 We're now going to take a look at the equations that describe the hydrostatic forces on surfaces submerged below a surface. So we'll be looking at plain surfaces to begin with. And so what we're going to do now, we're going to go through a couple of sections on a derivation for the equations here. And first of all, we'll be looking for the resultant force on the plain surface. And then the second thing that we'll be doing is trying to determine where does that resultant force act. So this is a lot of fluid statics that we're doing. It's very similar to course and statics. But what I'll do, I'll begin by drawing out a diagram that we'll be referring to as we go through the analysis. Okay, so this is a schematic for the plate that we're going to be deriving the equations for. A couple of things to note on here. First of all, what we have done is we've defined our coordinate system with respect to the center of area on the plate, which is right at that location. And so CA will denote the center of area. We've said that the center of area is acting at a depth HCA below the surface. The z-coordinate goes perpendicular to the plate as shown here on this part of the diagram. And then the x and the y-coordinates are there. We look at a differential element on the plate, and we will be deriving the equations from that differential element. The last thing to say is that the hydrostatic pressure averaged over the entire plate acts at a location that we referred to as being the center of pressure, Cp. And that is denoted by this vector Fr here. So what we're going to do, let's continue on, we'll refer back to this diagram as we work through it. But one of the things that we can also say about the resultant force is that it acts normal to the plate, as we said for hydrostatic forces earlier. And so although that vector looks a little crooked, it's supposed to be normal. And that's due to the fact that the fluid is not moving for hydrostatic. So what we're going to do, we're going to do a derivation for the forces on one side of the plate. So let's continue on with that and see what we get. So what we can begin by doing is writing out the differential force that would exist on our differential element. So this is the differential element here. Let's take a look at what the differential force would be. And we can write Df as being the pressure at that location multiplied by the differential area. And of course, if we want to get the resultant force, then what we would need to do is we would need to integrate that across the entire plate itself. Now we know the pressure distribution from hydrostatic. So let's introduce that. So with the pressure distribution, what we can do is we can write out the force. So we get this relationship here. And notice I've been using PA and P atmosphere. That's referring to the same thing. So just be aware of that. Now with this, what we know is the atmospheric pressure is not a function of location on the plate. Atmospheric pressure is a constant. So we can pull that out and integrate that first term without a problem. For the second one, what we need to know is we need to know how H, that is the depth of fluid at a given location. Let's look back at our diagram. H is shown here as being the depth of the differential element. We need to know how that varies as we move around and look at different differential elements. So the one thing that we do know is that H is only a function of y. If you move across the plate in the x direction, the depth does not change. It's only if you move in y that the depth will change. So we know that. And what we can do for the resultant vector is we can, first of all, for the first term, it is just the atmospheric pressure times the area of the plate. We can pull out the density and the gravitational constant and the second integral. And then that is HDA. So what we need to do, we need to handle this integral here. And so in order to do that, we're going to take a look at the relationship between depth and the area. And we'll re-express that in terms of sine value, sine theta. So let's take a look at that now. And I'll begin by drawing out the plate again. So here's our plate. And let's say this is our center of area located right there. And recall, that is where our coordinate system begins from. We've defined the coordinate system there. And if I sketch up, we said that the angle of the plate was theta. So if we are at a location y, and remember, coordinate system begins at the center of area. So at a location y, at this location, the center of area, we know that we have HCA. That is the depth at the center of area. But if we're at some location y, like right there, I'm going to draw a little right angle triangle. I know that this angle here is theta. And consequently, we know that this distance here can be determined through trigonometry to be y sine theta. So what we can write at any location where we're looking at our differential element y, we can write out that H at a given y location is the depth at the center of area minus y sine theta. And if we go lower along the plate, then y would be negative and that would be HCA and it turned into a plus y sine theta because it would be minus minus y. But that gives us a relationship for the depth at a given location where our differential element is. So what we can do is we can take that and plug it back into the integral for the resultant force on the plate. So we have this. Now what I'm going to do, I'm going to expand the integral. So HCA is a constant. This is varying with y. So let's rewrite that. And I've been able to pull HCA out because HCA is a constant. It's not changing. The center of area is defined. And then we have a minus, a minus that turns into a plus. We get rho g sine theta. And that will be a double integral over the area. So we get that there. Now, this is an integral that is rather well known. And what that refers to is the center of area for the plate itself. So that would be the center of area. But remember, we've defined our coordinate system. Let's go back to our schematic. We define the coordinate system to be about the center of area itself. Consequently, what that means is that where our coordinate system is defined, this is actually zero. And that last term disappears from our equation. So what we are then left with, what we're left with is this equation here. So the resultant force on the plate is nothing more than the pressure at the center of area. And that's what this is showing. Pressure center of area multiplied by the area of the plate. And I've shown it in vector notation here because we need to preserve the vectors. However, we can simplify that a little bit if we note the following. So if we note the pressure as being an outward force, so let's draw our plate again. We had our free surface up here. And what we note is Fr is an outward vector. So if we just say that it's an outward force and it's acting normal onto the plate surface, we can then neglect the vector notation. The other thing, let's say, I said Pa is P atmosphere here. Let's assume that we have a scenario where we have liquid in here and we have air down here, atmospheric air. Consequently, we can say here is also P atmosphere. If you have that specific case, if you have the outside pressure being P atmosphere and if you have knowing that the pressure or the force is acting normal to the plate, what we can do is we can simplify or get rid of the vector notation. And what we end up with, the pressure at the center of axis is nothing more than rho G H C A. And then that is multiplied by the area of the plate. And so in some books, you'll find this as being the expression for the pressure at the center of, or sorry, for the force that acts on the plate. And remember, this force is not at the center of area, but at center of pressure, which is different from the center of area. And so consequently, here we've derived what the force is. What we need to do in the next segment is we need to figure out where that center of pressure is. So that is hydrostatic forces and pressure distribution on a submerged plane surface will continue on by trying to figure out where the force is located at the center of pressure.