# 11.7 Archimedes’ principle  (Page 5/9)

 Page 5 / 9
$\text{apparent weight loss = weight of fluid displaced}$

or

$\text{apparent mass loss = mass of fluid displaced.}$

The next example illustrates the use of this technique.

## Calculating density: is the coin authentic?

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:

${\rho }_{\text{c}}=\frac{{m}_{\text{c}}}{{V}_{c}}=\frac{8\text{.}\text{630 g}}{0\text{.830 c}{\text{m}}^{3}}=\text{10}\text{.}4 g{\text{/cm}}^{3}.$

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!

## Phet explorations: buoyancy

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.

## Section summary

• Buoyant force is the net upward force on any object in any fluid. If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.
• Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid it displaces.
• Specific gravity is the ratio of the density of an object to a fluid (usually water).

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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