Archimedes' principle
/ ar-kih-MEE-deez /
Archimedes' principle is the rule that tells you exactly how strong the buoyant force is: it equals the weight of the fluid an object pushes out of the way. Lower your body into a full bathtub and the water that spills over weighs exactly as much as the upward push you feel. Legend says the ancient Greek thinker Archimedes discovered it in his bath and leapt out shouting Eureka. It answers the question: how much does a fluid lift an object, in numbers?
Precisely, the upward buoyant force on a body in a fluid equals the weight of the volume of fluid it displaces: F_b = rho_fluid × V_displaced × g, where rho_fluid is the fluid's density, V_displaced is the volume of fluid pushed aside (which equals the submerged volume of the object), and g is the acceleration due to gravity. From this follows the law of floating: an object floats when it can displace a weight of fluid equal to its own weight before it is fully submerged, which happens exactly when the object's average density is less than the fluid's.
This one idea explains ships, submarines, hot-air balloons, and hydrometers, and it lets you measure an object's volume just by the fluid it displaces. One honest clarification: the buoyant force depends on the weight of the displaced fluid, not on the weight of the object. A floating object displaces its own weight of fluid; a fully submerged object displaces its own volume of fluid, and whether that is enough to float it depends on how the two densities compare.
A wooden block floats with 60 percent of its volume under water. It displaces that submerged volume of water, whose weight equals the block's full weight, so the block's density must be about 0.6 times water's, roughly 600 kg/m^3.
A floating object sinks just far enough to displace its own weight in fluid.
The buoyant force equals the weight of the displaced fluid, not the weight of the object. A body floats only if its average density is less than the fluid's.