Hey, everyone. So now that we've seen the basics of pressure in air and in liquids, in this video, I'm going to show you how to calculate different kinds of pressures at different points throughout a liquid, and we'll do an example. Alright? So let's check it out. So remember that pressure in a liquid changes with depth. As you go down in a liquid, the pressure increases according to this equation here.
P = P bottom = P top + ρ g h
Now there's one super important thing about this equation that I really want you guys to understand. It has to do with this h term here. Normally, when we've used h, we use it as a height from the bottom. Right? We use it sort of like to represent heights, but this is actually wrong. Alright? So I want you to write this. This is wrong. You should never use height this way. And instead, height is always going to be a depth that is measured from the top down. Alright? So the correct way to use and represent your h in your diagrams is basically go from the top of whatever boundary it is that you're looking for and go down to whatever point that it is that you're interested in. This is going to be your H.
H, a sort of like an upside-down height. Alright. Now a lot of problems will look like this where you'll have multiple objects or multiple materials that are mixing together or sort of settling in layers like the problem they're going to work out down below. And in this case, the point or the boundary where two materials touch has a special name, it's called an Interface. So hopefully, you'll see here in this diagram that I've got a couple of interfaces. We've got one right here, which is where the yellow or the orange and the white liquid meet. There's another one where the white and the yellow liquid meet, and there's actually a third one which is where the yellow liquid will touch air. So interfaces are always between two kinds of fluids, whether it's liquid-liquid or liquid-air, but it's never going to be where some liquid hits something solid like glass or something like that. Right? So that's not really interface. You can forget about that. Alright?
Now at these points, what's really important is that the pressure of both materials is the same. So what do I mean by this? Well, if you had a molecule like a little tiny molecule, it was at the bottom of the glass, and you had one that was a little bit higher up, then the pressure of this bottom molecule would be higher than the pressure at the top. Right? That makes sense. We've learned that so far. But what this is saying is that if you grab the molecule that's sort of at the very, very top here and you grab the bottommost molecule of the layer on top of it, then these two things would basically have the same pressure. Alright? So, all this point is saying here is that in these kinds of problems or these kinds of situations, you're never going to have like crazy jumps in pressure from one boundary to the next. It's sort of going to be smooth, and the boundary is where they're equal. Alright?
Now one important consequence of this is that everywhere you have a liquid that's touching air like you have in this topmost boundary over here, what that means is that the pressures are equal. So in other words, the pressure of the liquid is equal to air. So this topmost molecule that's sitting right underneath the surface of the liquid is going to have the same pressure as standard atmospheric pressure. Right? And that's just a number that's the, you know, 101,000 pascals. All right. So let's just go ahead and jump straight into our problem here.
We're going to talk about something else called the gauge pressure, but I'm going to save it for later on. Alright. So in this problem here, we have these 2 liquids that are going to combine in this beaker, but they're not going to mix, and basically, so let's get started. So we have a 6 centimeter column of blue liquid, and we're told what the density of it is. So you pour some liquid into a cup, and it's going to be 0.06 meters. Right? That's a column like this. The density of this blue liquid is 1200. Then you're going to pour a 4 centimeter column on top of it. So this is obviously not to scale, this is going to be 0.04 and the density of this yellow liquid is 800. Now, we're told is that the entire beaker is 12 centimeters tall, so this whole entire thing here is 0.12. And what that means is if you kind of notice here, this is 46, which makes 10. All it just means here is this little tiny gap like this, which is 2, is going to be 2 centimeters, so 0.02 meters. Alright? Might need it. Not not not. No, I don't know. So let's go ahead and get started here.
For part a, we're going to calculate the absolute pressure at the blue-yellow interface. What does that mean? Well, if we're using the absolute pressure, right, we've got some densities, we're going to have to use our P bottom = P top + ρ g h equation. You're usually going to start off with that. Right? So what we have here is that we have that P bottom = P top + ρ g h . Now in these kinds of problems where you have lots of you know, we have different layers and you have different liquids that are mixing, it's very easy to get confused between what's the bottom and what's the top of what you're looking at. Alright? So the first thing you need to do is define what's the bottom and what's the top. Now in this problem here, what I'm asked for is I'm asked to calculate what is the pressure at this point over here. So this is the pressure at the blue-yellow boundary. Now to do that, I could set up an equation. I could set up this equation here and either use this interval from here down to the bottom, or I can use the top down to that boundary there.
Now what happens here is if I want to calculate this pressure, but I also don't know what that pressure is, then I can't use this interval because I have 2 unknown pressures. So you always want to combine a known pressure with an unknown pressure so that you can use this equation. Alright. So we're not going to use this bottom interval. Instead, we're going to use this interval over here because hopefully you guys realize that the pressure at the top right at this point over here is really just going to be the pressure of air because it touches air. Alright. So this is just going to be air like this. And this bottom, what I'm going to call the bottom, is really just going to be the blue-yellow interval. Alright? So even though it's not the real bottom of the cup, that's what I'm just calling my bottom for this interval. Okay? And so therefore, what you're going to use is you're going to use the pressure or the density of the yellow liquid and then the height of the yellow liquid. Alright? So in other words, that your P , blue-yellow interface is just going to be this is going to be 101,000 Pascals plus the density of the yellow, which is 800. Then we're going to use g for 10 to make things a little bit simpler.
Now what about the depth? What about h y ? It's basically just the height down and measured from the top down to wherever it is that you're measuring. So I'm measuring from here all the way down to the boundary. So this distance right here, which is just 0.04 , that's going to be my H y . Alright, that's going to be 0.04 . Now this just combines, if you multiply this out, it's going to be 320. So your answer is 101,320 Pascals. Alright? That's the first part. Now let's move on to the second part here.
Now they're not asking for the absolute pressure. They're asking for something else which is called the gauge pressure also at the blue-yellow interface. So this is going to be P gauge, at the blue-yellow interface . Alright. Now what is this? What's the gauge pressure? Well, basically, the gauge pressure is really just the difference between the bottom, the pressure, the absolute pressure you're trying to measure, and the atmospheric pressure. The gauge pressure is really just the pressure that you're measuring relative to the atmospheric pressure. So, for example, in this problem in part a, we were calculating the pressure at the bottom, and we used p , top , which is the air pressure. Basically, this term right here, the Rho g h term, is the difference between these two terms, right? If you move this p , top over, then all you're left with is this, and that's what the gauge pressure is. Your gauge pressure is really just the difference between these two pressures. Alright? It's the absolute pressure and atmospheric. Alright? So in other words, it's just going to be Rho g h , and we actually don't know what that is. That's just 320 Pascals. The reason it's called the gauge pressure is that if you actually took like an instrument and went down to this level here, you would not see this number. You would probably see 320 Pascals. It's a measurement done relative to the atmospheric pressure outside of the cup. Alright? Okay.
Now I got one last point, which is over here, but I'm actually just going to go ahead and continue on with part c. Alright. So part c, now we're going to calculate another absolute pressure, but now it's going to be at the very bottom of the blue liquid. Alright, so let's set up our equation again. So we got P bottom = P top + ρ g h . Alright, so what's the bottom? I'm trying to figure out this pressure all the way down here. So this is going to be P , blue . So in other words, this is going to be P , blue . Now, what's the top? Well, in this case, I also have 2 intervals where I can go from here all the way down or I can go from all the way at the top all the way down. Because I actually know both of these pressures now, I know what the blue-yellow boundary is, and I know what the pressure at the top is, I can actually use both intervals and both of them would be fine. But the simpler one is actually just to use this one all the way down, the blue-yellow one because I already have what that number is. It's right here. I calculated it in part a. So basically, what you're going to do is you're going to you're going to say your top is actually just going to equal the blue-yellow boundary. The bottom is going to equal the absolute pressure, and because now we know that this term here is really just this. So in other words, your p bottom is just going to be 101,320 plus, and now we've got the row for the blue, the g times the height of the blue. In other words, this is going to be 1200 times 10.
Now, what do we plug in for H B ? Well, you might be thinking you're going to measure from the top down, so you're going to go from here all the way down, but that's actually not what you're going to do. Your H B is actually just going to be the height of just the blue parts. And the reason for that is you're measuring from your interval, which is, in this case, this is the top, and this is the bottom. Okay? So what happens is, is in this interval, because you've called the top of this thing right here, the blue-yellow interface, then your h is going to be measured from the top down relative to that point. So in other words, this is going to be your H B , which is 0.06. It's not going to be the height all the way to the top of the liquids. Alright? Because you've already included, you've already counted the 0.04 when you calculated this first part here. So if you were to do it again, you'd be double counting. Alright. It's super important here. That's why it's really important to label everything. So it's going to be 0.06, and then, what you're going to end up with is 102,040 Pascals. Alright. Now basically, we're done here. You can go ahead and move on. What I want to do is show you real quick how we can actually calculate this in one quick step because some questions might actually just ask you right off the bat to do this. Alright? So I'm going to show you how to do this in one step. So I'm going to say that or you can also calculate this, your p blue as you can say this is going to be p top, in which case, you're going to go all the way up, and you're basically going to call a top the air. You're going to go all the way and say, well, this is actually going to be my interval now. So if I take this whole entire thing as my interval, the top is going to be air. And then what happens is you've got to add a Rho GH , except you've got to add it for the yellow liquid first. So they're going to be Rho yellow</