# Calculating and Understanding Buoyancy

James Reynolds, Elizabeth Friedl, Elaine Chan
• Author
James Reynolds

James Reynolds has trained oilfield geologist for two years. They have bachelor's degrees in biology and geology and have spent four years tutoring college students in these subjects.

• Instructor
Elizabeth Friedl

Elizabeth, a Licensed Massage Therapist, has a Master's in Zoology from North Carolina State, one in GIS from Florida State University, and a Bachelor's in Biology from Eastern Michigan University. She has taught college level Physical Science and Biology.

• Expert Contributor
Elaine Chan

Dr. Chan has taught computer and college level physics, chemistry, and math for over eight years. Dr. Chan has a Ph.D. in Chemistry from U. C. Berkeley, an M.S. Physics plus 19 graduate Applied Math credits from UW, and an A.B. with honors from U.C .Berkeley in Physics.

What is buoyancy? Learn the buoyancy definition and calculate it with Archimedes' Principle. See how to use the buoyancy formula with various example problems. Updated: 07/12/2021

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## Buoyancy Definition

Buoyancy is defined as the ability of an object to float. This happens when an object placed in a fluid is acted upon by an upward buoyant force. The gravitational force pushing down on a buoyant object in fluid will equal the force of the fluid pushing upward on the object.

For example, if a steel sphere were dropped into water, it would immediately sink to the bottom, but when a steel boat enters the water, it floats. The combined density of the heavy steel of the boat and the very light air inside the boat will be less than the density of water and allow the boat to float. To understand precisely why this happens, we need to understand Archimedes' principle, buoyant force, and the buoyancy equation.

### Buoyant Force

The buoyant force is a net upwards force acting on any submerged or partially submerged object. Think of a ball in the water. Gravity is pushing down on all the water, meaning as the ball falls deeper the weight of all the water above it is pushing in on the ball from all sides. Pressure always increases with depth in fluids (this is why our ears pop when we dive in a pool). Since the bottom of the ball is deeper under water than the top, the pressure on the bottom of the ball is greater than that on the top. The net pressure upwards on the ball is defined as the buoyant force. This force is fundamentally tied to gravity and because of this there is no buoyancy in space.

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## Buoyancy: Archimedes' Principle

Archimedes was a Greek mathematician. He was asked by King Heiron II of Syracuse to determine if the royal gold crown was made of pure gold. Archimedes noticed that when he put gold and silver that were the same weight as the crown in water, they displaced different amounts of water. The gold displaced less water than the silver. Yet, when Archimedes put the gold crown in the water, it displaced more water than the equivalent weight of pure gold. He determined that the crown was a mix of gold and silver.

Archimedes' principle states that when an object is placed in water an upward force ({eq}F_b {/eq}) will act on the object that is equal to the weight of water ({eq}W_f {/eq}) the object displaces. This is the buoyant force described earlier and is surmised in the following equation:

{eq}F_b = W_f {/eq}

If an object is only partially submerged, then the volume of water displaced will equal the volume that is partially submerged. If the object is completely submerged, then the volume of water displaced will equal the volume of the object.

## Buoyancy Formula

When using the formula for buoyancy, we are using the formula to solve for buoyant force ({eq}F_b {/eq}). The buoyancy formula includes the acceleration due to gravity of 9.8 meters per second squared (g), the density of the fluid (p), and the volume of the displaced fluid ({eq}V_f {/eq}). These factors are all multiplied together to give us the buoyant force in the equation {eq}F_b = gpV_f {/eq}.

This equation can be derived from Archimedes' principle because the weight of the displaced fluid can be described as the mass of the fluid multiplied by the acceleration due to gravity, so we have

{eq}F_b = W_f = m_fg {/eq}.

Now the mass of the displaced fluid can also be described by density multiplied by volume. This gives us the buoyancy formula:

{eq}F_b = pV_fg {/eq}

## Using the Buoyancy Equation

The buoyancy formula, {eq}F_b = gpV_f {/eq}, can be used to solve a variety of problems. Note that we must take into account whether the object is partially or fully submerged when applying this equation.

### Example 1:

Determine the buoyant force when a cube with an edge length of one meter is fully submerged in water.

We first need to determine the components on the right side of the buoyancy equation:

• When the cube is fully submerged, it will displace 1 cubic meter of water, so {eq}V_f = 1 \text{ m }^3 {/eq}
• Water has a density (p) of {eq}1000 \text{kg}/ \rm m^3 {/eq}.
• For g, we use Earth's acceleration due to gravity of {eq}9.8 \text{ m}/ \rm s^2 {/eq}

Now, we can determine the buoyant force ({eq}F_b {/eq}):

{eq}\rm F_b = (9.8 \text{ m}/ \rm s^2)\cdot(1000 \text{kg}/ \rm m^3)\cdot(1 \text{ m}^3) \\ F_b = (9.8 \text{ m}/ \rm s^2)\cdot(1000 \text{ kg}) \\ F_b = 9800 \text{ kg}\cdot \text{ m}/ \rm s^2 \\ F_b = 9800 \text{ N} \\ \text{N is a Newton of force which is 1 kg}\cdot \text{ m}/ \rm s^2 {/eq}

### Example 2:

Calculate the buoyant force when a cube with volume {eq}1 \text{ m }^3 {/eq} is half submerged in water.

• We can use the same values of p and g from Example 1.
• Since the cube is only half submerged, the volume of water displaced will equal half of the cube's volume, so {eq}V_f = 0.5 \text{ m }^3 {/eq}

Applying the buoyancy formula, we find that

{eq}\rm F_b = (9.8 \text{ m}/ \rm s^2)*(1000 \text{kg}/ \rm m^3) \cdot (0.5 \text{ m}^3) \\ F_b = (9.8 \text{ m}/ \rm s^2) \cdot (500 \text{ kg}) \\ F_b = 4900 \text{ kg}\cdot \text{ m}/ \rm s^2 \\ F_b = 4900 \text{ N} {/eq}

## Buoyancy Examples

There are several examples that will help illuminate the principles of buoyancy

• Ice cubes in water
• Birthday balloon in air
• A person in water

In the ice cube example, the ice floats because it is less dense than water. Most things constrict as they freeze, but water is one of a few substances that expands when frozen. When density goes down, the ice cube displaces more water thus increasing the buoyant force.

With the birthday balloon, it is important to remember that a fluid is any substance whose particles can move freely past each other. Air and any other gases fit this description. Most balloons are filled with gas of a lower density than air, which causes them to rise.

People are usually about 70% water. We have bones and muscles that are more dense, and air in our lungs that is less dense. The relative ratios of these things cause some people to float and others to sink in water. Scuba divers use metal weights and an air-filled buoyancy control device to manipulate their average density to rise and fall in water. An object that rises has positive buoyancy. One that neither rises nor falls is neutral, and one that falls has negative buoyancy.

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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.

#### What causes buoyancy?

The bottom of a submerged object is deeper under water than the top, so the pressure on the bottom of that object is greater than that on the top. The net force upwards on the ball is defined as the buoyant force. When this upward force is greater than the gravitational force (the object's weight), the object will rise in the fluid. Once the gravitational force and buoyant force are balanced, the object will stop rising.

#### How do you calculate buoyancy?

Fb = gpV

Fb is the buoyant force

g is the acceleration due to gravity of 9.8m/s2

p is the density of water

V is the volume of displaced fluid

#### What are the 3 types of buoyancy?

The three types of buoyancy are positive, neutral, and negative. Positive buoyancy is when an object floats in fluid. An object that neither rises nor falls has neutral buoyancy. When an object sinks, it has negative buoyancy.

#### What is a Newton of buoyancy?

N is a Newton of force which is one kg*(m/s^2). This is the standard way of quantifying the buoyant force.

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