http://www.ck12.org Chapter 12. Fluid Mechanics
silver. The problem was that the finished crown weighed the same as the gold given to the goldsmith. How could
Archimedes determine if the crown was pure gold without damaging it?
There is no way of knowing the exact amount of gold given to the goldsmith, nor is this particularly relevant. We
seek only a method whereby Archimedes could have made a conclusive statement regarding the honesty of the
goldsmith. There are several methods he could have used, none of which require any calculation. The most accurate
was given by Galileo, which we will use.
To make the problem interesting, we’ll choose a 500 g mass of gold. Furthermore, we’ll assume that the goldsmith
kept 20% (100 g) of the gold for himself and replaced it with an equal mass of silver. Note that this method assumes
that the volumes of gold and silver are additive.
a. Calculate the volume of 100 grams of gold. Gold has a density of 19. (^3) cmg 3.
Answer:
ρ=Vm→V=mρ= 19100. 3 gg
cm^3
= 5. 181 → 5. 18 cm^3.
b. Calculate the volume of 100 grams of silver. Silver has a density of 10. (^5) cmg 3.
Answer:
ρ=Vm→V=mρ= 10100. 5 gg
cm^3
= 9. 524 → 9. 52 cm^3
c. What is the volume of 500 g of gold and 400 grams of gold alloyed with 100 grams of silver?
Answer:
The method below is one of many.
Volume of 500 grams of gold=^5.^181 cm
3
100 g (^500 g) =^25.^905 →^25.^9 cm
3
Volume of 400 grams of gold + 100 grams of silver=
5. 181 cm^3
100 g
( 400 g)+
9. 52 cm^3
100 g
( 100 g)
= 30. 244 → 30. 2 cm^3
It seems clear the crown has a larger volume than the original volume of the gold given to the goldsmith. If a scale
balance in air had the crown on one side and the original amount of gold on the other, it would balance since the
weights are equal. But if the balance and its load were submerged in water, the buoyant force on the crown would be
greater since it displaces a greater volume of water. And since the buoyant force is greater on the crown, the scale
would sense a smaller weight on the side with the crown, and so this side of the scale would rise.
It has often been said that Archimedes filled a tank of water to its brim and placed the crown in it. He then either
measured the overflow or the amount the water level dropped in the tank compared to when it was full. He repeated
the experiment for an identical quantity of gold given to the goldsmith. The problem with this method is that unless
the original quantity of gold was substantial, the difference in volume would be difficult to measure. The difference
for the problem above is only 4. 3 cm^3. Galileo reasoned that it would be more accurate to simply see if the scale
became unbalanced. See if you can show that the buoyant force on the crown is 1. 166 → 1 .17 times greater than the
buoyant force on the original volume of gold given the goldsmith.
http://phet.colorado.edu/en/simulation/buoyancy