Buoyant Force Calculator

Steel ships float. People feel lighter in a pool. Both come down to buoyant force, the upward push a fluid applies to anything sitting in it. This calculator uses Archimedes' principle to work out how much that push amounts to, given the fluid you're in, the volume your object displaces, and local gravity.

What is buoyant force?

When you drop something into a fluid, it pushes some of that fluid out of the way. The weight of the fluid you've displaced is exactly the force pushing back up on whatever you dropped in. That's Archimedes' principle, and it's the whole of buoyancy in a single sentence.

If the buoyant force beats the object's weight, it floats. If the weight wins, it sinks. When they match exactly, the object hovers in place. That last state is what submarines and scuba divers aim for, since it's the only one where you don't have to do any work to stay where you are.

How to use this calculator

By default the calculator solves for buoyant force. Put in the fluid density (mass per unit volume), the displaced volume (how much fluid your object is shoving aside), and gravity at your location. For most places on Earth's surface, $9.81 \text{ m}/\text{s}^2$ is close enough. The result comes out in newtons.

To find density instead, switch the formula set. If you know the force, volume, and gravity, the calculator hands back the density of the fluid. That's useful when you're trying to identify an unknown liquid in a lab. Both metric and imperial units work throughout.

Understanding the formula

Here's the equation.

F_b = \rho \times V \times g

Where $F_b$ is buoyant force in newtons, $\rho$ (rho) is fluid density in $\text{ kg}/\text{m}^3$ , $V$ is displaced volume in $\text{m}^3$ , and $g$ is gravitational acceleration in $\text{m}/\text{s}^2$ .

Take a scuba diver who displaces $0.07 \text{ m}^3$ (about 70 liters) of seawater. Saltwater runs roughly $1025\text{ kg}/\text{m}^3$ , and gravity at the surface is $9.81 \text{ m}/\text{s}^2$ .

$F_b = 1{,}025 \times 0.07 \times 9.81 = 704.6 \text{ N}$

That's about 705 newtons pushing up, roughly the weight of a 72 kg person. A diver lighter than that (gear included) bobs up; heavier, and they sink. Divers carry weight belts to land right between those two states so they can hang motionless in the water.

The difference between fresh and salt water matters more than it looks. Freshwater clocks in around $1000 \text{ kg}/\text{m}^3$ and seawater around 1025. That $25\text{ kg}/\text{m}^3$ gap is small on paper, but for someone displacing 70 liters it works out to about 17 extra newtons of upward push. That's why swimming in the ocean feels easier than swimming in a lake.

Applications of buoyant force

Ship design is the most familiar example. Naval architects shape hulls so the displaced water weighs more than the ship itself. A loaded steel cargo ship floats because the hollow shape sweeps aside a huge volume of water, enough that the upward force outpaces every ton of steel above the waterline.

Submarines control depth by changing their own weight, not the buoyant force on them. Open the ballast tanks and seawater rushes in; the sub gets heavier and drops. Blow compressed air back into the tanks, water gets pushed out, the sub lightens and rises.

Hot-air balloons work the same way, except the fluid is air. The balloon displaces a volume of cool ambient air, and the upward buoyant force from that displacement lifts the lighter hot air inside the envelope along with the basket hanging off the bottom.

Hydrometers turn the principle into a measurement tool. A weighted float sits at a depth determined by the density of the liquid it's floating in. Brewers, winemakers, and battery shops use them constantly to read off specific gravity at a glance.

Tips for accurate calculations

Fluid density isn't a fixed constant. It changes with temperature and with whatever's dissolved in the fluid. Water hits $1000\text{ kg}/\text{m}^3$ exactly at 4°C and thins out as it warms. Salinity moves seawater density around. For anything precise, look up the value in a reference table rather than reaching for a round number.

Only the submerged part of an object counts toward displaced volume. A floating object displaces just enough fluid to match its own weight, not its total volume. The part above the waterline contributes nothing to the buoyant force.

Local gravity wobbles too. $9.81 \text{ m}/\text{s}^2$ is the standard value, but it's closer to 9.78 at the equator and 9.83 at the poles. For most calculations the standard is fine. If you're doing precision metrology or working at high altitude, look up the value for your latitude.

Frequently asked questions

Why do some objects float while others sink?

Average density. If the whole object, including any air pockets or hollow space, is less dense than the fluid around it, it floats. Solid steel is denser than water, but a ship is mostly air inside, so the average density drops below water's and the ship sits on the surface.

Does buoyant force depend on the object's depth?

For water and other practically incompressible fluids, no. Move a brick from 1 meter down to 10 meters down and the buoyant force on it is identical. The exceptions are gases, which compress noticeably with depth, and objects that themselves compress at extreme pressures (a submarine at the bottom of an ocean trench, for example). In those cases the displaced volume actually changes, and the buoyant force changes with it.

Can buoyant force be greater than the object's weight?

Yes. That's what makes things rise. A helium balloon climbs because the air it displaces weighs more than the balloon and its contents combined. Push a beach ball underwater and let go; it shoots back to the surface for the same reason. The upward acceleration equals the buoyant force minus the object's weight.

How does temperature affect buoyancy?

Heat the fluid and it thins out, which means slightly less buoyant force on whatever's in it. That's the entire trick behind a hot-air balloon. In the ocean, the same effect produces stratified density layers at different temperatures and salinities, which is part of why marine life adjusts buoyancy depending on what depth they're feeding in.

What is neutral buoyancy and why is it important?

Neutral buoyancy is the state where the buoyant force on an object exactly matches its weight, so it neither rises nor sinks. It just hovers wherever you left it. Submarines work hard to maintain it so they can sit at a target depth without constantly burning energy. Scuba divers chase the same state with weight belts and inflatable vests so they can glide instead of flail. Underwater photographers and marine researchers depend on it too, since drifting silently in place keeps them from kicking up silt or spooking the wildlife.