King Hiero had provided pure gold to a goldsmith to fashion a crown. When the crown was finished, it had the same mass as the gold supplied. Nonetheless, the king suspected that the gopldsmith had stolen some of the gold. The king believed the goldsmith had mixed a cheaper metal, silver, into the crown to replace the gold he had taken. The king asked Archimedes to analyze the crown and determine if the goldsmith had cheated him. Archimedes knew that silver was less dense than gold. He therefore asked for a mass of gold equal to the mass of crown. He then put the gold in a container, filled the container to the brim with water, and removed the gold.
(1) What happened to the water level in the container when the gold was removed? Archimedes the took the crown fashioned by the goldsmith and carefully lowered it into the container of water. (2) What would have happened to the water level if the crown had been made of pure gold? When Archimedes lowered the crown into the container, the water overflowed. (3) What did this indicate about the volume of the crown compared to the volume of the mass of gold? (4) Did this prove or disprove that the goldsmith had cheated the king?
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This question deals entirely with density and volume displacement. Density is a measure of the mass of a substance per unit volume, usually given in grams per milliliter. The more dense a substance is, the heavier 1 mL of it will be in comparison to 1 mL of a less dense substance. Since the volume of a rigid solid is not easy to directly measure, the easiest way to determine its volume is to measure how much water it displaces in a tub filled with water.
In the story above, when the mass of pure gold is removed from the container of water, the water level lowered some amount because the gold was no longer displacing any water. When the crown in question was lowered into the container, if the crown was made of pure gold then the water level would have risen to the top of the container just like it did with the pure gold sample since both objects contained the same mass. In other words, they would have displaced the same volume of water since they would have had the same density and mass. But when the crown was placed in the container, the water raised higher than the brim and overflowed. This means that the crown had a slightly larger volume than the pure gold and therefore displaced a larger amount of water. This could only happen if the crown was less dense than the pure gold, and this could only happen if the gold in the crown were mixed with some amount of a less dense metal like silver. So Archimedes did in fact determine that the king had been cheated by the goldsmith.
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