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>> Let's talk about fluids. Fluids are of course everywhere, right. Water is all over the earth. Water is inside of us. There is fluid in this pen which just came out onto the learning glass, right, which happened to frustrate the total internal reflection that's going on in the glass to pull that light out, reflect off this fluid and now you can observe it. Fluids are really important in physics because they govern a lot of our universe. So let's start at sort of a basic question that humans have asked many times. Will it float? And let's do the following. Let's say we have a box and we put it under water and now you ask the question, will it float? And you sort of probably already know the answer to this. Intuitively you probably know that this thing is going to float if it's made of certain material like Styrofoam or wood. But it's going to sink if it's made up of other materials like metal. So how do we answer this question from the physics point of view? To do that we need to go to Archimedes' Principle. And Archimedes' Principle is the following. The buoyant force equals the weight of the displaced fluid. Okay. So how much water did our box push out of the way? That weight of that water is exactly equal to the buoyant force on this box. So let's draw the free body diagram for this thing. There's always gravity here on earth acting down but now we have a buoyant force, B going up. This is the buoyant force that Archimedes was talking about. All right. What can we say about this stuff? Well, we know there's pressure here. There's some pressure on the bottom of the box. There's some pressure on the top of the box. The box has a cross sectional area A. And we know what pressure is. Pressure is just force divided by area. All right. So force is therefore pressure times area. And if I think about these pressures at the top of the box and the bottom of the box, what can we say? Well, we know that the buoyant force has got to be equal to mg if this whole thing is at rest. But it's also equal to the difference between the pressure at the top and the pressure at the bottom. Pb going up, Pt going down. And so this has to be equal to the weight of the displaced fluid And we know what the weight of the displaced fluid is. It's equal to how big is this box times the density of water. Density of water is rho. The box is what? Let's call the height of the box here h. And so we get rho g times A times h. Okay. Which is the rho of the box g times volume V of the box. Okay. But we also know that if that box has side L and side L, then that volume, which is area times height, just becomes L squared times h. So what's the buoyant force? It's rho, and we'll say we're in water, so this is the density of water, times g times h times L squared. And now let's go back to our original question, will it float? Well, we've got the buoyant force going up. We have mg going down. And we know that it's going to sink if that force down is bigger than B. And it's going to float if that force down is less than B. Let's think about this mg for a second. mg is the weight of the box and so it is the density of whatever that stuff is made of times the volume of the box times g. All right. But we also know that the buoyant force which is this. gh L squared, that is the same as volume of the box. And so now we're just going to see which one is bigger, okay. Is B going to be bigger than mg or is B less than mg? Rho H2O g volume of the box, is that bigger than mg which is rho for the box volume of the box times g? And look what happens. g cancels out. V box cancels out. And we end up with rho H2O bigger than the box or not? It's all about the densities. So it sinks if the density of the box is bigger than the density of the water. It floats if the density of the box is less than the density of the water. And this you already knew. If it's Styrofoam, the density of the box is certainly less than water, the box floats. If it's steel, then the density of the box is bigger than the density of the water and the thing sinks. Okay. So it comes back to something that you kind of intuitively knew already.

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