Foundations — Buoyancy — Archimedes' principle, derivation from pressure difference

Before we can derive Archimedes' principle in the parent note Buoyancy, we must own every letter it throws at us. This page builds each one from nothing — plain words, then a picture, then the reason the topic can't live without it.

0. What even is a fluid?

Look at the figure below. The little arrows are the fluid pushing outward on the walls of a tank and on a submerged object. Notice the arrows are always at right angles to whatever surface they hit — that is what "fluid pressure" looks like.

Why we need this: the whole topic is a fight between pushes from above and below. So we must first agree that a fluid does push, and always perpendicular to the surface.

1. Depth $h$ — how far down you are

In the picture, the top face of the block sits at depth $h_1$ and the bottom face sits lower, at depth $h_2$. The arrow labelled $h$ always points straight down, never sideways.

Why the topic needs it: pressure depends on depth. Two faces at different depths feel different pushes — that difference is the seed of buoyancy.

2. Density $\rho$ — how tightly packed the stuff is

Picture two identical boxes: one packed with many tiny balls (high $\rho$), one with a few (low $\rho$). Same box, different mass — that's density.

We meet two densities in this topic, and mixing them up is the classic trap:

• ${\rho }_f$ — the fluid's density (the water/air doing the pushing).
• ${\rho }_o$ — the object's density (the thing being pushed).

Why the topic needs it: pressure grows with the fluid's density, and floating/sinking is decided by comparing ${\rho }_o$ with ${\rho }_f$.

3. Gravity $g$ — the pull that stacks the fluid

Gravity is why deep fluid is squeezed: every layer of fluid has weight, and the layers pile onto each other. More fluid stacked above ⇒ more squeeze below.

Why the topic needs it: without gravity there's no weight, no stacking, no pressure-with-depth, and therefore no buoyancy. Every formula here carries a $g$.

4. Pressure $p$ — push spread over area

Think of pressing a drawing pin. Same push, but concentrated on a tiny point (small $A$) means huge pressure. Spread over your whole palm (big $A$), barely felt. Pressure is how concentrated the push is.

The figure below shows pressure acting over the flat bottom face of a block: uniform little arrows over area $A$ add up to one big force $F=pA$.

Why the topic needs it: the fluid gives us pressures on faces, but Newton's laws need forces. This equation is the bridge, used in Step 2 of the parent derivation.

5. Hydrostatic pressure — the depth formula

Now combine depth (§1), density (§2), gravity (§3) into the single most important formula behind buoyancy.

The figure shows two faces of a block at depths $h_1$ (top) and $h_2$ (bottom): the deeper face sits under a taller column of fluid, so it feels a larger pressure. That is the entire origin of buoyancy in one glance.

For the full story see Pressure in fluids — hydrostatic pressure.

Why the topic needs it: this is the engine. $p_{\text{bottom}}-p_{\text{top}}={\rho }_{f}g(h_2-h_1)$ is literally the buoyant force per unit area.

6. Volume $V$ and displaced volume $V_{\text{disp}}$

Key picture: a fully submerged object displaces its whole volume. A partly floating object displaces only the underwater part.

Why the topic needs it: Archimedes' result $F_B={\rho }_f{V}_{\text{disp}}g$ is just "weight of the fluid that used to fill that hole." Volume is what turns a density into a mass, and mass into a weight.

7. Net force & equilibrium — who wins the push-fight

For buoyancy we line up three vertical forces on a floating object: weight $W$ pulling down, buoyancy $F_B$ pushing up, and (if hung from a scale) tension. Balance means $F_B=W$. This is pure Newton's laws — equilibrium of forces.

Why the topic needs it: "floats vs sinks vs floats-partly" is decided entirely by comparing the up-push $F_B$ with the down-pull $W$. And Apparent weight and weighing methods uses exactly this: a scale reads $W-F_B$.

Prerequisite map

Read it top to bottom: gravity + density + depth build hydrostatic pressure; pressure over the deeper face beats the top face; turning that into force and adding up gives net upthrust; bring in volume and you have Archimedes' principle, which then decides floating.

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Pressure in fluids — hydrostatic pressure — where $p=p_0+\rho gh$ is built
• Density and relative density — the meaning of $\rho$
• Newton's laws — equilibrium of forces — balancing $F_B$ and $W$
• Apparent weight and weighing methods — a scale reads $W-F_B$
• Floating bodies and stability — metacentre — where buoyancy acts
• Pascal's principle — pressure in enclosed fluids