8.6 Archimedes’ principle  (Page 4/9)

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Specific gravity

Specific gravity is the ratio of the density of an object to a fluid (usually water).

Calculating average density: floating woman

Suppose a 60.0-kg woman floats in freshwater with $97.0%$ of her volume submerged when her lungs are full of air. What is her average density?

Strategy

We can find the woman’s density by solving the equation

$\text{fraction submerged}=\frac{{\overline{\rho }}_{\text{obj}}}{{\rho }_{\text{fl}}}$

for the density of the object. This yields

${\overline{\rho }}_{\text{obj}}={\overline{\rho }}_{\text{person}}=\left(\text{fraction submerged}\right)\cdot {\rho }_{\text{fl}}.$

We know both the fraction submerged and the density of water, and so we can calculate the woman’s density.

Solution

Entering the known values into the expression for her density, we obtain

${\overline{\rho }}_{\text{person}}=0\text{.}\text{970}\cdot \left({\text{10}}^{3}\frac{\text{kg}}{{\text{m}}^{3}}\right)=\text{970}\frac{\text{kg}}{{\text{m}}^{3}}.$

Discussion

Her density is less than the fluid density. We expect this because she floats. Body density is one indicator of a person’s percent body fat, of interest in medical diagnostics and athletic training. (See [link] .)

There are many obvious examples of lower-density objects or substances floating in higher-density fluids—oil on water, a hot-air balloon, a bit of cork in wine, an iceberg, and hot wax in a “lava lamp,” to name a few. Less obvious examples include lava rising in a volcano and mountain ranges floating on the higher-density crust and mantle beneath them. Even seemingly solid Earth has fluid characteristics.

More density measurements

One of the most common techniques for determining density is shown in [link] .

An object, here a coin, is weighed in air and then weighed again while submerged in a liquid. The density of the coin, an indication of its authenticity, can be calculated if the fluid density is known. This same technique can also be used to determine the density of the fluid if the density of the coin is known. All of these calculations are based on Archimedes’ principle.

Archimedes’ principle states that the buoyant force on the object equals the weight of the fluid displaced. This, in turn, means that the object appears to weigh less when submerged; we call this measurement the object’s apparent weight . The object suffers an apparent weight loss equal to the weight of the fluid displaced. Alternatively, on balances that measure mass, the object suffers an apparent mass loss equal to the mass of fluid displaced. That is

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