T HERE was once a king of Syracuse whose name was Hiero. The country over which he ruled was quite small, but for that very reason he wanted to wear the biggest crown in the world. So he called in a famous goldsmith, who was skillful in all kinds of fine work, and gave him ten pounds of pure gold. “Take this,” he said, “and fashion it into a crown that shall make every other king want it for his own. Be sure that you put into it every grain of the gold I give you, and do not mix any other metal with it.” “It shall be as you wish,” said the goldsmith. “Here I receive from you ten pounds of pure gold; within ninety days I will return to you the finished crown which shall be of exactly the same weight.” Ninety days later, true to his word, the goldsmith brought the crown. It was a beautiful piece of work, and all who saw it said that it had not its equal in the world. When King Hiero put it on his head it felt very uncomfortable, but he did not mind that—he was sure that no other king had so fine a headpiece. After he had admired it from this side and from that, he weighed it on his own scales. It was exactly as heavy as he had ordered. “You deserve great praise,” he said to the goldsmith. “You have wrought very skillfully and you have not lost a grain of my gold.” There was in the king’s court a very wise man whose name was Archimedes. When he was called in to admire the king’s crown he turned it over many times and examined it very closely. “ ‘Well, what do you think of it?’ asked Hiero. ” “Well, what do you think of it?” asked Hiero. “The workmanship is indeed very beautiful,” answered Archimedes, “but—but the gold—” “The gold is all there,” cried the king. “I weighed it on my own scales.” “True,” said Archimedes, “but it does not appear to have the same rich red color that it had in the lump. It is not red at all, but a brilliant yellow, as you can plainly see.” “Most gold is yellow,” said Hiero; “but now that you speak of it I do remember that when this was in the lump it had a much richer color.” “What if the goldsmith has kept out a pound or two of the gold and made up the weight by adding brass or silver?” asked Archimedes. “Oh, he could not do that,” said Hiero; “the gold has merely changed its color in the working.” But the more he thought of the matter the less pleased he was with the crown. At last he said to Archimedes, “Is there any way to find out whether that goldsmith really cheated me, or whether he honestly gave me back my gold?” “I know of no way,” was the answer. But Archimedes was not the man to say that anything was impossible. He took great delight in working out hard problems, and when any question puzzled him he would keep studying until he found some sort of answer to it. And so, day after day, he thought about the gold and tried to find some way by which it could be tested without doing harm to the crown. One morning he was thinking of this question while he was getting ready for a bath. The great bowl or tub was full to the very edge, and as he stepped into it a quantity of water flowed out upon the stone floor. A similar thing had happened a hundred times before, but this was the first time that Archimedes had thought about it. “How much water did I displace by getting into the tub?” he asked himself. “Anybody can see that I displaced a bulk of water equal to the bulk of my body. A man half my size would displace half as much. “Now suppose, instead of putting myself into the tub, I had put Hiero’s crown into it, it would have displaced a bulk of water equal to its own bulk. All, let me see! Gold is much heavier than silver. Ten pounds of pure gold will not make so great a bulk as say seven pounds of gold mixed with three pounds of silver. If Hiero’s crown is pure gold it will displace the same bulk of water as any other ten pounds of pure gold. But if it is part gold and part silver it will displace a larger bulk. I have it at last! Eureka! Eureka!” Forgetful of everything else he leaped from the bath. Without stopping to dress himself, he ran through the streets to the king’s palace shouting, “Eureka! Eureka! Eureka!” which in English means, “I have found it! I have found it! I have found it!” The crown was tested. It was found to displace much more water than ten pounds of pure gold displaced. The guilt of the goldsmith was proved beyond a doubt. But whether he was punished or not, I do not know, neither does it matter. The simple discovery which Archimedes made in his bath tub was worth far more to the world than Hiero’s crown. Can you tell why?

HIS BATH GAVE HIM THE TIP-OFF ---> H IS B ATH G AVE H IM THE T IP -O FF ARCHIMEDES was asked to check the suspected presence of silver alloy in the king’s gold crown. The solution which occurred when he stepped into his bath and caused it to overflow 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 bulk of the crown and cause the bowl to overflow. So delighted was Archimedes with his solution that he leaped from his bath and ran through the streets of Syracuse crying “Eureka!” Presumably you won’t be in your bath when you read NBC’s new facts, but we would not be surprised to hear you, too, shout “Eureka!”

## Case study - Archimedes Principal

Is the crown pure gold?

Archimedes was born in 287 BC in a Greek city of Syracuse in what is now Sicily. He was a mathematician and scientist. The king of the city, named Hiero II, provided a quantity of pure gold for the fashioning of a laurel wreath made of gold. It was meant as an offering to the gods of the city. The goldsmith presented the king with his crown. It looked magnificent, but to be sure the smith had not kept any of the gold for himself, the king weighed the crown and determined it weighed exactly the same as the pure gold given him for the project. So the smith was paid.

Later the king was informed that if the smith had replaced a small amount of the pure gold with an equal weight of silver and mixed the gold and silver, the product would look like pure gold. Not only was the king angered that the smith would cheat him, he worried that if the crown were not pure gold, but it would also offend the gods and bring misfortune to the city.

So he contacted Archimedes and tasked him with determining if the gold crown was indeed pure gold. Archimedes knew that pure gold would be denser than gold mixed with silver. All he had to do was to determine the density of the crown and he would have the king’s answer. But he had to figure out how to do this without melting down the crown to determine its volume.

The story goes that he was contemplating this problem and decided to think about it while taking a bath. After the tub was full of warm water, he stepped into it and as he lowered himself into the water, the water level rose and some of the water overflowed the tub. This gave him an idea on how to solve the problem. Allegedly he was so excited he ran naked through the streets to his study to begin conducting his tests, yelling I have it, which in Greek would be Eureka! Whether this last part is true, the secret to measuring the density of the crown (or any other oddly shaped object) is to submerge it in water.

Watch Video: https://youtu.be/ijj58xD5fDI

Your Assignment

Using information that would have been available to Archimedes at the time, you are going to perform some simple mass density calculations to determine the validity of his principle. In other words, you are going to determine for yourself whether or not the crown was made of pure gold.

Archimedes' Principle is based upon the approach that if you know the mass of an object, and you can determine the volume of water that it displaces, you can find the density of the object. Today, we calculate density as follows:

Density=Mass /Volume

Unit 3 Case Study: Archimedes and the Gold Crown

Download this document and Save it.

Complete the calculations in the spaces provided.

Upload the completed assignment.

The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

The first is to fill a pitcher to the brim with water, lower the object into the water, and catch the water that overflows in another container. Measure the volume of the water and divide it into the mass of the object to get the density.

To see for yourself what this might involve, use the following numbers to calculate the density of the crown. (While Archimedes would not use the units of grams and cubic centimeters, we’ll use those units for our calculations.)

Early Greeks could not measure volume to the same accuracy as can be done today, so this method may not have worked for Archimedes. A more practical approach is as follows:

Next, he tied the crown to a string tied to one end of the scale. Then he submerged the crown under water and found the mass required to exactly balance the scale. Let’s say the scale balanced at 2845.4 g.

The difference between the mass of the crown in air and its mass while submerged is the mass of the water displaced by the crown.

The mass of the water provides a way to calculate the volume of water, provided you know that the density of water is . Hint: rearrange the definition of Density to calculate volume.

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## COMMENTS

Archimedes first measured the mass of the crown (m 0 = 0.44 kg) and then its apparent mass, when the crown was immersed in water (m' = 0.409 kg). Using both masses he determined the density of the crown and realized it wasn't made of gold. How did he come to this conclusion? Givens: ρ Au = 1.93 10 4 kg/m 3; ρ H2O = 10 3 kg/m 3

This is where Archimedes' discovery came in useful. First, Archimedes took a lump of gold and a lump of silver, each weighing exactly the same as the crown, and filled a large vessel with water to the brim, precisely measuring how much water was contained in the vessel. He then gently lowered the lump of silver into it.

Unit 3 Case Study: Archimedes and the Gold Crown Download this document and Save it. Complete the calculations in the spaces provided. Upload the completed assignment. The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

Unit 3 Case Study: Archimedes and the Gold Crown Download this document and Save it. Complete the calculations in the spaces provided. Upload the completed assignment. The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

Unit 3 Case Study: Archimedes and the Gold Crown Download this document and Save it. Complete the calculations in the spaces provided. Upload the completed assignment. The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

Heiron asked Archimedes to figure out whether the crown was pure gold. Archimedes took one mass of gold and one of silver, both equal in weight to the crown. He filled a vessel to the brim with water, put the silver in, and found how much water the silver displaced. He refilled the vessel and put the gold in.

Archimedes' solution to the problem, as described by Vitruvius, is neatly summarized in the following excerpt from an advertisement: ... Silver has a density of 10.5 grams/cubic-centimeter and so the gold-silver crown would have a volume of 700/19.3 + 300/10.5 = 64.8 cubic-centimeters. Such a crown would raise the level of the water at the ...

The gold displaced less water than the silver. He then put the crown in and found that it displaced more water than the gold and so was mixed with silver. That Archimedes discovered his principle when he saw the water in his bathtub rise as he got in and that he rushed out naked shouting "Eureka!" ("I have found it!") is believed to be ...

Archimedes had to figure out if the crown was really pure gold; if it was not, Archimedes would have proof that the goldsmith had been dishonest and made the crown with a cheaper metal. 2 Archimedes knew that gold was a very heavy metal. He knew that it would be easy to find out if the crown was pure by calculating its density, or mass per unit ...

The best estimate of Archimedes' year of birth is 287 BC and a good estimate of the year of Hiero "gaining the royal power in Syracuse" is 265 BC. This would make Archimedes around 22 years of age when, according to Vitruvius, he solved the golden crown problem. Another genius of Archimedes' rank, Isaac Newton, was 22 and 23 years of ...

In the first century BC, Archimedes was asked by King Hiero to help solve a problem. The king had commissioned a goldsmith to create a crown from a quantity of pure gold, and the goldsmith complied. He delivered to the king a beautiful crown and the king was quite pleased. However, the king soon began to hear rumors that the goldsmith had ...

Unit 3 Case Study: Archimedes and the Gold Crown Tou Xiong Download this document and Save it. Complete the calculations in the spaces provided. Upload the completed assignment. The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

Archimedes' Principle: This is the principle that allowed Archimedes to solve the gold crown problem. The concept of Archimedes' principle is pretty complex, but, at its heart, it concerns the theory of buoyancy. When a solid body or object is placed into a liquid, it displaces an equal amount of liquid to the volume of the body immersed in it.

Unit 3 Case Study: Archimedes and the Gold Crown. Download this document and Save it. Complete the calculations in the spaces provided. Upload the completed assignment. The challenge is to determine if a gold crown is pure gold or gold mixed with another metal. There are three ways to do this.

Let's assume that the king gave 500 grams of gold to the goldsmith, and the goldsmith crafted a 500-gram crown. Archimedes determined the volume of the crown to be 9050 cubic centimeters, and he knew that densities of gold and silver are 19 g/cm 3 and 10 g/cm 3, respectively. How many grams of silver, if any, were used to replace gold in the crown?

For the case of the liquid being water at a temperature in the region of 20 C, the density is simply the ratio of the initial measurement to the difference in the initial and ﬁnal mass measurements. (At a temperature of 16 C the density of water is 999 kg m−3 and even at 22 C it has only fallen to 998 kg m−3:i.e., very close to 1 g cm−3 ...

Archimedes' Principle: This is the principle that allowed Archimedes to solve the gold crown problem. The concept of Archimedes' principle is pretty complex, but, at its heart, it concerns the theory of buoyancy. When a solid body or object is placed into a liquid, it displaces an equal amount of liquid to the volume of the body immersed in it.

Archimedes' Principle: This is the principle that allowed Archimedes to solve the gold crown problem. The concept of Archimedes' principle is pretty complex, but, at its heart, it concerns the theory of buoyancy. When a solid body or object is placed into a liquid, it displaces an equal amount of liquid to the volume of the body immersed in it.

crown were placed in a 20-cm-diameter bowl, and how much it would rise if the crown contained some silver. Cell C4 gives the mass of the crown. Cell C8 calculates the volume of the crown assuming it is gold (density = 19.3 g/cm3, Cell C7). The result, 51.81 cm3, is the volume of water that would be displaced by immersing the gold crown.

Unit 3 Case Study Archimedes Principle Is the crown pure gold? Background Archimedes was born in 287 BC in a Greek city of Syracuse in what is now Sicily. He was a mathematician and scientist. The king of the city, named Hiero II, provided a quantity of pure gold for the fashioning of a laurel wreath made