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A long, long time ago, King
Hiero II of ancient Greece
summoned a jeweler
to make himself
a crown out of pure gold.
However, the king soon became
distrustful of the jeweler
and wondered if the crown
was really made of pure gold.
To answer the question, the
king summoned his most trusted
adviser, Archimedes.
The king told
Archimedes to figure out
whether the crown was
indeed made from pure gold.
He knew that finding its
density would be the answer.
But how would he
find its volume,
which was needed to
calculate its density?
So what is density, and
why does density matter?
Well, density can be thought of
as how compact a material is.
In other words, how much
material is in a substance,
and how much volume does
that material take up?
We have right here a
pile of cotton balls
and a very small amount
of modeling clay.
If we put both of these
on a scale side by side,
we see that both the cotton
balls and the modeling clay
weigh 10.1 grams.
Because they have the
same mass, the material
in the modeling clay is much
more compact than the material
in the cotton balls.
Therefore, the material
in the modeling clay
has a higher density
than the cotton balls.
Now, if we look at the
actual equation for density,
density equals the mass,
which is measured in grams,
divided by the volume, which is
measured in centimeters cubed.
And volume is
traditionally found
by multiplying the length times
the width times the height.
We can also visualize density
with another experiment
that we have here.
We have four canisters
here of different materials
you can find around the house.
So you can try this experiment
at home if you like.
In this first one
we have cotton.
This one, we have tea.
In this one we have sprinkles.
And in this last one we have
honey, which is my favorite.
So if we put each of these
into the bath of water,
we should see them float
at different heights
because the density between
these materials is different.
So cotton floats very
high in the water.
Tea floats a little bit lower.
And we add in sprinkles,
and finally honey.
Whoa.
So because honey
sank to the bottom,
it has a very high density,
much greater than water.
The cotton balls, on the other
hand, are floating very high
and have a very low
density, less than water.
Now, density is a
materials property.
So it doesn't matter
the shape of an object
or the size of an object.
If it's the same material
between two different objects,
it's going to be
the same density.
For example, the cotton
balls that we have here
have the same density
as the cotton balls
from the first experiment.
So because the density doesn't
change between a material,
it's very good to
identify a material
if you know the density.
Finally, in this
last experiment,
I'll combine equal
parts of honey, water,
and oil into this glass, and
we see three layers form.
On the bottom is honey because
it has the highest density.
In the middle is water.
And on the top is oil because
it has the lowest density.
So how did Archimedes find
out the density of the crown?
Well, we know Archimedes must
know the mass of the crown,
because that's
relatively easy to find.
But how did Archimedes figure
out the volume of this crown
if he couldn't measure
the length, width,
and height of the crown?
Well, let's find out.
Archimedes thought
and thought about how
to measure the volume of the
crown to find its density.
After much
deliberation, he decided
to take a break by
taking a nice, hot bath.
However, as he
stepped into the bath,
it dawned on him
for the first time
that the water level rose to
offset the volume of his body.
He knew this was the
answer he was seeking.
And he said, aha, yes!
Now we know how Archimedes
found the volume
of an irregularly-shaped object.
That is, he realized that if he
placed this object in a liquid,
the amount of
liquid it displaced
is equal to its volume.
So now that we know
this, let's try
this experiment for ourselves
to see if Archimedes was right.
So first, we have a miniature
version of the king's crown
that Archimedes used.
First, we're going to determine
the mass of this object
by placing this on our scale.
This miniature crown has
a mass of 93.7 grams.
Now, let's find the
volume of this object.
And so first, we
filled up a glass
full of an orange-colored
water, and we filled it
all the way to the brim.
And now, when I place
this crown into the water,
now we see all the
water down here
overflowed and is collected.
And now I need to carefully
take out this inner glass.
So with the help of
a napkin to take up
some of the excess water, I'll
carefully remove the glass.
Presumably, the volume of
liquid that's in this container
equals the volume of
our miniature crown.
So now I'm going to very
carefully pour this water
into our graduated cylinder
so we can measure its volume.
And here we go.
So it looks like we are right
at about 64.2 milliliters.
And so now we know the volume
of our miniature crown.
The volume is 64.2
milliliters, which also
equals 64.2 centimeters cubed.
Let's see if Archimedes was
right by reshaping our modeling
clay, our crown,
into a cube shape
so that we can measure
its length, width,
and height with a ruler.
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So now that we've smashed
up our miniature crown,
let's take its dimensions.
Using our ruler, we
see that the length
is about 4.4
centimeters, its height
is about 4.4 centimeters,
and its width
is about 3 centimeters.
Remembering that volume is the
length times height times width
and multiplying all of
these values together,
we find the overall volume
is 58.1 centimeters cubed.
Now, accounting for
small experimental error,
this value is very
close to the value found
by the water displacement
method, which proves
that Archimedes was right.
So now Archimedes
realized the volume
of the gold crown
using this method.
And when he did--
He ran through the streets
shouting, eureka, eureka,
for he found the crown
was not made of pure gold.
The jeweler had
cheated the king,
and Archimedes proved it
by determining its density.
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