Buoyancy Equals the Weight of the Displaced Fluid
Gently push an object down into water. As it goes in, water is shoved aside and the surface rises, and at the same time you feel a force pushing up on your hand. The deeper you push, the harder it pushes back. This upward push is buoyancy. The remarkable thing is that its size is exactly equal to the weight of the water the object pushed aside. This is the very fact Archimedes realized in his bath 2200 years ago. What sets the buoyancy is not how heavy the object is, but how much water it displaces. That is why a lump of iron sinks, yet a ship made of the same iron floats.
First, drag how far the object is submerged. Whatever volume of the object sits underwater pushes aside the water that used to be there: this is the displaced volume. The more of it is submerged, the more water is shoved aside, and the larger the upward buoyancy. Submerge half and the buoyancy is half; submerge it fully and it is at its maximum. What creates buoyancy is not the object itself, but the water it elbowed out of the way. You can think of it as the water trying to reclaim the space the object took.
Why up, and not some other direction? Recall the hydrostatic pressure from the last lesson: pressure grows with depth. So the bottom face of a submerged object is deeper than its top face, and gets pushed up by a stronger pressure than the one pressing down from above. Because the upward push on the bottom beats the downward push on top, the difference leaves a net force pointing straight up. Drag the depth to lower the object further. Both the top and bottom pressures grow, but their difference is fixed by the object's height, so it stays the same. That is why buoyancy does not depend on depth, and why its size is exactly the weight of the displaced water.
Now drag the object's density. An object lighter than water rises until it settles at the depth where it displaces exactly its own weight in water. The fraction that sits underwater is just the object's density divided by the water's. At 0.6 times the density of water, it floats 60% submerged. This is why an iceberg sits mostly underwater: ice is about 0.9 times as dense as water, so 90% of it is below the surface. An object denser than water, on the other hand, cannot generate enough buoyancy even when fully submerged, so it sinks. Float or sink is decided not by weight, but by how the density compares with the fluid's.
Tie it all into one formula: buoyancy F = ρ g V. Here ρ is the density not of the object but of the surrounding fluid, g is gravity, and V is the submerged volume, that is, the displaced volume. Multiply the three and you get exactly the weight of the displaced fluid. Drag the fluid density ρ and, for the same volume, a heavier fluid gives more buoyancy. The key point is that ρ is the fluid's density. Seawater gives a little more buoyancy than fresh water, and in mercury even iron bobs at the top. What the object is made of never appears in the buoyancy formula: all you need is the fluid and the displaced volume.
Finally, look at a ship made of steel. Use the buttons to switch the same amount of steel between a solid lump and a hollow hull shape. As a lump it displaces little water, so buoyancy cannot beat its weight and it sinks. But spread out and hollowed into a hull, the same weight displaces far more water. The moment the weight of displaced water equals the ship's total weight, the ship floats. The key is average density: hollowing it out brings the average density, counting the space inside, below that of water. The secret to a giant cargo ship floating is exactly this empty space, and the enormous amount of water it displaces.