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# Finding height of fluid in a barometer

## Video transcript

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 kind of how highs the column of fluid above us times gravity let's see if we can use that to solve a fairly typical problem you'll see in your physics class or even on an AP physics test so let's say that I have a bowl a bowl let me draw the bowl this is my bowl a bowl and in that bowl I have mercury and then I also have this 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 so the test tube that I put in the bowl something like this and 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 and now we are actually on a on earth we're actually in Paris France at sea level because that's kind of you know what what an atmosphere is defined as so the atmospheric pressure so 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 and at sea level and in Paris France and in the bowl I have mercury and mercury is like a silverish color so I'll draw it as silver so I have a mercury mercury it's a flat I know I'm not drawing it completely flat whoops this is mercury I want to do it so that I can actually use the filled tool maybe after just make sure it has no holes in it I think I got rid of all the holes okay let me see let me try to use my fill tool there you go and let's say that that mercury there's no air in here it actually is going to go up this column a little bit and we're going to do the math as far as well one will see why it's going up and then we'll do the math to figure out how high up does it go so say the mercury goes up some distance all right this is all still mercury it's all still mercury and it's actually how a barometer works this is something that measures pressure so you can imagine up and over here at this part above the mercury but still within our our little test tube we have a vacuum there is no air vacuum one of my favorite words because it has to use in a row anyway so anyway what was I doing oh okay so we have this set up so my question to you is how high is this column of mercury going to go how high is this comma and first well let's just have the intuition as to why that there this thing is going up to begin with so we have all this pressure from all of the air above us and I know it's a little unintuitive for us because we were used to all of that pressure on our shoulders all of the time so we don't really imagine it but though there is literally the weight of the atmosphere above us and so that's going to be pushing down on the surface of this of this of the mercury on the outside of the test tube and essentially since there's no pressure here the mercury is going to go up words here but now is this the state that I've drawn is a static state we've assumed that all the motion is two optics that are etc etc so let's try to solve this problem oh and a couple of things we have to we have to know before we we do this problem it's mercury and we know that the mercury the the specific gravity and I'm using terminology because a lot of these problems the hardest part is the terminology the specific gravity of mercury mercury is 13.6 that's often a daunting statement on a test when you know how to do all the math and almost done you know what is specific gravity all specific gravity is is the ratio of how dense that object is that substance to water so all that means is that mercury mercury is thirteen point six times as dense as dense as water right and hopefully after the last video because I told you - you should have memorized the density of water is a thousand kilograms per meters cubed so the density of mercury let's write that down density that's the row or a little P depending how you want to view it of mercury is going to be equal to thirteen point six times the density of water so times one thousand kilograms per per meter cubed okay so let's let's go back to the problem so what we want to know is how high this this column of mercury is so we know that the pressure let's consider this point right here 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 the pressure at this point down pressure down has to be the same thing as the pressure up right because the mercury isn't moving we're in a static state and we learned several videos ago that the pressure in is equal to the pressure out of a on a liquid system so essentially I have one atmosphere pushing down here on the outside of the surface I must have one atmosphere pushing up here so the pressure pushing up at this point right here as you could imagine that we have that aluminum foil there again just imagine that you know what where the pressure is hitting the pressure there is one atmosphere so the pressure down right here must be one atmosphere and what's creating the pressure down right there well it's it's essentially this column of water or it's this formula which we learned in the last video so 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 the atmosphere that's pushing on the outside and pushing up here so let's see the density of mercury is this thirteen point six thousand so thirteen thousand six hundred 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 9.8 meters per second squared is going to be equal to one atmosphere and now you're saying Sal this this is strange I've never seen this atmosphere before we've talked a lot about it but how does an atmosphere relate to Pascal's or Newtons and all of that so this is something else you should memorize one atmosphere is equal to one hundred and three thousand Pascal's well that also equals one hundred and three thousand Newton's per meter squared right so 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 and one atmosphere is exactly this much equals one hundred and three thousand Newton's per meters squared so let's see let's let's if we divide both sides well we could just do all of the math so if we divide both sides by thirteen thousand six hundred and nine point eight we get the height is equal to one hundred and three thousand Newton per meter cubed over 13,000 600 kilograms per meter cubed times 9.8 meter per second squared and what does that get us do I notice make sure you always have the unit's right that's the hardest thing about these problems just to know that an atmosphere is a hundred thousand one hundred and three thousand Pascal's which is also the same as Newton's per meter squared now let's just do the math so let me type this in let me just so I'll use the pointer 103 thousand divided by thirteen thousand six hundred divided by nine point eight equals 0.77 and we're dealing all with Newton's so height is equal to 0.77 meters and you should see that the the units actually work out cuz 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 and then we have a Newton's up here but what's a Newton a Newton is kilogram meter squared per second so when you divide you know you have kilogram meter squared per second squared here you have kilogram meter per second squared when you do all the division of the unit's all you're left with is meters so we have point seven seven 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 oh well we make a little notch on this test tube and that represents one atmosphere and you can go around and figure out what what the atmosphere how many atmospheres different parts of the globe are anyway I've run out of time see in the next video