19. Fluid Mechanics

Buoyancy & Buoyant Force

# Is Crown Made of Gold? (Buoyancy)

Patrick Ford

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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.

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