**Concept:** How U-Shaped Tubes Work

Hey guys. So this week we're going to talk about the u-shaped tube also sometimes referred to as the u-tube and it's one of the more popular pressure gauges in physics, let's check it out. Alright, so a pressure gauge is simply a device an instrument a very simple instrument usually that uses height differences to calculate pressure. So the idea is you're going to have two liquids in here and it might look like something like this, don't draw this just yet I'm going to erase it you might have a liquid here and then put another liquid over here and you're going to be able to calculate pressures using these height differences here, okay? But for now, let me just erase this and we'll come back to it, you're going to calculate pressure using the pressure difference equation right here, and first I'm want to talk about these first two cases here, which are trivial, they're very silly but we're going to talk about them and then this is the important one. So, if you have one liquid in a u-tube 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 some other, some other gas, right? Whatever you have there, what's going to happen is that the height will be the same. So, you're going to have the same height here, okay? And that's because you have the same pressure, up here the pressure, let's call this pressure1 equals pressure2 and that's because they're both touching air. So, it's whatever the atmospheric pressure here at this point is because the pressures are the same you're going to have the same height, okay? In other words h1 is the same as h 2, we're going to call the left side 1 and the right side 2. Alright, so that's pretty straightforward, we know that the like 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 will the the liquid be on this other side over here? Do you think it'd be higher or do you think it'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 and that's because here there's 0 pressure, 0 Pascal and here there's some pressure of the atmosphere, right? It's pushing down. So, instead of being like this they're going to 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 write that. So, open side is always is lower than vacuum, cool? So, those things 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, 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 going to come here and I'm going to pour some blue liquid and I'm going to pour 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 going to 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 moves out, 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, let me draw the red liquid and, gotta get your colored pens out, cool? So, something like that, I was terrible. Alright, now what we're going to do is we're going to 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 going to call this A and B and on the right side the other two points that are 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 it as C and this height here that's the same height as B, 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 going to call this height 2 because it's on the right side and the distance between A and B, we're going to call this height1 because it's on the first side, okay?

Now, the top points are easy to remember A and B, A and C they're just the top of both columns but how are you going to remember, and D is pretty easy too, 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 are important, another distinct that's important, let's make this different color, let's make it green, is the distance between the top of C, the top of 2 and the top of 1, that's another thing you're going to get asked, it's a distance between these two, I'm going to 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 7 centimeters then obviously Delta h, if this is 7 high, if this here 7 high and this is 10 high then Delta h is of course 3 centimeters high, okay? So, let's write that equation, Delta h is just the distance between the difference between h1 and h2. Now, you don't know which one is bigger sometimes h1 might be actually higher than h2. So all these hs have to always be positive. So, I'm going to put this here, I'm going to say that it's the absolute value that just in case it comes out to be negative and I'm going to write here, this is going to come in handy later all hs must be positive, okay? So, this is the first share of equation, is not not even an equation, just eyeball this and you see that that's the difference, okay? There's another equation for this that's going to be very useful, I'm going to derive it later when we're solving a problem but for now I'm just going to give it to you real quick that it is that rho one so density of the first liquid times height of the first liquid, this height right here, equals density of the second liquid, the blue liquid, times height of the second liquid which is this right here, okay? And this is the most important equation for the u-tube, okay? The u-shaped tube with two liquids, this is the most important equation, you have to memorize this, it's going to come in handy to get a problem like this, cool? Let's do an example and see how you might see this in action, so it says that the u-tube shown above has two long sides both open at their ends. So, standard u-tube they're both open, I'm going to draw them real quick, the fact that says it has long sides doesn't really mean anything, that's just standard language, it just means that it's not going to overflow or nothing weird like that, so it says, you first pour water so that the height 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, you put water here, so that the height on both sides is 20. So, this here is 20 centimeters. Alright, if you look at our 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, this height doesn't matter, okay? So, that height there isn't going to matter, the fact that you give it is 20 isn't really going to be useful, I'll lightly scratch it then you pour enough of a particular oil on the right side of the column so that above the water, so that the column of oil above the water is five centimeters also, I'm going to put some oil here, let's make oil red and you're going to 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, cool? Now, we're going to be putting the water up a little bit. Remember, the water doesn't go above the oil and the reason for that is because water is lighter than oil, if you didn't remember that, if you didn't know that it's right here, right? Oil is lighter than water, water the density of water is 1,000. So the column of oil has to be higher, okay? So, oil higher because it is less dense, cool? So, we've drawn this, we're told that the column of oil is 5 centimeters high, this is the right side, I'm going to call this 2, this is 1. So, I'm going to that h2 is 5 centimeters, by the way I know that the density, density2 is the density of oil is 800 and I know that density one of water is 1000, cool? The question is, there's two questions here and you assume that the liquid don't mix, this a standard language, if the liquids mix, this is not going to work. So, in all these questions the liquids won't 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 A, we're asking, what is gauge pressure. So, what is p gauge at that point, okay? I hope you remember but if you don't, p, gauge, if you write the same equation p bottom equals p top plus Rho g. h, this is p gauge right here, okay? The gauge pressure is 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 is, okay? So, anyway to find p gauge we're going to just write rho g, h. Now, we want the p gauge here, the gauge pressure at that. 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 meters per second squared and then the height is 5 centimeters. So, 0.05 meters. Notice that all my units are standard units, right? For all these measurements which means I get standard units of pressure at the end which is Pascal. So, if you multiply all of this, I have it here, you are going to get 392, tiny little bit of pressure, 392 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 and the top of oil? So, Remember, this here is h2, we're going to call, we're going to come across this way here and this here is going to be h1 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 h2 and h1. So, guess what? To find out the h you have to have both h1 and h2. Now, we have h2 but we don't have h1. So, first we're going to find h1 and then we're going to be able to quickly calculate Delta h, to find h1 you're going to use the equations that I asked you guys to memorize which is Rho1, h1 equals Rho2. h2 and you can use this equation right here. So, what I want to do is I want to quickly solve for this and then I'm going to show you how this equation works, I'm going to show you how to arrive at this equation in case you need to but first, let's solve this. So, rho1 equals rho2, h2 divided by h1, I'm just solving for density1, density2 we have it right here, I'm sorry, we're looking for h1, I'm looking solving for the wrong thing here, circle the wrong thing, looking for h1. So, h1 is Rho2, h2 divided by Rho1. So, looks like this, the second height is 5 centimeters 0.05 meters and then the pressures are, pressure2 800, pressure1 is 1000, if you multiply this you get 0.04 meters or 4 centimeters. So, this height here, h1, h1 is 4 centimeters, if this is 4 and this is 5 this has to be 1 centimeter, okay? That has to be 1. So, Delta h is h2 minus h1 absolute value just in case it's negative and h2 is 5, this is 4 so this is 1 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 you use 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 teacher, professor, if you need to know this or not. So, here's how we're going to get to that equation, let me draw the tube again here and I'm going to put one liquid here and I'm going to put another liquid over here, I'm doing on a different, I'm flipping the sides here, So, I'm going to call this, I'm going to call this, I'm going to call this h1 right here, and then I'm going to go from the from the interface, I'm going to cross over to the other side and this is going to be h2, 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 going to be able to write that's the pressure between A and B. So, pressure B, p bottom equals p top plus Rho, g, h, pressure at the bottoms and B PB, pressure to top is going to the PA plus Rho, we're talking about the first liquid so it's Rho1 g and an h, h1 and from here I'm able to write the same equation but instead it's going to be p, let's call it C up here, D over here, p D at the bottom equals p C at the top plus Rho. Now this is a density of a second liquid g and then h2 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 go out of the way, cool? So, first thing you have to realize here is, if you look at B and D they are, you are within the same liquid at the same height, if you look under this line here, you have the red liquid, if you look right under this line here you have red liquid, okay? So, 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 is important, you're going to 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 going to write, I'm going to combine those two statements and I'm going to say, p A plus rho1 g, h1 equals p C plus rho2 g, h 2, almost there. Now, what about p A and p C? Now, if you look up here, p A is touching air and PC 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, thing number two is p A equals p air or p atm which is the same as p C. So, they're the same because they're the same you can just cut them from the equation, okay? And now you have rho1 g, h1 equals rho2 g, h2, you can cancel gravities and that's how you arrive at that equation, okay? So, it's actually very straightforward, it took a little bit here because I wanted 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 ps the big Ps 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 u-tube problems and you're going to see this a bunch, cool? So, let's keep going.

**Problem:** 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.

A simple U-tube that is open at both ends is partially filled with a heavy liquid of density 1000 kg/m^{3} . A liquid of density 568 kg/m^{3} is then poured into one arm of the tube, forming a column 12 cm in height, as shown.

What is the difference in the heights of the two liquid surfaces?

A. 5.54

B. 1.8582

C. 1.7507

D. 1.0318

E. 2.4066

F. 5.184

G. 9.288

H. 3.3512

I. 2.954

J. 5.97

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A barometer can be constructed with a U-shaped tube filled with some liquid, as sown in the figure below. One end of the barometer is closed, with all the air sucked out between the fluid and the cap, and the other end of the barometer is open to atmospheric pressure. If the density of the fluid is 2000 kg/m^{3}, and the difference in height between the ends of the barometer is 5 m, what is the ambient pressure?

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