Calculating mass of displaced liquid

- [Instructor] Let's solve a couple of problems to see how to calculate the amount of liquid displaced when something is floating on top of it and when something sinks into it. Here's the first one. An object of mass 30 grams and volume 40 cm cube is dropped in water. Calculate the mass of displaced water, given the density of water is one gram per cm cubed. Let's first try and draw the situation and then see how to solve it. So we're given an object of mass 30 grams. So let's draw an object. So let's say here is our object, imagine it's a stone. We know it's mass and we know it's volume, we'll write that down in a while, but it's dropped in water. So let's say we have a container with water and it's given that we take this stone and we'd drop it in water. We're asked to calculate the mass of displaced water. What does that mean? Well, whenever you drop any object inside any liquid, or any gas for that matter, then as that object gets submerged, some of that liquid has to move away to make space for that object, right? So that liquid that goes away is what we call the displaced water or the displaced liquid. And of course, it's that liquid that comes, moves over here and raises the level of water, and that's why we knew when you submerged something, the level of water rises because the water gets displaced. And we're asked to calculate what the mass of that displeased water is. That's what we need to figure out. Okay. So let's quickly get write on what is given to us. So what is given? We are given the mass of the object. So let's write that down over here somewhere. We know the mass of the object is 30 grams, we are given the volume of the object. So we know the volume that has given us 40 cm cubed, we are also given the density of water. So density, now instead of writing the word density, I usually like to use this symbol rho, which is just to save space. So anyways, density of water that is given to us as one gram per cm cube, and we need to calculate what the mass of the displaced water is. So I'm just gonna call that as Mw, w for water. That's what you need to figure out. Not the mass of the entire water, but how much water got displaced, that's what we need to figure out. Now, how do we do this? How do we figure that out? Well, first of all that value depends on whether our object is floating or sinking, right? I mean, think about it, if our object were to float like this, let's say somewhat this way, then because only a small amount of object is submerged, only a small amount of water would get displaced, right? In that case, we'll have a smaller value. On the other hand if that same stone where to sink, now, the entire body got submerged. That means that a lot of water must have gotten displaced, right? So this value would be more. So I guess the first thing we need to do is to calculate or figure out whether this object is going to float or sink, and then think about how to calculate this, right? So that's what we do first. Now, how do we do that? How do we figure this out, whether this is going to float or sink? Well, they've seen this before. The condition for floating is that the density of the object, again rho means density, so the density of the object should be smaller than the density of the fluid, in this case, the density of the water. So if this stone has smaller density than water, it will float, if it has larger density than water, it will sink. So first, and by the way, how do we calculate density? We calculate density as mass divide by its volume. And this is something that we talked a lot in previous videos. So if you need more clarity on why these things are there or you need more clarity on this, great idea to go back and watch those videos on densities and condition for floating. So anyways, the first step for us is to calculate what the density of the object is, our stone, and figure it out whether it is going to float or sink. In fact, can you try this first, before we do it over here. We know the mass of the stone, we know the volume of the stone, so can you go ahead and calculate what the density of the stone is, and check whether it's going to float or sink. Go ahead, give it a try. Okay, let's do it. So the density of our object is going to be, of our object, is going to be its mass divided by its volume. So it's 30 grams divided by 40 cm cubed, that will be the zero's cancel, 3 by 4. 3 by 4 is just 0.75, 0.75 grams per cm cubed. The density of our object that means is 0.75 grams per cm cube, density of water is one gram per cm cube. So object has a smaller density than water, which means the condition of a floating is satisfied, and that means our object is going to float. So here's a stone, that is going to float. So it might be somewhat like this. So if I put the stone in water, it's gonna look somewhat this way, it's gonna be floating there, okay? Now that we know it's gonna float, next is how do we calculate the mass of the displaced water? How do I do that? So in this example, in this case, this much water is going to get displaced, right? That much water is going to move up, right? Let me just take that much water and put it at the side. So here is the liquid that got displaced. It's this much amount of liquid that actually moves up, but I'm putting it to the right over here so we can see, and we need to calculate what this mass is. What is the mass of this water? How do we do that? How do we figure this out? Well, we can use Archimedes' Principle to figure this out. So Archimedes' Principle says that whenever an object is submerged in liquid, there's an upward force acting on it, let me use this color, upward force acting on it called the buoyant force, and that force equals the weight of this liquid, this displaced liquid. So whatever is the weight of this displaced liquid, whatever it's weight is, that weight equals the buoyant force. Which means if I displace more liquid, it'll have more weight and so the buoyant force will increase. That's what's Archimedes Principle, okay? Now, how does that help? Well, think about the forces on this stone. It's being pushed up by the weight of this liquid, the buoyant force, but it's also being pushed down, it's also being pushed down by its own weight by, due to gravity. So this is the weight of the stone and this is the buoyant force, which equals the weight of the displaced liquid. Now, we know that our was stone is floating, right? If the stone is floating and just staying there, it's stationary over there, what can we say about these two forces? A, they have to be equal to each other, right? Because if this force was larger than this, the store would be rising up. If this force was larger than this, the stone would be sinking. So the two forces must be exactly equal, which means the weight of this stone should equal the weight of this displaced liquid. So displaced liquid has the same weight as the stone, or the mass of that liquid should be the same as that of the stone. Because (indistinct) weight is just mass times G. So if the weights are the same, their masses must be the same as well. So from this, we can say, the liquid should have the same mass as the stone. And you know this mass of the stone is 30 grams and so from this, we can say the mass of that liquid or the mass of that water, that is this water, has to be 30 grams. So whenever objects are floating, this means that the displaced liquid or the displaced fluid should have the same mass as that of the object. Because the object is stationary, the two weights have to be equal to each other. Well, what if the object is sinking? Well, let's do one more problem for that. So here's the second problem, and the only difference in the second problem is that the mass of the stone and the volume of the stone has changed. Everything else is the same. It's dropped in water, we need to calculate the mass of the displaced water. Since the mass and the volume of the stone has changed, we have a different stone altogether, so that's it. Here is our new stone. And again, if we calculate its density, that's if we quickly go ahead and do that, which I'm pretty sure you can do, so I'm just directly writing it, if we directly go ahead and do that, we'll this time, we'll get it to be 15 grams (indistinct), five grams per centimeter cube. That means this time, the object has a larger density than water, it's denser than water. So this condition is not met, and so the object will sink. So if I put that object inside the water, it's going to sink. And as a result, it's going to displace that much amount of water. So whatever is the amount of water present over here, that will move away, and again, this is the displaced water. Okay? Now, can I do the same thing as I did before? Can we say that the stone is being pushed up with a buoyant force, which equals the weight of the displaced liquid, and it's being pushed down by its own weight and the two forces are balanced, so this should equal this, so that means the two weights should be equal and so their masses should be equal, can I do the same thing as before? No, I can't do that. Can you think why? Pause the video, this is super important, okay? Can you think why now I can say that their weights must be the same? Think about this. Okay, well, in the previous case, the object was floating, it was at rest. And that's why the forces were balanced, that the buoyant force was balancing the weight and that's why the two had the same weight. But this time object is sinking. If it is sinking, it's accelerating down, the forces are not balanced. Does that make sense? It's sinking, therefore the buoyant force is smaller than the weight of the object, otherwise it would never sing, it would just stay there. Therefore, this is smaller than this, and so the weight of the displaced liquid will be smaller than the weight of the stone. So the mass of the liquid will be smaller than the mass of the stone. Mass of the stone is equal to 50 grams, so most of the liquid won't be 50 grams. This time displaced water won't have 50 grams, it'll be less than that. Does that make sense? Okay, so how do we calculate this time? What do we do? Because the forces are not equal, I can't equate them, so this time what to do? Well, this time, let me write that down over here, this time because our object is completely submerged, we know that the volume of the displaced liquid should be equal to the volume of the stone, right? Think about it, because the opposite is completely submerged, water has to move away to make space for our entire object. And so this much, whatever is the volume of that stone, that much water must have gone away to make space for that. So I know for sure now that the volume, I don't know the mass of the liquid, but I know the volume of that displaced liquid volume of that water displaced, that should be the same as the volume of the stone, which is 10 cm cubed. Okay? And now I know the volume, I know the density of the water. can I calculate the mass? Yes, we can. Again, good idea to pause the video and see if you can try yourself, all right? So density is mass divided by volume. So mass will be density times volume, right? I'm pretty sure you can do that. So mass will be density of water, mass of the water times volume of the water, and that would equal now, density of water is given as one gram per cm cubed, times room of water, we just found out this is the displace water, okay? And so the cm cubed cancels out and we get 10 grams as the onset. And so the amount of liquid displaced, the mass of liquid displaced, we see is 10 grams, smaller than the mass of the stone. That's why the rate is smaller, buoyant force is smaller, and that's why it's sinking down. So long story short, when things are floating, we can say that the weight equals the buoyant force, which is the same as the weight of the displaced liquid. And so that means the two which should be equal, and so when things are floating, the mass of the displaced liquid should equal the mass of the object, okay? On the other hand, what happens when things are sinking? When things are sinking, the weight of the object is more than the buoyant force. So this weight is more than this weight, but because it's completely immersed, this time we can say that their volumes must be the same. So this time we say the volume of the displaced liquid equals the volume of the object. But of course, don't think of these as formulae, we can easily get confused, instead, always use logic to arrive at these steps.