The Archimedes' Principle

Archimedes of Syracuse  (287 BC – 212 BC, born in Syracuse, Sicily) is regarded as one of the leading philosophers, mathematician, engineer, and astronomer of ancient Greece.  He made significant contributions to the fields of mathematics, hydrostatics, principles of levers and various aspects of engineering. Experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors. He had estimated the value of 'pi' to a great accuracy and also made contributions to the methods used for estimating the areas and volumes of various geometric figures.

One of his most notable conributions is the Archimedes' Principle which states that the when an object of volume V is immersed in a fluid then the fluid exerts a force on the object, directed upwards and equal in magnitude to the weight of the volume V of the fluid displaced. Whether the object floats or sinks is determined by the difference between this upward force (also known as the bouyant force) and the weight of the object which is directed downwards. This difference is directly proportional to the difference in densities of the object and the fluid, which implies that if the density of the object is more compared to that of the liquid the object will sink, else it will float. What is not obvious here is that why this bouyant force should always be directed upwards and why should the force be equal to the weight of the water displaced?

The following is an explicit derivation of this relation by elementary mathematical treatment of some basic concepts of classical mechanics. The Blogspot does not have provision for writing mathematical expressions so I have posted here snapshots from the original PDF file that I had generated using LATEX software.