is said that Archimedes discovered his principle during a bath while thinking about how he could determine if King Hiero's crown was actually made of pure gold. While in the bathtub, he conceived the idea that he could determine the average density of an irregularly shaped object by weighing it in air and also in water. If the crown weighed 34.1 N in air and 31.4 N in water, determine if the crown is made of pure gold. The density of gold is 19,300 kg/m3. The density of water is 1000 kg/m3.

Answer: The crown is not made of pure gold.

Explanation:

Archimedes discovered that any solid, of any shape, when submerged in a liquid receives an upward force, equal to the weight of the volume of the liquid removed by the solid, which is equal to the solid volume.

So, if one body is weighed in air, the normal force will be equal to the gravity force (which we call weight) which can be expressed as follows:

Fg = m g = δ V g = 34.1 N

When submerged in water, the normal force is equal to the difference between the actual weight, and the upward force due to Archimedes' principle, called buoyant force, as follows:

Fn = Fg - Ep = δx. V. g - δH20. V. g = 31.4 N

Dividing Fg between Fn, and simplifying common terms, we have:

δx / (δx - δh20) = 34.1 / 31.4 = 1.09

Solving for δx, we get the following value:

δx = 12, 111 Kg/m3, less dense than pure gold, so we can conclude that the crown was not made of pure gold.