Mathematical Discovery - Archimedes' Principle

The discovery of the Archimedes’ Principle:

    Archimedes may have used his principle of buoyancy to determine whether the golden crown was less dense than solid gold.

 

     The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα!" meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.

 

     The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."

 

     So, with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object. Thus, in short, buoyancy = weight of displaced fluid. Archimedes' principle is true of liquids and gases, both of which are fluids. If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram (as a kilogram is unit of mass and not of force, the buoyant force is the weight of 1 kg, which is approximately 9.8 Newton.) It is important to note that the term immersed refers to an object that is either completely or partially submerged. If a sealed 1-liter container is immersed halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water, no matter what is in the container.

 

     If such an object is completely submerged, it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of force). If the container is completely submerged and does not compress, the buoyant force will equal the weight of 1 kilogram of water at any depth, since the volume of the container does not change, resulting in a constant displacement regardless of depth. The weight of the displaced water, and not the weight of the submerged object, is equal to the buoyant force.

 

     The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

 

     Suppose a rock's weight is measured as 10 Newton when suspended by a string in a vacuum with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 Newton. The force it then exerts on the string from which it hangs would be 10 Newton minus the 3 Newton of buoyant force: 10 − 3 = 7 Newton. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.