Pressure & Buoyancy · FloatLab

Liquid pressure and depth

A deep-sea submersible dives from the surface all the way to the bottom of the Mariana Trench. Press the depth buttons below to make it dive or surface.

The weight of the seawater presses on the water beneath it, so the deeper you go, the greater the pressure: p = ρgh. It depends only on the liquid's density ρ and the depth h — not on how wide the water is or what shape the container has. Every 10 m of seawater adds about 1 atmosphere of pressure.

A foam cup hangs outside the submersible; watch it get squeezed smaller on the way down. Watch the colour of the water: red light is absorbed first, a few tens of metres down only blue-green is left, and a few hundred metres down it is pitch black — only the submersible's own lights remain.

Communicating vessels: the ship lock

The harbour (sea) is 3 m lower than the river, so ships need a ship lock to get into the river. Culverts with valves connect the lock chamber to the water downstream and upstream.

Containers that are open at the top and connected at the bottom are communicating vessels. Filled with one liquid at rest, the liquid surfaces in all parts are always level. Open the downstream valve and the chamber and the harbour become communicating vessels: water flows out of the chamber until both surfaces are level, and only then can the lower gate open to let the ship in. Close the gate and the valve, then open the upstream valve: the chamber connects with the river, its water rises to the river level, and the ship is “lifted” up.

Try the steps in order. While the water levels on the two sides differ, a gate will not open: the water forces on its two faces are unbalanced by hundreds of tonnes.

Why things float

A cubic concrete caisson is under seawater, and the water pushes on it from every direction. The left/right and front/back faces are at the same depths, so their forces are equal and opposite and cancel out.

The top face is pushed down; the bottom face is deeper and is pushed up harder. The difference is the buoyant force:

F_b = F_bottom − F_top = ρg(h + a)a² − ρgha² = ρgV

Drag the caisson deeper: both forces grow, but their difference stays the same — once fully submerged, buoyancy does not depend on depth. Now let the bottom “sink into the seabed mud”: with no water underneath to push up, only the downward force is left and the caisson is pinned to the seabed (bridge piers feel no buoyancy either).

Measuring buoyancy by weighing

The crane on the quay carries an electronic crane scale with a concrete block hanging below it. In the air the scale reads the block's weight G. Drag the scale down to lower the block into the sea: the reading F drops — the water is pushing it up, and that push is the buoyant force:

F_b = G − F

The chart on the right plots buoyancy against how deep the block is in the water: the deeper it goes, the more water it pushes aside and the greater the buoyancy; once fully submerged, lowering it further changes nothing.

Switch to a steel block of the same size: it is much heavier, yet the buoyancy equals the concrete block's — buoyancy depends only on the volume of liquid displaced, not on the object's weight or material. Switch to the 2 m³ concrete block and the buoyancy doubles.

Archimedes' principle

On the quay stands a test tank filled with fresh water exactly up to its overflow spout. The crane lowers the block into the tank; the water it displaces runs out of the spout into a collecting tank standing on a platform scale.

Compare two numbers: how much the crane scale reading dropped (buoyancy F_b) and how heavy the water that ran out is (G_displaced). Lower the block to different depths and record a few rows — the two are always equal:

F_b = G_displaced = ρ_liquid g V_displaced

This is Archimedes' principle: an object in a liquid feels an upward buoyant force equal to the weight of the liquid it displaces. Legend says Archimedes saw the water overflow his bath and realised he could measure the volume of the king's crown by displacement.

Sink or float: the submarine

A submarine's shape and volume do not change, so its buoyancy is almost constant. It floats and dives by letting water into or out of its ballast tanks, changing its own weight:

G < F_b: it rises and ends up floating at the surface (partly out of the water, F_b = G); G = F_b: it hovers at any depth; G > F_b: it sinks and ends up on the seabed.

You can also compare densities: if the submarine's average density is less than seawater it rises, equal — it hovers, greater — it sinks.

Turn on the “seawater density layer”: below 60 m the water suddenly becomes less dense, buoyancy drops, and a hovering submarine plunges — a “sea cliff”. China's submarine 372 met one in 2014; the crew blew air into the tanks and recovered within three minutes. When it starts to plunge, try “Emergency blow”.

Ships and the load line

Steel is 8 times as dense as water, yet a ship of steel plate floats: made hollow, it displaces a lot of water and gets a large buoyant force. A floating ship has F_b = G_ship, so the mass of water it displaces equals the total mass of ship and cargo — its displacement.

The quay crane lifts containers onto the ship; the heavier it gets, the deeper it sits — watch the draft marks at bow and stern and the load line mark amidships. Load lines show how deep a ship may be loaded in different waters and seasons: S summer seawater, W winter, T tropical, F summer fresh water, TF tropical fresh water. Past the load line the ship is overloaded and may not sail.

Take the same ship from the sea into a river (fresh water): the density is lower, so it must displace more water to hold up the same weight and sinks a dozen centimetres or so — that is why the F line is above the S line. Load the cargo unevenly and the ship heels to one side.

Hydrometer and draft survey

A hydrometer is a glass tube weighted with lead shot at the bottom. In any liquid it floats, so the buoyant force always equals its weight. The denser the liquid, the less volume it needs to displace and the higher it floats. So its scale reads small at the top and large at the bottom, and the marks are uneven — wider apart at the top. Harbour surveyors use one to measure the density of the dock water.

Draft survey: bulk and container cargo is not put on a scale. Instead you read six drafts — bow, midships and stern on both sides — work out the mean draft, look up the ship's displacement table, multiply by the seawater density, then subtract the empty ship, fuel and stores: what remains is the cargo.

It is the same idea as Cao Chong weighing the elephant: put the elephant on a boat and mark the waterline on the hull; then load stones until the boat sinks to the same mark. The boat displaces the same water, so the stones together weigh as much as the elephant.

In-class testing