Archimedes was a Greek mathematician, inventor and physicist who lived from 287 - 212 B.C. A king once gave him a difficult task. The king had a crown. He had paid for pure gold. He was afraid that the crown-maker had mixed silver with the gold. He wanted Archimedes to figure out whether or not the crown was pure gold or mixed with silver. Now, you know that all he had to do was to figure out the density of the metal. Silver's specific gravity is 10.5 and the specific gravity of gold is 19.3. It should be easy to detect the difference. The problem was to determine the volume of the crown. The kind did not want Archimedes to harm the crown in any way. How do you calculate the volume of a crown?

Archimedes went home to think about the problem. He stepped into his bath, and watched the water rise and overflow when he sat. Suddenly, his problem was solved! The story tells that he ran out into the street, naked and dripping, shouting "Eureka!", which means "I have found it!"

What did he find?

When an object is submerged in water, the level of the water rises. This is because the object has moved some of the water out of the way, to make room for itself. When Archimedes sat down in his bathtub, his body took up so much room that the water overflowed onto the floor. He realized that the water that he pushed out of the tub was the same volume as his body.

If he submerged the crown in water, he could measure the volume of the crown. It was easy to measure the mass of the crown. He could compare the volume of the crown to the volume of an equal mass of pure gold. If the volume of the crown was greater than the volume of the pure gold, then the crown was a fake.

We will test to see if Archimedes' idea was correct.

Purpose: This experiment will test to see if an object does displace its own volume when submerged. We will then determine the volume of a small number of marble chips and find the density of marble. We will compare the density that we calculate with the known density.

Equipment: graduated cylinders, water, vernier calipers, three regularly-shaped objects, a quantity of marble chips, a balance.

• Determine the volumes of the three regularly-shaped objects. Choose a graduated cylinder. Make sure that at least one of the objects will fit into it. Partially fill it with water. Write down the level of the water in the cylinder. (Measure at the bottom of the meniscus. The meniscus is the bottom of the curved surface of the water.)
• Lower the object into the water slowly. Write down the new level of the water in the cylinder.
• Subtract the first level of water from the second level.
• Compare the change in the volume of water to the volume of the object.
• Record your results in a table:

Question 1: Compare the change in the volume of the water to the volume of the object. Are they the same?

Question 2: Suppose you dropped an irregular object into the graduated cylinder. The water rises 25.3 ml. What is the volume of the irregular object?

• Measure the mass of your marble chips.
• Partially fill the smallest graduated cylinder.
• Write down the volume of water in the cylinder.
• Drop the chips into the cylinder.
• Write down the new volume of the water in the cylinder.
• Subtract the beginning volume from the second volume.
• Write down the volume of the marble chips.
• Calculate the density of marble.
• Calculate the specific gravity of marble.
• The accepted specific gravity of marble is 2.6. Compare the specific gravity of your marble chips to the accepted specific gravity of marble. Is it the same or different?

Question 3: The marble chips displaced water. We assumed that the amount of displaced water was equal to the volume of the chips. If this assumption had been incorrect, would you have gotten the correct specific gravity for marble? Why or why not?

Question 4: What can you conclude about the displacement method for determining the volume of an irregular object?