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11.7 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).

A hydrometer has lead at the bottom and air on top. It floats on the fluid and specific gravity can be directly read from it.
This hydrometer is floating in a fluid of specific gravity 0.87. The glass hydrometer is filled with air and weighted with lead at the bottom. It floats highest in the densest fluids and has been calibrated and labeled so that specific gravity can be read from it directly.

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

fraction submerged = ρ ¯ obj ρ fl size 12{"fraction"`"submerged"= { { { bar {ρ}} rSub { size 8{"obj"} } } over {ρ rSub { size 8{"fl"} } } } } {}

for the density of the object. This yields

ρ ¯ obj = ρ ¯ person = ( fraction submerged ) ⋅ ρ fl . size 12{ { bar {ρ}} rSub { size 8{"obj"} } = { bar {ρ}} rSub { size 8{"person"} } = \( "fraction submerged" \) cdot ρ rSub { size 8{"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

ρ ¯ person = 0 . 970 ⋅ 10 3 kg m 3 = 970 kg m 3 . size 12{ { bar {ρ}} rSub { size 8{"person"} } =0 "." "970" cdot left ("10" rSup { size 8{3} } { {"kg"} over {m rSup { size 8{3} } } } right )="970" { {"kg"} over {m rSup { size 8{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] .)

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The weight of a person can be determined while submerged in a fat tank. Based on this, the percentage of body weight can be calculated.
Subject in a “fat tank,” where he is weighed while completely submerged as part of a body density determination. The subject must completely empty his lungs and hold a metal weight in order to sink. Corrections are made for the residual air in his lungs (measured separately) and the metal weight. His corrected submerged weight, his weight in air, and pinch tests of strategic fatty areas are used to calculate his percent body fat.

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] .

The density of a coin can be determined by measuring its weight in air and its weight submerged in a fluid of known density.
(a) A coin is weighed in air. (b) The apparent weight of the coin is determined while it is completely submerged in a fluid of known density. These two measurements are used to calculate the density of the coin.

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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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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