# Buoyancy Calculator

Created by Davide Borchia
Last updated: Jul 23, 2022

Submarines and ships stay afloat thanks to physics and engineering: learn how with our buoyancy calculator. This handy tool will stop you from sinking into despair even before the most difficult physics homework.

Keep reading: you will discover that something as simple as floating hides a more complex and interesting science than you would think. In our buoyancy calculator, you will learn:

• What is buoyancy;
• An explanation of buoyancy without math;
• How to calculate the buoyant force;
• How to use the results of the buoyant force formula to understand if an object will float or not; and
• More than water: the buoyancy in the atmosphere.

## What is buoyancy?

In terms of physics, buoyancy is a force experienced by bodies submerged (partially or entirely) in a fluid. We can observe this force in both liquid and gases, and, knowingly or not, it surrounds us. Swimming, partying, traveling: we owe many things to the buoyant force.

Buoyancy acts in the normal direction with regard to the surface of the Earth, effectively counteracting the gravitational pull. Its strength depends on the characteristic of the submerged object and the physical parameters of the fluid it's submerged in.

Before introducing the math behind the buoyant force formula, let's try to understand why objects float and why some others sink to the bottom.

## The buoyant force explained: why do some objects float while others sink

We owe our understanding of buoyancy to Archimedes: he discovered first the physical reasons objects stay afloat (or fail to do so).

🙋 Eureka! Or maybe not. The famous anecdote regarding the Greek polymath running in the streets of Syracuse — naked after interrupting a bath — is not related to the formulation of buoyancy, but rather to the discovery that an object displaces a volume of fluid equal to the volume of the submerged part: to understand buoyancy we need more steps.

Archimedes formulated the buoyant force equation using words:
Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

We can identify a couple of elements necessary to fully grasp the origin of buoyancy:

• "Buoyed up": buoyancy is always present as an upward force. It may fail in making an object float, but rest assured, it will try!
• "Weight of the fluid": the magnitude of the buoyant force depends on the fluid we're submerging objects into.
• "Displaced": in the buoyancy formula we will meet the volume of the displaced fluid.

We think you got it by now: the buoyancy equation is a game of volumes and masses, with a sprinkle of acceleration.

How to understand buoyancy? Take a cube, and submerge it entirely in water (or any fluid). You have six faces, each of them subjected to a certain pressure. Let's check them out:

• The four vertical faces** experiences a gradient of pressure (increasing with increasing depth), but the force acting on the faces doesn't have a component in the vertical direction: we can neglect them;
• The upper face lies at the lowest pressure. The pressure pushes from above.
• The lower face experiences the highest pressure, pushing in the upward direction.

The pressures, then, are not canceling each other as for the horizontal ones: the bottom one is always larger than the above one (due to gravity). What is the magnitude of this difference? Well, since the cube has faces with equal surfaces, we can equal the forces acting on the faces, not the pressures. The difference in force depends on the difference in the weight of the fluid column at the height of both faces, that is... the weight of a cube with the same volume of the submerged object.

And for non-cubic objects? Well, we can simply break them into cubes. The vertical faces will be ignored as before, while the horizontal faces will cancel each other when finding a correspondence, leaving out only the "exposed" areas.

## The buoyant force formula: how to calculate buoyancy

It's time to give a mathematical description of buoyancy: to calculate it; we will first find the magnitude of the buoyant force; its formula is:

$F_{\text{B}} = \rho\cdot V \cdot g$

Where:

• $\rho$ is the density of the fluid;
• $V$ is the displaced volume; and
• $g$ the acceleration due to gravity.

The product of $\rho$ and $V$ is the mass of the displaced fluid.

As a consequence, the buoyant force experienced by an object has its maximum value when the object is completely submerged (the displaced volume is at its maximum too). Try it: hang a weight to a rope, and lower it into the water until it's entirely submerged. Now lift it: until the object remains underwater, the force required to lift it doesn't change. However, as it gradually resurfaces, you will feel an increased weight. This doesn't happen because of the surface tension but because of a decreasing buoyant force.

## Will this float? The effects of the buoyant force equation

Now we know how to calculate the buoyant force. What about its consequences?

On Earth, every object is subjected to the gravitational pull, approximated on the surface by the acceleration value $g$. The weight we experience is nothing but the gravitational force:

$F_{\text{g}} = m\cdot g$

Where, of course, $m$ is the mass.

If an object is in a fluid, it will experience an upward force calculated accordingly to the equation for the buoyant force (uplift). We can identify three situations that allow us to calculate the buoyancy of an object:

• The buoyant force is smaller than the weight: the object sink. The higher the difference, the bigger the acceleration (and eventually velocity**.
• The buoyant force is larger than the weight: the object experiences a true uplift and moves upward.
• The weight and the buoyant force have the same magnitude: the fluid can't tell the difference between the body and the surrounding: the object floats in the true sense of the world, without changing its vertical position.

## Not only water: buoyancy in the atmosphere

The atmosphere is a fluid, and as a consequence, it applies an upward force (calculated in the same way as any other fluid with the buoyancy equation). The extremely low density of air, however, greatly reduces the magnitude of this force (the density of air at sea level is $1.2\ \text{kg}/\text{m}^3$, not far from a thousandth of seawater density): this is why we don't float around.

Helium-filled balloons, however, since filled with a gas with a lower density than the one of air, float freely. An abandoned balloon will start rising due to the buoyant force. The ascent will see air getting thinner and thinner: its density is slowly decreasing. At one point, the weight of the volume of air displaced by the balloon will equal the buoyancy, and the ascent will stop.

We know: the balloon would expand due to Boyle's law in an attempt to reach the outside pressure. In the case of a really stretchy balloon, this would prolong the ascent and eventually bring it to an end when the balloon itself rips. Here, assume that the balloon doesn't change shape!

The same phenomenon doesn't really occur in water: the density of a fluid remains almost constant even with an increased depth: if an object floats, it does so to the surface!

Davide Borchia
Fluid type (optional)
Sea water
Fluid density ρ
kg/m³
Volume V
Buoyant force B
N
Mass of displaced fluid
kg
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