Finding height of fluid in a barometer

In the last video, we learned that the pressure at some depth in a fluid is equal to the density of the fluid times how deep we are in the fluid, or how high is the column of fluid above us times gravity. Let's see if we can use that to solve a fairly typical problem that you'll see in your physics class, or even on an AP physics test. Let's say that I have a bowl. And in that bowl, I have mercury, and then I also have this kind of inverted test tube that I stick in the middle of-- this is the side view of the bowl, and I'll draw everything shortly. Let's say my test tube looks something like this. Let's say I have no air in this test tube-- there's a vacuum here-- but the outside of the bowl, this whole area out here, this is exposed to the air. We are actually on Earth, or actually in Paris, France, at sea level, because that's what an atmosphere is defined as-- the atmospheric pressure. Essentially, the way you could think about it-- the weight of all of the air above us is pushing down on the surface of this bowl at one atmosphere. An atmosphere is just the pressure of all of the air above you at sea level in Paris, France. And in the bowl, I have mercury. Let's say that that mercury-- there's no air in here, and it is actually going to go up this column a little bit. We're going to do the math as far as-- one, we'll see why it's going up, and then we'll do the math to figure out how high up does it go. Say the mercury goes up some distance-- this is all still mercury. And this is actually how a barometer works; this is something that measures pressure. Over here at this part, above the mercury, but still within our little test tube, we have a vacuum-- there is no air. Vacuum is one of my favorite words, because it has two u's in a row. We have this set up, and so my question to you is-- how high is this column of mercury going to go? First of all, let's just have the intuition as to why this thing is going up to begin with. We have all this pressure from all of the air above us-- I know it's a little un-intuitive for us, because we're used to all of that pressure on our shoulders all of the time, so we don't really imagine it, but there is literally the weight of the atmosphere above us. That's going to be pushing down on the surface of the mercury on the outside of the test tube. Since there's no pressure here, the mercury is going to go upwards here. This state that I've drawn is a static state-- we have assumed that all the motion has stopped. So let's try to solve this problem. Oh, and there are a couple of things we have to know before we do this problem. It's mercury, and we know the specific gravity-- I'm using terminology, because a lot of these problems, the hardest part is the terminology-- of mercury is 13.6. That's often a daunting statement on a test-- you know how to do all the math, and all of a sudden you go, what is specific gravity? All specific gravity is, is the ratio of how dense that substance is to water. All that means is that mercury is 13.6 times as dense as water. Hopefully, after the last video-- because I told you to-- you should have memorized the density of water. It's 1,000 kilograms per meter cubed, so the density of mercury-- let's write that down, and that's the rho, or little p, depending on how you want to do it-- is going to be equal to 13.6 times the density of water, or times 1,000 kilograms per meter cubed. Let's go back to the problem. What we want to know is how high this column of mercury is. We know that the pressure-- let's consider this point right here, which is essentially the base of this column of mercury. What we're saying is the pressure on the base of this column of mercury right here, or the pressure at this point down, has to be the same thing as the pressure up, because the mercury isn't moving-- we're in a static state. We learned several videos ago that the pressure in is equal to the pressure out on a liquid system. Essentially, I have one atmosphere pushing down here on the outside of the surface, so I must have one atmosphere pushing up here. The pressure pushing up at this point right here-- we could imagine that we have that aluminum foil there again, and just imagine where the pressure is hitting-- is one atmosphere, so the pressure down right here must be one atmosphere. What's creating the pressure down right there? It's essentially this column of water, or it's this formula, which we learned in the last video. What we now know is that the density of the mercury, times the height of the column of water, times the acceleration of gravity on Earth-- which is where we are-- has to equal one atmosphere, because it has to offset the atmosphere that's pushing on the outside and pushing up here. The density of mercury is this: 13.6 thousand, so 13,600 kilogram meters per meter cubed. That's the density times the height-- we don't know what the height is, that's going to be in meters-- times the acceleration of gravity, which is 9.8 meters per second squared. It's going to be equal to one atmosphere. Now you're saying-- Sal, this is strange. I've never seen this atmosphere before-- we've talked a lot about it, but how does an atmosphere relate to pascals or newtons? This is something else you should memorize: one atmosphere is equal to 103,000 pascals, and that also equals 103,000 newtons per meter squared. One atmosphere is how much we're pushing down out here. So it's how much we're pushing up here, and that's going to be equal to the amount of pressure at this point from this column of mercury. One atmosphere is exactly this much, which equals 103,000 newtons per meters squared. If we divide both sides by 13,609.8, we get that the height is equal to 103,000 newtons per meter cubed, over 13,600 kilograms per meter cubed times 9.8 meters per second squared. Make sure you always have the units right-- that's the hardest thing about these problems, just to know that an atmosphere is 103,000 pascals, which is also the same as newtons per meter squared. Let's just do the math, so let me type this in-- 103,000 divided by 13,600 divided by 9.8 equals 0.77. We were dealing with newtons, so height is equal to 0.77 meters. And you should see that the units actually work, because we have a meters cubed in the denominator up here, we have a meters cubed in the denominator down here, and then we have kilogram meters per second squared here. We have newtons up here, but what's a newton? A newton is a kilogram meter squared per second, so when you divide you have kilogram meters squared per second squared, and here you have kilogram meter per second squared. When you do all the division of the units, all you're left with is meters, so we have 0.77 meters, or roughly 77 centimeters-- is how high this column of mercury is. And you can make a barometer out of it-- you can say, let me make a little notch on this test tube, and that represents one atmosphere. You can go around and figure out how many atmospheres different parts of the globe are. Anyway, I've run out of time. See you in the next video.