density
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What is Density?
For any material, its density is defined as its mass per unit volume. Assuming
we are working with a homogeneous object, this amounts to the following
equation:
ρ = m/V
where ρ (the greek symbol "rho") is the density while m is the mass and V is the
volume.
Density is used in modeling many physical systems, one of which is the buoyancy
of objects. Buoyancy is defined as the upward force that keep things afloat.
Archimedes allegedly ran through the streets yelling "eureka! eureka!" after
having found the principles of buoyant force. His equation relates the densities
of objects and the fluids they occupy to their actual weight and the apparent
weight they have when immersed in a fluid. Mathematically, the Archimedes
principle is written as the following:
ρ_{Object}/ρ_{fluid} = F_{Object Weight}/(F_{Object
Weight }- F_{Apparent Immersed Weight})
The above equation can be used in a variety of ways. It was originally conceived
as a way to determine if a substance in question was actually gold or a fake.
One could rewrite the equation to solve for the density of
the object in question. If the density observed upon immersion in a fluid
matches the density of gold, which is a very dense metal, then the object in question
is likely to be gold because fake alloys are usually less dense.
The equation also predicts some fairly intuitive observations. For example, we
all know that a balloon will rise when filled with a hot or lightweight gas, but
why? This is because the weight of a displaced fluid is directly proportional to
the volume of a displaced fluid. If an object has a greater volume
for its mass, it will have a greater buoyant force acting upon it. A balloon is
a thin bubble of a lightweight material filled with a low-density gas
(e.g. helium or heated air). As a balloon inflates, it increases in volume while its mass stays
approximately the same (mostly the mass of the plastic shell as helium is very
light). At some point, the volume of the balloon increases so much that it
displaces enough outside air to increase the buoyant force enough to counter the
force of its own weight. Any increase in volume past this point will cause the
balloon to float upwards until it reaches a layer of air that isn't dense
enough for the same buoyant force to be present.
Measuring Density
Density can
be measured with a hydrometer, which uses the displacement of water to find an object's volume
and hence its density. Other methods of finding the density of objects include the density
integral. For some set of coordinates r, the function ρ(r) expresses the density
of the object at a certain coordinate. One would then be able to find the mass
of the object based on integrating the density function over the volume of the
object.
Once one knows the mass of the object for the
give volume, then according to the first
equation, one would also be able to calculate
the total density of the object. Aside from
these methods, one could use a pycnometer to measure the
density of a fluid or a gas pycnometer for a gas. A digital density meter is
available as well, based on an oscillating U-tube.