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Archimedes. ARCHIMEDES ((about 287 BC - 212 BC), an ancient Greek scientist, mathematician and mechanic, the founder of theoretical mechanics and hydrostatics. Developed anticipating integral calculus methods for finding areas, surfaces and volumes of various figures and bodies. works on statics and hydrostatics (the law of Archimedes) gave examples of the application of mathematics in natural science and technology. technical inventions(Archimedean screw, determination of the composition of alloys by weighing in water, systems for lifting heavy weights, military throwing machines), which won him extraordinary popularity among his contemporaries. He was the first to introduce the law of a body immersed in a liquid, which is now known to every schoolchild (fig. on the right, where F is the buoyant force and P is the force of gravity).

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Mechanics

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Archimedes (BC) Archimedes devoted himself to mathematics and mechanics. The devices and machines designed by him were perceived by contemporaries as miracles of technology. He discovered the law of specific gravity and studied the theory of lifting mechanisms. Archimedes devoted himself to mathematics and mechanics. The devices and machines designed by him were perceived by contemporaries as miracles of technology. He discovered the law of specific gravity and studied the theory of lifting mechanisms. Among his inventions is the Archimedes screw, a device for lifting water or loose materials such as sand. Archimedes spoke of the lever, the theory of which he was studying: “Give me a fulcrum and I will turn the whole world.” Among his inventions is the Archimedes screw, a device for raising water or loose materials such as sand. Archimedes spoke of the lever, the theory of which he was engaged in: "Give me a fulcrum, and I will turn the whole world."

Why is there a buoyant force? On the side walls the liquid presses with a force opposite in direction and equal in modulus, therefore the forces acting on the side walls of the body are mutually balanced, their resultant is zero. But the forces acting on the upper and lower parts are not equal. The column of water h 1 presses on the upper part with the force F 1. And on the lower part - the column of water h 2 with the force F 2. The value of the buoyancy force is equal to the difference F 2 -F 1.

Law of Archimedes: "A body immersed in a liquid loses as much in its weight as the weight of the displaced liquid." Archimedes' Law: "A body immersed in a liquid loses in its weight as much as the weight of the displaced liquid." the buoyancy force depends: on Vt, on the density of the liquid, but does not depend on the type of substance from which the body is made, the depth of immersion, on the shape of the object with an equal volume, the buoyant force depends: on Vt, on the density of the liquid, but does not depend on the type of substance which the body is made, the depth of immersion, on the shape of the object with an equal volume

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Prepared by the teacher of mathematics MOU secondary school No. 36 of Kaliningrad Kovalchuk Larisa Leonidovna

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Archimedes - the pinnacle of scientific thought ancient world. Subsequent scientists - Heron of Alexandria (1-11 centuries BC), Pappus of Alexandria (III century AD) - added little to the legacy of Archimedes.

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Archimedes was born in 287 BC in the Greek city of Syracuse, where he lived almost his entire life. His father was Phidias, the court astronomer of the ruler of the city of Hieron. Archimedes studied in Alexandria, where the Ptolemy rulers of Egypt gathered the best Greek scientists and thinkers, and also founded the world's largest library.

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The main works of Archimedes concerned various practical applications of mathematics (geometry), physics, hydrostatics and mechanics. In his essay "Parabola of Quadrature", Archimedes substantiated the method for calculating the area of ​​a parabolic segment, and he did this two thousand years before the discovery of integral calculus. In the work "On the measurement of a circle" Archimedes first calculated the number "pi" - the ratio of the circumference to the diameter - and proved that it is the same for any circle.

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The mathematical method of Archimedes, connected with the mathematical works of the Pythagoreans and with the work of Euclid that completed them, as well as with the discoveries of Archimedes' contemporaries, led to the knowledge of material space, to the knowledge of the theoretical form of objects located in this space, the form of a perfect, geometric form, to which objects are more or less approaching and the laws of which must be known in order to influence the material world.

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Archimedes studied the forces that move objects or bring them into balance, inventing a new branch of mathematics in which material bodies, reduced to their geometric form, retain at the same time their heaviness. This geometry of weight is rational mechanics, it is statics, as well as hydrostatics, the first law of which was discovered by Archimedes (the law that bears his name), according to which a force equal to the weight of the liquid displaced by it acts on a body immersed in a liquid.

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The famous "Eureka!" was uttered not in connection with the discovery of the law of Archimedes, but in connection with the law of the specific gravity of metals - a discovery that also belongs to the Syracusan scientist. According to legend, the ruler of Syracuse once approached Archimedes. He ordered to check whether the weight of the golden crown corresponds to the weight of the gold allotted to it. To do this, Archimedes made two ingots: one of gold, the other of silver, each of the same weight as the crown. Then he put them in turn in a vessel with water, noted how much its level had risen. Having lowered the crown into the vessel, Archimedes found that its volume exceeds the volume of the ingot.

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Archimedes tests and creates the theory of five mechanisms known in his time and called "simple mechanisms". These are the lever (“Give me a point of support,” said Archimedes, “and I will move the Earth”), a wedge, a block, an endless screw and a winch .. The invention of the endless screw led him to the invention of a bolt constructed from a screw and a nut.

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In 212 BC, while defending Syracuse from the Romans during the Second Punic War, Archimedes designed several war machines that allowed the townspeople to repel the attacks of the outnumbered Romans for almost three years. One of them was a system of mirrors, with which the Egyptians were able to burn the Roman fleet. Archimedes died during the siege of Syracuse: he was killed by a Roman soldier at the moment when the scientist was absorbed in the search for a solution to the problem set before him. Archimedes screw - water machine.

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Archimedes This is an amazing person whose name people remember for more than 2,000 years. He was a talented mathematician, mechanic and engineer. Every schoolchild is familiar with the number π, the rule of balance of the lever, the "golden" rule of mechanics, the law of floating of bodies, etc. The name of Archimedes lives on in legends.

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Biography Archimedes was born in 287 BC. in Syracuse on the island of Sicily. Archimedes' father, the astronomer and mathematician Phidias, was closely related to Hieron, the tyrant of Syracuse. The father instilled in his son from childhood a love of mathematics, mechanics and astronomy. In Alexandria of Egypt - scientific and cultural center that time - Archimedes met the famous Alexandrian scientists. He corresponded with Eratosthenes until the end of his life. It was here that Archimedes got acquainted with the works of Democritus, Eudoxus and other prominent Greek geometers. Leaving Alexandria, Archimedes returned to Sicily. In Syracuse, he was surrounded by attention and did not need funds. Due to the prescription of years, the life of Archimedes is closely intertwined with legends.

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The legend of the crown There is a legend about how King Hieron instructed Archimedes to check if the jeweler had mixed silver into his golden crown. The integrity of the product could not be violated. Archimedes could not complete this task for a long time. The solution came by chance, when he lay down in the bathroom and noticed the expulsion of fluid. Archimedes shouted "Eureka!" - "Found!", And ran naked into the street. He realized that the volume of a body immersed in water is equal to the volume of water displaced. Thus, Archimedes found out that silver was mixed into gold, exposed the deceiver and discovered the basic law of hydrostatics!

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Legends The legend says that built by Hieron as a gift to the Egyptian king Ptolemy, the luxurious ship "Syrokosia" could not be launched. Archimedes built a system of blocks (polyspast), with which he was able to do this work with the help of a few people.

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The second group consists of works on the geometric analysis of statistical hydrostatic problems: "On the equilibrium of plane figures." The famous law of hydrostatics, which entered science as the law of Archimedes, was formulated in the treatise On Floating Bodies. Any body immersed in a liquid is subjected to a buoyant force directed upward and equal to the weight of the liquid displaced by it. Archimedes' law is also valid for gases. FA = ρzh g∙VT = Рzh

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The siege of Syracuse The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse by the Romans in 212 BC. e. But at that time he was already 75 years old! The powerful throwing machines built by Archimedes threw heavy stones at the Roman troops. Thinking that they would be safe at the very walls of the city, the Romans rushed there, but at that time light short-range throwing machines threw a hail of cannonballs at them. Powerful cranes grabbed the ships with iron hooks, lifted them up, and then threw them down, so that the ships turned over and sank. According to legend, during the siege, the Roman fleet was burned by the defenders of the city, who, using mirrors and shields polished to a shine, focused on them Sun rays by order of Archimedes.

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Legends of Death According to the first, in the midst of the battle, he sat on the threshold of his house, reflecting in depth on the drawings he made right on the road sand. At this time, a Roman soldier running past stepped on the drawing, and the indignant scientist rushed at the Roman with a cry: “Do not touch my drawings!”. This phrase cost Archimedes his life. The soldier stopped and cold-bloodedly cut the old man down with his sword.

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The second version says that the Roman commander Marcellus specially sent a warrior in search of Archimedes. The warrior found the scientist and said: - Come with me, Marcellus is calling you. - What else Marcellus?! I have to solve the problem! The enraged Roman drew his sword and killed Archimedes.

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Mathematical works Archimedes was a remarkable practical mechanic and theorist, but the main business of his life was mathematics. According to Plutarch, Archimedes was simply obsessed with her. He forgot about food, did not care about himself at all. His works related to almost all areas of mathematics of that time: he owns remarkable research in geometry, arithmetic, and algebra. He found all the semi-regular polyhedra that now bear his name, significantly developed the theory of conic sections, gave a geometric method for solving cubic equations, the roots of which he found using the intersection of a parabola and a hyperbola. Archimedes spent and full study these equations, that is, he found under what conditions they will have real positive different roots and under which the roots will coincide. snub cube

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The surviving works of Archimedes can be divided into three groups: The first group is the determination of the areas of curvilinear figures or, respectively, the volumes of bodies. Archimedes found a general method to find any area or volume. Using his method, he determined the areas and volumes of almost all bodies that were considered in ancient mathematics. He considered his best achievement to be the determination of the surface area and volume of a sphere. The ideas of Archimedes formed the basis of integral calculus.

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The third group includes various mathematical works: For example, among cylinders inscribed in a sphere, how to find the cylinder with the largest volume? In On the Measurement of a Circle, Archimedes gave his famous approximation of the number pi: "the Archimedean number". He was able to estimate the accuracy of this approximation: To prove it, he built inscribed and circumscribed 96-gons for a circle and calculated the lengths of their sides.

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Archimedes screw Archimedes became famous for many mechanical designs. The endless screw he invented for scooping water moves water through a pipe to a height of up to 4m. It is still used in Egypt today.

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Archimedean spiral - a flat curve, the trajectory of point M moving from point O with constant speed along a beam rotating about the pole O with a constant angular velocity. Equation in polar coordinates: r = a∙f, where a is a constant.

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"Heavenly Sphere" of Archimedes Archimedes built a planetarium or " celestial sphere”, during the movement of which it was possible to observe the movement of five planets, the rising of the Sun and the Moon, the phases and eclipses of the Moon, the disappearance of both bodies behind the horizon line. After the death of Archimedes, the planetarium was taken by Marcellus to Rome, where for several centuries it aroused admiration.

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He discovered the "golden" rule of mechanics: how many times the mechanism gives a gain in strength, the same number of times it loses in distance "Give me a fulcrum, and I will turn the whole world"

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