19. Fluid Mechanics

Buoyancy & Buoyant Force

# Maximum Load on Floating Board

Patrick Ford

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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.

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