 We're now going to take a look at the application of liquid hydrostatics to manometry. And so if you recall we came up with an equation for the hydrostatic pressure within a liquid. And so this was assuming that the liquid was stationary, it was not moving. So that is the equation that we came up with. Now what we're going to do, and we did discuss application to manometry, we looked at mercury manometers, we looked at some old manometers. But what we're going to do now is we're going to consider a case where you might have a manometer with multiple fluids in it. And so how do you deal with that case? And that's what we'll be taking a look at. So let's assume that we have a liquid surface to begin with. And that at the top is some known pressure, p1. And underneath we'll have a layer of let's say oil. And the density of oil I will put rho with the subscript o for the oil. And what I'll say is the upper surface z is equal to z1. This is z2. And let's say the vertical axis here is z going up in that direction. And underneath let's say we have a layer of water, rho w for the water. And then let's say underneath that we have glycerin, rho g for the glycerin. And then finally let's say way down on the bottom we have mercury. So this is just some hypothetical situation where you might have a bunch of different liquids on top of one another. And then z3 for the water interface, the glycerin we'll say is z4. And then finally z5 for the mercury at the bottom. Now if we had this scenario, and if you wanted to measure the pressure at the bottom here at point p5, how would you do it? Well you'd have to look at all of the pressure distributions to all the fluids above. So let's take a look at that. So using our equation for the hydrostatic pressure distribution in a liquid, we can express what the pressure would be for this interface right here between the water and the oil. And then if we wanted to express the pressure at this interface, and then down here. So this here would be p1 is at the top. This would be p2. This is p3. This is p4. And this is p5 here. For p4 minus p3 it would be the density of glycerin times the gravitational constant times the difference in or the thickness of that layer. And you have to be very careful with making sure that you keep the convention two and two and one and one in your equation, which is what I'm doing for each of these expressions that I'm writing out. And then finally for the bottom p5 minus p4 is equal to minus the density of the mercury, gravitational constant, times the difference in thickness, which would be we have five. So that would be z5 minus z4. So we have all of these equations. But let's say we want to know p5. So the pressure at the very bottom, what is it? Well, what we would need to do is we'd need to combine all of these equations together. And that would then enable us to determine. So let's proceed and take a look at that. So if you want to know p5, and you would reference it to p1, which was the known pressure at the top, the way that we would do it is we would add all of these together. So we would take all of these here, and we would add them all together. And then similarly, with the right hand side, we would add all of these together. So let's go ahead and do that. So once we've done that, first of all, on the left hand side, we have the pressure. So let's take a look at what is happening there. We can see that the p2 is going to cancel out with that p2. p3 is going to cancel out with that p3. And p4 will cancel out with that p4. So what we're left with is p5 minus p1 on the left hand side. And for the right hand side, what I will do is I will express it out. There's no way to cancel out the term. So we'll just explicitly write it all out. So we get that for the expression. And this would then enable us to determine the pressure at 5, knowing the pressure at 1 on the surface. However, what do we need to know? Well, we need to know the depth of all those interfaces. So let's come back here. What we need to know is we need to be able to measure this interface, this interface, this interface, and that interface, and know the position of them with respect to our free surface at the top. That is the principle of manometry. In a manometer, what you're doing is you're measuring the fluid interface. And from that, knowing the density of the fluid, you can use the hydrostatic pressure distribution equation in order to determine what the pressure is at different locations within your system. So in order to solve for p5, you need to know the pressure at the surface, p1, and then you need to know all of those interfaces. And the other thing you need to know, you need to know the density of all your fluids within the system. So you need to know the density of oil, the density of water, the density of glycerin, and the density of mercury. So this is the principle behind which a manometer works. So the principle of a manometer, it's pretty simple. What you do is you measure the height of a known fluid. So that means you need to know the density of the fluid and you need to know where the interfaces are. And by doing that, you can then infer a pressure that you are trying to measure. So that is the principle of a manometer. We'll continue on and take a look at a couple of applications of that.