The tale of the scientist Archimedes, who cost an entire army. The legend of Archimedes and a brief biography of the scientist What was the first task given by the king to Archimedes

Archimedes

This is an amazing person whose name

people have been remembering for over 2,000 years.

He was a talented mathematician

mechanic and engineer.

Every student is familiar with c lo π ,

lever balance rule,

"golden" rule of mechanics,

the law of navigation of bodies, etc.

The name of Archimedes lives on in legends.

I was interested to learn something new about him.

With content:

• Biography
• Mathematical works
• Archimedean screw
• Archimedean spiral
• Celestial sphere "Archimedes
• Lever balance rule
• The golden rule of mechanics
• Block device
• legends
• Conclusion

Biography

Archimedes 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 - the scientific and cultural center of 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.

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 also carried out a complete study of these equations, that is, he found under what conditions they will have real positive different roots and under what conditions the roots will coincide.

snub cube

truncated tetrahedron

cuboctahedron

13 treatises of Archimedes have come down to us

• The treatise "On the ball and the cylinder" established that the ratio of their volumes is 2/3. The ball inscribed in the cylinder was carved on his grave.
• The essay "On the balance of plane figures" is devoted to the study of the center of gravity of various figures.
• In the treatise "On Conoids and Spheroids" Archimedes considers a sphere, an ellipsoid, a paraboloid and a hyperboloid of revolution and their segments and determines their volumes.
• In the essay "On Spirals" he explores the properties of the curve that received his name and is tangent to it.
• In the treatise "Measuring the circle" Archimedes proposes a method for determining the number pi, which was used until the end of the 17th century.
• In "Psammit" ("Calculation of grains of sand"), Archimedes proposes a number system that allowed writing super-large numbers, which amazed the imagination of contemporaries. "Counted" them up to 10 64 .
• In "Squaring a Parabola" determines the area of ​​a segment of a parabola first using the "mechanical" method, and then proves the results geometrically.
• Archimedes owns the "Book of Lemmas", "Stomachion" and discovered only in the 20th century. "Method" (or "Ephod") and "Regular Heptagon". In the Method, Archimedes describes the process of discovery in mathematics, making a clear distinction between his mechanical methods and mathematical proof.

The surviving writings of Archimedes can be divided into three groups:

The first group - 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.

The second group consists of works on geometric analysis of statistical hydrostatic problems:

"On the equilibrium of plane figures".

Famous law of hydrostatics ,

entered into science law of Archimedes , formulated in the treatise "On floating bodies".

For every body

immersed in liquid

buoyant force acts upward and

equal to the weight of the liquid it displaces.

Archimedes' law is also valid for gases.

F BUT = ρ well · g ∙ V T = P well

By the third group can be attributed various mathematical works: For example, as among the cylinders, inscribed into a sphere, find the cylinder with the largest volume?

In work "On the Measurement of a Circle" Archimedes gave his famous approximation of pi: « 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.

Archimedean screw

Archimedes 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.

Archimedean spiral -

flat Curve,

trajectory of point M,

moving from a point 0

at a constant speed along a beam rotating around the pole 0

with constant angular velocity .

Equation in polar coordinates:

r = a∙f, where a is a constant.

"Heavenly Sphere" of Archimedes

Archimedes built a planetarium or "celestial sphere", during the movement of which one could 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 removed

Marcellus to Rome

where throughout

several centuries

admired

In the treatise "On levers" Archimedes set

LEVER BALANCE RULE

opened "golden" rule of mechanics : how many times the mechanism gives a gain in strength, the same number of times it results in a loss in distance "Give me a point of support and I will move the whole world"

Archimedes was the first to invent

block device,

studied its mechanical properties

and put it into practice

Legend tells that the luxurious ship Sirokosia, built by Hieron as a gift to the Egyptian king Ptolemy, 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.

Crown legend

Exist the legend of 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!

Siege of Syracuse

engineering genius 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 the sun's rays on them on the orders of Archimedes.

Death Legends

For 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.

Second version says that the Roman commander Marcellus specially sent a warrior in search of Archimedes.

The warrior sought out 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.

About Archimedes in verse

And before us for many years In a difficult year native Syracuse Defended the scientist Archimedes.

He was engulfed by an unknown plan He did not know that there were enemies in the city, And in meditation on the hot earth He drew some circles.

He drew pensive, not proud, Forgetting current affairs, - And suddenly, in an incomprehensible chord, the Shadow of the spear crossed the drawing.

But calmly frightening the killers, He, without humiliation, without trembling, Stretched out His Hand, protecting Not himself, but the signs of the drawing.

One of the largest lunar craters (82 kilometers wide) was named after Archimedes

Used materials:

• http://class-fizika.spb.ru
• http://en.wikipedia.org
• http://www.home-edu.ru
• http://www.chrono.ru
• http://www.krugosvet.ru
• http://tmn.fio.ru
• http://edu.nstu.ru
• http://www.mirf.ru/archive.php
• Program "Physicon"

A native of the Greek city of Syracuse on the island of Sicily, Archimedes was an entourage of King Hieron who ruled the city (and probably his relative). Perhaps for some time Archimedes lived in Alexandria, the famous scientific center of that time. The fact that he addressed reports of his discoveries to mathematicians associated with Alexandria, such as Eratosthenes, confirms the opinion that Archimedes was one of the active successors of Euclid, who developed the mathematical traditions of the Alexandrian school. Returning to Syracuse, Archimedes remained there until his death during the capture of Syracuse by the Romans in 212 BC.

The date of birth of Archimedes (287 BC) is determined based on the testimony of a Byzantine historian of the 12th century. John Tzetz, according to which he "lived seventy-five years." The vivid pictures of his death, described by Livy, Plutarch and Valery Maximus, differ only in details, but they agree that Archimedes, who was engaged in deep thought in geometric constructions, was hacked to death by a Roman warrior. In addition, Plutarch reports that Archimedes “is said to have bequeathed to relatives and friends to install a cylinder described around a ball indicating the ratio of the volume of the described body to the inscribed one on his grave,” which was one of his most glorious discoveries. Cicero, who in 75 BC was in Sicily, discovered a tombstone peeping out of a thorny bush and on it - a ball and a cylinder.

Legends of Archimedes.

In our time, the name of Archimedes is associated mainly with his remarkable mathematical works, but in antiquity he also became famous as the inventor of various kinds of mechanical devices and tools, as reported by authors who lived in a later era. True, the authorship of Archimedes in many cases is questionable. So, it is believed that Archimedes was the inventor of the so-called. the Archimedean screw, which served to raise water to the fields and was the prototype of ship and air propellers, although, apparently, this kind of device was used before. Does not inspire much confidence and what Plutarch tells in Biography of Marcellus. Here it is said that in response to the request of King Hiero to demonstrate how a heavy load could be moved by a small force, Archimedes “took a three-masted cargo ship, which many people had previously pulled ashore with great difficulty, put a lot of people on it and loaded it with ordinary cargo. After that, Archimedes sat down at a distance and began to effortlessly pull the rope thrown over the chain hoist, which is why the vessel easily and smoothly, as if on water, “floated” towards him. It is in connection with this story that Plutarch cites the remark of Archimedes that “if there were another Earth, he would move ours by going to that one” (a better-known version of this statement is reported by Pappus of Alexandria: “Give me where to stand, and I will move the Earth "). The authenticity of the story told by Vitruvius is also doubtful, that King Hieron allegedly instructed Archimedes to check whether his crown was made of pure gold or whether the jeweler appropriated part of the gold by alloying it with silver. “Thinking about this problem, Archimedes somehow went into the bath and there, plunging into the bath, noticed that the amount of water overflowing over the edge is equal to the amount of water displaced by his body. This observation prompted Archimedes to solve the problem of the crown, and he, without a second's delay, jumped out of the bath and, as if he was naked, rushed home, shouting at the top of his voice about his discovery: “Eureka! Eureka!" (Greek "Found! Found!")".

More reliable is the testimony of Pappus that Archimedes owned the work About manufacturing[heavenly]spheres, which was probably about building a planetarium model that reproduced the visible movements of the Sun, Moon and planets, and also, possibly, a star globe with the image of constellations. In any case, Cicero reports that both instruments were captured in Syracuse as trophies by Marcellus. Finally, Polybius, Livy, Plutarch and Zetzes report on the grandiose ballistic and other machines built by Archimedes to repel the Romans.

Mathematical works.

The surviving mathematical writings of Archimedes can be divided into three groups. The works of the first group are devoted mainly to the proof of theorems on the areas and volumes of curvilinear figures or bodies. These include treatises About the ball and cylinder, About measuring a circle, About conoids and spheroids, About spirals and On the quadrature of a parabola. The second group consists of works on the geometric analysis of static and hydrostatic problems: On the equilibrium of plane figures, About floating bodies. The third group includes various mathematical works: On the method of mechanical proof of theorems, Calculus of grains of sand, Bull problem and preserved only in fragments Stomachion. There is another work The Book of Assumptions(or Book of Lemmas), preserved only in Arabic translation. Although it is attributed to Archimedes, in its current form it clearly belongs to another author (since there are references to Archimedes in the text), but perhaps evidence is given here that goes back to Archimedes. Several other works attributed to Archimedes by ancient Greek and Arabic mathematicians have been lost.

The works that have come down to us have not retained their original form. So, apparently, I book of the treatise On the equilibrium of plane figures is an excerpt from a larger essay Elements of mechanics; moreover, it differs markedly from Book II, which was obviously written later. The proof mentioned by Archimedes in the essay About the ball and cylinder, was lost by the 2nd c. AD Work About measuring a circle is very different from the original version, and sentence II in it is most likely borrowed from another work. Title On the quadrature of a parabola could hardly have belonged to Archimedes himself, since in his time the word "parabola" was not yet used as the name of one of the conic sections. Texts such as About the ball and cylinder and About measuring a circle, most likely underwent changes in the process of translation from the Dorian-Sicilian into the Attic dialect.

When proving theorems on the areas of figures and the volumes of bodies bounded by curved lines or surfaces, Archimedes constantly uses a method known as the "method of exhaustion". It was probably invented by Eudoxus (the heyday of activity c. 370 BC) - at least, Archimedes himself believed so. Euclid resorts to this method from time to time in Book XII Started. The proof by means of the method of exhaustion, in essence, is an indirect proof by contradiction. In other words, the statement "A is equal to B" is considered true in the case when the opposite statement, "A is not equal to B", leads to a contradiction. The main idea of ​​the exhaustion method is that in the figure, the area or volume of which is to be found, the correct figures are inscribed (or described around it, or they are inscribed and described at the same time). The area or volume of the inscribed or circumscribed figures is increased or decreased until the difference between the area or volume to be found and the area or volume of the inscribed figure is less than a predetermined value. Using various versions of the exhaustion method, Archimedes was able to prove various theorems that are equivalent in modern notation to the relations S = 4p r 2 for the surface area of ​​the ball, V = 4/3p r 3 for its volume, the theorem that the area of ​​a segment of a parabola is 4/3 of the area of ​​a triangle having the same base and height as the segment, as well as many other interesting theorems.

It is clear that, using the method of exhaustion (which is more a method of proving, rather than discovering new relations), Archimedes must have had some other method at his disposal, allowing him to find the formulas that constitute the content of the theorems he proved. One of the methods for finding formulas reveals his treatise On the mechanical method of proving theorems. The treatise describes a mechanical method in which Archimedes mentally balanced geometric figures, as if lying on the scales. Having balanced a figure with an unknown area or volume with a figure with a known area or volume, Archimedes noted the relative distances from the centers of gravity of these two figures to the suspension point of the balance beam and, according to the law of the lever, found the required area or volume, expressing them, respectively, through the area or volume of the known figure. One of the main assumptions used in the exhaustion method is that the area is considered as the sum of an extremely large set of "material" straight lines that are closely adjacent to each other, and the volume is considered as the sum of plane sections that are also tightly adjacent to each other. Archimedes believed that his mechanical method had no probative force, but allowed a preliminary result to be obtained, which could later be proved by more rigorous geometric methods.

Although Archimedes was primarily a geometer, he made a number of interesting excursions into the field of numerical calculations, although the methods he applied were not entirely clear. In sentence III of the essay About measuring a circle he found that the number p is less than and greater than . From the proof it is clear that he had an algorithm for obtaining approximate square roots from large numbers. It is interesting to note that he also gives an approximate estimate of the number , namely: . In a work known as Calculus of grains of sand, Archimedes sets out an original system for representing large numbers, which allowed him to write down the number , where itself R equals . He needed this system to count how many grains of sand it would take to fill the universe.

In labor About the spiral Archimedes investigated the properties of the so-called. Archimedean spiral, wrote down in polar coordinates the characteristic property of the points of the spiral, gave the construction of a tangent to this spiral, and also determined its area.

In the history of physics, Archimedes is known as one of the founders of the successful application of geometry to statics and hydrostatics. In book 1 of the essay On the equilibrium of plane figures he gives a purely geometric derivation of the law of the lever. In fact, his proof is based on the reduction of the general case of a lever with arms inversely proportional to the forces applied to them, to the particular case of an equal-arm lever and equal forces. The entire proof from beginning to end is permeated with the idea of ​​geometric symmetry.

In his essay About floating bodies Archimedes applies a similar method to solving hydrostatic problems. Based on two assumptions formulated in geometric language, Archimedes proves theorems (suggestions) regarding the size of the immersed part of bodies and the weight of bodies in a liquid with both higher and lower density than the body itself. In sentence VII, which refers to bodies denser than liquid, the so-called. Archimedes' law, according to which "any body immersed in a liquid loses as much compared to its weight in air as the weight of the liquid displaced by it." Book II contains subtle considerations regarding the stability of the floating segments of a paraboloid.

Influence of Archimedes.

Unlike Euclid, Archimedes was remembered in antiquity only occasionally. If we know anything about his works, it is only thanks to the interest that they had in Constantinople in the 6th-9th centuries. Eutocius, a mathematician born in the late 5th century, commented on at least three of Archimedes' works, apparently the best known at the time: About the ball and cylinder, About measuring a circle and On the equilibrium of plane figures. The works of Archimedes and the comments of Eutokios were studied and taught by the mathematicians Anthimius of Thrallus and Isidore of Miletus, the architects of the Cathedral of St. Sophia, erected in Constantinople during the reign of Emperor Justinian. The reform of the teaching of mathematics, which was carried out in Constantinople in the 9th century. Leo of Thessaloniki, apparently, contributed to the collection of works of Archimedes. Then he became known to Muslim mathematicians. We now see that the Arabic authors were missing some of the most important works of Archimedes, such as On the quadrature of a parabola, About spirals, About conoids and spheroids, Calculus of grains of sand and About method. But in general, the Arabs mastered the methods set forth in other works of Archimedes, and often used them brilliantly.

Medieval Latin-speaking scholars first heard of Archimedes in the 12th century, when two translations from Arabic into Latin of his work appeared. About measuring a circle. The best translation belonged to the famous translator Gerardus of Cremona, and in the next three centuries it served as the basis for many expositions and extended versions. Gerard also owned a translation of the treatise Words of the sons of Moses Arabic mathematician of the 9th c. Banu Musa, which cited theorems from the work of Archimedes About the ball and cylinder with a proof similar to that given by Archimedes. At the beginning of the 13th c. John de Tinemuet translated the essay About curved surfaces, which shows that the author was familiar with another work of Archimedes - About the ball and cylinder. In 1269 the Dominican Wilhelm of Moerbecke translated the entire corpus of Archimedes' works from Ancient Greek, except Calculus of grains of sand, method and short essays Bull problem and Stomachion. For the translation, Wilhelm of Moerbeke used two of the three Byzantine manuscripts known to us (manuscripts A and B). We can trace the history of all three. The first of these (manuscript A), the source of all copies made during the Renaissance, seems to have been lost around 1544. The second manuscript (manuscript B), containing Archimedes' work on mechanics, including the essay About floating bodies disappeared in the 14th century. No copies were made of it. The third manuscript (manuscript C) was not known until 1899, and began to be studied only from 1906. It was manuscript C that became a precious find, as it contained a magnificent essay About method, previously known only from fragmentary fragments, and the ancient Greek text About floating bodies, which disappeared after the loss in the 14th century. manuscript B, which was used in the translation into Latin by Wilhelm of Moerbeke. This translation was in circulation in the 14th century. in Paris. It was also used by Jacob of Cremona, when in the middle of the 15th century. he undertook a new translation of the corpus of works of Archimedes included in manuscript A (i.e., with the exception of the work About floating bodies). It was this translation, slightly corrected by Regiomontanus, that was published in 1644 in the first Greek edition of the works of Archimedes, although some translations by Wilhelm of Moerbeke were published in 1501 and 1543. After 1544, Archimedes' fame began to increase, and his methods had a significant influence on such scholars as Simon Stevin and Galileo, and thus, albeit indirectly, influenced the formation of modern mechanics.

Were devoted to mechanics, it would be natural to begin our conversation with a consideration of how the basic ideas of Greek mechanics arose and how they developed. The word "mechanics" itself comes from the Greek merhane- mekhane, which originally meant a lifting machine used in Greek theaters to raise and lower the Greek gods onto the stage, which were supposed to resolve the intricate course of the drama being presented; hence the often used saying: deus ex machina - god from the machine. Later, the word mechane began to be used to refer to military vehicles, and then to vehicles in general.

As the historian Diodorus Siculus says, Archimedes invents the cochlea, or Archimedes screw, which serves to raise water. The Archimedes screw (Fig. 1) is an invention with which, in the distant past, rivers were pumped or even completely drained.

Rice. 1 Archimedes screw

The catapult of Archimedes, or ballista (Fig. 2, Fig. 3) is an invention of Archimedes, which appeared presumably around 399 BC. The catapult was used as a weapon in various wars; antique two-arm torsion action machine for throwing stones. Later, in the first centuries of our era, ballistas began to mean arrow throwers.

Archimedes also proved that it was possible to pull heavy loads with less force than usual; the inventor ordered to pull a heavy ship ashore and fill it with cargo. Standing near the chain hoist (coil side), Archimedes began to pull the rope tied to the ship without any significant effort.

Fig.4. Paw of Archimedes

The paw of Archimedes (Fig. 4) is a prototype of a modern crane. Outwardly, it looked like a lever protruding beyond the city wall and equipped with a counterweight. Polybius wrote in World History that if a Roman ship tried to land near Syracuse, this “manipulator” under the control of a specially trained machinist grabbed its bow and turned it over (the weight of Roman triremes exceeded 200 tons, while the penter could reach all 500) , flooding attackers.

Rice. 5. Planetarium

Cicero wrote that after Syracuse was sacked, Marcellus took out two devices from there - "spheres", the creation of which is attributed to Archimedes. The first was a kind of planetarium, and the second modeled the movement of the stars across the sky, which suggested the presence of a complex gear mechanism in it.

The Romans were shocked to see Archimedes' machines in action. Plutarch writes that sometimes it came to the point of absurdity: when they saw some kind of rope or log on the wall of Syracuse, the invincible Roman legionnaires fled in a panic, thinking that another infernal mechanism would now be used against them.

Until recently, this evidence was considered doubtful, but in 1900, near the Greek island of Antikythera, at a depth of 43 meters, the remains of a ship were found, from which the remains of a certain device were raised - an “advanced” system of bronze gears dating back to 87 BC. This proves that Archimedes could well create a complex mechanism - a kind of "computer" of ancient times.

Archimedes holds the primacy in many discoveries from the field of exact sciences. Thirteen treatises of Archimedes have come down to us. In the most famous of them - "On the ball and the cylinder" (in two books), Archimedes establishes that the surface area of ​​​​the ball is 4 times the area of ​​\u200b\u200bits largest section; formulates the ratio of the volumes of the sphere and the cylinder described next to it as 2:3 - a discovery that he treasured so much that in his will he asked to put a monument on his grave with the image of a cylinder with a sphere inscribed in it and the inscription of the calculation.

In physics, Archimedes introduced the concept of the center of gravity, established the scientific principles of statics and hydrostatics, and gave examples of the application of mathematical methods in physical research. The main provisions of statics are formulated in the essay "On the equilibrium of plane figures." Archimedes considers the addition of parallel forces, defines the concept of the center of gravity for various figures, and gives the derivation of the law of the lever.

Using the principle of integration, Archimedes discovered the number pi. Subsequently, its meaning was constantly refined. In 1882, the German mathematician Ferdinand von Lindemann proved that pi is infinite. In the 20th century, computers were able to calculate about a billion decimal places. The computer made it possible to discover an exhaustive solution to the famous "bull problem". The smallest answer to it was found in 1880 and was expressed as a number consisting of 206,545 digits. One hundred years later, in 1981, computer scientists calculated over a billion decimal places. Modern Syracuse has almost no traces of its former greatness. Tourists are often taken to the so-called "Tomb of Archimedes" in the Grotticelli necropolis. In fact, this Roman burial does not contain the remains of the famous scientist.

The Archimedes Palimpsest is a Christian book compiled in the 12th century from "pagan" parchments from the 10th century. To do this, the old letters were washed away from them, and a church text was written on the material received. Fortunately, the palimpsest (from the Greek palin - again and psatio - I erase) was made of poor quality, so the old letters were visible in the light (and even better - under ultraviolet light). In 1906, it turned out that these were three previously unknown works of Archimedes.

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 suddenly noticed the effect of liquid displacement (he shouted: “Eureka!” - “Found it!”, 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, and this helped him to expose the deceiver.

There is a legend about how Archimedes came to the discovery that the buoyant force is equal to the weight of the fluid in body volume. He pondered over the task given to him by the Syracusan king Hieron (250 BC).

King Hieron instructed him to check the honesty of the master who made the golden crown. Although the crown weighed as much as the gold that was given to it, the king suspected that it was made of an alloy of gold with other, more cheap metals. Archimedes was instructed to find out, without breaking the crown, whether there is an impurity in it or not.

It is not known for certain what method Archimedes used, but we can assume the following: First, he found that a piece of pure gold is 19.3 times heavier than the same volume of water. In other words, the density of gold is 19.3 times that of water.

Archimedes had to find the density of the corona matter. If this density were more than the density of water is not 19.3 times, but a smaller number of times, which means that the crown was not made of pure gold.

Weighing the crown was easy, but how do you find its volume? That's what made it difficult for Archimedes, because the crown was a very complex shape. Archimedes was tormented by this task for many days. And then one day, when he, while in the bath, immersed himself in a bath filled with water, he suddenly came up with an idea that gave a solution to the problem. Exultant and excited by his discovery, Archimedes exclaimed; "Eureka! Eureka!" which means; "Found! Found!".

Archimedes weighed the crown first in the air, then in the water. From the difference in weight, he calculated the buoyancy force equal to the weight of water in the volume of the crown. Having then determined the volume of the crown, he was already able to calculate its density. And knowing the density, answer the question of the king: are there any impurities of cheap metals in the golden crown?

The legend says that the density of the material of the crown turned out to be less than the density of pure gold. Thus, the master was convicted of deceit, and science was enriched by a remarkable discovery. Historians say that the problem of the golden crown prompted Archimedes to take up the question of the floating of bodies. The result of this was the appearance of the remarkable work "On Floating Bodies", which has come down to us.

The seventh sentence (theorem) of this work is formulated by Archimedes as follows:

Bodies that are heavier than a liquid, being lowered into it, are all immersed deeper until they reach the bottom, and, being in the liquid, lose as much weight in their how much the liquid weighs, taken in the volume of bodies.

Ex. Assuming that the golden crown of King Hiero weighs 20N in air and 18.75N in water, calculate the density of the corona. Believing that to gold was only silver is mixed, determine how much gold was in the crown and how much silver. When solving the problem, the density of gold is rounded to be 20,000 kg/m3, the density of silver is 10,000 kg/m3.

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_1.jpg" alt="(!LANG:>Legend of King Hiero's Crown Archimedes About"> Легенда о короне царя Гиерона Архимед Около 287 – 212 г. до н. э. Сиракузы!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_2.jpg" alt="(!LANG:>Legend of King Hiero's Crown">!}

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Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_4.jpg" alt="(!LANG:>Association is a connection that occurs under certain conditions between two or more"> Ассоциация – связь, возникающая при определённых условиях между двумя или более мыслительными процессами (ощущениями, идеями, объектами, и т.п.)!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_5.jpg" alt="(!LANG:>Riddle for Mr. Sherlock Holmes">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_6.jpg" alt="(!LANG:>Matter density">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_7.jpg" alt="(!LANG:>LESSON PURPOSE: To form the concept of "density"; Determine what this physical quantity depends on">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_8.jpg" alt="(!LANG:>LESSON OBJECTIVES: Define a new concept for yourself" density» Enter the formula for calculating the density of a substance"> ЗАДАЧИ УРОКА: Определить новое для себя понятие «плотность» Ввести формулу для расчёта плотности вещества Ввести единицы измерения плотности Определить алгоритм расчёта плотности твёрдого тела Подумать, в каких профессиях необходимо знать как измеряется плотность тела!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_9.jpg" alt="(!LANG:>BODY WEIGHT">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_10.jpg" alt="(!LANG:>BODY WEIGHT">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_11.jpg" alt="(!LANG:>BODY VOLUME">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_12.jpg" alt="(!LANG:>BODY VOLUME">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_13.jpg" alt="(!LANG:>Bodies of the same volume but different mass There are three bodies on the desk."> Тела одинакового объёма, но разной массы Перед вами на парте лежат три тела. Чем они схожи друг с другом? Чем они отличаются друг от друга? Что можно сказать о веществах, из которых они изготовлены? Сравнить массы этих тел с помощью весов. Чем можно объяснить данный факт? Ваши предположения!!}

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Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_15.jpg" alt="(!LANG:>DENSITY OF THE SUBSTANCE">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_16.jpg" alt="(!LANG:>DENSITY OF THE SUBSTANCE">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_17.jpg" alt="(!LANG:>DENSITY OF A SUBSTANCE Density is a physical quantity that characterizes a property bodies of equal volume have different masses."> ПЛОТНОСТЬ ВЕЩЕСТВА Плотность – физическая величина, характеризующая свойство тел равного объёма иметь разную массу. ρ=m/v [ρ]=кг/м3!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_18.jpg" alt=">">

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_19.jpg" alt="(!LANG:>Working with tables in density table #1 Find the following solids: concrete, steel,"> Работа с таблицами Найдите в таблице № 1 плотности следующих твёрдых тел: бетон, сталь, железо, янтарь. Что означает численное значение плотности указанных твёрдых тел? Какое из этих твёрдых тел будет иметь наибольшую массу и наименьшую массу при равенстве объёмов?!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_20.jpg" alt="(!LANG:>First task from various substances: ice, water,"> Первое задание На рисунке перед вами три куба изготовленные из различных веществ: льда, воды, стали. Массы этих кубов одинаковы. Художник, когда рисовал эти кубы, перепутал таблички с названиями и просто наобум подписал их. Используя свой жизненный опыт, проверьте правильность надписей, сделанных художником.!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_21.jpg" alt="(!LANG:>ice steel water?">!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_22.jpg" alt="(!LANG:>Identify liquids! liquids that do not mix with each other"> Определите жидкости! В один сосуд налили три разнородные жидкости, которые не смешиваются друг с другом: ртуть, вода и нефть. Определите положение каждой жидкости и найдите по таблице № 3 учебника значение плотностей каждой из указанной жидкости № 1 № 2 № 3!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_23.jpg" alt="(!LANG:>Ingenious questions (addressing the descendants of Archimedes) As you know, when heated, bodies expand."> Вопросы на смекалку (обращение к потомкам Архимеда) Как известно при нагревании тела расширяются. Что происходит с массой тела и с плотностью при нагревании? Что изменится у твёрдого тела если его с Земли перенесут, не нагревая, не ломая на Луну? (Масса? Объём? Вкус? Плотность? Цвет?) Почему нельзя тушить горящую нефть (бензин, керосин) водой? А чем же тогда тушить?!}

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Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_25.jpg" alt="(!LANG:>Riddle for Mr. Sherlock Holmes balance scale Determine the volume of the body"> Загадка для мистера Шерлока Холмса Измерить массу тела на рычажных весах Определить объём тела с помощью мерного стакана (мензурки) Разделить полученное значение массы на измеренный объём Определить по таблице плотностей какому веществу соответствует полученное значение!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_26.jpg" alt="(!LANG:>LESSON OBJECTIVES: Define a new concept for yourself" density» Enter the formula for calculating the density of a substance"> ЗАДАЧИ УРОКА: Определить новое для себя понятие «плотность» Ввести формулу для расчёта плотности вещества Ввести единицы измерения плотности Определить алгоритм нахождения плотности твёрдого тела Подумать, в каких профессиях необходимо знать как измеряется плотность тела!}

Src="http://present5.com/presentacii-2/20171208%5C17718-plotnost_veshchestva_-_otkrytyi_urok.ppt%5C17718-plotnost_veshchestva_-_otkrytyi_urok_27.jpg" alt="(!LANG:>Where is it important to know what density is and how it is defined: In forensics B"> Где важно знать, что такое плотность и как она определяется: В криминалистике В медицине В минералогии В археологии В фармакологии В метеорологии На транспорте В пищевой и косметической промышленности И во многих других областях нашей жизни!}

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