3.3.3.1. Archimedes' Principle

Table of Contents

The basic idea

Archimedes' principle explains the upward force that a fluid exerts on an object placed in it. This upward force is called the buoyant force.

When an object is partly or fully surrounded by a fluid, the fluid pushes on it from all sides. Because pressure increases with depth, the fluid pushes more strongly on the lower parts of the object than on the upper parts. This difference creates a net upward force.

Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by that object.

For an object in a fluid, the buoyant force is
$$F_b = \rho_f g V_{\text{disp}}$$
where $\rho_f$ is the density of the fluid, $g$ is gravitational acceleration, and $V_{\text{disp}}$ is the volume of fluid displaced.
This is equal to the weight of the displaced fluid.

What "displaced fluid" means

If you put an object into water, the object takes up space that water could have occupied. The volume of water that would have filled that space is the displaced volume.

If the object is completely submerged, the displaced volume is the full volume of the object.

If the object is only partly submerged, the displaced volume is only the submerged part of the object.

This is why a floating object displaces less fluid than a fully submerged object of the same size.

Why the force is upward

The origin of buoyancy comes from fluid pressure. In a fluid at rest, pressure increases with depth. So the bottom of an immersed object experiences a greater pressure than the top.

That pressure difference produces a net upward force.

For a simple block of cross-sectional area $A$, with top at depth $h_1$ and bottom at depth $h_2$, the forces from the fluid are

$$F_{\text{top}} = p_1 A,$$

$$F_{\text{bottom}} = p_2 A.$$

Since $p_2 > p_1$, the net upward force is

$$F_b = F_{\text{bottom}} - F_{\text{top}}.$$

Using hydrostatic pressure, this becomes equal to the weight of the displaced fluid.

Rendering scientific sketch…

Pressure difference produces buoyant force

Mathematical statement

Suppose an object is submerged in a fluid of density $\rho_f$. If the displaced volume is $V_{\text{disp}}$, then the mass of displaced fluid is

$$m_f = \rho_f V_{\text{disp}}.$$

The weight of that displaced fluid is

$$W_f = m_f g = \rho_f g V_{\text{disp}}.$$

Archimedes' principle says that this weight equals the buoyant force:

$$F_b = W_f = \rho_f g V_{\text{disp}}.$$

This formula is one of the most important results in fluid mechanics.

Archimedes' principle does not say the buoyant force depends directly on the mass of the object.
It depends on the fluid density and on the displaced volume:
$$F_b = \rho_f g V_{\text{disp}}.$$

Fully submerged objects

If an object is completely underwater, then

$$V_{\text{disp}} = V_{\text{obj}}.$$

So the buoyant force becomes

$$F_b = \rho_f g V_{\text{obj}}.$$

In this case, the buoyant force depends on the object's volume, not on how deep it is, as long as the fluid density stays constant.

An iron ball and a wooden ball of the same volume, fully submerged in the same fluid, experience the same buoyant force.

Floating objects

For a floating object, the upward buoyant force balances the object's weight.

So if the object is floating at rest,

$$F_b = W_{\text{obj}}.$$

Since $F_b = \rho_f g V_{\text{disp}}$, we get

$$\rho_f g V_{\text{disp}} = m_{\text{obj}} g.$$

Therefore,

$$\rho_f V_{\text{disp}} = m_{\text{obj}}.$$

This means a floating object displaces a weight of fluid exactly equal to its own weight.

That is why large ships made of steel can float. Even though steel is dense, the overall shape of the ship makes its average density low enough that it can displace enough water before becoming fully submerged.

Rendering scientific sketch…

Floating object displaces fluid

Apparent weight

When an object is submerged in a fluid, it may seem lighter. This happens because the fluid provides an upward buoyant force.

If the real weight of the object is $W = mg$, then the apparent weight is

$$W_{\text{app}} = W - F_b.$$

So,

$$W_{\text{app}} = mg - \rho_f g V_{\text{disp}}.$$

This idea is often used to measure density or volume experimentally.

For a submerged object,
$$W_{\text{apparent}} = mg - F_b$$
and
$$F_b = \rho_f g V_{\text{disp}}.$$

Density and Archimedes' principle

Archimedes' principle helps compare the density of an object with the density of the fluid.

For a fully submerged object of volume $V$ and density $\rho_{\text{obj}}$, its weight is

$$W = \rho_{\text{obj}} V g.$$

Its buoyant force is

$$F_b = \rho_f V g.$$

Comparing these gives three important cases.

Case	Comparison	Result
Object sinks	$\rho_{\text{obj}} > \rho_f$	Weight is greater than buoyant force
Object floats	$\rho_{\text{obj}} < \rho_f$	It can balance weight before full submersion
Neutral buoyancy	$\rho_{\text{obj}} = \rho_f$	It stays suspended when fully submerged

Neutral buoyancy means the object neither rises nor sinks.

Simple example

Consider a block fully submerged in water. Let its displaced volume be

$$V_{\text{disp}} = 2.0 \times 10^{-3}\ \text{m}^3.$$

Take the density of water as

$$\rho_f = 1000\ \text{kg/m}^3,$$

and $g = 9.8\ \text{m/s}^2$.

Then

$$F_b = \rho_f g V_{\text{disp}}$$

$$F_b = (1000)(9.8)(2.0 \times 10^{-3})$$

$$F_b = 19.6\ \text{N}.$$

So the fluid pushes upward on the block with a force of $19.6\ \text{N}$.

Common misunderstandings

A common mistake is to think that a heavier object always experiences a larger buoyant force. This is not true. Two objects with the same submerged volume in the same fluid experience the same buoyant force.

Another common mistake is to think that buoyant force depends only on depth. In a fluid of uniform density, the buoyant force on a fully submerged object does not change with depth. The pressure does change with depth, but the pressure difference between top and bottom stays the same if the object and fluid remain unchanged.

A third mistake is to confuse floating with low mass. Floating depends on average density, not just mass.

Key idea:
Buoyant force depends on displaced fluid, not directly on the object's material.
$$F_b = \rho_f g V_{\text{disp}}$$

Physical meaning

Archimedes' principle connects motion in fluids with a very simple idea, fluids push upward by an amount equal to the weight of the fluid pushed aside. This principle explains why boats float, why helium balloons rise in air, and why objects seem lighter underwater.

It is one of the clearest examples of how pressure in a fluid leads to a measurable force on an object.

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