19. Fluid Mechanics

Buoyancy & Buoyant Force

# Buoyancy / Three Common Cases

Patrick Ford

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Hey, guys. So in this video, we're gonna keep talking about buoyancy, and I'm gonna show the three common cases that are gonna cover pretty much every possibility. Let's check it out. All right, So a knob checked floats or sinks, depending on its density compared to the liquid density. So if the object is denser than the liquid, it's gonna go, it's gonna sink. And if the object is less dense or lighter than the liquid, it rises to the top. So whoever is denser is going to be lower. Okay, so first situation is referring to this case right here. This here is sort of Ah ah, secondary case that I wanna talk about it sort of exception. Um, in here we have this object that floats above water so part of the object above water. And this is the part of the object that is underwater. And you can just tell by looking at the picture that the volume under is less than the volume total. So, for example, let's say that the volume totals 100 there's maybe 40 and 60 here. The volume under 60 is less than the volume total 100 because some of its above water. Pretty straightforward. What about the forces? Well, if the object is floating, just sits there. It is at equilibrium, right? Because it's just sitting there floating, which means the forces canceling the forces are FB going down. And and I'm sorry, FB going up. FB is always going up, an mg going down. So it must be the f b equals M g so that they cancel each other out and that's what happens there. What about the density? Which one of these two densities is greater? The object or the liquid? So think about it. I actually just mentioned it and hopefully you got it that the density of the liquid is greater and that's why the object floats because the object is lighter. So one of the one quick way to look at this that I like is to just look at the top of the objects in the top of the liquid. And because the liquid the top of the liquid, is lower than the top of the object, I think of it as being heavier. Therefore, it is denser, so liquid is lower, so it has a higher density. Okay, now, these three things here apply to this picture, and this here is just a different, slightly different situation, sort of an exception that I want to talk about. So here you have this object that that's in the middle here. It's floating with a cable, right? So think about this. What do you think would happen if I If I cut that cord right? It must be that the object would ride up because if the object was too dense to sink, it would just sink. The reason it sits there, it's because the cord is holding it. So what's happening here is that you have a tension pulling it down and then you have MG, also pulling it down, and then you have a boy and force pulling it up. It's still a equilibrium. It still sits there. But now you would write that the forces going down M G plus T equal the forces going up F B. The reason why I have this next to the other one is because in this situation you also have this be true that the object is less dense, even though it's underwater because it's on Lee underwater because of attention right. If you were to cut this, it would look a lot like this. Okay, so whenever you see a block being held underwater, you have to think what would happen or there's some tension or some like that. You have to think what would happen if I cut that tension and then you would know. Okay, well, here it goes to the top, which means that it is less dense than the liquid quote. The second situation is kind of trivial because it's very similar to this one. Um, here the object is floating, but instead of above water, it floats underwater. Right? So how does this happen? Well, this happens if you have an object and you put it underwater and you release it and it stays there. So in this case, the entire volume underwater is the entire volume of the objects. It is 100% underwater. In this case, we also have a cool Librium. Okay, because the two forces were gonna cancel f b and M G cancel. So they're still at equilibrium. Now, what's special here? The difference between these two situations one and two is that the density of the object is equal to the density of the liquid. Okay, these two situations are exactly the same. So if you have an object that's entirely underwater, it sinks. If it is denser, it's going to rise to the top if it's less dense. And if it does neither, if it stays in the middle, it's because its density is exactly equal to the density of the liquid. Now, how do you get this versus this? The difference is that in the first case, I manually I grabbed this object that I brought it just under the waterline, and I released it there. And because the density of the same it stays there on the second case, you just brought it lower and you released it. Okay, so these are identical situations. If an object floats entirely underwater and it doesn't sort of peek out outside of the water, the liquid and it doesn't sink, it is because the densities are exactly the same. Okay, so this is a simple case, but still important to know. Now, number three, what happens if the object sinks? What? What causes an object to sink is the fact that it is heavier. So in this case, you can see here the object is entirely underwater so the volume underwater is the same as a total volume. F B does not equal OMG f b does not equal mg. The reason why it sinks is because m g is actually going to be greater than f B. Okay, this is still at equilibrium. It's just sits there. But now there's a third force. So you're gonna have mg down. You're gonna have let me draw this bigger. You're gonna have mg down. You're gonna have enough be that's smaller. And because of this, this object is going to be Let me move this up. This objects pushing against the surface so the surface pushes back with the familiar force called normal and it's not gonna be that big. Um, they're both essentially gonna add up to cancel the mg. And here you can write that the forces going up F B plus normal equals the force going down mg. Okay, so there are three forces here, just like what we had here. So here we're going to say that the density of the liquid or the density of the object rather is greater. The object sinks because it's heavier. It goes all the way to the bottom and I pushes against the floor. So you have a normal force. Okay, so these three things have to do with this situation here and here. We have something very similar. Um, this is sort of a side case. Kind of like this one. We have a cable. Now look here. This is not sitting on the floor. It has a cable. But think about this. What do you think would happen if you cut the cable right? It must be that this object is not lighter than water. Otherwise, it would already have bubbled up to the top, and the string would have been sort of loose, right? If it stays there with the string taut with a tight rope, it's because it's trying toe fall. It's trying to sink. So if you were to cut that cable, what would happen is that it goes down. And that's because the density of the object is greater than the density of the liquid. So the forces here are You have a bigger MGI, then you have an f B, just like in the picture next to it. But now you have the help of a tension pulling you up and the way you would, right? This is very similar forces of top FB plus T equals M. G. It's the same thing. But now, instead of the tension instead of the normal force pulling up, you have the tension pulling you up, okay? And by the way, both the normal and the tension in this situation can be referred to as can be referred to as the appearance weights. Okay, so I'm gonna put here also known as in these problems apparent weight. So if you see a problem with attention or something sitting maybe on top of a scale or something and I ask you for a parent weight that is asking for normal or asking for tension opinion, which one you have. Cool. So that's plenty of talking. I wanna give you a shortcut, and then we're gonna go solve this. Um, there's a shortcut that says that the density oven object is the percent under times, times the density of the liquid. Okay. And I'm going to show you how to get to this equation. But I really just want to start this example here. Cool it says a block of unknown material, um, floats with 80% of its volume underwater. So let's draw that real quick. A bucket of water. 80% is under. Remember, the volume under is what you want. This is 20. But this is the useless one, right? You want the volume under? Okay, what is the density of the object? And I'm actually gonna calculate this really fast using this equation. The density of the object is the percent under times the density of the liquid and this object is 80%. Remember, 80% means 800.8 times the density of the liquid were underwater. So this is 1000. Which means I can quickly figured out that the density of the object is 800 kg per cubic meter. It should make sense that the density of the object is less than the density of water. That's why it's floating up top. Okay, now, that's how you can very, very quickly calculate this using the shortcut. Where does this shortcut come from? Let me show you really quickly. In case you're Professor doesn't like you using shortcuts. And he wants to see the full solution. Remember that all of these questions start with F equals M. A sum of all forces equals in May. And because we are at equilibrium that equal zero, the next step is to write the forces. And then you have that FB cancels out with MGI. The next step is to expand FB, which is gonna be density of the liquid, always gravity and then volume underwater. And that's gonna equal to M G. I can cancel the G's. Okay, Now, what I'm gonna do is I'm gonna rewrite m. And remember, density is mass over volume, So mass is density times volume. This is the density of the object and the total volume of the objects. So you're gonna have density liquid volume under equals, density of the object volume total, and we're solving for density of the objects. I'm gonna move volume total to the other side. Density object equals volume under divided by volume, total times density of the liquid. Now check the south. Let's say volume under is 800 and volume total is, uh, 1000. If you divide the true, you end up with 10000.8 and that's the percentage under. That means that this is 80%. So that's why this ratio here off partial over total is the percentage. So we can say that, uh, density of the object is the percent under, which is a decimal times density of the liquid. That's how we arrived at that. Okay, Last thing I want to show you with this shortcut before we go into example to is what happens if the object floats Does, by the way, only Only on Lee works if the object floats. Okay. Now, what happens if the object floats with 100% 100% underwater, which is situation number two over here. So if you have 100% underwater, 100% is one. This would be a one, which means you have density of the object equals density of the liquid. And we already knew that if you float entirely underwater, it's because of densities are the same. So this equation actually works for cases one and two, but it doesn't work for Case three. All right, let's check out example to real quick, and then we're gonna move on. So here we have an aluminum block. So the density of this object is going to be 2700. And it has these dimensions here. If I know all three dimensions, by the way I can calculate the volume. The total volume of this object is gonna be one times one times one, which is one cubic meter. And it sits on a scale at the bottom of a two by two by two Tank. So you have a tank like this. That is two by two. Okay, by true, let's make it three d here. Whoops. Two by two by two. And you have this box sitting here The box a little bit smaller. Cool. One by one by one. And it is filled with water to the very top Over here. How much does the scale read? Now, this this diagram is a little bit too complicated. You can just look at it as a block that is one on every side and it sits on top of a scale. The two by two by two just tells you that there's enough water here for this thing to be completely submerged. Okay. How much does the scale read? I hope you remember that the reading of the scale is the same thing is apparent weight, and it's the same thing as the normal force in this case. So ask you how much the scale read is the same thing is asking what is the normal force? And because normal is a force, we're going to start this question like we start every force question with F equals in May. So the sum of all forces on the object equals to m A. And because it sits there at equilibrium, that is zero. What are the forces on the objects? So the object gets pulled down with MGI, it gets pushed up with f B. It's always like that. But because the object is pushing against the scale, the scale pushes back against the objects. So we have a normal force going up. So the some of our forces is that you have the forces going up against the force going down so I can rewrite this to be normal. Plus, FB equals M. G. Because the forces air just canceling up equals down. Remember that stuff and we're looking for the reading of the scale. So we're looking for normal and to calculate normal. I just have to now move FB to the other side. So it's gonna go as the negatives you're gonna have. The normal is M G minus F B. And instead of writing FBI, I'm actually going to already go ahead and expand. FB FB is density of the liquid. Remember, it's always liquid, um, gravity and the volume under volume under. Okay, Now, before we start plugging the numbers here, let's take stock. Let's take inventory of what we have and don't have. So let's Z I have gravity. That's easy. I have density. The liquid. That's easy. We're in water, So the density of the liquid is just 1000 and the volume under Now this block is entirely submerged. So the volume under is the same as the volume total, which we calculated to be one cubic meter. So we have everything except mass, and that happens quite a bit. We're gonna use the fact that the density of the object is mass total divided by volume total. Okay. And we're gonna use that to our advantage, um, to be able to solve for mass. So mass equals mass of the object equals density of the object times volume total, and I can plug that in here or I could just go ahead and calculated. So the density of the object we have it here, it's aluminum. So it's 2700. The volume is one. So the masses 2700. So over here, I'm gonna put 2700 gravity. I'm gonna write it as a 10 just to make the multiplication little bit easier. Here, density of the liquid is 1000 gravity. 10 volume under one. Cool. So that's it? This is, um, actually already got ahead of myself here. This is 2700. Sorry about that. Hopefully caught that 2700. It's gonna be 27,000 when you multiply by 10. So this is gonna be 27,000 minus 10,000, which is 17 1,017,000. We're looking for normal, which is a force. So this has units of Newton's units of Newton 17,000, and that's it. We're done. Let's keep going

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