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Chapter 1: Problem 91

In the third century \(\mathrm{BC}\), the Greek mathematician Archimedes is said to have discovered an important principle that is useful in density determinations. The story told is that King Hiero of Syracuse (in Sicily) asked Archimedes to verify that an ornate crown made for him by a goldsmith consisted of pure gold and not a gold-silver alloy. Archimedes had to do this, of course, without damaging the crown in any way. Describe how Archimedes did this, or if you don't know the rest of the story, rediscover Archimedes's principle and explain how it can be used to settle the question.

Short Answer

Expert verified
Archimedes would weigh the crown and a sample of pure gold of the same weight separately and observe how much water each displaced when submerged. The pure gold sample displaces less fluid than a gold-silver alloy of the same weight. So, if the crown displaces more fluid than the pure gold, it suggests that the crown is not pure gold.

Step by step solution

01

Understand Archimedes's Principle

Archimedes's Principle states that the force of buoyancy on a submerged object is equal to the weight of the fluid displaced by that object. In simpler terms, this means that an object submerged in liquid is pushed upward by a force matching the weight of the liquid the object displaces.
02

Applying the Principle

To apply the principle, Archimedes would first need the weight of the crown and a sample of pure gold with the same weight. The pure gold is then submerged in a beaker full of water and the amount of fluid displacement is recorded. Next, the crown is submerged in a new beaker of water with the exact same volume as the first and the displacement is again recorded.
03

Compare Displacements

The pure gold and the crown have the same weight but they differ in their volumes if the crown is not made of pure gold. When they are each submerged, the pure gold displaces less water because it's denser than a gold-silver alloy. So, if the crown displaces more fluid than the pure gold sample, it signifies that the crown isn't pure gold and therefore the goldsmith hadn't used pure gold to craft the crown.

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