Archimedes' Principle

What the principle says. Any body immersed in a liquid or gas is acted on by an upward buoyant force — the Archimedes force. It equals the weight of the liquid displaced by the body: F_A = ρ_l·g·V_sub. The formula contains the density of the liquid, not of the body, and only the volume of the submerged part. What the body is made of and how much it weighs do not matter: aluminium and lead balls of the same volume are pushed up equally in water.

Where the buoyant force comes from. Pressure in a liquid grows with depth: p = ρ_l·g·h. The bottom face of a body is deeper than the top face, so the liquid pushes harder from below than from above. The pressure forces on the side faces are equal and cancel out. What remains is the difference: F_A = (p₂ − p₁)·S = ρ_l·g·(h₂ − h₁)·S = ρ_l·g·V. Example: a cube with a 10 cm edge is fully under water, its top face 5 cm deep. The pressure on top is 490 Pa, on the bottom (15 cm deep) 1470 Pa. The difference (1470 − 490)·0.01 m² = 9.8 N is exactly ρ_l·g·V = 1000·9.8·0.001. Turn on the pressure arrows in the simulator to see it for yourself.

The spring-scale experiment. In air the spring scale shows the body's weight P. When the body is lowered into a liquid the reading drops to P − F_A: the body seems to “lose” weight. The displaced liquid overflows through the spout into the cup, and the balance under the cup shows its weight — exactly the same F_A. For example, a 1 L aluminium cube weighs 26.5 N, and in water the scale shows 16.7 N. The 9.8 N difference is the weight of 1 L of water (1 kg) that has flowed into the cup.

Depth does not matter. While the body is entering the liquid, V_sub grows, so F_A grows and the reading falls. Once the body is fully immersed, F_A no longer changes however deep you lower it: the pressure grows equally on top and bottom, and their difference stays the same. You can see this on the left graph under the scene: after the “fully in” mark the lines run horizontally.

Sink, float or hover. Compare the weight P = ρ_b·g·V with the largest buoyant force ρ_l·g·V (body fully immersed). The volume and g are the same, so everything depends on the ratio of densities. If ρ_b > ρ_l, the body sinks and rests on the bottom, which pushes on it with N = P − F_A. If ρ_b = ρ_l, the body hovers inside the liquid at any depth. If ρ_b < ρ_l, the body rises and sits just deep enough for F_A = P: V_sub/V = ρ_b/ρ_l. Ice (900 kg/m³) in water is 90 % submerged, pine (500 kg/m³) 50 %. A steel ball (7800 kg/m³) floats in mercury (13,600 kg/m³) with 57 % under the surface, and cork in mercury with only 1.8 %.

Where you meet it. A steel ship floats because its hull is hollow: it displaces so much water that the average density of the ship, air inside included, is less than that of water. A submarine dives and surfaces by taking water into its ballast tanks or blowing it out with compressed air. A fish changes the volume of its swim bladder. A hydrometer floats deeper in a lighter liquid and is used to measure the density of milk, battery acid or antifreeze. The principle works in gases too: a hot-air or helium balloon rises because it is lighter than the air it displaces.

Common mistakes in problems. Using the density of the body instead of the liquid in F_A. Taking the full volume V of a floating body instead of the submerged part. Thinking the buoyant force is larger at greater depth. Confusing the weight of a body in liquid (the spring-scale reading P − F_A) with the buoyant force. Check yourself in “Test yourself” mode: after each answer the simulator runs the experiment and shows what really happens.

How the simulator works. The vessel is filled up to the spout, so everything the body displaces flows into the measuring cup. The liquid level is computed exactly from conservation of volume. The motion of the body accounts for weight, buoyancy, thread tension, liquid drag and impact with the bottom. The numbers on the readout and in the Measurements panel match the textbook formulas, with g = 9.8 m/s².