Buoyant Force Formula: F = ρVg, Archimedes' Principle

The buoyant force formula is $F_b = \rho_f V g$ : multiply the density of the fluid by the volume of fluid the object pushes aside and by g. The result is the upward force, often called upthrust, that every liquid and gas exerts on anything placed in it.

This is Archimedes' principle in symbols: the buoyancy force equals the weight of the fluid displaced. Notice that the object's own density does not appear. A steel cube and a wooden cube of the same size, both held fully under water, feel exactly the same upthrust. What differs is their weight, and the contest between weight and buoyancy decides whether something sinks or floats. It is one of the core results on the fluids formula sheet.

Quick facts

Formula
$F_b = \rho_f V g$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²], the same as any force
Direction
Vertically upward, opposite to gravity
Apparent weight
$W_{app} = W - F_b$
Floating
$F_b = W$ , with only part of the object submerged

Reading the buoyant force formula

Symbols in F_b = ρ_f V g
Symbol	Meaning	SI unit
$F_b$	Upward buoyant force	N
$\rho_f$	Density of the fluid, not of the object	kg/m³
$V$	Volume of the object below the surface	m³
$g$	9.8 m/s² near Earth	m/s²

The product $\rho_f V$ is the mass of displaced fluid, and multiplying by g turns that mass into a weight. So the formula really says: upthrust equals the weight of the fluid that used to fill the space now taken by the object.

For a fully submerged object, V is the whole volume. For a floating one, V is only the part under the waterline. Getting that choice right is half of every buoyancy problem.

Where the upthrust comes from

Pressure in a fluid grows with depth, as the pressure page shows with $P = \rho g h$ . Take a block of top area A and height h. The bottom face sits deeper than the top face by h, so the fluid pushes up on the bottom harder than it pushes down on the top:

$F_b = (P_{bottom} - P_{top})A = \rho_f g h A = \rho_f V g$

The side forces cancel in pairs. This short derivation is popular in AP Physics 2 and in Class 11 because it shows buoyancy is not a new force, just the net effect of fluid pressure.

Worked examples

Example 1: a block held under water

A sealed box of volume 0.0020 m³ is held completely under fresh water. Find the buoyant force.

$F_b = 1000 \times 0.0020 \times 9.8 = 19.6\ \text{N}$

If the box weighs less than 19.6 N it will rise as soon as it is let go.

Example 2: apparent weight of a stone

A stone of mass 500 g and density 2500 kg/m³ hangs from a spring balance, fully immersed in water. What does the balance read?

Volume: $V = \frac{m}{\rho} = \frac{0.500}{2500} = 2.0 \times 10^{-4}\ \text{m}^3$

Upthrust: $F_b = 1000 \times 2.0\times10^{-4} \times 9.8 = 1.96\ \text{N}$ . Weight: $0.500 \times 9.8 = 4.90\ \text{N}$ .

Reading: $4.90 - 1.96 = 2.94\ \text{N}$ . The stone seems to lose 1.96 N, exactly the weight of water it displaces.

Example 3: how much of an iceberg is hidden

Ice (917 kg/m³) floats in sea water (1025 kg/m³). What fraction of its volume is below the surface?

Floating means $F_b = W$ : $\rho_{sea} V_{sub} g = \rho_{ice} V g$ , so $\frac{V_{sub}}{V} = \frac{917}{1025} = 0.895$

About 89 to 90 percent is under water, leaving roughly a tenth on view.

Use the object's density only to find its weight or volume. The buoyant force itself always uses the fluid's density.

Sinking, floating and the force of buoyancy

Object denser than fluid: weight is bigger than the maximum upthrust, so it sinks.
Object less dense than fluid: it rises until just enough is submerged for $F_b = W$ , then floats.
Equal densities: it hangs at any depth, as a fish or submarine does when it adjusts its ballast.

A steel ship floats because its hollow hull encloses a large volume of air, so its average density is lower than that of water. Comparing the object's density with the fluid's is quick using the relative density formula. The force of buoyancy formula works in gases as well: hot air balloons use the same rule in air, with a fluid density near 1.2 kg/m³.

Common errors

Using total volume for a floating body. Only the submerged part displaces fluid.
Forgetting g. ρ_f V alone is a mass in kilograms, not a force.
Leaving volume in cm³ or litres. 1 cm³ = 10⁻⁶ m³ and 1 L = 10⁻³ m³.
Thinking deeper means more upthrust. Once fully under, the force does not change with depth for an incompressible liquid.

Frequently asked questions

What is the formula for buoyant force?

F_b = ρ_f V g: fluid density times the submerged volume times g. It equals the weight of the fluid the object displaces.

What is the SI unit of buoyant force?

Upthrust is measured in newtons like every other force; dimensionally it is [M¹ L¹ T⁻²].

Does buoyant force depend on the object's mass?

Not directly. It depends on the fluid density and the volume displaced. Mass matters only through the floating condition, where the upthrust must equal the weight.

What is the difference between buoyancy and upthrust?

They mean the same thing. Upthrust is the term common in Indian and British books, buoyant force in US books.

Is the buoyancy force formula the same in air and water?

Yes. Only the fluid density changes. Air is about 800 times less dense than water, so the upthrust in air is tiny unless the volume is very large, as with a balloon.

Why do you feel lighter in a swimming pool?

Water pushes up on you with a force equal to the weight of water your body displaces. Your apparent weight is your true weight minus that upthrust.