# Buoyancy: Calculating Force and Density with Archimedes' Principle

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• 0:01 The Buoyant Force
• 1:07 Archimedes' Principle
• 2:51 Calculating…
• 4:47 Floatation
• 6:36 Lesson Summary
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Instructor
Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

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 = ρf Vf g

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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## Buoyancy Practice Problems

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?

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.

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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