## Archimedes' Principle

Fig: http://physics.weber.edu/carroll/archimedes/images/buoyancy.gif

If the weight of the water displaced is less than the weight of the object, the object will sink

Otherwise the object will float, with the weight of the water displaced equal to the weight of the object.

Archimedes' Principle explains why steel ships float Fig: http://physics.weber.edu/carroll/archimedes/images/exboy.gif

Archimedes and the Law of the Lever

"Give me a place to stand on, and I will move the earth."

Fig: http://physics.weber.edu/carroll/archimedes/images/levermid.gif

The basic principle of the law of the lever and - possibly - the concept of center of gravity were established on a mathematical basis by scholars earlier than Archimedes.

What did Archimedes do?

Archimedes proved the law of the lever, starting with these three assumptions.

Assumption 1. Equal weights at equal distances from the fulcrum balance. Equal weights at unequal distance from the fulcrum do not balance, but the weight at the greater distance will tilt its end of the lever down.

Assumption 2. If, when two weights balance, we add something to one of the weights, they no longer balance. The side holding the weight we increased goes down.

Assumption 3. If, when two weights balance, we take something away from one, they no longer balance. The side holding the weight we did not change does down.

How would you use these to prove the law of the lever?

A Key Step of Archimedes' Clever Proof of the Law of the Lever Fig : http://physics.weber.edu/carroll/archimedes/images/leverprf.gif

Archimedes Nine Surviving Treatises

On the Sphere and Cylinder (in two books)

shows the surface area of any sphere is 4 pi r2, and the volume of a sphere is two-thirds that of the cylinder in which it is inscribed, V = 4/3 pi r3

Measurement of the Circle

shows that pi, the ratio of the circumference to the diameter of a circle, is between 3 10/70 and 3 10/71

On Conoids and Spheroids

finds the volumes of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis

On Spirals

develops many properties of tangents to the spiral of Archimedes

Centers of Gravity of Planes (in two books)

finds the centers of gravity of various plane figures and conics, and establishes the "law of the lever"

Quadrature of the Parabola

finds, first by "mechanical" means (Archimedes' "Method") and then by rigorous geometry, the area of any segment of a parabola

The Sand Reckoner

attempts to remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number - the number of grains of sand that it would take to fill the whole of the universe by creating a place-value system of notation, with a base of 100,000,000 (and contains the most detailed surviving description of the heliocentric system of Aristarchus of Samos)

Method Concerning Mechanical Theorems

describes the process of discovery in mathematics

On Floating Bodies (in two books)

finds the positions that various solids will assume when floating in a fluid, and establishes Archimedes' principle (that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object)