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>> The idea is the following. Hydraulic lifts are really great because by applying just a little bit of force you can in fact generate a whole bunch of force to move heavy objects. So whenever you look at a tractor, you'll see a big piston. And that piston is able to move a lot of dirt, big rocks, and so forth. How do they do it? Well, the way you do it is the following. We'll draw a simplified model of a hydraulic lift. Okay. This is what a hydraulic lift looks like. There's some sort of oil in here, everywhere. And when you push down on one side, say with a person, you can lift something up on the other side, say an elephant. Okay. Here's my elephant. Let's draw the head of the elephant, big trunk, okay, floppy ears. You can in fact lift up an entire elephant with your weight if you have a hydraulic lift. Now how does this work? How do we take advantage of this? Well, when you look at this picture, what's the first thing you notice? What's the difference between the right side and the left side, other than the fact that there's an elephant in the room. What else is different between those two sides? Who has a thought? Somebody hand the mic to Ben. Can you pass the mic back to Ben? Ben, what's the difference between these two sides? >> The diameter of the platform. >> Yeah. This one is a very small diameter and this one is a very big diameter, right. So we'll say that this has diameter d1. This has diameter d2. And clearly from the picture, d2 is much bigger than d1. And if you think about any sort of hydraulic like in a tractor, that piston that is going to move stuff like dirt or rocks, it's very big diameter. The hose that's going into the piston is very small diameter. And so that's how hydraulics work. Small diameter goes to big diameter and somehow this lets you lift a lot of weight. All right. So let's see how that works. Well, if I draw a dash line right across there, I can say these things are at the same height, okay. If they're at the same height, then what can I say about the pressure in this region one versus the pressure in that region two, the pressure of the oil? I can say the following. Bernouilli's equation still holds. And Bernouilli's equation tells us the following. The pressure in region one plus the kinetic energy of any moving fluid in region one plus the height of the fluid times rho times g has to be a constant and so all of that has to equal that stuff in region two, P2 plus one-half rho V2 squared plus rho gy2. Now this equation holds for any regions of connected fluid. So the region up here in the oil and region over there in the oil, they have to satisfy Bernouilli's equation. But this thing is just balanced there. And so we don't have any speed of oil. Nothing's moving. They're also at the same height so y1 is equal to y2. I can get rid of those. And look, all I'm left with is pressure in region one has to be equal to pressure in region two. Whatever the pressure is over in the oil on the left side, that has to be equal to the pressure in the oil on the right side. All right. But we know a little bit about pressure, right. What is pressure is force divided by area, so F1 over A1 has to equal F2 over A2. And in fact I know what that force is. The force is the weight of the person versus the weight of the elephant. Okay. We'll call this m of the person. And so F1 just becomes m of the person times gravity. A1 is the cross sectional area of that piston. And that is just pi r square. They don't give us r. But they do give us d1 and so it's pi times d1 over 2 squared. That has to be equal to F2 which is the weight of the elephant. m sub elephant times g. And I'm going to divide that by the cross sectional area of the piston on that side, pi times d2 over 2 quantity squared. And now a bunch of stuff is going to cancel out, right. We want to solve this thing for d1. We would give you d2, the diameter over here. You need to find out what d1 is. And so I want to solve this thing for d1 but before I do that, I can cross a bunch of stuff out. g on the left and the right, cross that out. pi on the left and the right, cross that out. A1 over 2 squared here, A1 over 2 squared there, cross those out. And now we can quickly solve this thing for d1. What do we get? d1 is going to be, I'm going to have to multiply it up over there and that means that this me is going to come down beneath the mp. And then I've got to multiply across by a d2 squared but I'm going to take the square root of all that stuff. And so we get that. What's the diameter of piston one? It's this. Square root of mp over me, all that times d2. Now, again, when you solve these problems, make sure you keep the variables all the way through because by keeping the variables all the way through you immediately can see if it makes sense. First of all do the units make sense? Yeah. Diameter two, if it's in meters, then diameter one will be in meters because the mass over mass, those kilograms, will cancel out. It also makes sense that if the mass of the person was equal to the mass of the elephant, then that would be the case where both diameters would have to be the same. And that's in fact what we find here. mp over me would be 1. d1 would equal d2. But we know the mass of the elephant is much bigger than the mass of a person. And so this number is much smaller than 1. So d1 is in fact less than d2, which makes sense. That's how we drew it. Small piston over here, big piston over there. Okay. This is how you attack this problem for the first part, part A. The second part is going to be let's put another person on here of unknown weight. The elephant moves up some amount. Can we figure out what that weight of the unknown person is? You do it the same way using Bernouilli's equation except now your heights don't drop out. That's the only difference. Okay. So we're not going to do that here but you can approach it very similar. All right. Questions about that one? Everybody okay with that one? Anybody have a hydraulic lift at home or a hydraulic jack for their car? Most jacks for your car a now screw jacks, right. It's this big, threaded bolt and you sit there like this and you crank it for a while and the car goes up. But occasionally you buy a hydraulic lift, a hydraulic jack, and it looks something similar to this. a hydraulic jack, and it looks something similar to this. It's a big piston and then there is a little lever arm attached to a very small inlet that forces liquid into that big piston. And so you sit there and you put oil in there very slowly and it lifts up your entire car. And it's a very powerful way to do it because of the size difference of those two cylinders.

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