Archimedes' principle - The gold crown

Enunciado:

According to the legend, Hiero II of Syracuse asked Archimedes to determine without damaging it if a crown he has ordered was really made of gold. He suspected it was made of a cheaper metal. Archimedes first measured the mass of the crown (m0 = 0.44 kg) and then its apparent mass, when the crown was immersed in water (m’ = 0.409 kg). Using both masses he determined the density of the crown and realized it wasn’t made of gold. How did he come to this conclusion?

Givens: ρAu = 1.93 104 kg/m3; ρH2O = 103 kg/m3

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Solution:

The easiest way to determine if the crown was made of gold is to place the crown on one dish of the balance and the same amount (mass) of pure gold on the other, immersing everything in water (see figure). If the beam of the balance remains horizontal (left) the crown is made of gold. This is so because in this case both objects have the same volume and therefore they displace the same volume of water, so the buoyant force acting on both objects is the same. But if the balance tilts towards the mass of gold (right), the density of the crown will be smaller, its volume will be larger the buoyant force acting on it will be larger as well.

We can determine the density of the crown ρC using the same balance applying Archimedes’ principle.

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As shown in the figure below, we can measure both the mass of the crown m0 (left) and its apparent mass m’ when it is underwater (right).

The magnitude of the apparent weight P’ of the crown will be its weight minus the buoyant force F. And the latter equals the weight of the displaced water:

Now, we can express the volume of the crown in terms of its density ρC. Isolating and substituting the givens of the problem:

Which is smaller than the density of gold, so the crown cannot be made of gold.

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