force reactions, orthogonal to each other. I'm going to replace it with b so y in the y direction and b sub x in the x direction. And as long as they are orthogonal 90 degrees to each other. I can draw them in any direction. I could have drawn one this direction and one this direction. They have to just cover the entire 2D space. Okay, the only other constraint we have is this roller constraint, which is at point a, and so let's go see what kind of force reactions we get from a roller constraint. So, now I've, I've removed the five pound force from the left hand side. I've removed the pin constraint. All that's left is the roller constraint. And you'll see that the roller doesn't prevent motion left or right, it doesn't prevent rotation, so there's no moment reaction that stops rotation. The only thing it does is it stops motion down. And an ideal roller would not only stop motion down, it would also stop motion up. You can think of an ideal roller as the, the roller mechanism that's on the top of the closet doors perhaps in your home, where it rolls along the track and it, and it can't go up or down. So there's only one direction that is constrained from motion and so on my free body diagram over here, I would draw just a force reaction in the y direction. I'll go ahead and put a sub y. So that's complete step four. The last step we have to do is add dimensions. That's relatively straight forward. We have six inches from the left to point a, the weight is in the center since the entire beam is 23 inches long. That would be 11 and a half inches to the, to the weight. And then finally we have 17 inches on the right-hand side to the, back to the roller reaction. So this is 17 inches. And so that's how we draw a free body diagram. We've identified the body of interest of being. We've sketched the body free of constraints. We applied external forces. There was no moments in this case. The external forces being the five pound weight, and the weight of the body itself. We replaced the pen constraint and the roller constraint with forced reactions and we added dimensions. And so, that's, that's it. See you next module.