Photo by Mark Deneyer.
You may have heard that 90% of an iceberg lies below the water.
Why is that? What determines whether something sinks or floats, and,
if it floats, how much of it remains above the surface? This is the
question of buoyancy, which is governed by a
simple principle that Archimedes figured out, supposedly while taking
a bath, after which he made his famous shout 'Eureka!'.
Buoyancy
Buoyancy is the upward force exerted on an object immersed in a
fluid. Of course, water is the most common fluid, but
buoyancy also applies to hot air balloons (where the fluid is the
surrounding air) and many other situations. What's the basic idea?
Archimedes figured out that the key to buoyancy is how much volume the
object displaces compared to its weight.
Archimedes Principle of buoyancy states that the upward force
on an object in a fluid is equal to the weight of the fluid
that is displaced. If this bouyant force is less than the weight
of the object itself, the object will be left with a net downward
force and will sink. If the object floats, it floats enough that the
bouyant force exactly balances its weight.
For solid, uniform objects
like an iceberg, this boils down to the object's mass density,
its mass divided by its volume, usually represented by the Greek
letter .
For something like a boat hull,
which is hollow, not uniform, you have to just look at the total weight and
the volume of displaced water.
Example: Icebergs
So let's take the case of the iceberg.
Lets say it has mass M_{i} and volume
V_{i}.
Their ratio is given by the mass density of ice:
M/V = _{i} = 0.917
g/cm³. Since we already know it floats, lets say that the
volume below the surface of the water is
V_{w}. This is the volume of water displaced, and
the buoyant force is equal to the weight of that displaced
water, which has mass M_{w} =
V_{w}_{w}.
The mass density of liquid water was originally used to define the
gram, so it has the convenient metric value
_{w} = 1
g/cm³ (or 1000 kg/m³).
The weight of an object is given by its mass times the
acceleration of gravity, g = 9.8 m/s²:
W = Mg
The iceberg has weight W_{i} = M_{i}g and
the buoyant force is equal to the weight of the displaced water,
W_{w} = M_{w}g. Furthermore, since the
iceberg is floating, its weight exactly balances the buoyant force:
W_{w} = W_{i}
M_{w}g = M_{i}g
V_{w}_{w}g =
V_{i}_{i}g
V_{w} = _{i}/_{w} V_{i}
So, the fraction of ice underwater,
V_{w}/V_{i}, is given by the ratio of
densities _{i}/_{w}=0.917. Over 90% of
an iceberg's volume (and mass) is underwater.
As you can see, the convenient definition of the gram gives us a quick
way to see how much of a floating substance lies below the surface of fresh water:
the fraction is equal to that substance's mass density in
g/cm³. Ice has mass density 0.917 g/cm³, so
91.7% lies below the surface of water. In fact, it isn't quite that,
because icebergs are actually found on seawater, which is more
dense than freshwater. But that's a minor detail.
Summary
 Archimede's Principle of bouyancy states that the bouyant
force on an object is equal to the weight of the fluid displaced by
that object.
 The underwater fraction of a substance floating on water is given
by that substance's mass density in g/cm³.
