PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
Buoyant force is the force that a fluid exerts on a object that is immersed within it. It is called buoyant force because this force is a lifting force, often making the object buoyant. Buoyant force can be calculated using Archimedes' Principle.
Let's work some problems on buoyancy, buoyancy is the net force that a fluid like water or air exerts on a solid that's immersed inside of it. So as we saw on Archimedes Principle we can calculate the buoyant force fairly easily just by asking how much does the fluid that I've displaced by the solid weigh? Alright so let's work some simple problems to begin with. Find the buoyant force on 2 times 10 to the minus 4 cubic meters of iron that's immersed in water. Now notice that it doesn't tell me what this 2 times 10 to the minus 4 cubic meters tells me about the iron. But because it's got a unit I immediately know that's got to be the volume. So we'll go ahead and use Archimedes Principle f buoyant equals density of the fluid volume displaced times the acceleration due to gravity.
Now the density of the fluid it's water, so I know the density of water is 1,000 kilograms per cubic meter. So we'll go ahead and write that, the volume displaced well jeez that's 2 times 10 to the minus 4 cubic meters. And remember I know that immediately just from the unit. The teacher can't avoid giving you a unit because otherwise it'll be wrong, right? So we can always use that key to tell us what this piece of information is. And then I got to multiply by the acceleration due to gravity which of course is 9.8 meters per second squared and when I multiply all those things out it will give me the buoyant force 1.96 Newtons okay notice Newtons not kilograms is the unit of force. Alright let's go to the next one, find the buoyant force on 20 kilograms of iron immersed in water. Alright I'm going to do this the same way but I run into a problem immediately. I need the amount of volume that's displaced but I don't know that. They didn't give me the volume they gave me well let's see kilograms so that means they gave me mass.
Alright I know the mass and I know it's iron, so I can look up the density of iron and relate the mass to the volume using that physical quantity. Alright and you will be expected to do this, you'll have a table somewhere in your book that will give you a bunch of densities you can always just look at that table and see what's the density of iron. Well it turns out that the density of iron which is mass divided by volume is 7,874 kilograms per cubic meter okay. So then I'll go ahead and I'll solve for the volume by swapping these 2, so I'll have volume equals mass 20 kilograms divided by density 7,874 kilograms per cubic meter and when you work out that value you'll get 2.54 times 10 to the minus 3 cubic meters. And now it just becomes like number 1, so I'm not going to go through all that, the answer is 24.892 Newtons.
Alright so pretty good, pretty good, let's see what happens when they give us another piece of information. So over here it's asking for the buoyant force on 75 Newtons of copper immersed in water. Alright 75 Newtons, again we're going to use that unit to determine what piece of information the problem is giving us. Newtons is a force, and so since they've told us that this piece of copper is 75 Newtons that means that its weight is 75 Newtons. Now of course we could turn into the mass by dividing by 9.8 meters per second squared and then turn it to problem number 2 and then do all that business and turn it into problem number 1 again, but seems like a lot of work and I'm tired. I don't want to do that much work. So let's do it a different way, it turns out that there's a wonderful property of buoyant forces that we can use whenever the object is completely submerged in the fluid.
Alright so let's just look and see how we get this, f buoyant I'm not going to calculate this directly because I know how it's going to go and after you watch it you'll know how it's going to go too. I'm going to calculate the ratio of the buoyant force to the whole weight. Watch how this works because it's really neat, the buoyant force is the density of the fluid times the volume displaced times the acceleration due to gravity. What's the weight? Well the weight is the whole mass times the acceleration due to gravity but the whole mass is the density of the solid times the volume of the solid times the acceleration due to gravity.
Now the wonderful thing is, of course the acceleration due to gravity cancels but if it's completely immersed the volumes cancel too. And so that gives me this wonderful fact that the ratio of buoyant force to total weight is just the ratio of our densities which is beautiful because then I'll just fill in the densities 1,000 for water, for copper it's 8,960 I think yep 8,960 and now all I need to do to get the buoyant force is just multiply by the weight. f buoyant equals 1,000 over 8,960 times 75 Newtons and when we carry through that multiplication we end up with 8.37 Newtons which is wonderful. It was quick, there was very little opportunity for us to make a mistake which is of course not the case that we would go back through all that business in problems 1 and problem 2 and it just gives us the answer immediately very, very nice.
Alright let's do this last one, this last one is actually associated with what our Archimedes originally used buoyancy to determine. So a king for example would give him a crown and would ask him what is it made of? Is this pure gold? Is this pure silver? And what Archimedes would do is he would weigh the crown in air and he would also weigh it when it was immersed in water. And he would look at that difference to determine what the density is. Alright let's see how this goes, so when I'm weighing it in air I have 35 Newtons and then the weight. So if it's an equilibrium then the weight is 35 Newtons easy enough. When I'm weighing it in water, I have a buoyant force, I have its weight in water which we said was 32 Newtons, and then I have its actual weight, which we already determined was 35 Newtons.
Alright so that means that the buoyant force must make up the difference between the weight in water and the weight in air. So the buoyant force must be 3 Newtons f buoyant equals 3 Newtons. And now we're going to use our ratio result from number 3, if we divide by the weight 35 Newtons then that will give us the ratio of the densities. Density of the fluid divided by the density of the solid, so then again we'll solve for the density of the solid and it'll give us density of the solid equals 35 over 3 density of the fluid. And when we plug in all the numbers we end up with 11,670 kilograms per cubic meter. Now that's pretty to the density of lead, so I'm sorry king your crown isn't made of gold, alright that's buoyancy.