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Lecture 13
An Introductory Physics Course with Peter Eyland
Lecture 12 (Archimedes)
In this lecture the following are introduced:
• Archimedes' achievements
• Archimedes' death
• Hiero's crown
• Archimedes' principle
Archimedes' achievements
Archimedes lived from 287 to 212 BCE in Syracuse, Sicily, during the reign of king Hieron II. He is credited with
• working out the principles of lever operation,
• inventing the compound pulley,
• inventing the hydraulic screw, for raising water from a lower to a higher level (used in concrete trucks),
• evaluating the limits of the number π (Greek: pi)  the ratio of a circle's circumference to its diameter,
• inventing the catapult,
• constructing a mirror system that burned the invading Roman's ships sails by focusing the sun's rays on them,
• discovering the law of hydrostatics that bears his name (Archimedes' Principle).
Archimedes' death
Plutarch says (in one account), that during the Second Punic War with Rome,
"Archimedes, as fate would have it, was intent upon working out some problem by a diagram,
and having fixed his mind and eyes upon the subject of his thought, didn't notice the incursion of the Romans,
nor that the city was taken.
A soldier, unexpectedly coming upon Archimedes (who was lost in thought and reasoning), commanded Archimedes to follow him to Marcellus
[the conquering Roman general]. Archimedes declined to do so before he had worked out the proof of his problem.
The enraged soldier drew his sword and ran him through."
Hiero's crown
When a gold crown was made for Hiero, he asked that Archimedes determine whether it was pure gold or mixed with silver.
When he was meditating on the problem in the public baths, he allegedly hit on the solution,
shouted ευρηκα, and dashed home naked.
The (alleged) solution was to put a weight of gold equal to the crown, and known to be pure, into a bowl which was filled with water to the brim. Then the gold would be removed and the king's crown put in, in its place. An alloy of lighter silver would increase the volume of the crown and cause the bowl to overflow. The goldsmith was (allegedly) beheaded when the crown was found to be not pure gold.
Chris Rorres, at Drexel University, criticises this solution, because:
• it doesn't use Archimedes' principle, (the apparent loss in weight equals the weight of fluid displaced) and
• could not have been done with the precision of their instruments.
His solution to Archimedes and Hiero's crown
Archimedes' principle by considering pressures
Take a mass with constant crosssectional area, floating partially submerged in water. 

For equilibrium, the weight and force of the air pressure downwards, are balanced by the upward force from the water pressure. 

Since it is floating, it has lost all of its weight. There is then an upward force balancing weight and air pressure. The upward bouyancy force is called the Archimedean upthrust. 
The Archimedian upthrust (or buoyancy) of a partially or fully submerged object is equal to
the weight of fluid displaced by the object.
Example 1:
An object of mass 0.5 kg is made from material of density 4000 kg.m^{3} and suspended by a string so that it is totally
immersed in a liquid of density 1500 kg.m^{3}. Find the tension in the string
Example 2:
An iceberg of density 917 kg.m^{3} and regular crosssectional area floats in salt water of density
1030 kg.m^{3}. Find the fraction that is submerged
Example 3:
A frog in a hemispherical cockleshell finds that she just floats without sinking in a peagreen sea
of density 1350 kg.m^{3}. Given that the cockleshell has a radius of 60mm and has negligible mass, find the mass of the frog.
Example 4:
The supertanker Globtik London has a mass of 2.2x10^{8} kg when empty and can carry 0.5x10^{6} m^{3} of oil
with density 875 kg m^{3}. The density of sea water can be taken to be 1020 kg m^{3}.
Assume that the shape is a rectangular prism 380 m long, 60 m wide, and 40 m high. Find how deep the hull is submerged in the water.
Example 5
Sea water has a density 1025 kg.m^{3}. A submarine has a mass of 22.5x10^{6}kg when floating in sea water so that 10% of its volume
is above the water. Find the mass of water that must be taken into its tanks so that it can fully submerge.
on the surface:
weight of submarine plus empty tank =weight of water displaced
22.5 x 10^{6 } x g = (0.9 x volume of sub) x 1025 x g
divide by g
22.5 x 10^{6 }= 0.9 x volume of sub x 1025
volume of sub x 1025 = 22.5 x 10^{6 }/0.9 = 25 x 10^{6} kg
under water:
weight of submarine plus water in tank =weight of water displaced
(22.5 x 10^{6} + mass of water) x g = volume of sub x 1025 x g
divide by g
22.5 x 10^{6} + mass of water = 25 x 10^{6}
mass of water = (25  22.5) x 10^{6}
= 2.5 x 10^{6} kg
Example 6:
A person can walk on water (density 1000 kg.m^{3 }) if they wear very big shoes.
A pair of shoes are shaped like a rectangular boxes.
Each shoe weighs 2 kg and is 200 mm wide and 300 mm deep.
If a person of mass 80 kg wearing these shoes, slides on water so that the top of the shoes are level
with the surface of the water, find the length of shoes needed.
weight of water displaced = weight of person + shoes
2 x (length x 0.2 x 0.3) x 1000 x 9.8 = (80 + 2 x 2) x 9.8
divide by g
120 x length = 84
length = 0.7m
Summarising:
Archimedes lived from 287 to 212 BCE in Syracuse, Sicily, and constructed levers, pulleys, screws, catapults etc.
Archimedes' principle says that the apparent loss in weight of a body partially or totally immersed in a fluid is equal to the weight of fluid displaced.
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