Density plays a crucial role in Archimedes’ principle. The average density of an object is what ultimately determines whether it floats. If its average density is less than that of the surrounding fluid, it will float. This is because 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 is related to that of the fluid. In Figure 4, for example, the unloaded ship has a lower density and less of it is submerged compared with the same ship 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 = VsubVobj=VflVobj.fraction submerged = VsubVobj=VflVobj. size 12{ { {V rSub { size 8{"sub"} } } over {V rSub { size 8{"obj"} } } } = { {V rSub { size 8{"fl"} } } over {V rSub { size 8{"obj"} } } } } {}
(8)The volume submerged equals the volume of fluid displaced, which we call VflVfl size 12{V rSub { size 8{"fl"} } } {}. Now we can obtain the relationship between the densities by substituting ρ=mVρ=mV size 12{ρ= { {m} over {V} } } {} into the expression. This gives
VflVobj=mfl/ρflmobj/ρ¯obj,VflVobj=mfl/ρflmobj/ρ¯obj,
(9)where ρ¯objρ¯obj size 12{ { bar {ρ}} rSub { size 8{"obj"} } } {} is the average density of the object and ρflρfl size 12{ρ rSub { size 8{"fl"} } } {} is the density of the fluid. Since the object floats, its mass and that of the displaced fluid are equal, and so they cancel from the equation, leaving
fraction submerged=ρ¯objρfl.fraction submerged=ρ¯objρfl. size 12{"fraction"`"submerged"= { { { bar {ρ}} rSub { size 8{"obj"} } } over {ρ rSub { size 8{"fl"} } } } } {}
(10)
We use this last relationship to measure densities. This is done by measuring the fraction of a floating object that is submerged—for example, with a hydrometer. It is useful to define the ratio of the density of an object to a fluid (usually water) as specific gravity:
specific gravity=
ρ
¯
ρw,specific gravity=
ρ
¯
ρw, size 12{"specific"`"gravity"= { {ρ} over {ρ rSub { size 8{w} } } } } {}
(11)where
ρ
¯
ρ
¯
size 12{ρ} {} is the average density of the object or substance and ρwρw size 12{ρ rSub { size 8{w} } } {} is the density of water at 4.00°C. Specific gravity is dimensionless, independent of whatever units are used for ρρ size 12{ρ} {}. If an object floats, its specific gravity is less than one. If it sinks, its specific gravity is greater than one. Moreover, the fraction of a floating object that is submerged equals its specific gravity. If an object’s specific gravity is exactly 1, then it will remain suspended in the fluid, neither sinking nor floating. Scuba divers try to obtain this state so that they can hover in the water. We measure the specific gravity of fluids, such as battery acid, radiator fluid, and urine, as an indicator of their condition. One device for measuring specific gravity is shown in Figure 5.
Specific gravity is the ratio of the density of an object to a fluid (usually water).
Suppose a 60.0-kg woman floats in freshwater with 97.0%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
fraction submerged
=
ρ
¯
obj
ρ
fl
size 12{"fraction"`"submerged"= { { { bar {ρ}} rSub { size 8{"obj"} } } over {ρ rSub { size 8{"fl"} } } } } {}
(12)for the density of the object. This yields
ρ¯obj=ρ¯person=(fraction submerged)⋅ρfl.ρ¯obj=ρ¯person=(fraction submerged)⋅ρfl. size 12{ { bar {ρ}} rSub { size 8{"obj"} } = { bar {ρ}} rSub { size 8{"person"} } = \( "fraction submerged" \) cdot ρ rSub { size 8{"fl"} } } {}
(13)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⋅103kgm3=970kgm3.ρ¯person=0.970⋅103kgm3=970kgm3. 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} } } } } {}
(14)
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 Figure 6.)
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.
