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The next example illustrates the use of this technique.
The mass of an ancient Greek coin is determined in air to be 8.630 g. When the coin is submerged in water as shown in [link] , its apparent mass is 7.800 g. Calculate its density, given that water has a density of $1\text{.}\text{000}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$ and that effects caused by the wire suspending the coin are negligible.
Strategy
To calculate the coin’s density, we need its mass (which is given) and its volume. The volume of the coin equals the volume of water displaced. The volume of water displaced ${V}_{\text{w}}$ can be found by solving the equation for density $\rho =\frac{m}{V}$ for $V$ .
Solution
The volume of water is ${V}_{\text{w}}=\frac{{m}_{\text{w}}}{{\rho}_{\text{w}}}$ where ${m}_{\text{w}}$ is the mass of water displaced. As noted, the mass of the water displaced equals the apparent mass loss, which is ${m}_{\text{w}}=8\text{.}\text{630 g}-7\text{.}\text{800 g}=0\text{.}\text{830 g}$ . Thus the volume of water is ${V}_{\text{w}}=\frac{0\text{.}\text{830 g}}{1\text{.}\text{000 g}{\text{/cm}}^{3}}=0\text{.}\text{830}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ . This is also the volume of the coin, since it is completely submerged. We can now find the density of the coin using the definition of density:
Discussion
You can see from [link] that this density is very close to that of pure silver, appropriate for this type of ancient coin. Most modern counterfeits are not pure silver.
This brings us back to Archimedes’ principle and how it came into being. As the story goes, the king of Syracuse gave Archimedes the task of determining whether the royal crown maker was supplying a crown of pure gold. The purity of gold is difficult to determine by color (it can be diluted with other metals and still look as yellow as pure gold), and other analytical techniques had not yet been conceived. Even ancient peoples, however, realized that the density of gold was greater than that of any other then-known substance. Archimedes purportedly agonized over his task and had his inspiration one day while at the public baths, pondering the support the water gave his body. He came up with his now-famous principle, saw how to apply it to determine density, and ran naked down the streets of Syracuse crying “Eureka!” (Greek for “I have found it”). Similar behavior can be observed in contemporary physicists from time to time!
When will objects float and when will they sink? Learn how buoyancy works with blocks. Arrows show the applied forces, and you can modify the properties of the blocks and the fluid.
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