Archimedes principle

Why does a small steel nail sink while a massive steel ship floats? Both are the same metal, so mass alone cannot be the answer — the shape and the fluid matter too. The clue is that any fluid, sitting still, pushes back on whatever it touches, and that push grows the deeper you go. Slide an object into water and the water below it pushes up harder than the water above pushes down, so there is a net upward shove. Archimedes worked out exactly how big that shove is, over two thousand years ago, reportedly while stepping into an overflowing bath. His rule — the buoyant force equals the weight of fluid displaced — is the single idea behind floating ships, sinking stones, submarines, and the hydrometer that checks your car battery's acid. This article builds that rule from pressure and depth, then uses it to decide, for any object, whether it sinks, floats, or hangs suspended.

The core insight

A fluid pushes up on an immersed body with a buoyant force exactly equal to the weight of the fluid the body displaces, because fluid pressure increases with depth.

The core idea

Thrust and pressure. Thrust is a force acting perpendicular to a surface; pressure is thrust spread over area:

P = \dfrac{F}{A}

where: $P$ is pressure in pascals ( $\text{Pa} = \text{N/m}^2$ ), $F$ is thrust in newtons, and $A$ is the area in square metres. A fluid at rest presses on every surface it touches, in every direction, and this pressure rises with depth — the reason a dam is built thicker near its base than at the top.

Why the push is upward. Imagine a submerged block. Fluid presses down on its top face and up on its bottom face. Because the bottom face sits deeper, the fluid there is under greater pressure, so the upward push on the bottom exceeds the downward push on the top. The two forces do not cancel; a net upward force survives. This net force is the buoyant force, also called upthrust.

Archimedes' principle. When a body is partly or fully immersed in a fluid, the buoyant force on it equals the weight of the fluid it displaces:

F_{\text{buoyant}} = \text{weight of fluid displaced}

Notice what the rule does not say — it never mentions the material the body is made of. Only the volume actually pushed into the fluid, and the fluid's own density, decide the upthrust.

Figure

A submerged cuboid showing smaller downward pressure arrows on the top face and larger upward pressure arrows on the deeper bottom face, with a net upward buoyant force arrow

Sink, float, or hang suspended? Compare the body's own weight, $W = mg$ , with the maximum buoyant force possible — the weight of fluid displaced if the body were fully submerged.

Body's density greater than the fluid's: weight beats the maximum upthrust, so it sinks.
Body's density less than the fluid's: upthrust balances weight before full submersion, so it floats, part above the surface.
Densities equal: the body stays suspended anywhere inside the fluid, neither rising nor sinking.

A solid iron block sinks, but the same iron shaped into a hollow ship's hull floats: the hull pushes aside a far greater weight of water than the solid block ever could, so it needs to submerge only a little to generate enough upthrust. A hydrometer — weighted at the bottom, marked along its stem — floats upright, sinking until the weight of liquid it displaces matches its own weight; in a denser liquid it floats higher, so the stem reading gives the liquid's density directly.

Derivation ABuoyant force equals the weight of fluid displaced

The goal is to show, starting only from the fact that fluid pressure increases with depth and the definition thrust equals pressure times area, that the net upward force on a submerged block must equal exactly the weight of the fluid the block has pushed out of the way — not some other multiple of it, and not a force that depends on what the block itself is made of.

Step 1. Picture a rectangular block of horizontal face area $A$ , fully submerged with its top face at depth $h_1$ and its bottom face at depth $h_2$ , so $h_2 > h_1$ . The fluid pressure at each face pushes perpendicular to that face: downward on the top, upward on the bottom. Using thrust equals pressure times area, the downward force on top is $F_1 = P_1 A$ and the upward force on the bottom is $F_2 = P_2 A$ .

Step 2. Because pressure increases with depth and $h_2 > h_1$ , the bottom pressure exceeds the top pressure, $P_2 > P_1$ , so $F_2 > F_1$ . The sideways forces on the four vertical faces are equal and opposite in pairs, by symmetry, and cancel completely — only the top-bottom mismatch survives as a net force.

Step 3. The net force is upward and equal to the difference, $F_{\text{net}} = F_2 - F_1 = (P_2 - P_1)A$ . This quantity $(P_2 - P_1)A$ can be shown, using the standard relation between fluid pressure and the weight of the fluid column above a depth, to equal precisely the weight of fluid occupying the block's own volume — that is, the weight of the fluid the block displaced.

Step 4. So the net upward force, the buoyant force, is $F_{\text{buoyant}} = \text{weight of fluid displaced}$ , exactly Archimedes' rule. Nothing about the block's own material entered the argument — only its submerged volume and the surrounding fluid's pressure. This is why the same rule applies whether the immersed object is wood, iron, or a hollow steel hull.

Worked example

A block of wood of mass 2 kg is lowered gently into a tub of water and comes to rest floating, partly above the surface. Find the buoyant force on the block. (Take $g = 9.8\ \text{m/s}^2$ .)

Step 1. Find the block's weight: $W = mg = 2 \times 9.8 = 19.6\ \text{N}$ .

Step 2. Apply the floating equilibrium condition. The block floats without accelerating, so the net force on it is zero. Weight (down) and buoyant force (up) are the only two forces present, so they must be equal in size:

F_{\text{buoyant}} = W = 19.6\ \text{N}

Step 3. Interpret using Archimedes' principle. Since the buoyant force equals the weight of water displaced, the water pushed aside by the submerged part of the block also weighs 19.6 N — just enough to hold the block up, with the rest of the block staying above the surface.

Answer: the buoyant force is 19.6 N, equal to the weight of water displaced.

Common traps

Calling a floating object "weightless." It still has weight; it floats only because the buoyant force exactly balances that weight.
Assuming heavy objects always sink. Density decides it, not mass alone — a massive steel ship floats because its hollow shape displaces a large weight of water.
Forgetting partial immersion. A floating body is only part-submerged; use only the volume actually under the surface when reasoning about displaced fluid.
Mixing up thrust and pressure. Thrust is a plain force, in newtons; pressure is force per unit area, in pascals — doubling the contact area halves the pressure, not the thrust.

Ask the tutor: curious why the same iron sheet sinks as a block but floats as a boat hull — ask the AI tutor to walk through the displaced-water comparison with real numbers.

Why does the buoyant force not depend on what the object is made of?

Because the derivation only ever uses the submerged volume and the surrounding fluid's pressure at each depth — the pressure difference between the top and bottom faces comes entirely from the fluid, not from the object. Two objects of identical shape and volume, one lead and one wood, displace the same volume of fluid when fully submerged and so feel the same buoyant force. What differs is their own weight, which decides whether that same buoyant force is enough to keep them afloat.

If a submarine can control whether it floats or sinks, how does Archimedes' principle still apply?

A submarine has ballast tanks that it fills with water or empties using compressed air, changing its own weight without changing its submerged volume much. Filling the tanks makes its weight exceed the buoyant force, so it sinks; emptying them makes weight fall below the buoyant force, so it rises. At any instant, Archimedes' principle still governs the buoyant force exactly as usual — the submarine is simply adjusting the other side of the comparison, its own weight, to tip the balance either way.

Does Archimedes' principle work in air as well as in water?

Yes — air is a fluid too, and any object in air experiences a small buoyant force equal to the weight of air it displaces. A helium balloon rises because helium is far less dense than air, so the weight of air displaced exceeds the balloon's own weight plus the helium inside it. For everyday solid objects the effect is usually too small to notice, since air is roughly a thousand times less dense than water, but the same rule applies without any change.

Archimedes principle

The core idea

Worked example

Common traps

Before this

Where this leads