Physics

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Unit 3 Case Study: Archimedes and the Gold Crown

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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.)

Density Calculation	Result	Units
=
Is this pure gold?

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:

First, Archimedes would confirm that the crown had the mass of the gold provided originally: 3014 g.

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.

Calculation	Result	Unit
Mass of water displaced =Mass of Gold – Mass of Gold under water
Volume of water with this mass = =
=
Is this pure gold?

A third way of doing this would be to balance the crown with an equal mass of pure gold, melted into an object and tied to the balance beam. Then lower both crown and pure gold into a water. Hint: which should displace more water, pure gold or a gold mixed with another metal?

If such a test were performed and the crown were made with gold mixed with another metal, which way would the balance tip? Explain why the balance should tip this way below.