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19. Fluid Mechanics

Pressure Gauge: U-shaped Tube


How U-Shaped Tubes Work

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Hey, guys. So this figure, we're gonna talk about the U shaped tube, also sometimes referred to as the YouTube. Andi, It's one of the more popular pressure gauges in physics. Let's check it out. All right, So a pressure gauge is simply a device and instrument. A very simple instrument. Usually, um, that uses high differences to calculate pressure. So the idea is, you're gonna have to liquids in here, and it might looks like something like this. Don't draw this just yet. I'm gonna erase it. Might have a liquid here, and then put another liquid over here and you're gonna be able to calculate pressures using these high differences here. Okay, but for now, let me just erase this and we'll come back to it. Um, you're gonna calculate pressure using the pressure difference equation right here. And first, I want to talk about these first two cases here, which are trivial. Um, they're very silly, but we're gonna talk about them, and then this is the important one. So if you have one liquid in a YouTube shape, this is what it looks like. A U shaped tube and the two sides are open. So it's open here and it's open here. Which means that it's exposed to air on both sides. Usually air or or some other, Um, some of the gas, right? Whatever you have there, what's gonna happen is that the height will be the same. So you're gonna have the same height here, okay? And that's because you have the same pressure up here. The pressure, Um, let's call this pressure. One equals pressure to, and that's because they're both touching air. So it's Whatever the atmospheric pressure here at this point is because the pressures air the same, you're gonna have the same heights. Okay. In other words, H one is the same as H to. We're gonna call the left side one in the right side to all right, so that's pretty straightforward. We know that the liquids will level up if they have the same pressure, and they'll be at the same height. What if you have one liquid but one vacuum? Well, you can only have a vacuum if one side is closed. Let's close over here closed. And let's say that on this side the liquid goes up to here, right, and this is a vacuum. Then How high would the other would the liquid be on this other side over here? Do you think it be higher or do you think you'd be lower and think about how pressure is applied to both sides? And the answer is that it should be higher. Sorry, it should be lower, Andi. That's because here there's zero pressure zero plus cow. And here there's some pressure of the atmosphere, right? It's pushing down. So instead of being like this, they're gonna do this. Okay, so that's that Different pressures will mean different heights, and the open side is always going to be lower. Let's right that so open side is always is lower, then vacuum cool. So those two are trivial, the most common. The one that you actually would get most of the time is when you have two liquids and both sides are open. So here's the basic idea. You put some liquid here. Let's put red liquid, right. Uh, let's put this up over here and you pour enough liquid, and if it's by itself, it's going to balance out. So this is the same height. Now I'm gonna come here, and I'm gonna pour some blue liquid and I'm gonna put some blue liquid. And let's say I have this much blue liquid that I pour, but the blue liquid is heavy, so it's gonna push down on the red liquid. So it actually is going to move sort of this way, right? So the blue liquid will be here, Okay, because it moved out Well, if it moves down by this little bit, then this thing has to move up by that same little bit. And now the red liquid is up here. So let me draw the red liquid. Can I get your color pens out? Cool. So something like that, it looks terrible. All right, now what we're gonna lose is we're gonna look at There's four specific points that are important in this problem, and one of them is going to be the top here. The other one is going to be this point here. So if you start from blue and you cross over to the other side, this is another point that's important. Let's call this. Let's make it all red because it's on the left side. So we're gonna call this A and B and on the right side, the other two points. That important is the very top of the liquid up here. Just like the top of this liquid. You have the top of this liquid as well. A B Let's call this C and this height here, that's the same height as be Let's call that d those four points important. You have to memorize those four points. I want to point out that the distance between C and D we're gonna call this height to because it's on the right side and the distance between A and B. We're gonna call this height one because it's on the first side. Okay, now the top ones, they're easy to remember A and B A and see, they're just the top of both columns. But how are you going to remember and D is pretty easy to D is the interface between the two liquids. It's the interface where the two liquids touch. Now, the bottom of the other one here, How do you know where it goes? Well, you just go from the interface and you cross over to the other side. You cross over to the other side. Okay, Those are the points that important. Another distance that's important. Let's make us different color. It's make it green is the distance between the top of C, the top of two in the top of one. That's another thing you're gonna get asked. It's a distance between these two. I'm gonna call this Delta H. So let's give it some numbers. Just as an example. Let's say that this is 10 centimeters and this is seven centimeters, then obviously Delta H. If this is seven high, if this year's seven high and this is 10 high on Delta H is, of course, three centimeters high. Okay, so let's write that equation. Delta H is just the distance between the difference between H one and H two. Now you don't know which one is bigger. Sometimes H one might be actually higher than H two. So all these ages have to always be positive. So I'm gonna put this here. I'm gonna say that it's the absolute value that just in case it comes out to be negative and I'm gonna right here, this is gonna come in handy later. All h is must be positive. Okay, so this is the first sort of equation. It's not even an equation. Just eyeball this and you see that? That's the difference. Okay, There's another equation for this. That's gonna be very useful. I'm gonna derive it later when we're solving a problem. But for now, I'm just gonna give it to you real quick that it is that row one. So density of the first liquid times heights of the first liquid this height right here equals density of the second liquid. The blue liquid times heights of the second liquid, Which is this right here. Okay. And this is the most important equation for the YouTube. Okay. The U shaped tube with two liquids. Um, this is the most important equation. You have to memorize this. It's gonna come in handy if you get a problem like this. Cool. Let's do an example and see how you might see this in action. So it has a YouTube has shown above has too long sides. Both open at their ends. So standard YouTube, they're both open. I'm gonna draw them real quick. Um, the fact that it says it has long sides doesn't really mean anything. That's just standard language. It just means that it's not gonna overflow. Nothing like that. So it was your first pour water s O that the heights of the column of water columns on both sides is 20. So you put enough water, let's put water right around the middle here, and let's make water blue. Um, you put water here so that the height on both sides is 20. So this here is 20 centimeters. All right, if you look at the original diagram up here, you notice that this height here was never mentioned. Let's make this a different color. This height here is never mentioned. And I'll just tell you right away that that height doesn't matter, okay, this height, um, this height doesn't matter. Okay, So that height there isn't gonna matter. The fact that you're giving this 20 isn't really going to be useful are likely. Scratch it. Um, Then you pour enough of a particular oil on the right side of the column so that above the water s so that the column avoid above the water is five centimeters tall. You're gonna put some oil here. Let's make oil red, and you're gonna pour some oil. Now. It's not as simple as putting oil here. And then the water has to go up a little bit as a result of the oil. Right? So let's actually move the water up a little bit. Whoops. Cool. Now we're gonna move on the water up a little bit. Um, remember, the water doesn't go above the oil. And the reason for that is because water is lighter than oil. And if you didn't remember that, if you didn't know that it's right here. Right? Water Oil is lighter than water. Water. The density of water is 1000. So the column of oil has to be higher. Okay, So oil higher because it is less dense. Cool. So we've drawn this. Um, it's We're told that the column of oil is five centimeters high. This is the right side. I'm gonna call this to This is one I'm gonna say that h two is five centimeters, by the way. I know that's the density density to is, the density of oil is 800 and I know that density one of water is 1000. Cool. The question is, uh, there's two questions here, and you assume that the liquids don't mix. This is standard language. if the liquids mixed this this is not gonna works. And all these questions the liquids will mix. And you should also know that water and oil don't mix. So what is the gauge pressure at the water oil interface. So, water oil interface Is this point right here and for part, they were asking, what is, um, gauge pressure? So what is p engage at that point? Okay, I hope you remember, But if you don't pH if you write the full equation people, autumn equals P top plus road GH. This is P gauge right here. Okay. The gauge pressure. Just the additional pressure that shows up in this equation. So you can think of p gauge as P bottom without the P top. That's kind of what it iss. Okay, So anyway, to find p gauge, we're gonna just right row G h. Now we want the P gauge here the gauge pressure at that point. So we want the density of the liquid that's on top of it. That's actually applying the pressure. And that's the red liquid, which is oil and the density there is 800 kilograms per cubic meter, and then you have gravity, which is 9.8 m per second square. And then the height is five centimeters, 0.5 m. Notice that all my units are standard units, right for all these measurements. Which means I'm gonna get standard units of pressure at the end, which is pass cow. So if you multiply all of this, I have it here. Um, you are going to get 3 92 Tiny little bit of pressure. 3 92 Pascal. Cool. So that's it for part A. Let's look at part B. Part B is a little bit more involved. It asks for what is the height difference between the top of the water in the top of oil. So remember this year's H to we're gonna call. We're gonna come across this way here, and this year is going to be a change one. And the difference between the two is this gap here, which is Delta h. Okay, Delta H. So this question is asking us for what is Delta H? Well, I told you earlier the Delta H is the difference between H two and H one. So guess what? To find out a h. You have to have both h one N h two. Now we have hte too, but we don't have a choice one. So first, we're gonna find each one, and then we're gonna be able to quickly calculate Delta H to find each one you're gonna use the equation that I asked you guys to memorize, which is Row one h one equals wrote to h two, and you can use this equation right here. So what I wanna do is I want to quickly solve for this, and then I'm going to show you how this equation works. I'm gonna show you how to arrive at this equation in case you need to. But first, let's solve this. So row one equals road to H two, divided by H one. I'm just solving for for for for density. One density too. We have it right here. I'm sorry. We're looking for each one I'm looking solving for the wrong thing. Here. Circle the wrong thing. We're looking for each one. So each one is road to H two divided by row one. So looks like this. Um, the second height is five centimeters, 0.5 m. Onda, Then the pressures are pressure to 800. Pressure one is 1000. If you multiply this, you get 0.4 um, meters or four centimeters. So this hike here, h one, um, H one is four centimeters. If this is four and this is five, this has to be one centimeter. Okay, that has to be one. So Delta H is H two minus h one absolute value, just in case. It's negative. And H two is five. This is four. So this is one centimeter, okay? And the question wanted it in centimeters. So that's why I left it like that. Now, how did I arrive at this equation? Here, let me show you. And this is a little bit involved. If your if your teacher professor wants you to know how to solve from scratch, then you have to do this if he or she is okay with you starting off from this equation from the shortcut, then you don't really need to know this part. Okay? So check with your with your teacher. Professor, if you need to know this or not, um So here's how we're gonna get to that equation. Let me draw the tube again here, and I'm gonna put one liquid here and I'm gonna put another liquid over here. Um, I'm doing this in a different I'm flipping the sides here, So we're gonna call this. I'm gonna call this. I'm gonna call this H one right here, and then I'm gonna go from the from the interface. I'm gonna cross over to the other side, and this is going to be ht to Okay, So remember, every time you have an H every time you have a height of column you can write, you can write the pressure difference equation, so I'm gonna be able to write. That's the pressure between A and B. So pressure be P bottom equals P top. Plus Roger Aah! Pressure The bottom is gonna bpb pressure The top is going to be p a plus row. We're talking about the first liquid. So it's role one g and then h h one. And from here I'm able to write the same equation, but instead it's gonna be P Let's call it C appear d over here. P d at the bottom equals P C at the top plus row. Now this is the density of the second liquid G and then h two, which is this here? Okay, so you first start by writing these two equations, and now we're going to do a bunch of stuff to merge them together. Let me get out of the way. Cool. So first thing you have to realize here is if you look at B and D, they are with you are within the same liquid at the same heights. If you look under this line here, you have red liquid. If you look right under this line here, you have red liquid. Okay, So because because you are within the same liquid, if you are within the same liquid and at the same height, I would recommend writing this because it's important you're gonna have the same pressure. Because of this, you're able to say that the pressure of B is actually the same as the pressure of D. Okay, so that means that this equals this, which means that this equals this. Okay, If the left side here equals the left side here, then the right side have to be the same as well. So what I'm going to do is I'm gonna right um I'm gonna combine those two statements. I'm gonna say p a plus row one g h one equals P C plus road to GH tube. We're almost there. Um, now what about p a N p. C? Well, if you look up here, p a is touching. Air NPC is touching air. Therefore, they both have atmospheric pressure, so they both have the same pressure. Okay, so this is thing number one. You have to realize the number two is, um p a equals P air or P A T m. Which is the same as PC. So they're the same because they're the same. You can just cut them from the equation. Okay. And now you have row one. G H one equals road to GH two. You can cancel gravity's. And that's how you arrive at that equation. Okay, so it's actually very straightforward. I took a little bit here because I want to explain all the steps, but if you look at it, you start with these two equations, and then you set the yellows equal to each other. The piece, the big piece cancel and the GS cancel and you are left with this which is the same as this over here. Cool. So you can always use this equation in these YouTube problems, and you're gonna see this a bunch, So let's get going.

Water and oil are poured into a u-shape tube, as shown below. The column of oil, on the right side, is 25 cm tall, and the distance between the top of the two columns is 9 cm. Calculate the density of the oil.