The Historical Legend
According to historical legends, Hieron II of Syracuse, an ancient Greek king in the third century BC, received a gold crown, but suspected that it might be merely gold plated rather than solid gold. The king did not want to be cheated, but also did not want to destroy what might be a solid gold crown. Hieron therefore asked Archimedes if he could figure out a way to tell if the crown was solid gold or gold plated without destroying the crown.
Archimedes pondered the problem and was puzzled at first. When Archimedes went to take a bath, he noticed the water rising around him as he stepped into the bath. The solution hit him in a flash of insight. Archimedes was so excited that he jumped out of the bath and ran naked through the streets of Athens shouting Eureka - (I have found it).
Archimedes had discovered the principle that tells us if an object will float.
Archimedes's Principle
Archimedes's principle states that the buoyant force acting on an object immersed or floating in a fluid equals the weight of the fluid displaced. What does that mean?
First what is a fluid? Ordinarily most people think of a fluid as a liquid. The scientific definition of a fluid however includes either a liquid or a gas. Any liquid or gas is a fluid.
The buoyant force is the upward force on an object immersed in or floating on a fluid. If the object weighs less than the buoyant force, it will float on top of the fluid. If it weighs more, it will sink.
Because the buoyant force equals the weight of the fluid displaced, an object must displace a greater weight of fluid than its own weight in order to float. That means that to float an object must have a density (mass divided by volume) less than the density of the fluid. If the object's density is greater than that of the fluid, it will sink.
Examples of Archimedes's Principle
Ships, even those made of metal, float because they have enough empty space inside that their total average density is less than the density of water. Helium or hot air balloons fly because their average density is less than the density of the air around them. In both cases the fluid, water or air, creates an upward buoyant force equal to the weight of the fluid displaced. The upward buoyant force is greater than the downward gravitational force.
If you swim in the ocean, you may have noticed that you float more easily in salt water than in fresh water. That is because salt water is more dense than fresh water. A swimmer therefore displaces a greater weight of salt water than fresh water and experiences a greater buoyant force. In addition fat is less dense than bone or muscle. That means that people with a higher percentage of body fat float more easily than lean or muscular people who have little body fat.
With the formulation of his principle, Archimedes began the study of hydrostatics.
Further Reading
Hecht, E. Physics:Algebra/Trig 2nd ed., Brooks/Cole, 1998.
