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

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

Complete the calculations in the spaces provided.

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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#### IMAGES

1. Unit 3 Case Study Archimedes Principle.docx

2. PHY120 Unit 3 Case Study Archimedes Principle.pdf

3. The Real Story Behind Archimedes' Eureka Experiment And The Gold Crown

4. The Real Story Behind Archimedes' Eureka Experiment And The Gold Crown

5. Archimedes' principle of buoyancy (crown of Archimedes)

6. PHY120 U3 Case Study Archimedes Principle 2 Fuhrer.rtf

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1. Archimedes' principle

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

2. Archimedes and the Golden Crown

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.

3. Archimedes and the Gold Crown.docx

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.

4. PHY120 U3 Case Study 3.4 Archimedes Principle.docx

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.

5. Unit 3 Case Study Archimedes Principle.docx

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.

6. Archimedes' principle

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.

7. The Golden Crown (Introduction)

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 ...

8. What led to Archimedes' discovering his principle?

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 ...

9. PDF Archimedes and the Golden Crown

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 ...

10. The Golden Crown (Sources)

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 ...

11. PDF Archimedes and the Golden Crown

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 ...

12. Unit 3 Case Study Archimedes Principle-TouXiong.docx

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.

13. Archimedes and the gold crown problem

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.

14. Case study

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.

15. The Golden Crown ( Real World )

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?

16. PHY120 U3 Case Study Archimedes Principle 2.docx

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.

17. Archimedes and the golden 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 ...

18. Archimedes and the gold crown problem

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.

19. Archimedes and the gold crown problem

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.

20. PHY120 UNIT3 CASE STUDY ARCHIMEDED X.docx

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.

21. PDF Computational Geology 30 Archimedes' Bath

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.

22. Unit 3 Case Study Archimedes Principle

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