Archimedes supposedly was asked to determine whether a crown made for the king consisted of pure gold (density of gold is 19.3 * 103 kg/m3). According to legend, he solved this problem by weighing the crown first in air and then in water. Suppose the scale read 8.90 N when the crown was in air and 7.80 N when it was in water. Find the density. What should Archimedes have told the king?

First, we can find the mass of the object in air

Since W = mg

m = W/g = (8.9) / (9.8) = 0.908 kg

Then, by Archimedes principle, we can find its volume. The volume is found by the weight of the water displaced by the formula

W = Vρg

The Weight is the difference in scale readings. The density of water is 1000 kg/m3

(8.9 - 7.8) = V (1000) (9.8)

Thus V = 1 X 10-4 m3

Then, since density is mass / volume

ρ = 0.908/1 X 10-4

ρ = 8254 kg/m3 which is 8 X 103 kg/m3

The density of gold is 19.3 X 103 kg/m3

Since those densities are not the same, the crown is either hollow or not pure gold