 Let's create for ourselves a chart of the different types of support and the reactions that correspond to them. We're familiar with the first type, a cable. A cable acts in one dimension in whatever direction the cable is actually aligned in. If I draw a picture of a cable and I slice it off, then we know that that cable is acting in just one dimension. That dimension is normal, in other words, perpendicular to the cut across the cable. But it is only in a direction that's out away from the body. In other words, it can only act in what we call tension. And that value has to be greater than 0. And there is no tangential component. You can think about this as if you take a string, you can pull on the string and it will resist. But if you try to move the string up and down, it doesn't resist. And if you try to push on the string, it doesn't resist either. So this is a very simple case where you know the only force that can possibly be held by the cable or the string is along the length of the string. Notice there's a special case of something like this, which is a cable over a pulley where you may have a system. Let's consider a pulley. And you have a cable running over the top of the pulley. And if you're going to cut that cable in both locations, you can actually make an assumption that the tension in the cable, that the tension is the cable is the same value, but it's just applied in different directions. It's sort of a special case of a cable, but you can often make use of that. Notice in a case like this, there will obviously be some other force applied where the pulley itself is being supported. So cables are one of our simplest cases of a support and its corresponding reaction. Let's consider another one. This is called a simple support. A simple support is as easy as setting a book down on a surface. All the support does is it gets in the way. And it supplies a normal force, pushing back on whatever is sitting upon it. The simple reaction is again in one dimension, but this time it's normal, but in only. In other words, if it's being applied to the body, it's applying into the body, it's pushing on the body. This is what we call a compressive force. C for compressive force, and that also has to be greater than 0, but greater than 0 in the direction that's pushing into the body. Now there are a couple of cases in here. What's really simple is if it's actually considered to be a frictionless system, where we simplify and assume that the block itself would slide side to side without any resistance, in that case there is no tangential force, no tangential reaction. But you can sometimes consider the case of friction, in which case the tangential force, the tangential reaction, I should say, will be proportional to the normal reaction. In other words, if there is some force in the tangential direction, it's proportional to the force in the normal direction. Or the other way we could write that is that the tangential is equal to some coefficient times the force in the normal direction. Another typical type of support is called the roller support. In the case of a roller support, we usually represent it something like this with a big wheel sitting on top of something. The roller support is very similar to the simple support in that you're basically presuming only one dimension, some sort of pushing. However, we're assuming that somehow that it's also potentially being latched in there or held in there, so you could actually go in either direction. You do not necessarily have to be a compressive force. That there's also some means of it holding it in place, even if it's being lifted off, if the body's being lifted up off of the surface. So in other words, it can resist either in the positive or the negative direction. However, there is no side to side resistance. So this is in one dimension, normal only. In other words, perpendicular to the surface. And there is no tangential component. Notice this is also true, the same sort of system. For something we call a pinned bar, think about a bar as being something like a cable, but instead of a cable, something like a dowel or a long rod. In the case of the pinned bar, if you take something like that, what it means by pinned is it means at the far end, it's allowed to rotate. And we'll talk more about a pin in a second. But at the far end, it's allowed to rotate. So it can move side to side, or it's not going to resist side to side because it'll tend to spin. However, if you're able to push or pull on it directly in the direction that the bar is, if you slice across a bar like that, you'll get the same sort of force. You can only consider the normal force. You can't consider any tangential resistance because that pin at the end means that it could spin around that point. Let's consider that pin that we're talking about. This is another sort of support. The idea of a pinned support, you can think about it as if you took, for example, two straws and ran a pin through the middle of the two straws to connect them. Two straws could spin around one another. They can pivot, but they can't slide off of one another. In other words, let me draw the straws here. If I had a long straw right here and I took another straw and I ran it next to it and then I ran a pin through the two straws to connect them, you would notice that you could change the angle between the two straws, but they would still stay connected. That's the idea of a pin support. Something like this, it's represented with a picture that looks something like this. Usually it's shown as something attached to the ground with a little circle on it and a little triangle representing that pivot point around which anything that's attached to it could pivot. We replace that in two dimensions. It's a two-dimensional system because it can resist both normal and tangential. In other words, we need to think about it by replacing it with a force that can either be a vertical force or a horizontal force if we're oriented this way. Notice that we don't necessarily have to do them normal and tangential. We can do four other two-dimensional orientation. So for example, we could consider it as being a set of forces that are in some other basis as long as there's two dimensions. Another type of support called a fixed support. A fixed support actually accounts for three dimensions where there's dimensions are normal force, tangential force, and moment. In this case, what we represent this with some sort of rod or something sticking into a surface. What this essentially means is that not only does this support push back or pull back or prevent you from sliding side to side, it also prevents rotation. It supplies a moment that prevents you from rotating the body about that particular point. It holds it up and down. It holds it side to side. And it prevents it from rotating, unlike a pin which would allow it to rotate freely, even if it wouldn't allow it to change its position. And the last example we're going to talk about is kind of a rare example. It's very rarely used in lower level courses, but occasionally it might come up. It's something known as a fixed roller. It's usually represented with a picture that looks something like this where we have the block. And it's fixed inside the block, but then we put some sort of rollers along the bottom, which means that it is allowed to slide side to side, but it's prevented from moving up and down, and it's prevented from pivoting. But it is allowed to move side to side so there is no reaction side to side. In this case, this is two dimensions where those two dimensions are normal and moment. So if I replace something that acts like a fixed roller, I would replace it with a normal force and a moment around that point. So how does this relate to our static equilibrium? Well, in two dimensions, we have three static equilibrium equations. Can you remember what they are? Sum of forces in the x direction is equal to 0. Sum of forces in the y direction is equal to 0. Sum of moments about some point is equal to 0. So if we have these three pieces of information, these three equations, to relate our various forces, we can handle three unknown supports. Three equations, three unknown support reactions. If we know the loads, the forces that are being applied to the system, we can use these three equations to find three unknown support reactions. With our most typical cases, something called simply supported, where we consider some sort of beam or bar, we attach one side with a pin and one side with a roller, pin and roller. The pin can support one force vertically, or one force normal, one force tangential, and the roller can support one force normal. That's three unknowns. And then we can apply our three equations. Sum of forces in the x, sum of forces in the y, and sum of moments. Another very typical one is called the cantilever. In this case, we have a system where we might have something extended out, and we have it fixed into the wall. Well, for that case, that's a three-dimensional support. It can supply a reaction horizontally. It can supply a reaction vertically, and it can support a moment. And to solve for each of those three things, we need to apply sum of forces in the x, sum of forces in the y, and sum of moments. So how does this fit into our process? When establishing your reactions, you're going to identify a point of application. In other words, determine where you're removing the force from or where you're cutting it from. Figure out the point of application. Identify a local basis. In other words, figure out what is normal and what is tangential. Identify the reaction type and direction. Then we establish a global basis, which may line up with your normal and tangential forces. In fact, it's usually wise to pick one that does line up with at least one set of normal and tangential forces, and then finally consider the reaction in that global basis. So to give an example of something like that, let's say, for example, we have some sort of system where you have a plank that might be leaning on an inclined surface. And we decide to recognize that as being a, well, we could consider that to be simply supported. Well, in a case like that, we identify the point of application. That's the point where we're going to apply our force. We identify a local basis. We see that this is at some sort of angle. We can identify that angle, and we sort of determine that there is a normal in that particular direction and a tangential along that direction. A normal that's perpendicular to tangential that's parallel to that surface. We identify the reaction type. We see here that if it's a simple support, the only reaction type is going to be normal. And then we might talk about this as being in a global basis. We see that there's no tangential because it can slide here, so we say the only component is normal. But then we might decide that we want to talk about this in a global basis where up is our y direction and side to side is our x direction. Because maybe somewhere else we have some loads that are perpendicular, I mean, that are horizontal or vertical. And if that's the case, then we end up thinking about what our reactions are, the components of our reaction here, which would be there's the x component of the reaction and the y component of the reaction. So if this is the reaction normal, we have the two components, rx and ry. Notice in this case, the angle we're talking about is actually this angle here. That the angle there, if we rotate everything, is going to end up being the same angle here. So rx is going to be equal to rn sign of that angle. And our y is rn cosine of the angle. And then we might use those particular values in our static equilibrium.