The Archimedes Principle: Once upon a time, a man named Archimedes was tasked with figuring out how to tell if a crown was made of gold or if it was a fake - without ruining it. Because of the crown's odd shape, he didn't know its volume to tell how dense it was. One day, Archimedes took a bath and noticed that the bathwater rose according to how much of his body was in the water. The volume of the submerged part of his body was the same volume as the water that rose out of the way - or, in other words, was displaced. He realized that he could determine the crown's volume this way and compare its weight to the same volume of pure gold to see if they weighed the same. This idea struck him so much that he ran naked through the town shouting, "Eureka!" This lightbulb moment led to Archimedes' Principle.

In simple form, the Archimedes law states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. Mathematically written as:

Fb = ρ × V × g

... where Fb is the buoyant force, ρ is the density of the fluid, V is the volume of the object submerged, and g is the acceleration due to gravity

Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces.Archimedes' principle is a law of physics fundamental to fluid mechanics.

Archimedes' screw solved one of the greatest practical problems of antiquity, which was finding an easy way to lift liquids. Archimede created a machine that allowed this operation to be carried out with relative simplicity: the Archimedean screw. The machine is made up of a large screw and placed inside a tube, not necessarily welded watertight. The lower part of the tube is immersed in a liquid and, by rotating the screw, each step collects a certain quantity of substance which is raised along the spiral until it exits from the upper part, to be discharged into a storage basin.— with Tasty Besty Food 1M.
The energy for rotation can be provided by a handle, by animals, by windmill propellers or by agricultural tractors. The Archimedean screw is attributed to Archimedes on the basis of the testimonies of Diodorus Siculus and Athenaeus. Recent studies, however, indicate that it may have already been invented before Archimedes, as it is thought to have been used to irrigate the hanging gardens of Babylon. Archimedes may have studied the screw during his stay in Alexandria in Egypt and may have imported an instrument into Italy that was therefore already known in the Middle Eastern country.
Archimedes' studies have a notable influence on the history of science both in antiquity, when the rigor of his demonstrations is taken as a model, and in the Renaissance when his works, published in versions or in the original text, are the subject of great interest for those who founded modern experimental science. Galileo Galilei takes up Archimedes' screw in his work Le Meccaniche: in the passage "On Archimedes' screw for removing water", he demonstrates how it works. “It does not seem to me that in this place Archimedes' invention of raising water with the screw should be passed over in silence: which is not only marvelous, but miraculous; for we will find that the water ascends in the vine, continually descending.”
Even today, Archimedes' screw is used in various contexts to lift substances in the solid, liquid and gaseous states. Furthermore, the hydraulic auger can be applied to uneven water levels, as it exploits the potential energy in a stationary position. At the highest point, the potential energy of the water is maximum and as a result of the consequent fall towards the lowest point, it is conveyed into a rotor connected to a generator which transforms the kinetic energy given by the movement of the screw into electrical energy . The fluid enters the cochlea, namely its three or four compartments, at the highest point, while a motor, started by an electrical impulse, sets it in motion. The different compartments form individual chambers in which the incoming water pushes, thanks to the earth's gravitational force, creating a rotation principle. The energy produced by the rotation of the auger shaft is transmitted, through a belt multiplier, to a generator; the rotation speed is minimal, in fact what wins in this technology is not the speed, but the thrust force.