Buoyancy
Archimedes of Syracuse: The Father of Buoyancy
People have been aware of objects floating on water (or sinking) since before recorded
history. It was not until Archimedes of Syracuse came along, that the theory of flotation and
the buoyancy principle were defined.
Archimedes was born at Syracuse on the island of Sicily in 287 BC. His father is thought to
have been an astronomer, and as a young boy, Archimedes developed a life-long interest in the
study of the heavens. As a youth he traveled to Egypt where he studied at the great Library of
Alexandria.
Archimedes is often described as being absentminded, self-absorbed, and somewhat eccentric.
Despite these personal attributes, he was recognized in his own time as a genius, and is
revered today as one of the greatest figures in the history of science and mathematics.
Archimedes' first love was mathematics. He would often spend days so intently fixed on
solving a problem that he neglected both food and his person to the point that his friends
would carry him kicking and fighting to the bath. He often stooped to the ground to work
mathematical problems by drawing figures in the dirt. He is even said to have carried a small
wooden tray filled with sand, which he used to draw his figures and work on his mathematical
problems. This tray would have been Archimedes' version of the modern lap top computer. Of
course, such a device is not without its problems: A strong wind could blow away a brilliant
proof; a bully could kick a theorem into your face, and should a cat wander into the tray, the
outcome could be too disgusting to contemplate.
Archimedes' accomplishments in mathematics are many, the most notable being:
- The determination of circular area; he approximated more precisely than anyone to date.
- The near development of the Calculus.
- The quadrature of the parabola.
- The sand-reckoner.
- The determination of the surface area and volume of a sphere.
His brilliant methods and proof for the sphere problem are described in
his treatise On the Sphere and the Cylinder.
Archimedes and the Kings Crown
Despite his mathematical prowess, Archimedes is perhaps best remembered for an incident
involving the crown of King Hiero.
As the story goes, the king of Syracuse had given a craftsman a certain amount of gold to
be made into an exquisite crown. When the project was completed, a rumor surfaced that the
craftsman had substituted a quantity of silver for an equivalent amount of gold, thereby
devaluing the crown and defrauding the king. Archimedes was tasked with determining if the
crown was pure gold or not. The Roman architect Vitruvious relates the story:
While Archimedes was considering the matter, he happened to go to the baths.
When he went down into the bathing pool he observed that the amount of water which flowed
outside the pool was equal to the amount of his body that was immersed. Since this fact
indicated the method of explaining the case, he did not linger, but moved with delight, he
leapt out of the pool, and going home naked, cried aloud that he had found exactly what he was
seeking. For as he ran he shouted in Greek: Eureka! Eureka! (eureka translated is "I have
found it").
Although there is speculation as to the authenticity of this story, it remains famous.
Probably no other tale in all of science combines the elements of brilliance and bareness
quite so effectively. Whether the story is true or not, there is no doubt to the truth of
Archimedes understanding of buoyancy.
Archimedes and the Principle of Buoyancy
Here is what Archimedes had found. Since an object immersed in a fluid displaces the same
volume of fluid as the volume of the object, it was possible to determine the precise volume
of the crown by immersing it in water. After determining the volume of water, a piece of pure
gold could easily be made to match the volume of the water, and thus the volume of the crown.
In theory, if the volume of the crown and the volume of the gold block are the same, they
should also have the same mass. The only reason they would not have the same mass is if one
of them was not pure gold. When the two objects were placed in a balance they did not have
equal mass. Faced with this evidence the craftsman confessed to his crime.
Extending this idea further, if the mass of the water displaced is greater than the mass of
the object, the object will float (Note: this calculation will require that the object be
forcibly submerged). If the mass of the water is less than the mass of the object, the object
will sink. If by chance the two masses are equal, the object will be suspended in the water
at varying depths depending on the initial depth of the object and the water's temperature and
turbidity. Every vessel that has ever sailed on water, every submarine that has ever launched,
and in short, all objects that come in contact with a body of water, are governed by the
principle of buoyancy defined by the great mind of Archimedes.
Archimedes and Simple Machines
Archimedes was also well recognized for his mastery of simple machines. Though he appears
to have considered machines as a lesser discipline to mathematics, he used them to serve his
purposes when needed.
The first example of his engineering ability came when King Hiero's ship could not be
launched. Though usually modest, Archimedes is claimed to have said "Give me a place to
stand on and I will move the world". The case of the king's ship was soon to prove he was
correct. All the men in Syracuse had tried to launch the ship, but their combined strength and
ingenuity had not been equal to the task. Archimedes set about building a system of levers
and pulleys into a compound machine. When he was done he had the ship fully loaded with cargo
and crew, put the end of a rope in Hiero's hand and told him to pull. The king did so, and the
ship steadily eased into the water. The king, previously being somewhat skeptical of such
machinery, immediately issued a proclamation stating: "From this day forth Archimedes is
to be believed in everything that he may say". There is no evidence that he took unfair
advantage of this edict, and it probably gave him no further authority with his wife.
As the years went on king Hiero died and Hieronymus took his place. Hieronymus joined
forces with Carthage against Rome. Marcellus, the Roman leader, soon blocked the harbor with
Roman ships and led his army to the gates of the city. It is easy to believe that Archimedes
had little interest in military matters, but with a Roman army threatening the city he drew upon his
genius to help his countrymen. And help he did. Almost singlehanded he kept Marcellus at bay.
He first project was to design and built cranes, similar to the ones used today. With his
cranes he grappled the Roman ships, raised them high in the air, and let them fall to smash
against the water; or he swung them over the wall of the city and lowered the ship and crew
to the waiting Syracusians.
He next built catapults that would hurl huge stones against the ships that
were out of his cranes' reach. He also used these catapults to shower metal
and small rocks at the Roman
soldiers who patrolled outside the city walls. His machines and contraptions
were so effective that simply throwing a rope over the city wall was enough
to send the Roman soldiers running
in panic. Marcellus did the only wise thing; he with drew his assault and settled
down to starve out the enemy. Even the great Archimedes could not solve this
problem. He quietly went
back to his home and to his beloved mathematics.
The Death of Archimedes
In 212 BC Syracuse surrendered to Rome. Before sending his men to sack the city Marcellus
told them "Spare that mathematician". Plutarch records what happened next:
As fate would have it, intent upon working out some problem by a diagram,
and having fixed his mind alike and his eyes upon the subject of his speculation, he
[Archimedes] never noticed the incursion of the Romans, nor that the city was taken. In this
transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him
to follow Marcellus; which he declining to do before he had worked out his problem to a
demonstration, the soldier, enraged, drew his sword and ran him through.
As much as anyone, Marcellus lamented the death of Archimedes. In the midst of his triumph
he provided Archimedes with an honorable burial and befriended his surviving relatives.
Thus ended the life of Archimedes. For all of his great weapons, for all of
his practical invention, his true love was pure mathematics. His levers and pulleys and
catapults were mere trifles compared with the beautiful theorems he discovered. It was his
mathematics that would be his greatest legacy. In this arena, Archimedes stands unchallenged
as the greatest mathematician of antiquity. It would take some 2000 years to produce his equal
in the form of Isaac Newton.
In accordance with the expressed desire of Archimedes, his family and
friends inscribed on his tomb the figure of his favorite theorem; the sphere and the
circumscribed cylinder, and the ratio of the containing solid to the contained. When
Cicero was in Sicily at Quaestor in 75 BC, he discovered the neglected and forgotten tomb of Archimedes near the Agrigentine
Gate. Realizing what he had found, he piously restored the burial site.
Links for Additional Study
In Class Activity
Click here for instructions on building foil boats
Objective: Given the stated materials, build a vessel that will hold the most pennies and still float
Materials Needed:
- aluminum foil (10 x 10 cm)
- Scotch tape (30 x 1.9 cm)
- water container
- pennies (50)
Rules: Using the aluminum foil and tape,construct a vessel that will float on water and hold a load of pennies.
You do not have to use all the materials, but you may not use any more than listed . The boat
that holds the most pennies wins, and the penny that causes boat to sink, does not count
