According to legend, the following challenge led Archimedes to the discovery of

Question

According to legend, the following challenge led Archimedes to the discovery of his famous principle: Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense) metal had been substituted for some of the gold, then the crown would displace more water than the pure gold.

Part B:

yes

no

Explanation / Answer

The change in apparent weight is due to the buoyant force of the water. The crown displaces a volume of water when it is submerged; the water then acts with this amount of force against the crown, causing it to weigh less. Using standard values for g (9.8, and remembering that newtons correspond to kg), we have: 4.5 N = 458.9 g 5.0 N = 509.9 g Using the density of gold, we find 509.9 g of gold should displace 505.9/19.32 = 26.4 cm^3 of water. This amount of water has a mass of 26.4 g, which pulls upwards with a force of 0.26 N. So the apparent weight under water of the crown, if made of pure gold, would be 5.0 - 0.26 = 4.74 N. The displaced water has mass 49.0 g, so 49.0 cm^3 water are displaced by the crown. Thus, the crown actually has density = 509.9 /49.0 = 10.4 g/cm^3. Either way you look at it, the crown is not pure gold.

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