PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
Archimedes' principle states that the buoyant force on a fluid is equal to the weight of the displaced fluid. To calculate the buoyant force, we use the equation buoyant force = density of fluid x volume of displaced fluid x acceleration due to gravity. In a completely submerged object, the volume of displaced fluid equals the volume of the object. If the object is floating, the volume of the displaced fluid is less than the volume of the object but the buoyant force = the weight of the object.
Alright let's talk about Archimedes Principle. Archimedes Principle has been around for a really, really long time and it was very important to rulers of like Greece and Egypt years and years and years ago, and we'll talk about why in just a minute. So what Archimedes said was that the buoyant force on something is equal to the weight of the displace fluid. Now what does a buoyant force mean? The buoyant force, is the net force that is exerted on an object that's immersed in a fluid. Alright so we've got a force from the pressure on the bottom and a force from the pressure on the top we add them together, we want the net force and that's the buoyant force. And so what Archimedes said was that, that buoyant force is the weight of how much fluid the object is displacing. So buoyant force equals weight of the displaced fluid. Well what is that? Well it's the density of the fluid times the volume of the volume of the fluid that's benen displaced. So that's the volume of the object that's immersed in the fluid times the acceleration due to gravity.
Alright now we have 2 major situations in which we can use Archimedes Principle, if the object is completely submerged, so the entire object is immersed in the fluid then the volume displaced equals the volume of the object. Alright, now so that makes it really, really simple. All I need to know is what's the volume of the object and I'm done. If the object is floating on the other hand, then what that means, is that it's not all the way immersed. What that means is that the volume displaced is actually less than the whole volume, because some of it, is floating on the top, we said it was floating. So in the floating situation we always have the buoyant force is equal to the weight of the object because the object is floating. And that means that its weight has to be canceled by an upward force, what's that upward force? That's the buoyant force, so these are the 2 separate situations and they're entirely separate.
We'll see that there're equations that you can derive that are perfectly valid when the object is fully submerged but that are not true for floating objects. So you just need to be careful about that distinction. Alright, so why is Archimedes Principle true? Well we can look at a situation in which we've got an object immersed in a fluid like this, alright now what are the forces acting on this object we're going to draw a free body diagram because we're good Physicists so we've got weight and then we've got 2 forces that are acting from the fluid. The fluid has a pressure in it, so there's a pressure at the bottom and a pressure at the top. Now pressure is force per unit area, and it points in any direction. So the pressure at the bottom, the force that's acting on the object is pushing up because that's what direction, I mean if it's going to push on the object well what's, it's going to push up right? So the upward force is pressure at the bottom times the area, because force is pressure times area.
Same way pressure at the top is pushing down. So the downward force is pressure at the top times the area. Now the buoyant force is the net force exerted on the object by the fluid, so that means that it's equal to pressure at the bottom times the area minus pressure at the top times the area. So the area I can take out because it's the same for top and the bottom, so that means that it's the change in pressure times the area. Now we know that as you descend into a fluid the pressure increases. How much is the increase? Well the change in pressure is given by the density of the fluid times acceleration due to gravity times how far you went down. Now here we went down a distance h, so look what we got here. h times a that's the volume displaced, so that means that the buoyant force is equal to density of the fluid times acceleration due to gravity times the amount of volume that was displaced. So Archimedes Principle alright why do people care? Well one of the most enigmatic uses of this principle in history has been to determine the density of valuable objects like crowns and vase and things like that. Are they made of gold or is somebody trying to cheat me?
Alright here's the idea, all you need to do is weigh the object in air and then weigh the same object when it's emerged in water. So why does this work? Well here's the idea, when I weigh an object basically what I'm doing is I've got a scale and I'm trying to measure how much force it requires for me to suspend the object? So I've got a tension force going up from my scale and then I've got a weight going down and this needs to equal each other. So when I weigh it in air, the tension force is equal to mg, is equal to the weight. What about when I weigh it in water? Well when I weigh it in water I've got a buoyant force now and this buoyant force depends on the volume of the crown and the density of water. So that means that this buoyant force inside of it has information about the density. So when I look this tension here, this tension is the weight of the crown in water. It's going to be smaller than the weight of the crown in air and that difference is this buoyant force. And that buoyant force contains information that I can pull out about the density of the crown. And that's Archimedes Principle.