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Archimedes’ principle refers to the force of buoyancy that results when a body is submerged in a fluid, whether partially or wholly. The force that provides the pressure of a fluid acts on a body perpendicular to the surface of the body. In other words, the force due to the pressure at the bottom is pointed up, while at the top, the force due to the pressure is pointed down; the forces due to the pressures at the sides are pointing into the body.

Since the bottom of the body is at a greater depth than the top of the body, the pressure at the lower part of the body is higher than the pressure at the upper part, as shown in [link] . Therefore a net upward force acts on the body. This upward force is the force of buoyancy, or simply buoyancy .

The exclamation “Eureka” (meaning “I found it”) has often been credited to Archimedes as he made the discovery that would lead to Archimedes’ principle. Some say it all started in a bathtub. To read the story, visit NASA or explore Scientific American to learn more.

Density and archimedes’ principle

If you drop a lump of clay in water, it will sink. But if you mold the same lump of clay into the shape of a boat, it will float. Because of its shape, the clay boat displaces more water than the lump and experiences a greater buoyant force, even though its mass is the same. The same is true of steel ships.

The average density of an object is what ultimately determines whether it floats. If an object’s average density is less than that of the surrounding fluid, it will float. The reason is that the fluid, having a higher density, contains more mass and hence more weight in the same volume. The buoyant force, which equals the weight of the fluid displaced, is thus greater than the weight of the object. Likewise, an object denser than the fluid will sink.

The extent to which a floating object is submerged depends on how the object’s density compares to the density of the fluid. In [link] , for example, the unloaded ship has a lower density and less of it is submerged compared with the same ship when loaded. We can derive a quantitative expression for the fraction submerged by considering density. The fraction submerged is the ratio of the volume submerged to the volume of the object, or

fraction submerged = V sub V obj = V fl V obj .

The volume submerged equals the volume of fluid displaced, which we call V f l . Now we can obtain the relationship between the densities by substituting ρ = m V into the expression. This gives

V fl V obj = m fl / ρ fl m obj / ρ obj ,

where ρ obj is the average density of the object and ρ fl is the density of the fluid. Since the object floats, its mass and that of the displaced fluid are equal, so they cancel from the equation, leaving

fraction submerged = ρ obj ρ fl .

We can use this relationship to measure densities.

Figure A is a drawing of an unloaded ship floating high in the water. Figure B is a drawing of a loaded ship floating deeper in the water.
An unloaded ship (a) floats higher in the water than a loaded ship (b).

Calculating average density

Suppose a 60.0-kg woman floats in fresh water 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

for the density of the object. This yields

ρ obj = ρ person = (fraction submerged) · ρ fl .

We know both the fraction submerged and the density of water, 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 .

Significance

The woman’s density is less than the fluid density. We expect this because she floats.

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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