 Today we're going to talk a little bit about free body diagrams, what a free body diagram is, and give you some hints about how to go about drawing one. So why do we want to draw free body diagrams? Well let's consider first a condition called static equilibrium. Static equilibrium is a condition where forces on an object are balanced, in other words, they are equal, so that they don't change the motion of the object. In other words, they are static. Although this is a bit of a misnomer because the body could be in motion and still be in equilibrium. It doesn't have to necessarily be not moving. However, in civil engineering and in structures, we are most often going to be considering things that aren't moving or that we don't really want to move. If we can identify a body in static equilibrium, we can apply a balance of forces to determine the force values that we may want to know. And that's what free body diagrams are useful for. Down below we have a particular example of, well, this would be considered a free body in the fact that she does represent freedom, but that particular connotation aside, we're going to consider a body in its environment, in this case the Statue of Liberty. So first step in a free body diagram is to define the body of interest. And we're going to do so by drawing a boundary around it. Okay? So it doesn't have to be a real detailed boundary, but we're going to go ahead and draw a boundary here around the Statue of Liberty, separating her from the environment. Now that we've separated the body of interest from the environment, our next step is to identify body forces. There are a set of forces of certain type that act over a distance. These are forces like gravity as the primary one that we will use at the scale that we're talking about. Although electromagnetic forces are also part of the type of forces that can be identified as body forces. And if we were at a much, much smaller scale, atomic forces such as the strong or the weak force could also be considered as being body forces. These are forces that are acting over a distance and can be considered as acting on the entire body. Generally we are going to apply those body forces to the center of the body, something called the center of gravity or the center of electromagnetic force. These centers can be found in a multitude of different ways. We'll identify some center at which the gravity acts. Okay? And in this particular case with something like this, we call the action of gravity the weight of an object. So we've identified the body forces here. Here we have the weight of the Statue of Liberty. So once we've identified body forces, our next action is to identify boundary forces. And our goal here is to free the body by replacing all contact parts on the boundary with a force. And in fact a special kind of force, a distributed force. So we go around the entire body and try to think about what is actually touching or contacting the body at this point. In the case of the Statue of Liberty, we kind of have two types of boundaries here. First of all, we have all around the upper part of the Statue of Liberty, we have contact with the air. And this air is pushing on all the places where the air is touching the Statue of Liberty. And then along the bottom, we have a distributed force of the pedestal pushing on the bottom of the Statue. Notice that all boundaries will have some force upon them. However, in some of our problems, we may choose to neglect particular forces because they are either too small, unimportant, or somehow cancel out as part of our analysis. Once you've identified all these body forces with a distributed force, we usually simplify by representing this distributed force with a concentrated or point force. For example, all along the base of the pedestal, you have lots of little small forces, but those little small forces are acting over their all small areas. So really it's a form of pressure, it's force applied over an area. And instead of thinking about that as a whole bunch of small forces, we can consider it as being one large concentrated force acting on the center of that. For those of you in calculus class, we gather that by doing the integral of those small forces over the area. So in this case, we have the force represent this Fp, force Fp of the pedestal. In the case of something like air pressure, you'll notice that the air pressure is pushing both up and down and side to side in all different directions. It's actually pushing every direction around the body. And if you think about it, if we removed the Statue of Liberty, there would be air pressure that would have been would be pushing down on the pedestal. Well, when we measure the weight of something, we measure it within this air pressure. And so the air pressure is sort of ubiquitously pushing on everything in every direction. So usually, not always, but usually we are going to neglect air pressure or consider it to be a net force of zero because it's pushing in all directions around the body that we're concerned about. So in this case, we're not going to include it in our larger representation. All we really have here are the weight and the force. And we've recognized that the air pressure exists, but we are going to neglect it in this free body diagram. As long as you make the choice and recognize what's there, then you're good. So now that we've actually identified our boundary forces, our body forces and our boundary forces, let me sketch this again here. So now we have our Statue of Liberty here sketched a little less carefully with its weight and the force of the pedestal. We want to consider our force components. We're going to, we're going to consider our force components that are what are called normal to the surface. We'll call normal, which are perpendicular to the surface and tangential, which are parallel to the surface. In this case, we see that our force of our pedestal, it's easy to think about it pushing up, but there may also be something potentially because it's fixed to that pedestal that might be pushing sideways, parallel to the surface that's connecting the Statue of Liberty to the pedestal. So we want to break our boundary forces into two parts, the parts that are perpendicular and the parts that are parallel. Now it may turn out in this case that if there's nothing pushing side to side on the Statue of Liberty, that the tangential portion doesn't have any value, but we want to identify that it exists. This would be in what we consider a local basis or a local coordinate system that depends on what the surface looks like at that contact point. Once we've identified all the components of our forces, of our boundary forces, our last piece is to create a global basis. In other words, we're going to orient ourselves in a fashion that allows us to consider all of the forces in the same basis or the same coordinate system. In this case, it's pretty straightforward. We'll go ahead and orient ourselves so that in our two-dimension, we have a vertical dimension that's pointed directly up and we'll call positive as up. And although it's unimportant in this problem, we'll go ahead and create a horizontal component that's to the right and called to the right being positive, to the right as we face the Statue of Liberty as being positive. So there's our list of actions. Define the body of interest with the boundary, identify body forces, identify boundary forces, and simplify them with a concentrated force. Consider components of that force that are parallel and perpendicular to the surface, and then finally create a global basis that we can describe things in that gives us direction for the entire system. Let's apply this to another example here. Here's a picture of a horse and a cart. So let's follow our steps, shall we? Let's define the body by drawing a boundary around it. Now there are multiple ways that we can go about doing this. I've decided that my first system I want to consider are the horse and cart together. Then we need to go ahead and identify body forces, forces that act over a distance here, and the only one that we're going to be concerned about here, the forces that are due to gravity. In other words, the weight. I'm going to think about the horse and cart as being separate weights. We have the weight of the horse and the weight of the cart that are both part of the body forces within our system. We haven't separated them. They're both within the system, but they both contribute to the balance of forces in our system. And we try to locate these in what we consider to be the center of gravity of each of those components. Now that we've identified body forces, let's consider the boundary forces. Most of our boundary is going to consist of air pressure, which we're going to neglect. But we should also consider the other places that we have contact with something besides the air. And in this case, we have five different locations that are in contact with something. We have the location where the wheel is touching the ground. We have the location where each of the horses hooves are touching the ground. And along those little locations, there are forces acting. And it's hard to draw a distributed force here. So we're going to go ahead and just think about these as being concentrated forces at each of those contact points. But notice those forces don't necessarily just hold the horse up. If the horse decides that it wants to strain and push forward and start moving, then it's going to push against the ground in order to start moving the cart. If it does so, for every action there's an equal and opposite reaction, the ground must push upon the horse as well. So each of these particular locations may have components that are perpendicular to the surface, our normal components, and components that are parallel to the surface, our tangential components. So now we've identified each of those components, and we can give them names. We might call this the force on the cart. We call this force on hoof 1, force on hoof 2, force on hoof 3, and force on hoof 4. Notice we've represented this with two component forces. You could also represent this with a single force drawn in that direction, but almost always you're going to immediately want to break that down into parallel and perpendicular components. We notice that in this system almost all our forces are either acting vertically or perfectly horizontally, so it's wisest for us to create a global basis. That's also vertically and horizontally, and we're going to go ahead and call a positive direction as being up, and our positive direction as being right in our global basis. Now we've created a free body diagram for the horse and the cart. However, let's say our problem asked us to determine how much the horse was pulling on the cart. Well, in this case, this free body diagram doesn't do us a lot of good because we've kept the horse and cart both inside the free body. If we were interested in how much the horse was pulling on the cart, what we might need to do instead is to separate the horse from the cart. So now let's say with our horse and cart problem that what we really wanted to know is the force between the horse and the cart, how much the horse was pulling on the cart, for example, as it's moving or as it's straining to move. In order to do that, it didn't help to have a free body diagram that has the horse and cart inside. What we need to do is separate them by our free body diagram. So I can now consider either of these possibilities. So let's go back and look at the blank screen again and redraw our free body diagram. We have two choices here. One choice is to draw our free body diagram around the cart. Another choice is to draw our free body diagram around the horse. In both of those cases, the forces we were considering before still come into play and we can still neglect the air pressure. However, there is now one more force that we need to consider. We need to look at this space, the place where the cart connects to the horse. And in that particular location there, we need to represent it with a force. Well, if we're thinking about the cart, we can think about it this way. The cart is being pulled by the horse. So there is a force by the horse, of course, on the cart. Notice, depending on how we actually drew the line, we might want to draw that line, that boundary, through the connector of the two. And in which case we can redraw that force in two components, basically as parallel. And, I mean, that's perpendicular to the surface and it's parallel to the surface, our normal and tangential components. Because notice, the horse is going to be pulling forward, but there might also be some component of the horse lifting and keeping the car vertical. So that force there, if we consider the balance of forces on the cart, you have the weight of the cart, our body force. You have the force on the wheels. And we call this the force of the cart, I believe. Two components of the force of the cart. And then you also have two components of, and now we have to determine what we want to call this. Do we want to call this force of the horse? Well, we already called the footprints on the ground the force of the horse. We could also call this the tension. Tension would be the pulling, the tension between the cart and the horse. And now we've drawn a free body diagram that includes just the cart, but would give us information about the relationship between the cart and the horse. Notice if we've chosen to do that in the other direction, because for every action, there's an equal and opposite reaction. If we have a tension T, and it's two components where the horse is pulling the cart, if we consider how the cart is pulling on the horse, we must have an equal and opposite reaction, T. And notice I draw the components equal and opposite to what we had before. That that same tension, but pointing in different directions, must be being applied on the horse. Now notice again this is useful that if we describe our global basis where up is positive and to the right is positive, you're going to get equal and opposite. Both of the values of the horse pulling on the cart are going to be positive, whereas the values of the cart pulling on the horse are going to both end up being negative. Well, that will conclude our example of free body diagrams. You'll have plenty of opportunity to practice this. The key idea is as you're trying to decide what free body diagrams you're drawing, is to think of if there are any forces that you're going to wish to identify and make sure that the boundary of your free body slices through something that would be supplying those forces.