Archimedes (of the famed buoyancy principle) was asked by a wicked monarch to in

Question

Archimedes (of the famed buoyancy principle) was asked by a wicked monarch to investigate a crown that, was supposedly made of pure gold. Actually, the equally wicked artisan mixed gold and silver in the manufacture of the crown. You may assume that the volume, of the mixture is the combined volume of the components. Archimedes knew that gold has a density of 19. 3 g/cm3. and that Silver has a density of 10. 5 g/cm3. He weighed the crown in air and found it to weigh 9. 81 N. He weighed it a second time in water and found it to 9. 221 N. Based on these measured weights, Archimedes was able to determine the fraction, by weight, of gold in the crown. Now. I would like you to do the same calculation.

Explanation / Answer

For air: [V_g*_g + V_s*_s]*g = 9.81

So, [V_g*19.3*10^3 + V_s*10.5*10^3]*9.81 = 9.81

19300*V_g + 10500*V_s = 1................eqn 1

For water: [V_g*_g + V_s*_s]*g - (V_g+V_s)*_w*g = 9.221

9.81 - (V_g+V_s)*1000*9.81 = 9.221

V_g + V_s = 6.004*10^-5.................................eqn 2

Solving eqns 1 and 2, 19300*(6.004*10^-5 - V_s) + 10500*V_s = 1

V_s = 1.804397*10^-5 m^3 = 18.04397 cm^3

V_g = 4.1996*10^-5 m^3 = 41.996 cm^3

Weight of gold = 41.996*19.3 = 810.54 g

Weight of silver = 18.04397*10.5 = 189.46 g

% of gold = 810.54/(810.54+189.46)*100 = 81.05 %

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