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Ch 20: Fluid MechanicsWorksheetSee all chapters

# Buoyancy & Buoyant Force

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Sections
Density
Intro to Pressure
Pascal's Law & Hydraulic Lift
Pressure Gauge: Barometer
Pressure Gauge: Manometer
Pressure Gauge: U-shaped Tube
Buoyancy & Buoyant Force
Ideal vs Real Fluids
Fluid Flow & Continuity Equation

Concept #1: Intro to Buoyancy & Buoyant Force

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Example #1: Comparing Buoyant Forces

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Practice: When an object of unknown mass and volume is fully immersed in large oil (800 kg/m3 ) container and released from rest, it stays at rest. Calculate the density of this object.

Concept #2: Buoyancy / Three Common Cases

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Hey guys. So, in this video we're going to keep talking about buoyancy and I'm going to show the three common cases that are going to cover pretty much every possibility, let's check it out. Alright, so an object floats or sinks depending on its density compared to the liquid density. So, if the object is denser than the liquid it's going to go, it's going to 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 a secondary case that I want to talk about sort of exception, in here we have this object that floats above water. So, part of the object is above the water and this is the part of the object that is under water 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 the volume total is 100 and 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 force is canceled and the forces are FB going down and, I'm sorry, FB going up FB is always going up in m, g going down. So, it must be the FB equals m, g so that they can't cancel each other out and that's what happens there, what about the density? Which one of these two then sities 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 quick way to look at this that I like is to just look at the top of the object and 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 slightly different situation sort of an exception that I want to talk about. So, here you have this object that, that's in the metal here, it's floating with a cable, right? So, think about this, what do you think would happen if I cut, 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'll have m, g also pulling it down and then you'll have a buoyant force pulling it up, it's still at equilibrium it still sits there but now you would write that the force is going down m, g plus T equal the force is going up FB, the reason why I have this next the other one is because in this situation you also have this be true, that's the object is less dense even though it's under water because it's only under water because of the tension, right? If you were to cut this it would look a lot like this, okay? So, whenever you see a block being held under water you have to think, what would happen or there's some tension reason 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, cool?

Practice: 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?

Example #2: Is Crown Made of Gold? (Buoyancy)

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Hey guys. So, let's check out this buoyancy problem, here we want to verify if a 100 gram crown is in fact made of pure gold, almost all these questions the way you're going to validate that is by checking the density of the object, okay? So, the density of gold is 19.32 grams per cubic centimeter or 19,320 kilograms per cubic meter, okay? Kilograms per cubic meter and what we're going to do is we're going to calculate the density of gold, of this object and figure out if 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, I will attempt to draw a crown it's probably going to come out terrible, there you go, I told you. So, it's completely submerged, it's got a little string here, you measure the tension to be 0.88. So, there is a tension here, 0.88 Newtons there is a buoyant force always up and there's an m, g always down and we want to calculate the density of the object, all of these questions are going to start with F equals m, so the sum of all forces equals m, a the next step is to just write that, that equals 0 by the way, the next step is just to write the forces. So, all the forces going up equal all the forces going down. So, FB plus T equals m, g. Alright, and if you look at this real quick you're going to 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 going to plug in a little bit, equals mass times gravity, we need density of the object so I'm going to rewrite mass of the object 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 write that we have gravity 9.8, I'm going to round it to 10, actually let's make a 9.8 because we're trying to be very precise in our calculation, 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 going to 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 and gravity 9.8. So, we've 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 going to 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 going to get mass total divided by 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 grams and though you don't know the density at least that is your target so this is a little tricky but I'm going to rewrite it and you're going to see what's going to happen, you're going to have 1,000 times 9.8, instead of V total I'm going to have mass which is 100 grams so 0.1 hundred kilograms divided by the density of the object plus 0.88 equals the density of the object times volume which we're rewriting is mass. 100 divided by the density of the object 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 cancelled here. So, you could have known ahead of time, not to do that because you have to undo it anyway but the chances are that you wouldn't know that, right? You wouldn't know that that was coming. So, I try to solve problems in a way that most people would do which is sort of systematic and not already knowing things 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 so 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 going to move a bunch of stuff around.

So, if you multiply all of this, let's see, this is going to be, this is going to be 9, this is only 980 over here divided by Rho of the object, density of the object plus 0.88 and on this side, this is going to be 0.98. So, when you move this over here you're going to get 98, 0.98 minus 0.88 which is 0.1. So, 980, density of the object equals 0.1. Now, I'm going to move the density of the object up here. So, 980 equals 0.1 density of the object, finally I can move the. One over here. So, 980 divided by 0.1 is the density of the object therefore the density of the object must be 9800 kilograms per cubic meter and this is a problem because this is nowhere near gold, gold is 19,320 kilograms per cubic meter, this is like less than half or a little bit more than half of this guy's so it's way off from gold, by the way, if you get something that was very close, if you got something that was like 19,200, 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 here would be not, 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've, 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 19320 and what would have happened is that the left side of the equation and the right side of the equation would not equal to each other, right? You would end up with something, I'm just going to make it up, you'd 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 are done here, let's keep going.

Practice: An 8,000 cm3 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?

Example #3: Maximum Load on Floating Board

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Hey guys. So, in this video we're going to start talking about fluids in motion or fluid flow and the first thing we're going to cover is the distinction between real fluids and ideal fluids, let's check it out. Alright, so the motion of fluids or the flow of fluids can get pretty complicated it's actually an area, that's still under active research in physics but we're going to simplify things by using a model called ideal fluid. So, the idea that fluids are pretty complicated but if we eliminate a ton of the complexities, the more complicated complexities of real fluids you end up with something that's called ideal fluids, they don't exist but they are a way to make things more manageable, cool? And there are some things you should know first you should know that these fluids will always be, ideal fluids will always be incompressible, that's a simplification, incompressible means constant density. Remember, density has to do with how tightly packed molecules are and in ideal fluids that's going to be constant, in real fluids molecules could get tighter or less dense, less tightly packed depending what's going on, okay? So, incompressible constant density, that simplifies things a bunch and the second thing you should know is that ideal fluids are always going to have what's called laminar flow, and laminar flow just means steady flow. So, if you look at water going to a pipe, if it's let's say see-through pipe or something like that you would see that there's just a constant stream of water that looks very clean and neat as opposed to real fluids that could have what's called turbulent flow, turbulent flow or turbulence, you could have turbulence if the liquid is moving or the fluid more generally is moving too fast, okay? So, imagine, if water is going way too fast that it goes through a little a little clot type thing here, some sort of constriction, then it could be that the water starts going all over the place and this is turbulence and this is generally bad news, lucky for you, you're probably not going to see any turbulence questions, you may just have to know this conceptually, cool? And the third distinction between them and I'm going to start over here is whether this motion is going to have viscosity, whether the fluid is going to have viscosity or not, okay? So, the defining characteristic, this is the most important of the three here, the defining characteristic of real fluids is that they have what's called viscous flow, viscous flow, in other words, the liquid has viscosity and viscosity has to do with the thickness, thickness of the fluid. So, for example, honey, right? If you get a cup full of honey and you turn it like this it's going to move very, very slowly it's because there's a lot of viscosity, ideal fluids have no viscosity at all they have no resistance, viscosity is essentially fluid friction, it's essentially fluid friction and it works very similar to air resistance or a kinetic friction and that it slows it down, okay? So, real fluids could have viscosity and they could have a viscous friction, ideal fluids are always going to have what's called non viscous, not very creative name, non viscous flow, in other words, no friction. So, that's the big difference, if you have an ideal fluid it's going to flow, it's going to flow smoothly with no friction, a real fluid could have turbulence and it has viscosity, okay? Now, lucky for you, most problems you see and maybe even all of the problems you see will be about ideal fluids, in fact a lot of professors don't even get into real fluids. So, if yours doesn't your life is simple, we're going to also assume ideal fluids unless something says explicitly that this is a real fluid or if they refer to viscosity which is fluid resistance, right? So, if they say that there's some sort of viscosity then that means there's resistance which mean it's a real, which now it means that you have a real fluid.

In most real fluid problems you're going to have viscosity because it's the defining characteristic but you're not going to have turbulence and you're also not going to have compression of the fluid, okay? So, then you're really not going to see this most of the time, you're not going to see this most of the time but you are going to see this quite a bit. So, you're going to have no turbulence and no compression of the fluid, okay? So, really it's going to come down to whether or not it has viscosity. So it is a quick intro so you can know some of the terminology and that's it, let's keep going.

Practice: 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? 