means that an all-silver crown would have to have nearly twice the volume
as an all-gold one to weigh the same. A smaller fraction of silver mixed with
the gold, though, would have been tricker to detect. See the sidebar for the
details of the calculation.
Archimedes could have tested this by putting a known gold block of the right
mass underwater, checking the water level, and then looking to see if he got
a different result when he lowered the crown into the water. Of course, if the
crown was more delicate or a smaller fraction of the gold was a different
metal, then the difference might have been undetectable by the technologies
of the time. Air bubbles, either trapped in a hollow space within the metal as
it was being shaped or captured on the surface of the crown as it was being
submerged, would be added to the apparent volume.
FINDING DENSITY FROM DISPLACEMENT
If you want to find out the density of something oddly shaped, like a handful of
screws, you can proceed as follows.
First, weigh the object(s) of interest on a postal or kitchen scale, and write
down the mass (we’ll assume we are using grams). In our case (Figure 8-26)
our screws weighed 24.29g.
Fill a graduated cylinder, or at least the most precise measuring cup you
have, and fill it with enough water to cover the object you are analyzing (but
don’t put the object in yet). A tall, narrow container can be read more pre-
cisely because adding a small amount of volume will make the water level
rise more. Note how much water is in the measuring cup (30ml in our case,
in Figure 8-27).
Now put in your objects and see what the difference in volume is (being
careful with the units!) In other words, see how much the water level went
up. If your container is marked in ml, that is the same as cm
3
. If it is marked
in cups, 1 US cup equals 236.588cm
3
so you’ll need to multiply your result by
that to get cm
3
. In our case (Figure 8-28) the water rose by about 4 ml.
The mass divided by the volume difference (with and without the object
in the water) will give you the average density of the object. With precise
enough measurements, you might be able to look up densities of common
materials and deduce what your submerged object might be made of. This
will work well for things like metal, as long as you are putting in some-
thing big enough so that it is easy to see. In our case with our screws of an
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unknown steel alloy, we got 24.29g/4cm
3
or about 6g/cm
3
. Steel is about 8g/
cm
3
, so this is a little inaccurate because of our limited ability to measure
water volume within a milliliter, and the tendency of air to get trapped on the
screws. Adding more (identical) screws will give you larger mass and volume
numbers, which may be easier to measure accurately.
This method will require care for anything porous that absorbs water. If you
completely saturate the porous material, you’ll be measuring its density
when it is full of water, not as it would be on land. Also, this technique only
works for things that are heavier than water. In the next section, we’ll see
how to measure something lighter than water.
ARCHIMEDES’ PRINCIPLE
Despite the charm of the crown story, Archimedes is usually remembered for his
realization that the situation is different for a body that is floating in (or on) a fluid.
For our counterfeit crown test, we assumed the crown sank and was resting on
the bottom. But floating is a little more complicated to think about.
Why does a boat float at all? Some ship-building materials, like oak and other
types of wood commonly used in ships, are less dense than water, and so it’s not
all that surprising. However, what about metal ship hulls, including ones that are
carrying many tons of cars or electronics across the Pacific? Archimedes real-
ized that a floating body will displace an amount of water (or other fluid) of equal
weight. Today, this is called Archimedes’ Principle. Note that the word “weight” is
appropriate here, since we are measuring a force. In zero gravity, the boat, water,
deck chairs, and passengers would all be floating around in a big mess. Gravity
is required for orderly floating.
FIGURE 826: Screws on a postal scale FIGURE 828: Measuring cup with screwsFIGURE 827: Measuring cup before
adding screws
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BUOYANCY FORCE
Assuming we are on a planet, anything placed in water is being pushed and pulled by several forces.
First, gravity pulls down on it. If its average density is higher than that of water (1g/cm
3
), then it sinks.
(It’s not an accident that 1cm
3
of water has a mass of 1g; the metric system was designed that way since
it is so often more convenient compared to needing to divide by 62.4 lbs/ft
3
all the time.) When some-
thing sinks, it has to push water out of the way. You can think of this as the water exerting a force equal
to the weight of the water pushed out of the way, called buoyancy. If the density of an object is exactly the
same as that of water, it will displace exactly the same amount of water as itself and will be fully sub-
merged, but will not rise or sink unless some other force acts on it.
If, however, its average density of an object is lower than that of water (say, a metal-hulled ship with
thousands of cars on board but a lot of air in the hull) it will float. But some of it will be below the
surface of the water. Ships are rated by their maximum displacement, which is how much the boat, fuel,
cargo, passengers, and anything else can weigh in total before the ship sinks. Or, in other words, the
weight of the water pushed aside by the underwater part of the ship.
Let’s take a step back now to realize that we can deduce the volume of an object if it sinks on its own,
but not its weight. We can find the weight of something if it floats, but not its volume. (Assuming, that is,
that we are deducing these quantities purely from the amount of water displaced.)
What I have to measure
another way
What I can deduce
Floating ship Volume Mass
Sunken treasure Weight Volume
However, if you have something that does not float, you can drop it in water, find out its volume, then put
it in a very lightweight container that will float to get the weight. The average density of the container, the
air, and the object will let it float, you will be measuring the combined weight of the enclosed air and the
container as well as your object of interest.
If we are in a known gravity field (like the surface of the earth), the mass and weight measurements can
easily be deduced from each other, as we noted in the Density section of this chapter. Once we have both
mass (or weight) and volume, we can calculate the density to find out if our crown is made of gold or
something else. We’d find the measurement by looking up the density of candidate materials or calcu-
lating likely mixtures until we got a match.
When we talk about objects floating in “a fluid” we can also mean floating in the air, like a blimp or
a hot-air balloon. Hot air is less dense than cold air, so heating the air in a hot-air balloon will let the
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buoyant force become bigger than gravity, which makes the balloon rise.
It’s obvious that hot-air balloons need to be light to fly. However, now that we
understand ship displacement, we see how important it is for ships to be light,
too. Otherwise, the average density for the ship plus a useful amount of cargo
would get higher than the density of water (and would sink). Wood is significantly
less dense than water (oak is around 70% the density of water, plus or minus
10%) and therefore has been a successful shipbuilding material for millennia.
For ships, it also matters whether you are in freshwater or seawater. Seawa-
ter is denser than freshwater, and so a ship will displace less seawater than
it would freshwater. Some large cargo ships have a horizontal line, called a
Plimsoll line, painted on their hull to mark the waterline under various condi-
tions and guard against overloading.
BUOYANCY EXPERIMENT
It is tricky to measure displacement unless the object is big. 10mL is usually
just a fine line on a measuring cup, and seeing differences any smaller than
10mL is hard without equipment beyond what most people will have at home.
With that said, let’s try to see if we can measure the density of a gold ring
with a precise measuring cup.
First, we will fill the measuring cup with enough water to float a ring in a
soda bottle cap, which we will use as a miniature boat (Figure 8-29).
Now, we will carefully take a gold ring and put it in a bottle cap. It just barely
floats. The water rises about 10mL above where it was before we put in the
ring (Figure 8-30). That would imply that the collective mass of the bottle cap,
FIGURE 829: Measuring cup with just
water
FIGURE 830: Measuring cup with “boat”
and ring
FIGURE 831: The ring’s displacement
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the ring, and the air around them should be about 10g, since each displaced
mL is equal to 1g of water.
We can then check our number by putting the bottle cap plus ring on a
kitchen or postal scale and seeing if we get roughly the same number. The
bottle cap plus the ring was 9.79g. The ring alone was 7.79g.
When we let the ring sink (Figure 8-31) it displaced less than 1 mL, so that
would imply that the density of this ring (which is marked as being composed
of 14K gold) would be something more than 8g/cm
3
. The density of 14K gold
(which is defined as being 58.3% gold, or 14/24 gold; 24K gold is essentially
pure gold) is about 13g/cm
3
. We can see it is pretty challenging to do these
measurements for something small with home equipment; if you happened
to have a big block of gold lying around, it would be a lot easier to try this
measurement.
SUMMARY AND LEARNING MORE
Volume, density, and displacement have enormous practical value, and this
chapter got you started with all three. Now that you have some formulas for
finding the density of basic shapes, you might try estimating values of more
complicated ones by imagining them as two or more basic ones smushed
together, or consider how you might use Cavalieri’s Principle to find a simpler
equivalent to a complex volume. For example, the volume of ice cream in a
cone plus a scoop might be a half-sphere plus a cone. If your local ice-cream
place is particularly generous, It might be closer to a full sphere plus a cone,
or it may be a cylinder, depending on the type of ice cream scoop they use.
Wikipedia has good articles under “Volume,” “Density” and “Displacement
(ship)” that parallel the ideas in this chapter.
Shipbuilding is another area to explore, and we encourage you to play around
with floating plastic cups, weighted with pennies, in a bowl of water to get a
sense of how displacement (and stability of a floating object) work.
You might also explore and experiment with floating objects more generally,
like a 3D printed part with infill and see if you can think about what their average
density might be, based on how much of the object is below the waterline. Calcu-
lating this is complicated for a 3D print, since the print has outer shells and infill.
The density of PLA is around 1.24g/cm
3
, although it can vary with additives.
You can also try to estimate the percentage of an iceberg that is underwa-
ter. For example: the density of freshwater ice is 0.92g/cm
3
, and seawater is
about 1.0273g/cm
3
. The ratio between the two is 0.92/1.03 or 89%. Therefore
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