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Calculating Pressure in Liquids

Patrick Ford
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Hey, guys. So in this video, I'm gonna show you how to calculate pressure at different points within the liquid using to common examples. Let's check it out. Cool. So, first, remember that the pressure in the liquid changes with depth. As you go down in a liquid, the pressure is going to increase, and the exact calculation is used is done using this equation the pressure equation. You should also know that in between two liquids, whenever to liquids, touch or two materials, they don't have to be liquids whenever to matures touched. We have something called an interface. So, for example, over here in this little cup beaker, uh, this line right here, where these two things touches called interface. There's another interface here, and I want you to see if you can figure out where it goes. And hopefully you're thinking that the other interfaces over here, and that's true. So this is the interface between air and the yellow liquid. Those are two different materials, and then there's an interface between yellow and blue liquid. Now, technically, there's an interface between, like down here between blue liquid and glass. Those are two different materials, but you're never gonna use that. So just forget about it. What's important about interfaces that at these points, the pressure of both materials is the same. So what does that mean? So imagine you have a molecule all the way at the bottom of the liquid here, blue molecule and a molecule all the way at the top. These two molecules, you're gonna have a different different pressures because they're different heights. And the molecule at the bottom is gonna have a greater pressure in the molecule at the top because he has more stuff on top of it. Now what? This what this over here saying is that the top most molecule, the one that's like all the way at the top, has the same pressure as the bottom most molecule in the yellow liquid. I do it right here. But imagine this is a yellow molecule. Okay, so this has some consequences. But I want you to remember that the pressure is the same. The first consequence is that everywhere a liquid touches air, the pressure of the liquid equals the pressure of the air. So what does that mean? So if you look at the molecule, all the way at the top of the Yellow column. This is all the way. The topic the top most molecule. It is right under air. Therefore, the pressure at the top of the yellow is the same as the pressure of air, and this is gonna be very useful. So let's do example one to see how this might work. So it's, I suppose the density of a particular beer is 10 50. So density Rho is 10 50. This is using the the standard units. I don't have to do anything weird and we want to know what is the gauge pressure at the bottom of a solo cup if it is field filled with beer to its very top. So I have a solo cup. Looks something like this. Um, it's not a perfect straight cylinder, but here doesn't make a difference. And it just has beer. I'm gonna make beer blue, um, has beer all the way to the tops. It's got 12 centimeters. Hike 0.12 meters and it's got beer everywhere on Do we wanna know what is the gauge? Pressure gauge pressure. Okay, Now, almost every time you asked for a pressure on do you have a height, you're going to use the pressure equation right here, which combines links, pressures with heights. Okay, so let's write this real quick. So we're gonna say because we have ah, height and I wanna pressure. I'm gonna write that P bottom equals Pete Top plus row G h okay. And P bottom is down here. This is the P bottom, and Pete Top is the top of the liquid peat top. Now, because the top of the liquid touches air because it interfaces with hair. The pressure at the top here will be the pressure of air. Or more precisely, the atmospheric pressure, the standard atmospheric pressure, which is one a t. M. Now, if you remember, we can't really use 1 18 and we have to use the Paschal version. So the pressure at the top is 1.1 times 10 to 2/5 Paskowitz, Just a no number. You're supposed to memorize that plus row G bitch. Now, before we get further here, I wanted to talk about this, but you may have noticed that the question says that we want the gauge pressure and this is a little tricky and I did this on purpose. I hope you remember that. What gauge pressure means is this right here? Okay, so let me make that yellow as well. Meaning we don't want the pressure at the bottom. That's not what we want. We want the pressure difference. So I don't want p bottom. I just want I just want row G h. Okay, so I was almost going the wrong way. I wanted to talk about how the pressure at the top was gonna be the pressure of air. So you made that made that connection. But this question, actually, once just the gauge pressure. Which means you don't even use p top. You just use row G h. Okay. Ro is density, which we have. It's 10 50 gravity. I'm going to use 10 instead of 9.8 to make things a little simpler. And the heights is points 12 m and all the units air standards of the pressure will come out in Moscow and you multiply everything. The answer is 12 60. But scow, this is the engage pressure here. Quote. Let's do another example. And on the second example, we have two liquids on top of each other, which is kind of common. So you're going to get something like this with two liquids and the first liquid here, It says it's blue. So I got some blue liquid here and the second liquid is yellow. So I'm just gonna sort of do this and I'm gonna put a little line here. Cool. And it says, uh, the blue liquid you pour six centimeter column of blue liquid so you pour liquid essentially until it reaches a six, uh, centimeters of heights. That's what it means by six centimeter column. So this height here is 0.6 m, and the the density of blue liquid is 1200 standard units, and then you add a four centimeter column, so of yellow liquid. So this is four centimeter column, and the density of the yellow liquid is 800. Also standard units, and this is on a 12 centimeter beaker. So this entire thing is 12 centimeters. So 0.12 m. By the way, if you look at this quickly, you got 6 10 64 So that's 10 which means there's sort of a two centimeter gap here. I don't know if we're gonna need this. Let's right here. Just in case. Cool so the liquids don't mix. This is very important. If they look liquids, Miss mix, this would be a mess. And you you wouldn't be able to solve it like this on we went to calculate the absolute pressure at the blue yellow interface. So there is blue here. Just make this blue. Andi, I want to know what is the pressure here? So part A is asking for this. What is the pressure in the blue yellow interface? How do we do this? Well, there's two ways we can set this up. We can set up an equation that goes from here to here. And then we would use the Six Centimeter Heights. Or we can set up an equation from here to here, and then we would use the Four Centimeter Heights. Okay, so we have a choice there, but we don't really have a choice. Because if you have an unknown pressure, this is my unknown pressure. I have to couple it with a known pressure. I don't know the pressure here. I also don't know the pressure here, but I do know the pressure at this point. The pressure at this point because the top of the yellow liquid touches air. The pressure at the top of the yellow is the same as the pressure of air or the atmospheric pressure, which is 1.1 times 10 to the fifth. So therefore, I have to use the top interval to calculate this. Okay, so let's do that. We're gonna say that now. We're using this piece here, the top someone to write that P bottom. The equation is always the same. Equals P top plus row. G h. Now, when we do this, um, this is the density because we're talking about this interval here this interval, this height of liquid. Um, the density here is the density of yellow. And it's the height of the yellow column. And P bottom means means blue yellow interface. And Pete Top is air. I hope you see this. I'm just doing this slowly so we don't screw this up. Bottom is over here. So this is the interface between the two, which, by the way, is what we want. And top is where we touch air. Okay, so let's calculate it. Just a matter of plug and stuff in. We already know that the pressure of areas 1.1 times 10 to the fifth. The density of yellow is 800 gravity. I'm gonna round it to 10 just to make life easier. And the height here is 0. meters. Okay? And if you do all of this and I have it here, you this piece here is this entire thing here is just 3 Not much. Okay? And if you do this whole thing, you end up with 101 3 20. Which if you were to round 22 significant figures or three significant figures, you would basically end up with this number. In other words, there's negligible difference here. And that's because it's on Lee Four centimeters, which is a tiny column of air or liquid. But that is the pressure in that middle point. This blue, this thick blue line over here. Quote for part B. Part B asks us. Read out of the way Part B asks us for the gauge pressure. Remember? Absolute pressure means p bottom right right there. Between those two gauge pressure, it's just a difference. So if I'm asking for the gauge pressure from the blue and yellow interface. It's just I hope you see it. It's just this part here. It's how much higher is the pressure of blue relative to the pressure directly on top of it. Okay, so when I say pressure gauge, we're just saying, um is the pressure gauge between between these two these two things here. So it's between blue and the gap between between the air and the blue over here. So it's the pressure gauge across the yellow is gonna be row yellow G, height, yellow. And we already did this the entire thing here. If you multiply its just 3. 20 ph Just asking how much higher is the pressure of the blue line relative to the water on top of it? Okay, so this is related to this, uh, and see is about a new point. We want the absolute pressure. So we want the absolute pressure you can also think of this is P bottom right at the bottom of the blue liquid. So all the way at the bottom of blue. Hm. Cool. And again, we're gonna use the P bottom equation. So P bottom equals p top. Plus row G h. But now we're going to set up a new interval from here to here, right across the six centimeters. Why? Because I know this pressure. I now know this pressure over here. And to find this pressure, I'm going to write an equation from here to here. I hope that makes sense. So when I write p bottom, this is all the way at the bottom of blue. The P top is gonna be the blue yellow interface. This is going to be we're gonna go through. Think of it as you start at the top of the liquid, you're gonna go through the liquid, you're gonna go through the blue liquid, and the heights will be the height of the blue liquid, which is six. I'm doing this very slowly so that hopefully everyone can follow that. So, um, P bottom is what we are looking for. Pete Top would just calculated it's this number right here, Tank. So 101 3 20 Plus the density of blue, which is 1200 gravity, which I'm going around us 10 and the height of blue right here. 60.6 And if you do all of this. You get that? The answer is 102 040 busk out. Very similar. Not a big difference, but a little bit higher pressure than before. Okay, so if I want the pressure at the bottom of the second liquid, I confined the pressure in the middle, right? So I started the top. I know this one. I can find the next one, and then I can find the next one. Now, if you didn't want to do this one at a time, let's say this question asked you directly foresee which sometimes they'll do that. It was just gonna say, Hey, find the bottom, the pressure all the way at the bottom. Can you do this in one step? And the answer is yes. So or you could have done this in one shot. You could have said the pressure. Let me rewrite this just to make this a little bit cleaner. Let's say you have a liquid here. You have another liquid here and let's call this pressure a pressure. Be in pressure, see? And by the way, if this is a rare pressure, is the pressure of their because it's touching air But if you have a situation like this, which is what we have, we could see the pressure. See, Is the pressure a plus plus row G H for the first liquid? Let's call that row one G H one. So we're adding that tiny pressure plus the next column of air or a column of liquid row G H. But now it's gonna be the density of the second liquid, plus the heights of the second liquid here. Okay, so this is one. This is two. If you were to do this, you would get something like this. Pressure of air, which is 101000 plus the first density. Which up here was 800 gravity. 10 in the first hype, 0.4 Then we do the same thing for the second column. So 1200. Let me get out of the way. 1210 in the heights is 12100.6 If you do this, you would end up with the exact same answer. And you can try it if you want. You end up with the same exact answer here. So point I wanna make here. That's very important. Is If you have two columns or three columns or four columns of liquid on, do you want to find the pressure at the very bottom? You can just keep stacking. Row G. H is for every column you have, and that is a faster way of solving this. If you have to do it in one step, Cool. That's it for this one. Let's keep going.