Archimedes' Principle

Picture an object sitting inside a fluid, fully or partly submerged. It takes up space that the fluid used to occupy, so it displaces some volume of fluid. Archimedes' principle says the fluid pushes back with an upward force, called the buoyant force, equal in size to the weight of that displaced fluid: F_b = rho_fluid x V_displaced x g. You can prove this by imagining swapping the object for an equal-sized blob of fluid. That blob just sits there, neither rising nor sinking, so the surrounding fluid must be pushing up on it with a force exactly equal to its weight. Swap the real object back in, and the fluid around it has not changed, so it still pushes up with that same force.

For a floating object like a boat, the object sinks into the water only until the weight of water it displaces equals its own weight. A 4500 kg boat does not need to displace some fixed volume tied to depth; it settles until the buoyant force balances gravity, so the displaced water weighs 4500 kg worth, regardless of the lake's total depth. For a fully submerged object like a submarine, the buoyant force depends only on its volume and the fluid density, both of which barely change as it dives, so diving deeper does not increase buoyant force even though pressure around it rises sharply.

This connects directly to pressure with depth, since the pressure difference between the top and bottom of a submerged object is exactly what generates the buoyant force in the first place. It also sets up Newton's second law problems where buoyant force, gravity, and sometimes an applied force must balance or produce acceleration for floating, sinking, or rising objects.