Without the buoyant force, fish could not swim, boats could not float and your dreams of flying away with a handful of helium balloons would be even more impossible. In order to understand this force in detail, you must first understand what defines a fluid, and what pressure and density are.

In your everyday conversations, you likely use the words ​fluid​ and ​liquid​ interchangeably. However, in physics there is a distinction. Liquid is a particular state of matter defined by a constant volume and ability to change form to flow or fit the bottom of a container.

A liquid is a type of fluid, but fluids are defined more broadly as a substance that has no fixed shape and that can flow. As such, it includes both liquids and gases.

Density is a measure of mass per unit volume. Suppose you have a cubic container, 1 meter on each side. The volume of this container would be 1 m × 1 m × 1 m = 1 m³. Now suppose you fill this container with a particular substance – water, for example – and then measure how much it weighs in kilograms. (In this case, it should be about 1,000 kg). The density of the water is then 1000 kg/1 m³ = 1000 kg/m³.

Density is essentially a measure of how tightly concentrated the matter is in a substance. A gas can be made more dense by compressing it. Liquids do not compress as easily, but slight density differences in them can be generated in a similar manner.

Now what does density have to do with buoyancy? That will become more apparent as you read on; however, for now, consider the difference between the density of air and the density of water and how easily you “float” (or not) in each. A quick thought experiment and it should be obvious that denser fluids will exert greater buoyant forces.

Pressure is defined as force per unit area. Just as mass density was a measure of how tightly packed the matter was, pressure is a measure of how concentrated a force is. Consider what happens if someone steps on your bare foot with a sneaker, versus if they step on your bare foot with the heel of a stylish pump. In both instances, the same force is exerted; however, the high-heeled shoe causes much more pain. That’s because the force is concentrated on a much smaller area, so the pressure is much greater.

This same principle underlies the reason sharp knives cut better than dull ones – when a knife is sharp, the same force can be applied to a much smaller surface area, causing much greater pressure when used.

Have you ever seen images of someone resting on a bed of nails? The reason they can do this without pain is because the force is being distributed over all of the nails, as opposed to a single one, which would cause said nail to puncture your skin!

Now, what does this idea of pressure have to do with fluids? Suppose you have a cup filled with water. If you poke a hole in the side of the cup, the water will begin flowing out with an initial horizontal velocity. It will fall in an arc much like a horizontally launched projectile. This could only happen if a horizontal force were pushing that liquid out sideways. That force is a result of the internal pressure of the liquid.

All fluids have internal pressure, but where does it come from? Fluids are made up of lots of small atoms or molecules that are all moving around and bumping into each other constantly. If they’re bumping into each other, they’re certainly also bumping into the sides of any container they are in, hence this sideways force pushing the water in the cup out the hole.

Any object submerged in a fluid will feel the force of these molecules bumping around. Since the total amount of force depends on the surface area that is in contact with the fluid, it makes sense to talk about this force in terms of pressure instead – as a force per unit area – so that you can speak of it independently of any object it might be acting on.

Note that the force that a fluid will exert on the sides of its container or on a submerged object depends on the fluid that lies above it. You can imagine the water in the cup above the hole is pressing down on the water below it due to gravity. This contributes to the pressure in the fluid. As a result of this, not surprisingly, in a fluid pressure increases with depth. That’s because the deeper you go, the more fluid is sitting on top of you, weighing you down.

Imagine lying at the bottom of a swimming pool. Consider the sheer weight of the water above you. On land, that amount of mass would crush you entirely, but under the water it doesn’t. Why is this?

Well, it is also due to pressure. The pressure of the water that is all around you contributes to “holding up” the water above you. But also, you have your own internal pressure. As the water applies a pressure to you, your body applies an outward pressure keeping you from imploding.

The buoyant force is a net upward force on an object in a fluid due to the pressure of the fluid. The buoyant force is the reason some objects float and all objects fall more slowly when dropped in a liquid. It is also why helium balloons float in the air.

Because pressure in a fluid depends on depth, the pressure on the bottom of a submerged object will always be slightly greater than the pressure on the top of a submerged object. This pressure difference results in a net upward force.

But how big is this upward force and how can it be measured? This is where Archimedes’ principle comes into play.

Archimedes' principle (named for the Greek mathematician Archimedes) states that for an object in a fluid, the buoyant force equals the weight of the displaced fluid.

Imagine a submerged cube of side length ​L​. Any pressure on the sides of the cube will cancel with the opposite side. The net force due to the fluid will then be the difference in pressure between the top and bottom multiplied by ​L​², the area of one cube face.

The pressure at depth ​d​ is given by:

where ​ρ​ is the fluid density and ​g​ is the acceleration due to gravity. The net force is then

Well, ​L​³ is the volume of the object. The volume of the cube multiplied by the density of the fluid is equivalent to the mass of the fluid displaced by the cube. Multiplying by ​g​ makes it a weight (force due to gravity).

An object in a liquid, such as a submerged rock or a floating boat, will feel an upward buoyant force, but also a downward gravitational force and possibly a normal force due to the bottom of the container, and even other forces as well.

The net force on the object is the vector sum of all of these forces and will determine the objects resulting motion (or lack thereof). If an object is floating, it must have a net force of 0, hence the force on it due to gravity is exactly cancelled by the buoyant force.

An object that is sinking will have a net downward force due to gravity being stronger than the buoyant force on the object. And an object at rest at the bottom of a fluid will have the force of gravity countered by a combination of the buoyant force and the normal force.

A consequence of Archimedes’ principle is that, if the density of the object is less than the density of the fluid, the object floats in that fluid. This is because the weight of the fluid it is able to displace if fully submerged would be greater than its own weight.

In fact, for a fully immersed object, the weight of displaced liquid being greater than the force of gravity would result in a net upward force, sending the object to the surface.

Once at rest on the surface, the object will only sink deep enough into the fluid until it has displaced an amount of equivalent to its own mass. This is why floating objects are generally only partially submerged, and the less dense they are, smaller the fraction that ends up being submerged. (Consider how high a piece of Styrofoam floats in water versus a piece of wood.)

If the density of the object is more than the density of the fluid, the object sinks in that fluid. The weight of water displaced by the fully submerged object is less than the weight of the object, resulting in a net downward force.

The object will not, however, fall as fast as it would through air. The net force will determine the acceleration.

An object with the same density as a particular fluid is considered neutrally buoyant. When that object is completely submerged, the buoyancy force and gravitational force are equal regardless of what depth the object is suspended at. As a result, a neutrally buoyant object will stay where it is set within the liquid.

​Example 1:​ Suppose a 0.5-kg rock of density 3.2 g/cm³ is submerged in water. With what acceleration does it fall through the water?

​Solution:​ There are two competing forces acting on the rock. The first is the force of gravity acting downward with a magnitude of

The second is the buoyant force, which equals the weight of the displaced water.

In order to determine the weight of displaced water, you need to find the volume of the rock (this will equal the volume of water displaced). Because density = mass/volume, then volume = mass/density = 500/3.2 = 156.25 cm³. Multiplying this by the density of water gives the mass of displaced water: 156.25 × 1 = 156.25 g, or 0.15625 kg. So the buoyant force acting in the upward direction has a magnitude of ​F_b​ = 1.53 N.

The net force is then 4.9 – 1.53 = 3.37 N in the downward direction. Using Newton’s second law, you can find the acceleration:

​Example 2:​ The helium in a helium balloon has a density of 0.2 kg/m³. If the volume of an inflated helium balloon is 0.03 m³ and the latex of the balloon itself weighs 3.5 g, with what acceleration does it float upwards when released from sea level?

​Solution:​ Just as with the rock in water example, there are two competing forces: gravity and the buoyant force. To determine the force of gravity on the balloon, first find the total mass. The balloon’s mass is density of helium × volume of balloon + 0.0035 kg = 0.2 × 0.03 + 0.0035 = 0.0095 kg. Hence the force of gravity is F_g = 0.0095 × 9.8 = 0.0931 N.

The buoyant force will be the mass of displaced air times the acceleration due to gravity.

So the net force on the balloon is F_net = 0.36 – 0.0931 = 0.267 N. So the upward acceleration of the balloon is