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# Pressure at a depth in a fluid

## Video transcript

in the last video we showed that any external pressure on a on a liquid in a container is distributed evenly through the liquid but that was only that only applied to and that was called was a Pascal's principle but that only applied to external pressure let's think a little bit about what the internal pressure is within a liquid and we're all familiar I think with the notion of you know the deeper you go into a fluid or the deeper you dive into the ocean the the higher the pressure on you so let's let's see if we can think about that a little bit more analytically and get a framework for what the pressure is at any depth under the water or really in any fluid so here I've I've drawn a cylinder and in that cylinder I have some fluid let's not assume that its water it's some some fluid and that's the blue stuff and I'm also assuming that I'm doing this on a planet that has the same mass as Earth but it has no atmosphere so this is there's a vacuum up here there's no air and we'll see later because the atmosphere actually adds pressure on top of this so assume that there's no air but it's on the same mass as a planet of the same mass so the gravity is the same so one there is gravity so the liquid will fill this this container on the bottom part of it and also the gravitational constant would be the same as Earth so we could imagine this is a horrible situation where Earth has lost its magnetic field and the solar winds have gotten rid of Earth's atmosphere that's that's very negative so we won't think about that but anyway let's go back to the problem let's say within this cylinder I have a I don't know it's like a thin thin a piece of foil or something that takes up the entire the entire area of the cross sectional area of this cylinder and I did that just because I want to I want that to be an indicator of whether the fluid is moving up or down or not so let's say I have that in the fluid at some depth H and since the fluid is completely static nothing's moving that object that's floating right at that level that a depth of H will also be static and in order for something to be static or it's not moving what do we know about it well we know that the net force is on it must be zero in fact that tells us that it's not accelerating and obviously if something's not move it has a velocity of zero and that's a constant velocity so it's not accelerating any direction so it's net forces must be zero right so this force down this force down must be equal to the force up right so what is the force down what is the force down acting on this cylinder well it's going to be the weight of the water above it right because we were in a gravitational environment and so this water has some mass and and and so whatever that mass is times the gravitational constant will equal the force down so let's figure out what that is so like the force down which is the same thing as a force up it's going to equal the mass of the of this water right so the mass of this water is he going to equal the mass of that water times the gravitational constant oh and actually I shouldn't say water I'm using the the let me let me change this because I said this is going to be some random liquid the mass of the liquid so the force down it's going to be equal to the mass of the liquid times gravity and what is that mass of the language well now I'll introduce you to a concept called density and I think you understand what density is it's kind of how much there is of something in a given amount of volume how much mass per volume and that's the definition of density so and the the letter people use for density is Rho let me do it in a different color up here or down here so Rho which looks like a pee to me that equals mass per volume mass per volume and that's the density so that the units are kilograms per meter cubed that is density and and so you know I think you might have an intuition of if I have a cubic meter of lead that is more what one let lead is more dense than say I don't know marshmallows and so because of that if I have a cubic meter of it will have a lot more mass and in a gravitational field way a lot more than a cubic meter of marshmallows and of course there's always that that you know that trick people say is you know what weighs more a pound of feathers and or a pound of lead but those obviously weigh the same the key is the volume of a cubic meter of lead is going to weigh a lot more than a cubic meter of feathers well anyway making sure that we now know what density is let's go back to what we were doing before so we said that the the downward force is equal to the mass of the liquid times the gravitational force right and so what is the mass of the liquid well we could use this formula right here density is equal to mass times volume so we could also say that mass is equal to density times volume all right I just multiply both sides of this equation times volume so let's the volume of the so in this situation see the force down is equal to let's substitute this with this the mass of the liquid is equal to the density of the liquid times the volume of the liquid I could get rid of these ELLs times gravity and what's the volume of what's the volume of the liquid well the volume of the liquid is going to be the cross-sectional area of the cylinder right times the height so let's call this cross-sectional area air a for area that's that's the area of you know the cylinder or the foil that's floating within the water so we could write that the downward force is equal to the density of the fluid I'll stop writing the L or the F or whatever I was doing there the density of this liquid times the volume of the liquid and the volume of the liquid is just the height times the area of the liquid so that is just times the height times the area and then times gravity times gravity fair enough so let's we've now figured out if we knew the density this height the cross-sectional area and and the gravitational constant we would know the force coming down but that's you know that's kind of vaguely interesting but let's try to figure out what the pressure is because that's what what started this whole discussion is you know what is the pressure when you go to two deep parts of the ocean etc so this is the force what is the pressure on this on this on this foil that have floating well it's the force divided by the area of this for pressure this of this foil so it's I would take the force and divide it by the area which is the same thing as a so let's do that so let's divide both sides of this equation by area so the pressure coming down so that's P sub D almost sounds like a rappers name P sub D the downward pressure at that point is going to be equal to and keep in mind that's going to be the same thing as the upward pressure right because the upward force is the same and the area of what you're going upwards or downwards is going to the same thing so the downward pressure is going to be equal to the downward force divided by area we're just going to be equal to this expression divided by area so essentially we can just go to the area here right so it equals P H a G divided by a so get rid of the A's in both situations so the downward pressure pressure downward is equal to the density of the fluid times the depth of the fluid essentially or the height of the fluid above it times the gravitational constant P Hg and as I said the downward pressure is equal to the upward pressure and how do we know that because we know the upward force is the same as the downward force if the upward force were less this this little blue this little piece of foil would actually accelerate downwards the fact that it's static it's in one place lets us know that the upward force is equal to the downward force so the upward pressure is equal to the downward so let's use that in an example if I were on the same planet and let's say this is water and so the density of water so the density of water and this is something good to memorize is 1,000 kilograms per meter cubed 1,000 kilograms per meter cubed so let's say that you know we have no atmosphere but I were to go I don't know 10 meters under the water roughly 30 feet under the water what would be the pressure on me so my pressure would be the pressure would equal the density of water a thousand kilograms per meter cube make sure your units are right I'm running out of space I don't the units times the height 10 meters times the gravitational acceleration 9.8 meters per second squared it's a good it's a good exercise for you to make sure the units work out that's what 10,000 times 9.8 so the pressure is going to be equal to 98 thousand ninety eight thousand Pascal's which actually isn't that much it just sounds like a lot we'll actually see that this is almost one atmosphere which is the pressure at sea level in France I think anyway I'll see you in the next video