Pressure and Pascal's principle (part 2)

Welcome back. To just review what I was doing on the last video before I ran out of time, I said that conservation of energy tells us that the work I've put into the system or the energy that I've put into the system-- because they're really the same thing-- is equal to the work that I get out of the system, or the energy that I get out of the system. That means that the input work is equal to the output work, or that the input force times the input distance is equal to the output force times the output distance-- that's just the definition of work. Let me just rewrite this equation here. If I could just rewrite this exact equation, I could say-- the input force, and let me just divide it by this area. The input here-- I'm pressing down this piston that's pressing down on this area of water. So this input force-- times the input area. Let's call the input 1, and call the output 2 for simplicity. Let's say I have a piston on the top here. Let me do this in a good color-- brown is good color. I have another piston here, and there's going to be some outward force F2. The general notion is that I'm pushing on this water, the water can't be compressed, so the water's going to push up on this end. The input force times the input distance is going to be equal to the output force times the output distance right-- this is just the law of conservation of energy and everything we did with work, et cetera. I'm rewriting this equation, so if I take the input force and divide by the input area-- let me switch back to green-- then I multiply by the area, and then I just multiply times D1. You see what I did here-- I just multiplied and divided by A1, which you can do. You can multiply and divide by any number, and these two cancel out. It's equal to the same thing on the other side, which is F2-- I'm not good at managing my space on my whiteboard-- over A2 times A2 times D2. Hopefully that makes sense. What's this quantity right here, this F1 divided by A1? Force divided by area, if you haven't been familiar with it already, and if you're just watching my videos there's no reason for you to be, is defined as pressure. Pressure is force in a given area, so this is pressure-- we'll call this the pressure that I'm inputting into the system. What's area 1 times distance 1? That's the area of the tube at this point, the cross-sectional area, times this distance. That's equal to this volume that I calculated in the previous video-- we could say that's the input volume, or V1. Pressure times V1 is equal to the output pressure-- force 2 divided by area 2 is the output pressure that the water is exerting on this piston. So that's the output pressure, P2. And what's area 2 times D2? The cross sectional area, times the height at which how much the water's being displaced upward, that is equal to volume 2. But what do we know about these two volumes? I went over it probably redundantly in the previous video-- those two volumes are equal, V1 is equal to V2, so we could just divide both sides by that equation. You get the pressure input is equal to the pressure output, so P1 is equal to P2. I did all of that just to show you that this isn't a new concept: this is just the conservation of energy. The only new thing I did is I divided-- we have this notion of the cross-sectional area, and we have this notion of pressure-- so where does that help us? This actually tells us-- and you can do this example in multiple situations, but I like to think of if we didn't have gravity first, because gravity tends to confuse things, but we'll introduce gravity in a video or two-- is that when you have any external pressure onto a liquid, onto an incompressible fluid, that pressure is distributed evenly throughout the fluid. That's what we essentially just proved just using the law of conservation of energy, and everything we know about work. What I just said is called Pascal's principle: if any external pressure is applied to a fluid, that pressure is distributed throughout the fluid equally. Another way to think about it-- we proved it with this little drawing here-- is, let's say that I have a tube, and at the end of the tube is a balloon. Let's say I'm doing this on the Space Shuttle. It's saying that if I increase-- say I have some piston here. This is stable, and I have water throughout this whole thing. Let me see if I can use that field function again-- oh no, there must have been a hole in my drawing. Let me just draw the water. I have water throughout this whole thing, and all Pascal's principle is telling us that if I were to apply some pressure here, that that net pressure, that extra pressure I'm applying, is going to compress this little bit. That extra compression is going to be distributed through the whole balloon. Let's say that this right here is rigid-- it's some kind of middle structure. The rest of the balloon is going to expand uniformly, so that increased pressure I'm doing is going through the whole thing. It's not like the balloon will get longer, or that the pressure is just translated down here, or that just up here the balloon's going to get wider and it's just going to stay the same length there. Hopefully, that gives you a little bit of intuition. Going back to what I had drawn before, that's actually interesting, because that's actually another simple or maybe not so simple machine that we've constructed. I almost defined it as a simple machine when I initially drew it. Let's draw that weird thing again, where it looks like this, where I have water in it. Let's make sure I fill it, so that when I do the fill, it will completely fill, and doesn't fill other things. This is cool, because this is now another simple machine. We know that the pressure in is equal to the pressure out. And pressure is force divided by area, so the force in, divided by the area in, is equal to the force out divided by the area out. Let me give you an example: let's say that I were to apply with a pressure in equal to 10 pascals. That's a new word, and it's named after Pascal's principle, for Blaise Pascal. What is a pascal? That is just equal to 10 newtons per meter squared. That's all a pascal is-- it's a newton per meter squared, it's a very natural unit. Let's say my pressure in is 10 pascals, and let's say that my input area is 2 square meters. If I looked the surface of the water there it would be 2 square meters, and let's say that my output area is equal to 4 meters squared. What I'm saying is that I can push on a piston here, and that the water's going to push up with some piston here. First of all, I told you what my input pressure is-- what's my input force? Input pressure is equal to input force divided by input area, so 10 pascals is equal to my input force divided by my area, so I multiply both sides by 2. I get input force is equal to 20 newtons. My question to you is what is the output force? How much force is the system going to push upwards at this end? We know that must if my input pressure was 10 pascals, my output pressure would also be 10 pascals. So I also have 10 pascals is equal to my out force over my out cross-sectional area. So I'll have a piston here, and it goes up like that. That's 4 meters, so I do 4 times 10, and so I get 40 newtons is equal to my output force. So what just happened here? I inputted-- so my input force is equal to 20 newtons, and my output force is equal to 40 newtons, so I just doubled my force, or essentially I had a mechanical advantage of 2. This is an example of a simple machine, and it's a hydraulic machine. Anyway, I've just run out of time. I'll see you in the next video.