Archimedes purportedly used his principle to verify that the king's crown was pure gold, by weighing the crown while it was submerged in water. Suppose the crown's actual weight was 14.0 N. The densities of gold, silver, and water are 19.3 g/cm^3, 10.5 g/cm^3, and 1.00 g/cm^3, respectively.

(a) What would be the crown's apparent weight if it were made of pure gold?

(b) What would be its apparent weight if it were made of 80% gold and 20% silver, by volume?

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  1. Gold, being denser than silver, will have less volume and less buoyant force acting upon it.

    (a) Crown mass = W/g = 14.0/9.8
    = 1.428 kg
    Crown volume = V = 1428 g/19.3 g/cm^3
    = 74.0 cm^3 = 74.0*10^-6 m^3
    Buoyant force = V*g*(water density)
    = 74*10^-6*9.8*1000 = 0.725 N
    Apparent weight = 14.0 N - 0.725 N
    = 13.3 N

    (b) density of alloy = 0.8*19.3 + 0.2*10.5 = 17.54 g/cm^3
    Volume will be 1.10 times as large to achieve the same crown weight.
    Buoyancy force will be 10% greater. or 0.798 N
    Apparent weight = 13.2 N

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