Whenever an object (partially or fully submerged) displaces a fluid, the fluid surrounding the object exerts a vertically upward buoyancy force aka upthrust U on the object.
Archimedes’ Principle states that the magnitude of upthrust U is exactly equal to the weight of the displaced fluid. In terms of the density of the fluid ρf and the displaced volume Vf, we have the formula
For clarification, the “displaced fluid” refers to the fluid that has been pushed out of the way by the object. The “displaced volume” Vf thus refers to the space that is now occupied by the object instead of the fluid.
Origin of Upthrust
Consider a fully submerged sphere as illustrated below. The fluid, being pressurized, will exert pressure forces on the sphere. To be more exact, at every point on the spherical surface where the fluid is in contact with the sphere, the fluid will be exerting a pressure force that is directed normally (i.e. perpendicularly) into the sphere. The resultant of all these pressure forces is the force of upthrust.
Because hydrostatic pressure increases with depth, the pressure forces pushing the bottom surface upward is always stronger than the pressure forces pushing the top surface downward. This explains why these pressure forces do not cancel out, but instead sums up to be the vertically upward force of upthrust.
Proof of Archimedes’ Principle
First, we must realize that upthrust is the resultant of all the pressure forces acting on each point on the object in contact with the fluid. Next, we must realize that the magnitude and direction of these pressure forces depend only on the hydrostatic pressure and the orientation of the object’s surface at each point. Crucially, they do not depend on whether the displacement is caused by an iron sphere or a Styrofoam sphere. Any sphere causing the same displacement results in the same amount of upthrust!
Here comes the genius bit! Why not we consider the scenario when the object is also made of the same fluid that it is displacing. For example, let’s imagine a water ball submerged in water. Since a water ball should be hovering in equilibrium in water, the water ball’s weight must be supported by the upthrust exerted by the surrounding water! Get it?
When a fluid is displaced by an object, the surrounding fluid will continue to exert the same upthrust as before. The upthrust that was exerted on what used to be the displaced fluid is now exerted on the object instead. We know that the displaced fluid’s weight used to be balanced by the upthrust it received from the surrounding fluid. Hence, we can deduce that upthrust is equal (in magnitude) to the weight of the displaced fluid.
Archimedes’ Principle by Reasoning
Derivation of Upthrust Formula for a Cuboid