Buoyancy: Crates That Float

Not every drop lands on a deck. When a crate misses and splashes into the harbour, players expect it to bob back up, settle at the waterline and wait to be fished out. Getting that right uses nearly everything in this course so far: the water plane, forces, mass, integration and even torque.

The rule

Water pushes up with the weight of the water you shove aside.

This is Archimedes' principle. The upward buoyant force is

$F_b = \rho\, V_{\text{sub}}\, g$

where $\rho$ (rho) is the water's density, about 1000 kg/m³ for fresh water (sea water is a little denser); $V_{\text{sub}}$ is the volume of the body below the surface; and $g = 9.8$ m/s².

Weigh that against the body's weight, $mg$:

• A body floats when its average density is less than the water's. It settles at the depth where the submerged part lifts exactly its weight, and there the fraction under water equals the ratio of the two densities.
• A body denser than water sinks, because even fully under it can't displace enough water to hold itself up.

The crate in our example is a 0.5 m cube (0.125 m³) with a mass of 50 kg, so its density is 400 kg/m³, and it floats with 40 % of its height under water.

In a game, you measure the submerged part every step. For a box, that is how deep its bottom face sits below the water plane (the negative of its signed distance, from the geometry lesson), clamped between 0 and the box's height, times its base area. Two extras make it believable:

• Water drag, a force opposite to the velocity, so the crate settles instead of bobbing forever.
• Applying the lift at the center of the submerged part, not the crate's center of mass. That creates torque, which rights a tipped crate or rolls it over.

The shortcut

Densities first. The crate's density against the water's tells you "float or sink" instantly, and their ratio tells you how much sits under. Then confirm with forces.

The trap is using the crate's whole volume even while half of it is in the air. That fires crates out of the water like corks. Only the submerged part pushes.

Drills

Answer fast — trust the rule.

Remember

• Buoyant force = water density × submerged volume × g, pushing up.
• Densities first: lighter than water floats, and the density ratio is the fraction under.
• Use only the submerged part, apply it at that part's center, and add drag so crates settle.