19. Fluid Mechanics

Density

1

concept

## Intro to Density

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Alright, so, liquids and gasses are types of fluids, types of fluids. So liquid is a fluid and a gas is a fluid. So we're going to use the term fluids to refer generally to both liquids and gasses. And the reason we do this is because liquids and gasses behave very similarly in a lot of different situations. So instead of saying liquids and gasses, liquids and gasses all the time, we're just gonna say fluids, which refers to both both things. Cool, so density is the first big concept you have to understand and you may remember density from chemistry class, the density of the material has to do with how tightly packed the molecules are. So, for example, here you got the same sort of volume, this sort of blue um cup. And it's got these little green balls here, they're not very packed together. So what I'm gonna say that this is low density and here they're very tight together. So this is going to be high density. Okay, so the more compressed things aren't the higher density you have density in physics is given by the letter, by the greek, letter wrote row, which is a little p a curvy P and if you remember, it's simply mass divided by volume, mass divided by volume. So mass in physics is always kilograms and volume is a three dimensional length. So it's going to be cubic meter since kilograms per cubic meter. Remember if you have the three dimensions of an object like a rectangle or something, then the volume of a rectangle would be the width of the rectangle times the height times the depth. Right? And sometimes you see length instead of one of these three measurements. And because each one of these guys is a meter, you got to meet him in the meter. You have cubic meter. Cool. Now sometimes you are given the density and you're given these dimensions, right? So you're given density rho and you're given the three dimensions. Whenever you're given the three dimensions, you're able to find volume. And if you have row and volume, if you have rho and volume, then you're going to be able to find the mass. And that's because of the equation rho equals mass divided by volume. Therefore if I move the v up here, I hope you see that right away, you get m equals rho volume. So let's put this this over here, mass equals rho volume. Alright. And they try to trick you with this but it's very straightforward um It's just a play on this original definition here of density. Sometimes you see something that says that objects have the same material in density problems. This usually means that they have the same density. Okay, so if you have a if you have two pieces of wood and we say it's the same kind of wood, you can also infer that they have the same, you can conclude that they have the same density. Cool. And then the last quick point I would make in the world. One example is if you have liquid in a container to liquids or more liquids. Two or more liquids in a container. The liquid of the higher density will be at the what do you think? Top or bottom? Higher density liquid will be at the bottom. Okay. The high density liquid will be at the bottom and you can think of this as higher density being heavier. Now I'm putting this in quotes because it's not necessarily heavier. It's gonna be heavier depending on whether you have more or less volume of it. But on a per molecule basis it is heavier or or per small area or per small volume. It is heavier. Therefore it's gonna go to the bottom because liquids can sort of move around. So you might have seen something like this where they put all kinds of different things and you can see them becoming very different sort of a heterogeneous mixture here. Um And honey is all the way at the bottom which means honey is the highest density out of all these things that are here. Cool. So we'll see some stuff like that later. Let's do a quick example here. Um What is the total weight of of air molecules inside a large warehouse? And I give you the dimensions here. So I want the total weight of air. Right? So air does have a weight. And so first let's start with wait wait remember is just MG Mass times gravity. And I know gravity I'm gonna use here just for the sake of keeping this simple, I'm gonna say gravity is approximately 10 m/s square. So I'm gonna use 10. So if it's asking me for weight and I know gravity, all I really need is mass. So this question is really about finding the mass of air in this space. Now, I'm given this right, so you can sort of draw this, it's 100 wide, 100 deep. So it looks something like this not to scale. Okay, so this is 100 m here, 100 m here And 10 m high. Whenever you're given three measurements, you can right away find the volume. vol is just those three measurements together, multiplied 100 times 100 times 10. And here you can just count the zeros or five zeros. So this is 10 to the fifth and I got meter times meters times meters so cubic meter. Right? Or you can write it out if you want 12345. So it's 100,000 cubic meters. Okay, now I have the volume. I have the density right here. So I can find the mass because remember density is mass over volume, I have the volume. I have the density. It was given right here. So we can just find the mass which is the equation I showed you just a few moments ago. So roe V and then you're gonna multiply the two row is going to be 1.225 kilograms per cubic meter. I highly recommend you put the cubic meter down here right? Like don't put it over here, if you put it in the bottom, it's going to be easier to play with it Times The volume, which is 100,000 Q B commuter. And then notice what happens here right away. This cancels with this. And you just got this big multiplication. The mass therefore is going to be if you put this in the calculator, you're going to get 00. You're left with kg. A little bit of dimension analysis here. Cool, are we done? No, because we're getting mass so that we can plug it in here and get the weight. But that's the last step I'm gonna do here. Weight is mass times Gravity. And I just have to multiply those two. We're gonna use gravity as 10. So we just have to add an extra zero here. And the unit for weight, since it's a force is Newton's. So this is a million newtons of weights. Cool. So the air in this entire thing is actually pretty heavy. If you would put that entire air on top of you, it would crush you in a very small amount of time. Alright, cool. Let's do this example here. If you want you can pause the video and give this a shot yourself. I'm gonna keep rolling here. It says the density of whole blood. So whole blood means it's all the different parts you have of your blood plasma. Everything else is nearly this now in physics, whenever they see a value is nearly or approximately, we're just going to use that value. So density is rho of whole blood is 1.06 kg per leader. I'm gonna write it like this, notice that it didn't say kilograms per cubic meter instead of said per leader. Um and these two are not equivalent but they are related and we'll talk about this in a future video. So we're just gonna leave it like that for now. And then it says, how many kilograms are in a pipe pints of whole blood. So ask me how many kilograms kilograms is the units for mass? If I say how many kilograms I'm asking for the mass. So what is the mass? And then I'm given the volume here, the volume is 4 73 mL. Now you can't really use milliliters, you're supposed to use leaders, but let's leave it alone for now, let's not sort of prematurely convert units here. Alright, so this is very straightforward. I have three variables that are related by this equation by the definition of density, which is mass over volume. I want to know mass, I have the other two, I just have to move things around this question is a little bit more straightforward than the other one PV or Rovi And this is 1. kg per liter times of volume, which is 473 millie leaders can't really do this without changing either leaders into milliliters or milliliters into leaders. Hope you remember this is very straightforward. One leader is 1000 mL, so I'm actually just gonna scratch this and put 1000 milliliters right here. Middle leaders will cancel and then you're left with kilograms, which is what you want. So all we gotta do here is multiply this big mess. And if you do that, you get 1.6 times 4 73 Divided by 1000. And this is gonna be in kg. And if you do this in the calculator, you get 0.5 or one kg. That's how many kilograms. Or how much the mass of the mass of blood, of whole blood. If you have one pint of it with this density, cool, that's it for this one, let's go to the next one.

2

concept

## Density Values & Specific Gravity

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everyone. So now that we've seen density, we're gonna talk about a slightly less familiar term, which is called specific gravity. Now, if you've never heard that or seen that in your classes, you can probably go ahead and skip this video, but some professors might want you to know it. So it's a good thing to know. It's actually very straightforward. So let's just jump in. Right? So we have the specific gravity which doesn't really have a variable, so we just we just abbreviated as S. G. Is a term that's sort of related to the density of a material, but it's not exactly density. The specific gravity of any material is defined as the density of that material divided by the density of freshwater. Now, freshwater is always going to be on the bottom and we know that there is a fixed value. So really quickly here, what happens here is that if you're dividing two densities, you're gonna divide the units right, kilograms over meters, cubed, kilogram over meters cubed. And whenever you divide two variables that are the same, they cancel out. So this is just a number that has no units. It's just like two or five or 0.5 or something like that. Alright, so just to put into an equation, you know, just make a little bit simpler, the specific gravity of any material, which I'm just gonna call X is equal to the density of that X. That material divided by the density of fresh water. Now again, like we said, the density of fresh water is always a fixed value. It's 1000. So one really simple way we can sort of phrase this or write this equation is the density over 1000. Alright, so pretty straightforward. So let me just go ahead and run it through some quick examples if you know that the density of some material is equal to 2000 kilograms per meters cubed and that means that the specific gravity of that material is 2000 over 1000. So in other words it's just too, it's twice as dense as water. It also works the other way around. If you know what the specific gravity for material is, I'm going to say that the specific gravity of some material y is equal to 0.7, right? Less than one or more than one, then that that means that the density of that material is equal to 0.7 times 1000 which is equal to 700. Right? So it works both ways. If you know the density, you can figure out the specific gravity and then vice versa. Alright, so really what we can see here is that the specific gravity doesn't really have anything to do with gravity. It's kind of one of those unfortunate terms That later on we figured out it's not really it doesn't really have to do with gravity. Um but it's sort of that's what we stuck with. So, you know, we're kind of stuck with it. Right? But it's just a relative number. Right, so two or 0.7 or something like that for how many times denser a material is relative to freshwater. Alright, so that's all it is. It's just a number that says how much times denser it is than water. So, that's all there is to it. Let's let's go ahead and just work out this example. All right, so, we're gonna calculate the volume of some wooden cubes. In other words, we're gonna calculate V. That's the volume for a specific gravity of 0.8. So, we have S. G. That equals 0.8 and it weighs 16,000 newtons. Now. Be careful here because we have newton's. So you might think it's mass, but it's actually wait some of the words, the W which is equal to M. G. Is equal to 16,000 newtons. Alright, so then how do we calculate the volume? We're gonna have to relate this back to the density equation. We have row that equals M over V. If you re arrange for this, you see that V is equal to M over row. Right? These two things swap places. So, in order to calculate the volume, I'm gonna need the mass, which I don't know that's the mass and I'm gonna need the density. So, the way I'm gonna do this is that this s G here is gonna give me density, Right? Because remember that's the relationship between specific gravity and density and then this equation here for weight is gonna give me the mass. Alright, so let me go ahead and just do this one first. Um So I'm just gonna bring this down here. So we have that M equals W. Over G. Which is 16,000. And we're just gonna use 10 Fergie. Just make it really simple here. So you can knock off one of the zeroes. And just this just becomes uh this is actually 100 kg. Alright, so that's your mass. And then on the other side we have the specific gravity. So we have S. G. Is equal to 0.8. So what that means here is that the density Is going to be 0.8 times 1000 right? It's 0.8 times the density of freshwater. So in other words, you're gonna get 800 kg Per meter. Cute, this makes sense. We've got a number that was less than one, which means that it's less dense than water. And we got 800. So that makes sense. Alright, actually I did this wrong, so this is yellow and this should be blue. Alright, so now we're just gonna plug both of these into our equation here and we're gonna get that V is equal to 1600 divided by the density which is you just get to and the units for that are going to be in meters cubed. Right? So we have kilograms and then kilograms grams per meter cube. So you have S. I. Units so you should end up with are also S. I. Units, so that is your final answer. So pretty straightforward, let me know if you guys have any questions.

3

Problem

ProblemA wooden door is 1 m wide, 2.5 m tall, 6 cm thick, and weighs 400 N. What is the density of the wood in g/cm^{3}? (use *g* = 10 m/s^{2})

A

0.0267 g/cm

^{3}B

0.267 g/cm

^{3}C

267 g/cm

^{3}D

2670 g/cm

^{3}4

Problem

ProblemSuppose an 80 kg (176 lb) person has 5.5 L of blood (1,060 kg/m^{3} ) in their body. How much of this person's total mass consists of blood? What percentage of the person's total mass is blood?

A

0.0519 kg

B

0.0583 kg

C

5.19 kg

D

5.83 kg

5

Problem

ProblemYou want to verify if a 70-g crown is in fact made of pure gold (19.32 g/cm^{3} ), so you lower it by a string into a deep bucket of water that is filled to the top. When the crown is completely submerged, you measure that 3.62 mL of water has overflown. Is the crown made of pure gold?

A

Yes

B

No

Additional resources for Density

PRACTICE PROBLEMS AND ACTIVITIES (36)

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