Archimedes principle & buoyancy

- [Instructor] How is it that an extremely heavy ship made of metal floats on water, but a tiny piece of metal like a spanner easily sinks into it? Why is that some people who know how to swim can easily float, but others, like me, can't? To answer questions like these, we need to understand the principle of flotation. And this principle was discovered by a Greek mathematician called Archimedes. And as the legend goes, one day, when Archimedes, well, I'm using Hulk, as I don't have an Archimedes action figure. Anyways, when he stepped into his bathtub, he saw the water spilling out. A very common sight. But that day, by seeing it, something clicked in his head. He got so excited that he jumped out of the tub and started running through the city shouting, "Eureka, eureka!" Which meant, "I have found it, I have found it." But what did he find? Whatever he found out is today famously called Archimedes' principle of flotation. And it basically says that the buoyant force acting on any submerged object equals the weight of the displaced fluid. So we'll first try to understand what this statement means, and then we'll see if we can answer our original question. So let's start with the buoyant force. What does that mean? Well, you might be actually familiar with this. For example, whenever you are inside a swimming pool or underwater, you might know that you feel a little lighter, right? And this can be actually experimentally verified. So over here I have Archimedes who's hanging by a weighing scale, and right now the weighing scale is showing 160 grams. It's a toy, all right? But we'll see what happens with that weight as I lower him inside water. Let's see what happens to it. Look at that weighing scale. As I dip him under water, look, the reading becomes lower because Archimedes is feeling lighter and lighter. And this is not just true for water, this would be true for any liquid. So you submerge a body inside a liquid and that body will feel light. But what does that mean or why is this happening? Well, we know that his weight is nor really changing because his mass is the same, so the gravitational force acting on him is the same. So that's not changing. So what could it mean? Well, what could mean is that something must be pushing up on him to balance some of his weight, making him feel lighter. Right? So what's pushing on him? Well, it has to be the water, or the liquid. And we'll talk about why water is pushing up on him a little bit later, but it turns out this is true not just for liquids. This can also happen inside gases. My favorite example for this is the helium balloon. We know that when you let go of a helium balloon, it starts rising up. Which means, again, there must be a force acting upwards on it. Who's pushing it this time? It must be the air. So this means whenever we have objects submerged inside liquids or gases, which are collectively called fluids, by the way, a fluid means anything that can float, liquids or gases, they have a natural tendency to push up on things. And that force is what we call buoyant force. And the word buoy means to float. I think it has a Dutch origin. But it's called so because this force is literally what makes them float. This is what's pushing them towards the surface trying to make them float. But we know not everything floats. Things can also sink, right? And that's why we are interested in knowing what does this buoyant force depend on so that we can predict whether things will float or sink. And that's what Archimedes' principle tells us. It tells us what buoyant force depends on. It tells us that this buoyant force should equal the weight of the displaced fluid. Okay, what does that mean? Well, again, if we come back to our Hulk. Sorry, Archimedes. We see that right now this much of his muscular body is underwater, right? But before he stepped inside, that space was occupied by water, right? So this means once Archimedes goes underwater, that much water should move out to make space for his body to come over there. So it should move out. Where does it go? Well, if there's space inside the container, it'll just go up, because water can easily flow. It'll just go up. But if there's no space, water will just fall out. That's what we saw earlier. This is what we call the displaced liquid. The liquid that moves out or moves up to make space for the submerged body is what we call the displaced liquid. And Archimedes' principle is saying the weight of this displaced liquid equals the buoyant force. Whatever is the weight of this liquid, that equals the buoyant force. Meaning the more liquid you displace, the more weight of liquid you displace, more is the buoyant force. And the same thing is gonna happen over here as well. Before the helium balloon came over here, it was occupied by air, which I'm showing by green so that we can see. But once the helium balloon comes over there, that air must have moved somewhere else to make space for the helium balloon. Now of course the air and the liquid they will not maintain their shape. Of course they won't maintain their shape, I'm just showing it this shape. But anyways, the air must have moved, right? So again, this is the displaced air. And Archimedes' principle says whatever is the weight of this displaced air, that will be the buoyant force acting on the balloon. So now let's see if the earlier experiment makes sense. You see, as our muscular Archimedes gets lowered underwater, more and more liquid gets displaced to make space for his submerged body. And as more and more liquid gets displaced, more weight of liquid gets displaced, and as a result the buoyant force starts increasing, becomes bigger, and so he feels lighter and lighter, and so the weighing scale reads lower and lower. Makes sense, right? Now, before we explore why Archimedes' principle is true, let's quickly go ahead and see if we can answer our original question. So why does a metallic ship float? Let's concentrate only on the base of this ship so that it becomes easier to analyze. So if I only look at the base of that ship, notice because there is no water inside that ship, that means this much amount of water must have been displaced. Now, that is a lot of water, if you think about it, because this ship is pretty big. And since this is a lot of water, it has a lot of weight, and therefore, from Archimedes' principle, the buoyant force acting on this ship must be very large, large enough to support the weight of that entire ship. Now let's say we take the same amount of metal, that same amount of metal and we flatten it. Now you might now this will sink, but why? Well, because now you see it is only displacing this much amount of water. Only that much. It's no longer displacing the water on top of it because the shape has changed, can you see that? And since the displaced water is little weight, its weight is little weight, so the buoyant force acting on that same piece of metal is little and so the whole thing would sink. So you can now see the secret behind ships. Ships have a lot of empty space such that their metal occupies a large volume underwater, because of which they will displace a lot of water, making sure the buoyant force is large enough to support its weight. That's the secret. On the other hand, if you have flat things or things which do not have empty space, they will not displace enough water or enough liquid in which case they can easily sink. And that's why even if you take a tiny piece of metal which is pretty light, it will sink, because it's not able to displace enough liquid. Now let's try and answer why would I panic underwater and sink. Well, when you're trying to float in water, when you breathe in, that's when your lungs expand, your body expands. Of course I'm exaggerating over here. But as a result the volume of your body underwater increases, meaning you displace more water, and so the buoyant force on you starts increasing and that can support your weight. But when I am in water, I will panic and I will start screaming. As a result, I will let all that air go, and so my body shrinks, and so I will displace less fluid, and so the buoyant force decreases, and there are good chances that I will sink. Which is why I always wear a life jacket when entering not-so-shallow water. Okay, finally, we might be wondering, why is Archimedes' principle even true? What's the logic behind this? To answer that, we need to first understand where the buoyant force even comes from. Well, for that let's imagine we have Archimedes completely submerged inside water. Now because water has pressure, it starts pushing on Archimedes from all the directions. I'm not showing all the arrow marks. It actually has to push from all the directions. But what's important is that the pressure increases with depth. As you go deeper, the pressure increases because water has to carry more weight on top of it. And because of this, the forces from the bottom becomes larger than the forces from the top. And if you need more clarity on why the pressure increases with depth and why it puts forces in all the directions, we've talked a lot about that in the a previous video called "Pressure in Liquids." Feel free to check that out. Anyways, because the forces from the bottom is more that these forces don't cancel out, if you add them, we'll get a net force acting upwards. And that force itself is what we call the buoyant force. So it comes from the pressure of the water. But how do we calculate it? Well, here's the insight. Imagine I took Archimedes out and I filled that space with some other material. Let's say I fill it with super-heavy gold. My question is, do you think that the buoyant force will change or do you think it'll remain the same? Think about this for a while. Well, let's think about this. The buoyant force comes due to the pressure from the surrounding liquid, right? Now, putting some other material, does that change the pressure? No. Because the pressure in the liquid only depends upon how deep you go. And that depth has not changed, everything has remained the same. And therefore the pressure remains the same, so the forces at every point remain the same, which means the buoyant force should remain the same. That means regardless of what material I put in this space, whether I put heavy gold or even if I put super-light styrofoam, the buoyant force will not change. It does not depend upon what comes in this space. Does that make sense? Okay, now you may be asking, okay, fine, but how do I calculate that buoyant force? Well, here comes the eureka moment. Since the buoyant force does not depend on what I put in the space, what if I just put the same water? Even now the buoyant force should remain the same. But now I know that this piece of water is stationary, it's not moving. That means the forces on it should be balanced. In other words, the buoyant force should equal the weight of that liquid, this piece of liquid. Only then this piece of liquid would stay stationary. Because think about it, the whole water is actually stationary, isn't it? But what liquid is that? Hey, when I put Archimedes back in the water, it's that same liquid that gets displaced, isn't it? That means our buoyant force should equal the weight of the displaced liquid. The Archimedes' principle, eureka. Now, it did take me some time to wrap this logic around my head, so if you don't get this the first time, don't worry. Ponder upon it for some time and I'm pretty sure eventually you'll get it. So what did we learn in this video? We saw that whenever objects are immersed in liquids or in gases, which are collectively called fluids, then they have a natural tendency to push up on those things. And we call this force the buoyant force. This occurs because in fluids, due to gravity, the pressure at the bottom is always more than the pressure at the top. And as a result, when you add up these forces, there will always be a net upward force. And how do we calculate this buoyant force? Well, you figure out how much fluid gets displaced when you submerge these bodies. And then, according to the Archimedes' principle, the weight of this displaced fluid will equal the buoyant force acting on them.