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## AP®︎/College Physics 2

### Unit 1: Lesson 2

Buoyant Force and Archimedes' Principle

# Archimedes principle and buoyant force

Introduction to Archimedes' principle and buoyant force. Created by Sal Khan.

## Want to join the conversation?

• shouldnt we take into account the weight of the cube? i mean the forces down shouldnt they be from water pressure at the top + m*g?
(38 votes)
• What he calls the net force in this video should be called the buoyant force. The net force is the sum of the buoyant force and the weight of the cube. So, for example, if the cube is made of pure water, its weight will equal the buoyant force in magnitude (opposite direction), and the net force will be zero. If the cube is made of steel (denser than water) the weight will be greater than the buoyant force, so it will sink. If its made of wood (less dense that water) it will float to the top.
(63 votes)
• I am confused because I thought at the beginning of this lecture Sal said that the pressure around a submerged object was equal from all directions but at the end of his lecture he says that the pressure is greater underneath the object than on top and that is the buoyant force. Could someone help clear this up for me? What is Sal actually saying?
(54 votes)
• His first example was a point, which has a volume of zero. Around this point the pressure is equal because no volume of liquid is displaced. Also, a point has no "height", hence no difference between pressure up and bottom. In the cube example, it has volume, which generates difference in pressure up and bottom.
(2 votes)
• does that mean the pressure in me helps me not to be squashed by the atmosphere
(31 votes)
• yes!
(37 votes)
• This question appeared in my physics final paper and there is confusion amongst students over its answer: "A wooden block is lying on the bottom of the tank sticking (with glue) to it. When water is poured into the tank, water does not enter below the block. Is there a buoyant force acting on the block? Explain."
Please help.
(12 votes)
• Short answer, no. Remeber how suction cups works. because the rubber doesn't let any air under the cup, the pressure of atmosphere is what making it stick to a surface. Same rules apply here
(3 votes)
• Does an object (such as a hot air ballooon) float because it weighs less than a volume of normal air equivalent to the volume of space it takes up? Also, as the air is heated up by the balloon, it becomes less dense and it should float. But what exactly is pushing it up--is it the hot air itself pushing upwards on the inner walls of the balloon?
(7 votes)
• There are two explanations as to what is pushing up the balloon. One is that atmosphere (which is a fluid in static equilibrium) cannot distinguish between the balloon and an equivalent amount of normal air in its place. Therefore, it provides an upward force due to difference in hydrostatic pressure at the bottom of balloon and the top, which is equal in magnitude to the weight of normal air the size of the balloon. Had there been normal air there it would have been static as the upward force would have been equal to its weight but since the weight of the balloon is less than the upward force acting on it, it will move up. This is what concept of floating and upthrust is. The second explanation is considering the tendency of the entire system to lower its gravitational potential energy which can be done if the balloon were replaced with air (due to its greater mass) and therefore all elements of air above the balloon, try to reduce the net energy by coming down in place of the balloon and in the process providing an upward push on it.
(3 votes)
• At the beginning, after Sal draws the cup, he puts a dot with arrows pointing to it. What is that? Pressure?
(2 votes)
• Its to illustrate that even though pressure increases with depth, the pressure at a given depth acts through all directions. If you were shrunk down to the size of a point, and placed at a particular depth, you'd feel the same pressure at that depth all around you. It wouldn't be coming from just one direction.
(13 votes)
• I've seen people write things like "Archimedes' principle says that the buoyant force acting on an object is equal to the weight of the liquid displaced. This simply means that if something is denser than the liquid, it will sink." I've tried figuring out how they came to that conclusion and did a considerable amount of research on it, but I could never figure it out. How did they come up with that?
(3 votes)
• This is something difficult to visualize. But here is how to get there:

The force of water above the object is given by rho*g*h, and the buoyant force underneath the object is equal to the (pressure at the bottom of the object)*(surface area of the bottom of the object). Let's look a little closer at that surface area. The surface area is related to the volume; generally, the greater the total surface area of an object, the greater the object's total volume. For example, an empty balloon has a much smaller surface area than a balloon filled with air. Why did the surface area change? Well, that's because we increased the balloon's volume!

Now, if an object has a greater density, that means that, per amount of surface area, that object also has a greater mass for that given area. If that amount of mass on the surface of the object is greater than the mass of the area of the water (or any liquid) underneath it, then the gravitational force pulling downwards on the mass of the object will cause the object to "push aside" the liquid in its way.
(6 votes)
• if I exhale completely, I sink to the bottom of a pool. If my lungs are full of air, I float. Is this simply due to the fact that the volume of my body is greater when my lungs are full of air, and thus my overall density is less? Is there not some other buoyant property afforded by the fact that my lungs are full of a low-density gas?
(3 votes)
• When you inhale, you increase your volume, which makes you displace more water, which increases the buoyant force on you. The air you inhale has very little mass, so it doesn't really add anything to your weight. The net upward force increases.
(5 votes)
• At , to summarize Archimedes principle- for every submerged object the weight of water displacement equals the object's weight? Is that right? Thanks.
(0 votes)
• No. Nothing was talked about the object's weight. It summarizes saying that the buoyancy force acting on the submerged object is equal to weight of displaced liquid, which depends only on the volume of the object. That is not the only force acting in the object though, there is also the object's weight but it was not mentioned so far.
(6 votes)
• how does he know that the pressure of the cube at the bottom is higher than the pressure on top?
(3 votes)
• Pressure is directly proportional to the depth below the surface of the liquid. The deeper we go down, the higher the pressure. The larger the cube, the more the pressure difference between the top and bottom.
(3 votes)

## Video transcript

Let's say we have a cup of water. Let me draw the cup. This is one side of the cup, this is the bottom of the cup, and this is the other side of the cup. Let me say that it's some liquid. It doesn't have to be water, but some arbitrary liquid. It could be water. That's the surface of it. We've already learned that the pressure at any point within this liquid is dependent on how deep we go into the liquid. One point I want to make before we move on, and I touched on this a little bit before, is that the pressure at some point isn't just acting downwards, or it isn't just acting in one direction. It's acting in all directions on that point. So although how far we go down determines how much pressure there is, the pressure is actually acting in all directions, including up. The reason why that makes sense is because I'm assuming that this is a static system, or that the fluids in this liquid are stationary, or you even could imagine an object down here, and it's stationary. The fact that it's stationary tells us that the pressure in every direction must be equal. Let's think about a molecule of water. A molecule of water, let's say it's roughly a sphere. If the pressure were different in one direction or if the pressure down were greater than the pressure up, then the object would start accelerating downwards, because its surface area pointing upwards is the same as the surface area pointing downwards, so the force upwards would be more. It would start accelerating downwards. Even though the pressure is a function of how far down we go, at that point, the pressure is acting in every direction. Let's remember that, and now let's keep that in mind to learn a little bit about Archimedes' principle. Let's say I submerge a cube into this liquid, and let's say this cube has dimensions d, so every side is d. What I want to do is I want to figure out if there's any force or what is the net force acting on this cube due to the water? Let's think about what the pressure on this cube is at different points. At the depths along the side of the cube, we know that the pressures are equal, because we know at this depth right here, the pressure is going to be the same as at that depth, and they're going to offset each other, and so these are going to be the same. But one thing we do know, just based on the fact that pressure is a function of depth, is that at this point the pressure is going to be higher-- I don't know how much higher-- than at this point, because this point is deeper into the water. Let's call this P1. Let's call that pressure on top, PT, and let's call this point down here PD. No, pressure on the bottom, PB. What's going to be the net force on this cube? The net force-- let's call that F sub N-- is going to be equal to the force acting upwards on this object. What's the force acting upwards on the object? It's going to be this pressure at the bottom of the object times the surface area at the bottom of the object. What's the surface area at the bottom of the object? That's just d squared. Any surface of a cube is d squared, so the bottom is going to be d squared minus-- I'm doing this because I actually know that the pressure down here is higher than the pressure here, so this is going to be a larger quantity, and that the net force is actually going to be upwards, so that's why I can do the minus confidently up here-- the pressure at the top. What's the force at the top? The force at the top is going to be the pressure on the top times the surface area of the top of the cube, right, times d squared. We can even separate out the d squared already at that point, so the net force is equal to the pressure of the bottom minus the pressure of the top, or the difference in pressure times the surface area of either the top or the bottom or really any of the sides of the cube. Let's see if we can figure what these are. Let's say the cube is submerged h units or h meters into the water. So what's the pressure at the top? The pressure at the top is going to be equal to the density of the liquid-- I keep saying water, but it could be any liquid-- times how far down we are. So we're h units down, or maybe h meters, times gravity. And what's the pressure the bottom? The pressure at the bottom similarly would be the density of the liquid times the depth, so what's the depth? It would be this h and then we're another d down. It's h plus d-- that's our total depth-- times gravity. Let's just substitute both of those back into our net force. Let me switch colors to keep from getting monotonous. I get the net force is equal to the pressure at the bottom, which is this. Let's just multiply it out, so we get p times h times g plus d times p times g. I just distributed this out, multiplied this out. That's the pressure at the bottom, then minus the pressure at the top, minus phg, and then we learned it's all of that times d squared. Immediately, we see something cancels out. phg, phg subtract. It cancels out, so we're just left with-- what's the net force? The net force is equal to dpg times d squared, or that equals d cubed times the density of the liquid times gravity. Let me ask you a question: What is d cubed? d cubed is the volume of this cube. And what else is it? It's also the volume of the water displaced. If I stick this cube into the water, and the cube isn't shrinking or anything-- you can even imagine it being empty, but it doesn't have to be empty-- but that amount of water has to be moved out of the way in order for that cube to go in. This is the volume of the water displaced. It's also the volume of the cube. This is the density-- I keep saying water, but it could be any liquid-- of the liquid. This is the gravity. So what is this? Volume times density is the mass of the liquid displaced, so the net force is also equal to the mass of liquid displaced. Let's just say mass times gravity, or we could say that the net force acting on this object is-- what's the mass of the liquid displaced times gravity? That's just the weight of liquid displaced. That's a pretty interesting thing. If I submerge anything, the net force acting upwards on it, or the amount that I'm lighter by, is equal to the weight of the water being displaced. That's actually called Archimedes' principle. That net upward force due to the fact that there's more pressure on the bottom than there is on the top, that's called the buoyant force. That's what makes things float. I'll leave you there to just to ponder that, and we'll use this concept in the next couple of videos to actually solve some problems. I'll see you soon.