19. Fluid Mechanics
Intro to Pressure
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Hey, folks. Hopefully, you tried this practice problem here. There's a little bit of a classic in physics when it comes to fluid dynamics. Some questions will ask you about the deepest known point in the ocean because it's kind of cool. We know it as the Challenger Deep, which is just an area at the very bottom of the Marianas Trench, somewhere out in the middle of the ocean, and it's at a depth of 11,000 meters. It's insanely deep. So basically, from sea level, you have to go all the way down a depth. So I'm gonna call this an h of 11,000 to hit the bottom. Alright. So here's gonna be me floating around somewhere like this. I'm all the way down at the bottom of the trench. This actually would be taller, this would be deeper than Mount Everest is tall. Alright? So insanely deep. Now the surface area of the average human ear is 20 centimeters squared. So basically, what this means here is that if I have my face like this and my ear, the surface area is gonna be 20 centimeters squared. Now remember, this is in centimeters squared. We're gonna want, an area in terms of meter squared, so we have to do a conversion. We have to cancel out the centimeters. Hopefully, you guys realize that's 1 over 100, but we have a squared, in the units. So we have to apply the conversion factor however many times we have the exponent. So in other words, you need to cancel out the centimeters squared and basically, which you end up with is 0.002 meters squared. Alright, so just a little bit of housekeeping with the units. So in other words, what we're trying to find here is we're trying to calculate the average force that's gonna be applied on your ear at that depth. So in other words, we want f. So what is the relationship? What's the relationship between force and area? Well, remember, we're just gonna use the definition of pressure. So pressure is equal to force divided by area. Right? So if we have the area, then we can rearrange and solve for the force. This is just pressure times area. Alright, so we have A. So really all this problem is about is calculating what the pressure is at that depth. So that's what we're going to go and do over here. So in other words, how do we calculate the pressure? Alright, well, in this problem here, we've got that the top basically just all the way up at the at the surface, we have air and all the way at the bottom, we need to figure out what the pressure is. We can use the pressure equation. So in other words, I have this as the top and this is all the way at the bottom, and I have this h here. So what this means is that we're gonna use the p bottom equals ptopequationplusrho, and then this is gonna be the the density of saltwater times g h. Alright? So this ptop because it's touching air, right, the surface is exposed to air, then that means that ptop is equal to p, the pressure of air. So in other words, this is just air and this is the pressure that I'm looking for. Right? So this pressure over here. So p bottom is just gonna be, the pressure of air, which is 101,000 plus the density of saltwater is not a1000, it's actually a little bit more, it's a 1030. Remember, we saw the the common units or common density values times g, which is going to be 9.8, it's not going to use 10, and then the height, which is going to be 11,000. Now, if you go ahead and rearrange this and work this out, a different way to express this number is that it's 1.01 times 10 to the 5th. And then if you work this number out over here, what you're going to get is this is 1.11 times 10 to the 8th. So we can see here is it just looking at the units or so just looking at the exponents, this is a 5th exponent and this is an 8th exponent, which means this number is like a 1000 times bigger than this number over here. So this number is like a 1000 times bigger. And because of that, what that means is that when you add this number over here, it actually is almost like insignificant. This would actually just be like a this would be like as if you add like a 1 in, like, the 4th decimal place over in that number. So basically, what that means here is that your pressure at the bottom is effectively 1.11 times 10 to the 8th, and that's in Pascals. Now we're not done yet. Remember, we came over here looking for pressure because we have to plug it into this equation over here and now we have it. Alright, so to finish things off, we have the force is equal to 1.11 times 10 to the 8 times the area, which is gonna be 0.002, and your force is going to be equal to 2.22 times 10 to the 5th in Newtons. Now, this might just seem like an ordinarily big number, this is your final answer. But I wanna remind you, this is the amount of force that would be acting just on the tiny surface area of your ear. So if you to put it into perspective, this would be like if you had a 22,000 kilogram object pressing up against your ear, which is about £50,000. So you have £50,000 of force that's acting on the tiny little surface area of your ear. This is in like an insanely enormous amount of force and an insanely enormous amount of pressure that would be acting at death. It would obviously just crush you unless you had some kind of a submersible to keep you safe. Alright. So that's it for this one folks. Let me know if you have any questions.
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