Physics · Mechanics · Grade 10

Why doesn't a ship sink?

A steel ship floats, but a steel nail sinks — why?

The situation

A ship made of a tonne of steel floats easily on water, while a small steel nail sinks at once. A piece of ice floats, but most of it hides under the water. Lower various bodies into a vessel and measure the force with which the liquid pushes them up — then fill a steel barge with load and find out when it sinks.

Topic

Archimedes' principle and conditions for floating. Cargo ship, a piece of ice and a steel nail

Key ideas

Archimedes' principle
Density
Conditions for floating

Formulas

F_{A} = ρ_{s} g V_{b}

Archimedes' force: density of the liquid (or gas) × $g$ × volume of the submerged part of the body.

m g = ρ_{j} g V, ρ_{j} = ρ_{s} \frac{P_{1}}{P_{1} - P_{2}}

The gravitational force on the body. Hydrostatic weighing: $P_{1}$ is the newton meter reading in air, $P_{2}$ the reading when fully submerged in the liquid.

\frac{V_{b}}{V} = \frac{ρ_{j}}{ρ_{s}}

Submerged fraction of a floating body (in equilibrium $F_{A} = m g$ ).

Δ h = \frac{V_{b}}{S_{v e s s e l}}

The liquid level in the vessel rises because of the displaced volume.

d = \frac{m_{h u l l} + m_{l o a d}}{ρ_{l} L B}

Draught of the barge. If $d$ exceeds the height of the side, the ship sinks.

Theory

Where does the force come from?

Pressure in a liquid increases with depth: $p = ρ g h$ . A greater pressure acts on the lower surface of the body than on the upper — the pressure difference pushes the body up. This is the Archimedes force, and it depends only on the submerged volume, not on what the body is made of.

Condition for floating

If the density of the body is less than that of the liquid ( $ρ_{b} < ρ_{l}$ ), it sinks in just far enough that the weight of the displaced liquid equals the weight of the body. If it is greater, even the whole volume under the surface is not enough and the body sinks; on the bottom the support force $N = m g - F_{A}$ acts on it.

Why does a steel ship float?

A ship's hull is hollow: its average density (steel + load + the air inside) is less than that of water. The more load is added, the deeper the ship sits. If water comes in over the side, it takes the place of the air inside, the average density approaches that of steel — and the ship sinks.

River and sea

Sea water is salty and denser (≈ 1025 kg/m³). That is why, with the same load, a ship floats slightly higher in the sea. On the load line mark on a ship's side, the fresh-water (F) line is drawn higher than the sea (S) line.

Archimedes' force in air too

Archimedes' law holds for gases too: in air every body is acted on by a buoyant force $F_{A} = ρ_{a i r} g V$ ( $ρ_{a i r} \approx 1.2$ kg/m³). Usually it is very small — about 0.04 N for a 15 cm cube (the “air buoyancy” in the HUD). But a body whose average density is less than that of air rises: a 1 m³ balloon filled with helium can lift about 1 kg ( $1.2 - 0.17$ kg/m³), and a hot-air balloon flies because the air inside is heated and becomes less dense. Water and air transport rely on the condition for floating: ships, submarines (which change their average density by taking water into the ballast tanks or emptying them with air), aerostats and airships.

Hydrostatic weighing (lab)

We hang the body from a newton meter, first in air ( $P_{1} \approx m g$ ), then fully immersed in the liquid ( $P_{2} = m g - ρ_{l} g V$ ). The difference is the Archimedes force: $P_{1} - P_{2} = ρ_{l} g V$ ; from it we get the volume of the body, and then its density: $ρ_{b} = ρ_{l} P_{1} / (P_{1} - P_{2})$ . The method is accurate even for irregular shapes and for small bodies — no measuring cylinder is needed. In “Hydrostatic weighing” mode, find the density of the “Mystery alloy” cube by this method. The body must not touch the bottom or the wall of the vessel; air buoyancy is ≈ 0.01–0.05 % of the weight for metals — it can be neglected.

Did you know?

An iceberg floats in sea water: $917 / 1025 \approx 0.9$ — about 90 % of it is under water.
The load-line mark on ships is named after the English member of parliament Samuel Plimsoll: the British Merchant Shipping Act of 1876 made this mark compulsory.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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