Hi, welcome back to Applications in Engineering Mechanics. As I mentioned before, this course is a follow-on to my earlier course, Introduction to Engineering Mechanics, and you'll need the principles from that course to be successful in this course. So for module two, our learning outcomes are to go back and review some items from that earlier course. We're going to look at expressing the 2D and 3D static equilibrium equations. We're going to go back and remember and recognize and how to apply the principle of transmissibility. And we're going to explain the relationships between sums of moments. So let's go back and look at the fundamental static equilibrium equations from my earlier course. So first of all, let's start with the static equilibrium equations. In vector form, the static equilibrium equations say that, you have to have a balance of forces and a balance of moments about any point on a body to have the body remain in static equilibrium. And so in scalar form, what that means is, if we're looking in just [COUGH] Two dimensions here in the X and Y plane. That means that we will not have any acceleration of our body. And I've got my i-beam here. I won't have any acceleration in the X direction. I won't have any acceleration in the Y direction. And I won't have any moment about point p. and so, let's say that p is at the origin at this point. But p can be any point that I choose on the body. Also on my, some of my forces in the X and the Y direction, it has to be two orthogonal directions. And so if I had a, a body on a slope, for instance. And let's say that the X, the X axis was in my parallel to the slope and my y axis was perpendicular to the slope, I could also use some of the forces in the para, perpendicular direction equal to zero and some of the moments equal to zero to enforce static equilibrium. Now when I go into 3D, I've also got to include some of the forces in the X direction equaling zero, so I don't get any acceleration, excuse me, in the Z direction, so I don't get any acceleration in the Z direction. In addition, now, to the sum of the moments, about point p in just the plane around the Z axis, now I also have to have some of the moments about the, [COUGH] excuse me, about the X axis equal to zero and the sum of the moments about the Y axis equal to zero to enforce static equilibrium. So we have three independent equations for static equilibrium in 2D and six independent equations for static equilibrium in 3D. Let's talk briefly now about the principle of transmissibility. If I have a force acting on a body and I can either, let's say it's a, a half a pound force, and I'm pushing with my finger. That half a pound force, if I push, over here, or if I pull with a half pound force over here, the effect on the body is the same as long as we're talking about a rigid body. So as long as the force is along the same line of action, and it's magnitude is the same, then externally it doesn't matter with a rigid body of whether it's a push or a pull. Now internally it would matter for it's formable bodies, but that's a future course. Also for the relationships between sums of moments, if I have sum of the forces, sums of the forces equal to 0 so I'm at forcing static equilibrium, and sum of the moments about point p is equal to 0 for static equilibrium. As I mentioned before, then the sum of the moments about A, which is any other point, as long as the body is in static equilibrium will also be equal to zero, and that's for any arbitrary point that we choose on or off the body. So here's a picture of that. this picture's of a cantilever beam, and it's fixed on the, on the left-hand side, and we might have a point P where we sum moments, and it's equal to zero, but that means that the sum of the moments about any other point, maybe A1, A2, even off the body, the moments have to be equal to zero. So we may sum moments about any, any point to enforce static equilibrium. And so that leads to an, an alternative set of equilibrium equations, as well. So let's just talk about two dimensions. There's three independent equations, we talked about some of the forces in the X and Y and some of the moments equal to 0 about a point P. But now, you could also use an alternative set of equilibrium equations, which would be sum of the forces in any direction equal to 0, so maybe it's some arbitrary direction in the XY plane, and then you can also sum moments about point A1 and then A2, so as long as you use three different equations, you get three independent equations. And so. you can even solve these problems with static equilibrium equations by summing moments about three different points. And we'll, we'll have examples of where we'll do problems using the standard some of the forces in the XY and some of the moments equal to 0 and then where we use two moment equation or three moment equation and mix them up. So in my earlier course, Introduction to Engineering Mechanics, we started to look at real world problems. In this course, Applications of Engineering Mechanics, we're going to take that knowledge even further and, and study some exciting engineering structures. So let's get started.