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Knowledge of the buoyant force is important when trying to understand why some objects float while other objects sink. In this lesson you'll learn about this unique force and how we apply it to various situations using Archimedes' Principle.

The Buoyant Force

I want you to try a little experiment. Find a swimming pool and jump on in. While you're underwater, take note of how easy it is to lift the entire weight of your body. You can do somersaults, flips, and jump really high! Now, try all of those things out of the water. It's a lot harder to lift yourself like that on land, right?

This is because of something you experience in the water called the buoyant force, which is the upward force of a fluid. Buoyancy is an easy concept to understand if you know a little about pressure in a fluid. In a fluid (either a gas or a liquid), pressure increases with depth. So when an object is submerged in water, meaning that it is completely in that fluid, the pressure on the bottom of the object is greater than on the top. This creates a net upward force on the object, so the object is buoyed upward against gravity.

When you jumped in the pool, the pressure against your feet was greater than on your head because your feet were deeper in the water. Therefore, the buoyant force acted upward, pushing you upward and making it easier to lift yourself in the water.

Archimedes' Principle

Think that's cool? It gets even better! Not only does the buoyant force create an upward lift on an object in a fluid, but it's also equal to the weight of the fluid displaced by that object. This was discovered by Archimedes back in the 3rd century B.C., so we call this Archimedes' Principle. Again, it's important to remember that we're talking about fluids, so both liquids and gases, like water and air.

Imagine that you have a full glass of water sitting on the counter. It's so full that if you put anything else into it, the water will spill over the top of the glass and on to the counter. If you were to collect the water that spills out, you would find that this is the same volume as that of the object you put into the glass.

This is what we mean by displacing the fluid, and it's a simple way to measure the volume of an irregularly shaped object since we can easily measure the fluid it pushes out of the way. And remember, the buoyant force is equal to the weight of this displaced fluid, NOT the weight of the object itself.

This means that if the weight of the submerged object itself is equal to the buoyant force (the weight of the displaced fluid), then the object will neither sink nor float. But if the weight of the object is greater than the buoyant force (the weight of the displaced fluid), then the object will sink. And, if the weight of the object is less than the buoyant force (still the weight of the displaced fluid!) then it will rise to the surface and float.

Fish don't float or sink because their weight is equal to the buoyant force. But a heavy boulder sinks to the bottom of a lake because its weight is more than that of the fluid it displaces. And a piece of wood floats on the surface because its weight is much less than that of the fluid it displaces.

Calculating Archimedes' Principle

Archimedes' principle describes the relationship between the buoyant force and the volume of the displaced fluid, but also the density of the displaced fluid.

We can write this principle in equation form as:

FB = ρfVfg

where FB is the buoyant force, ρf is the density of the displaced fluid, Vf is the volume of the displaced fluid, and g is the acceleration due to gravity, 9.8 m/s2. It's very important to remember that the density and volume in this equation refer to the displaced fluid, NOT the object submerged in it.

This equation is helpful because you can use it to determine the buoyant force on an object. For example, say you submerge an object in water and find that the object displaces 1.0 liter of water. Water has a density of 1.0 kg/L, so now we have everything we need to determine the buoyant force acting on the submerged object because we have the volume and density of the displaced fluid. Consequently, we also have the volume of the object because this is the same volume as that of the displaced fluid!

To calculate the buoyant force, simply plug in the numbers. Now our equation reads: FB = 1.0 kg/L * 1.0 L * 9.8 m/s2. Once we do the math, we find that the buoyant force equals 9.8 kg-m/s2, which is the same as 9.8 Newtons.

If the weight of the object is more than 9.8 N, then the object will sink. If it is less than 9.8 N, the object will float. But if the weight of the object is exactly 9.8 N, then the object will neither sink nor float because it is the same as the buoyant force.

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We've touched on flotation already, but floating objects are special enough to deserve just a little more time and explanation. Have you ever seen a large ship traveling through the water? It floats on the water, even though it's heavy enough that you may think it should sink. In this case, it's the shape of the object that determines whether it will float or not.

If you take an entire iron ship and melt it into a solid block, it will take up less volume because it fills a smaller area. But this also means it displaces a smaller volume of water, which in turn decreases the buoyant force. A block of iron will sink, but an iron ship will float because its wide bottom takes up more space in the water, displacing more water and weight, and therefore increasing the buoyant force pushing upward against it.

In fact, a floating object will displace a weight of fluid equal to the weight of the object. This is known as the principle of floatation, and engineers take this into account when designing objects that need to float. Be it a giant cargo ship or a hot air balloon, the object must displace a weight of fluid equal to its own weight in order to float.

This also means that the buoyant force will be greater on objects in denser fluids than fluids that are less dense. You are more likely to float in salt water than freshwater because saltwater is denser than freshwater. But the reverse is also true: less dense objects float more easily than denser ones. For example, women float more easily than men because men are more muscular (and therefore more dense) than women. You can also try this with soda cans - a can of diet soda will float in water, but a regular soda will sink. This is because the diet soda is less dense than the regular soda, so the buoyant force pushes it upward to the surface.

Lesson Summary

Objects submerged in fluids have forces acting on them from all sides, but the upward force in a fluid is a special one, known as the buoyant force. Objects are buoyed up from below because the pressure in a fluid increases with depth, so the force on the bottom of the object is greater than that on the top of it.

Archimedes' Principle tells us that the buoyant force is equal to the weight of the fluid displaced by an object. If the weight of the object is greater than the buoyant force (the weight of the displaced fluid), then the object will sink. If the weight of the object is less than the buoyant force, the object will float. But, if the weight of the object is the same as the weight of the displaced fluid, then the object will neither sink nor float.

Floating objects are special cases because they displace a weight of fluid equal to their own weight. This is known as the principle of flotation, and it tells us why a block of iron will sink, but a wide iron ship will float on the surface.

If we weigh an object in air, and also weigh that object while it is submerged in water, the difference in the weights is the buoyant force.

Problem 1:

An object of weighs 14.7 kg in the air. The same object weighs 13.4 kg when submerged in water. What is its density?

Problem 2:

A helium balloon is going to lift a load of 180 kg. What volume of helium is needed to lift the load, including the weight of the empty helium balloon?

Answer 1:

The buoyant force equals the difference between the weight in air and the weight in water. This is equal to the density of the fluid multiplied by the acceleration due to gravity multiplied by the volume of the object. The weight of the fluid in air is equal to the density of the object multiplied by the acceleration due to gravity multiplied by the volume of the object. Dividing the weight of the object in air by the buoyant force gives the ratio of the density of the object divided by the density of the fluid equal to 14.7/(14.7-13.4) = 11.3. This corresponds to a density of 11.3 kg per meter cubed.

Answer 2:

The buoyant force on the helium balloon equals the weight of the helium plus the weight of the load. Write this equation in terms of densities and solve for the volume of helium.

ρair V g = (ρHe+ 180 ) g.

V is the volume of helium.

g is the acceleration due to gravity.

The density of air near the Earth's surface is 1.29 kg per meter cubed.

The density of helium is .18 kg per meter cubed.

Solving for V in the above equation:

V = 180 / ( 1.29 - .18 ) = 160 meters cubed.

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