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19. Fluid Mechanics

1

concept

11m

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Hey, guys, In this video, we're going to start talking about buoyancy, which is the thing that causes objects to float or be pushed up when in a liquid, Let's check it out. All right, So if you have an object that is immersed in a liquid, the object will be pushed up by a force called buoyant Force. And buoyant force comes from the fact that the physical phenomenon responsible for this is called buoyancy right, which usually is associated with floating. Um, and because this is buoyant force with the B, we're gonna call this F B. Now. The reason this happened is because of a pressure difference between the top and the bottom of an object that's under a liquid. So let's say you have a box that is completely submerged completely underwater, and we know the water is gonna push on all sides, right? The water is going to apply pressure on all sides, and pressure is associated with with force. So if there is a pressure, there is a force. Now the force on the left will cancel the force on the right because they are at the same height for every force on the left. I can draw the equivalent force on the other side on the right. That will exactly cancel it. So the side forces don't do anything. But the top and bottom forces will be different because the top and bottom pressures air different. And we know that the deeper you go under liquid, the greater the pressure. So the pressure at the bottom over here at this point is greater than the pressure at the top. Therefore, the force with which the liquid pushes up on this box is greater at the bottom than it is at the top. And the difference between these two forces is the buoyant force. Okay, so you're always gonna have a stronger force up than a force down. So the Net buoyant force is going to be up cool. And so that's the number one thing opportunity to know, and you probably should memorize. This is our committee's principal. Um, he said he figured out that the magnitude of this buoyant force is the same as the weight of the liquid displaced. Okay, so you should memorize that quote because that could show up in some way. And it's important to know that it's a big principle in physics. Now, the problem is that this quote here doesn't really help you calculate this force. So I'm just gonna give you the equation. If you were to sort of translate this quote into physics equation, you would get that the buoyant force is the density of the liquid density of the liquid. Um, gravity 9.8 or whatever planet your own, um, in the volume, um, of the object that is underwater. So I'm gonna call it Roll Liquid G and V under. So there's two really important things here. Distinctions to make in this equation that the density that matters, the density that you should be looking into is the density of the liquid. Even though the object is inside. And this is the force on the object, the density that matters, the density of the liquid don't get that messed up. Okay, in the volume that matters is the volume of the object. But on Lee, the amounts of volume that is underwater. So it's the volume under water or under the liquid. Okay. Or the volume submerged of of the object that is submerged or immersed in the liquid. Okay, I'm just gonna call it under to keep these words simple. And it's not necessarily the entire volume. Unless, of course, you are entirely under. So let's think about this real quick. Um, the idea here is that the Maurer of the oven object is underwater, the stronger the fourth. And you might play this out in your head and just sort of imagine that if you're getting a piece of Styrofoam as you push it down under water, um, there's more resistance, there's more force pulling it up. And that's because the more volume you have, the more buoyant force you have. Okay, so that's that. And then one last thing I want to remind you because you're gonna be using this lot is the density of an object. The definition of density is mass over volume, so the density of the object will be the mass of the object and the total volume of the objects. I'm gonna write V I'm gonna write, actually, mass total. It's the entire mass sweeps. You won't go away. It's the entire mass of the object divided by the entire volume of the objects and the reason I'm being careful. That distinction is because you have volume under which isn't necessarily the entire thing. And you have volume total, which is the entire thing, and we're going to be using these two equations all the time to solve these questions. Okay, the last point I wanna make and we're gonna do an example here is that almost every single buoyancy force a problem is just a force problem. So it's just a good old f equals. They made problem. We could have done this right after you learn. If it was in May. In some ways, um, and also that most of the time it's gonna be even simpler than Africans in May, because most of the time the objects will be at rest meeting the velocity zero. And there will be, um, in equilibrium, which means the acceleration is zero and the Net force zero. So it's really an f. Some of our forces equals zero problem. Most of the time, just equilibrium forces. They're just gonna cancel coal. So let's do a quick example here. So it says when an object of unknown mass, so we don't know the mass and volume unknown volume. We don't know the volume great is fully immersed, fully immersed. Let me draw that. Fully immersed means you're entirely underwater. Remember I told you there's two kinds of volumes, right? There is volume under, and there's volume total because you are fully immersed. This means that volume under equals volume total. Okay, uh, in a large water tank and released from rest, it accelerates up. So when you release the object that's going to accelerate up on, Bennett says, once the object reaches equilibrium, 30% of its volume is above water. So here's the object it's going to, um, eventually it stops up there. So when you release, it accelerates up. Eventually it stops, and 30% of the object is above water. Okay, Now, if you remember on this equation here, the only two volumes that matter are the total volume and the volume under so 30% above doesn't do anything for us. What you really care about is the volume under. So you're gonna change that into 70% under. That's one of the things that you have to do in these questions is you always want to know the volume under. That's what matters. Okay, Now, with just that information, we want to calculate the density of this object. So what is wro object? Cool. Now again, all these questions they're going to start with f equals m A. Equals zero because you are at equilibrium. So I'm going to write that the some of our forces equals zero. And remember the way to solve force problems has always been to draw free body diagram and then write f equals in May. So let's draw everybody diagram. What's going on here with forces? Well, this object is here, so there has to be an mg pulling it down. And the new thing here is that because it's under liquid, it's going to be pushed up by a buoyant force and FB is always going to be up. Okay, let me write that here. Just so you have it always up because as, um because the density down here is stronger as you go lower quote. So if the system is the equilibrium of the objects equilibrium, it means that the forces cancel so you can simply write that f b equals M g. The next step is always going to be almost always going to be to expand this FB what do you mean by that? We're gonna rewrite FB based on its equation, which is this. So instead of writing FBI, I'm gonna rewrite it as Rogov quote. So let's do that density. Remember of the liquid gravity and volume under equals to M. G Right away, G cancels. And this is gonna happen quite a bit. Not always, but quite a bit. And we wanna know the density of the objects. Now, if you stop for a second, you look here, you will see that density Rho object isn't anywhere in this equation, right? It's not here. You have density, the liquid, But that's not what we want. So it actually is here. But you have to look a little bit more carefully. It is going to be inside hiding inside of one of these things here, right? Are there any of these things that you can rewrite that you can rewrite to make this variable show up? And there is. And I warned you about this, that we were gonna be using this equation all the time. So very often you're gonna rewrite things to make variables show up. Okay, so, um, the ray we're gonna do this is density of the object is mass total, divided by volume total. Therefore, mass total is density of the object times volume total. All right, so what we're gonna do is we're gonna replace mass with density volume, and now we're gonna have our variable there. Okay, So the density of the liquid times volume under equals density of the object, which is what we're looking for. That's good, uh, times volume total. Cool. Now realize also that we know we know the volume under. Okay, we know that we are 70% underwater, so I can rewrite volume under as 70% of or 700.7 times volume total. Okay, it's rewriting this here. So if you do that, you can cancel these two outs and then we are basically done. The density of the object is 20.7. The density of the liquid we are in water were in water. So this is just 1000. So the density of the object is 700 kilograms divided by her per cubic meter. Okay. It makes sense. It should make sense that the object is floating above water because it is less dense than water. Therefore, it rises to the top. I wanna talk about one last thing here. Um, notice here that I had said that the volume under the water is the same as the total volume. But that was before That was before he released in the object moved this way here. Now the volume under is less than the volume total because some of it is above cool. That's it for this one. Let's keep going.

2

example

3m

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Hey guys. So let's check out this conceptual buoyancy question, and I chose this one because it's pretty tricky and there's quite a few of these kinds of tricky buoyancy questions I wanna help you think through, so let's check it out. So here we have a piece of wood and a piece of metal, and they both have the same volume. So I'm gonna say volume would equals volume metal. They're both place in a large water tank. The wood is going to float. Let me draw the wood over here and the metal sinks to the bottom. And that's what you would expect would almost always floats. And metal almost always sinks. And we want to know what is the greater buoyant force on it. So which one has the greater FB? And before I go, I want you actually think about this for a second, maybe positive video. If you need to and pick one, I want you to select an answer. Commit to it. Don't be scared. Pick one and then keep going to see if you got it right. And what most people do here is they try to think about this logically, which sounds like a good idea. But the problem is these questions exist because they are tricky and because they sort of defy common sense. Right. So what I like to do is I always like to delegates the decision making to the equation. What I mean by that is that instead of me thinking through it logically or using common sense because I know that that fails and I'm ready for that, I'm going to instead write the equation and let the equation this side. So what is the equation for buoyant force? The equation for buoyant force is density of the liquid gravity and the volume of the object that is under the liquid. Okay, now check this out. Both of these guys are underwater, so the density of the liquid is the same because they're in the same liquid. So this is a tie, which means that this is not going to help you determine who Which one has the more more buoyant force? A greater buoyant force. They also both experienced the same the same little G because they're both, you know, the earth there next, each other. Now it's all gonna come down to the volume under and they both have a total volume of 50. But the wood is is, um, has some of its volume staking out. So the volume under, I'm gonna make up some numbers. It's gonna be, like 10 and 40 for example, and this is gonna be all 50 of it are gonna be underwater. So the the metal has mawr. The metal has mawr of its total volume underwater, um, than the wood does. Therefore, the stronger, buoyant force will be meadow. And hopefully you got this right. But a lot of people actually incorrectly pick would because they think it's floating. So it must be if it's floating, it must be that the force pushing it up is stronger. It must be that the force pushing it up is stronger than the other one. That's not why it floats. It floats because the M G is much, much smaller. Okay, so here the forces, actually, let's draw some arrows here. The F B is actually stronger than here, but the M G is much, much stronger than here. Okay, so don't get caught up right the equation and use that to determine the writing. So let's keep going

3

Problem

When an object of unknown mass and volume is fully immersed in large oil (800 kg/m^{3} ) container and released from rest, it stays at rest. Calculate the density of this object.

A

800 kg/m^{3}

B

80.0 kg/m^{3}

C

8000 kg/m^{3}

D

8.00 kg/m^{3}

4

concept

16m

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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

5

Problem

A block floats with 40% of its volume above water. When you place it on an unknown liquid, it floats with 30% of its volume above. What is the density of the unknown liquid?

A

857 kg/m^{3}

B

600 kg/m^{3}

C

400 kg/m^{3}

D

571 kg/m^{3}

6

example

8m

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Hey, guys. So let's check out this buoyancy problem here. We want to verify if a 100 g crown is, in fact made of pure gold. Almost all of these questions, the way you're gonna validate that, is by checking the density of the object. Okay, so the density of gold is 19.32 grams per cubic centimeter. Or, um, 19,000 320 kg per cubic meter. Okay, kilograms per cubic meter. And what we're gonna do is we're gonna calculate the density of of this object and figure out is the density of the object is the density of the object 19,320. And if it is, this is gold. So if yes, if this is true, then it is gold. Okay, So if I ask you, is this gold? What you're doing is you're calculating density, so let's do that. So here you lower it by a string into a deep bucket of water and then when the crown is completely submerged. So let's draw this, uh, I will attempt to draw crown. It's probably gonna come out terrible. There you go. I told you so. It's completely submerged. Got a little string here. You measured attention to be 0.88. So there is a tension here. 88 Newtons. There is a buoyant force, always up, and there's an MG always down. Um and we want to calculate the density of the object. All of these questions. They're gonna start with that because it may. So the sum of all forces equals in May. The next step is to just write that, um that equals zero. By the way, the next step is just to write the forces. So all the forces going up equal the forces going down so f b plus t equals M G. All right. And if you look at this real quick, you're gonna notice that the density of the object is nowhere here, but you have to have a little faith as you start to change some variables around. As you start to expand some of these variables, the density of the object should show up, and you don't stop until it does so. FB is density of the liquid, so that's not good enough. Yet times gravity times, volume under plus tension. We have that we're gonna plug in a little little bit equals mass times gravity. Now we need the density of the object. So I'm gonna rewrite mass of the objects into density of the object. Remember, that density is mass divided by volume. Therefore, mass is density times volume. So it's the mass of the object. So it's the density of the object, and it's always going to be the total volume. Okay, when you're rewriting mass, it's the total volume because you're looking at the total mass and that times G. Okay, so we are looking for this, and if you take inventory here, we have the density of the liquid because it's water. Let's right that we have gravity. 9.8. I'm gonna round it to 10. Actually, let's make a 9.8 because we're trying to be very precise in our calculation. Um, the volume under this object is entirely underwater. So volume under is the total volume is the total volume. But I don't have that either. I don't have that either. So that's gonna be a problem. Let's just leave it like that for now. Volume total plus tension. Tension is 0.88. Let's go to the right side density of the object. That's what we're looking for. Density of the object. Cool. Leave it alone. Volume total. We don't have that, um, and gravity 9.8. So we got a little bit of a problem, which is this is my target, but I actually don't have this either. So again, you're gonna have to rewrite some stuff. Okay? So back to this equation here, If you solve for volume total, if you solve for volume total, you're gonna get mass total divided by the density, divided by the density of the object. So if you write this the good news, the good news is that you know, mass, it's 100 g. And though you don't know the density, at least that is your target. So this is a little tricky, but I'm gonna rewrite it, and you see, you're going to see what's gonna happen. You're gonna have 1000 times 9.8. Instead of vetoed, I'm gonna have mass, which is 100 kg. So 1000.100 kilograms divided by the density of the object plus 0.88 equals the density of the object times volume which were rewriting his mass 0.100 divided by the density of the objects. Um, times 9.8. Now, at this point, you might be freaking out just a bit, but notice that this cancels with this, and then you end up having just one unknown out of this entire thing. So it's a little bit messy because we expanded, right. We rewrote mass so that the density of the object shows up, and it turns out that it actually canceled here. So you could have known ahead of time not to do that, because you had to undo it anyway. Um, but the chances are that you wouldn't know that, right? You wouldn't know that that was coming. So I tried to solve problems in the way that most people would do, which is sort of systematic and not already knowing things, Um, in advance. So you have to be able to be good at manipulating these things, Which is why I wanted to show you this question as an example, you have to be back and forth. You have to be very fluent, if you will, with this little equation, so you can move some stuff around and just keep going and keep changing some stuff until you're left with one target. So this is just good, solid physics hustle to get to the target. Variable. Now we're gonna move a bunch of stuff around. So if you multiply all of this, um, let's see, this is going to be this is going to be nine points. This is gonna be 9 80 over here, divided by a row of the object. Then you get the objects plus 800.88. And on this side, this is gonna be 0.98. So when you move this over here, you're going to get 98.98 minus 0.88 which is 0. so 9 80 density of the object equals 0.1. Now, I'm gonna move the density of the object up here. So 9 80 equals 800.1 density of the object. Finally, I can move the 0.1 over here, So 9 80 divided by 800.1 is the density of the object. Therefore, the density of the object must be kg per cubic meter. And this is a problem because this is nowhere near gold. Gold is 19 3 20 kg per cubic meter. This is, like less than half or Ah, little bit more than than half of this guy's. It's way off from gold. By the way, if you get something that was very close, if you got something that was, like 19 uh, 200 if it's that close, then whoever wrote the question meant for it to have been gold. So even if it's not exactly the same, if it's really close, that's gold. If they didn't mean it for it to be gold, then they'll make a number that is very, very different. That's the case here. This is clearly not gold. Okay, so the answer, he would be not, uh Let's get out of the way. The answer here is not gold. Okay, Now I want to quickly talk about something else. There's another way you could have solved this question. I don't like it, but it works. Which is, once you get to the big equation right here, you could have I mean, this is kind of a hack, but what you could have done is you could have plugged in the density right here. You could have plugged in the density of gold. 19,000, 320. And what would have happened is that the left side of the equation on the right side of the equation would not equal to each other. Right? You would end up with something. I'm just gonna make it up. You don't end up with something like 20 equals 40. And then you would say 20 is not equal 40 because this didn't turn out to be true. It must be that the density of this thing is actually not the density of gold. Which means that this thing is not gold. Okay, long story short. This is not gold. We're done here. Let's keep going.

7

Problem

An 8,000 cm^{3} block of wood is fully immersed in a deep water tank, then tied to the bottom. When the block is released and reaches equilibrium, you measure the tension on the string to be 12 N. What is the density of the wood?

A

1153 kg/m^{3}

B

988 kg/m^{3}

C

847 kg/m^{3}

D

960 kg/m^{3}

8

example

6m

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Hey, guys. So let's check out this buoyancy example. So here I have a wooden board, have a wooden board. Let's make it thick. And let's call this M one m one. And I want to know what is the maximum out of mass that I can put on top of the board. So I wanna know how much am too. I can put here in such a way that M two does not get wet, okay? And this is a very common kind of question. What is the maximum load you can put on top of a board or a container so it doesn't get wet, Okay. And the way to solve this is very similar to how we've solved other other buoyancy questions, which is just using f equals M A equals zero. It's an equilibrium question. What's special about here is this idea of not getting wet. So let's think about this real quick. Let's say you put some amount of mass here, and the water level is here. Now, if you put more mass, what's gonna happen is it gets heavier. So you would imagine that it goes down a little bit. So if you add more mass, The water level might rise up to here, right? I mean, the water doesn't rise up the object lower relative to the water. And if you keep adding mass, that's gonna happen until you get to this maximum point over here, where if you add any more mass, um, the mass on top is going to get wet. So this this idea of not getting wet has to do with your going all the way up here. So what's the conclusion? What happens when the when the water's all the way to the top? Well, what happens is the volume of the object of M one that is underwater is the entire volume of one. In other words, it is 100% submerged. So that's what this is going to be that whatever thing, whatever container or board is is holding things on top of it is going to be 100% under water, Okay, but not any. But it's not going to be any lower than that. Okay, so we're gonna start with it. Um and that's it. Other than that, we're gonna write that some of our forces equals it. May equals zero because this is an equilibrium problem. Now one distinction here is that there are two objects and every time you write at because they made you write if he goes in for a target objects. In other words, when you write the sum of all forces, you say that some of our forces on objects one which is the board, equals M A, which equals zero. So the only forces that matter are the forces acting on object one and the forces acting on M one are there is its own MGI right, which is M one g. There's also the weight of the M two, which is on top of it, which pushes down on it. So this is m two g, right? Because it's supporting that weight and there is a buoyant force. FB Now you might be thinking isn't there a normal force here? There is if you look at em too. M two is being pulled down by M two g, its own weight and it's being pushed up by n, which is the reaction force of them one. But we're not looking at him to were writing some of our forces for the one objects which means that the only forces that matter are the forces on board one or object one. Okay, so if this is an equilibrium, which it is because it's not moving, Andi, it's not accelerating. It means that the forces canceled. So forces going up, f b equals forces going down m one g plus m two g. And what we're looking for here is M two. So let's go after that. As usual, we're gonna rewrite this as density of liquid gravity and in in volume under. And just to be clear, volume under is for the object that is actually underwater. So its volume one cool and then we have this stuff here. Now we don't know what m one is. We don't know the mass of the board. And whenever you don't know the mass and these problems, we're just gonna rewrite it. So then city one is mass one volume one total volume one total. So M one is densely one volume, one total. So let's be right that so then city one volume one total, um, times gravity, by the way, gravity's gonna cancel. But for now, let's just leave it here, plus M to M two is what we're looking for. So leave it alone. G notice that gravity cancels. You can Onley cancel a gravity or any variable if it shows up in every single one of the terms. Here you have one term, another term plus another term and gravity shows up in all three of them so we can cancel. We're looking for em too. Do I know the density of the liquid? Yes. Now be careful here it says salt, Salt Water Lake. What does that mean? Well, freshwater freshwater has density of 1000 and you may remember saltwater has a higher density. One way to remember is that you add some salt so it gets heavier. Um, the density of salt water is gonna be 10. 30 or 10. 35. You should stick with whatever number you're. Professor likes for that. Some people use it a little bit different. Cool. So that's this here 10. 30. The volume under what is the volume under? Well, volume under is actually the same thing here as volume total. The whole point of not getting wet is that the amount of of of the object that's underwater is the entire object but we don't have that either. So we have to calculate. And I want to remind you that volume is if you have a three dimensional objects or sort of a rectangle, it is just base or with times, death times heights. We don't have all these measurements, but you can also write this as you can replace these two guys with the area times the heights. And that's what we have. Okay, the area here, it says one square meter and 10 centimeters. So 100.1 m. And if you see you have meters square with meters, there's 3 m here. So this is 30.1 cubic meter is our volume. So there's gonna be 0.1, um, equals density one, which is the density of the board. We have that 700 volume one, which is calculated 10.1. So do this carefully plus mass to Okay, Now we can solve this. Uh, this is gonna be 103 minus. We're gonna move this over to the other side. 700 times. 7000.1 is the same thing in 700 by 10. So this is 70 and so mass to is the difference there mass to is 33 kg and we are done. Hopefully you start seeing the pattern here of all of these questions being solved just by writing that all the forces going up equals all the forces going down and then getting kind of clever with making your target variable show up on the equation and moving some stuff around. So you get everything that you need. So just good physics hustle, Let's keep going.

9

Problem

You want to build a large storage container, with outer walls and an open top, as shown, so that you can load things into it, while it floats on fresh water, without any water getting inside. If the bottom face of the container measures 3.0 m by 8.0 m, how high should the side walls be, such that the combined mass of container and inside load is 100,000 kg?

A

5.0 m

B

2.9 m

C

9.1 m

D

4.2 m

Additional resources for Buoyancy & Buoyant Force

PRACTICE PROBLEMS AND ACTIVITIES (8)

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- A 950-kg cylindrical can buoy floats vertically in sea-water. The diameter of the buoy is 0.900 m. Calculate t...
- A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the...
- A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surfa...

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