Buoyancy and Archimedes Principle
The buoyant force on an object is equal to
the weight of the fluid displaced by that object
Archimedes principle states
A body wholly or partially immersed in a fluid is buoyed up by
a force equal in magnitude to the weight of the fluid displaced
by the body.
The volume of an object is
equal to its mass divided by density. v = m/d
Following are three
applications of buoyancy and Archimedes principle.
1. Given two identical buckets filled to the brim with
water. One bucket has a floating object in it, such as
a block of ice or a piece of wood. Both buckets with content
weigh the same. The floating object displaces a volume of water
equal to its weight.
2. If a
rock in a boat floating in a pond is tossed into the water, and
the rock sinks, the water level of the pond will drop.
This is because the rock
displaces its mass while in the boat, and displaces its volume
in the water. This assumes the rock sinks and is therefore more
dense than water.
This can be proven mathematically as follows below.
The increase in volume of the
pond, when the boat, with a rock in it, is placed in the pond,
is equal to the mass of the boat divided by the density of
water, plus the mass of the rock divided by the density of
water.
The buoyant force on the boat
with the rock in it is equal to the weight of the rock and the
boat. When the boat, with the rock in it, is floating in the
pond, the volume of water displaced is the combined mass of the
boat and rock divided by density of water.
The floating boat and with rock displace a volume of
water that can be expressed as:
mass of
boat
mass of rock
V1 =
+
density
of
water
density of water
Remember that volume of any
material is equal to mass over density
When the rock is in the
water, it displaces its volume, not its mass. The volume
of water displaced by the submerged rock and floating boat is
the mass of the boat divided by the density of water, plus the
volume of the rock. The volume of the rock can be expressed as
its mass divided by density. The rock is more dense than the
water, evidenced by the fact that it sinks.
The volume of water now
displaced by the floating boat and rock submerged in water is
mass of the boat divided by density of water, plus mass of the
rock divided by density of the rock.
The floating boat and
submerged rock displace a volume of water that can be expressed
as:
mass of
boat
mass of rock
V2
=
+
density of
water
density of rock
Subtract the second
expression from the first and the result is
mass of
rock
mass of rock
V1 - V2
=
+
density of
water
density of rock
Look at the right side of
this equation. On the right hand side of the expression, the
mass of rock is numerator in both fractions. Density of rock is
larger than density of water, meaning the second fraction is
smaller than the first, and the result of the subtraction is a
number greater than zero and therefore positive. This means that
V1 - V2 is a positive number, which can only be true of V1 is
larger than V2. V1 represents the volume of water displaced by
the boat with the rock in it, and V2 represents the volume of
water displaced by the floating boat and submerged rock, which
has now been algebraically proven to be a smaller value.
Therefore the water level of the pond goes down when the rock is
thrown out of the boat and into the water.
3. Just
over ten percent of an iceberg sits above water. This is because
ice is less dense than water.
The density of water is about
997 kilograms per cubic meter. Salt water, according to most
physics text books, weighs about 1,024 kilograms per cubic
meter. When water freezes it squeezes out most of the salt brine
and becomes relatively pure frozen water, with a density of
about 917 kilograms per cubic meter. Ice floats because it is
less dense than water.
The weight of a floating
iceberg, or any other floating object, is equal to its density
times volume times gravity.
Weight of
iceberg = Density of ice * Volume of iceberg * g
The volume of water displaced
by a floating object has the same weight as the submerged
portion of the floating object. This is equal to the buoyant
force on the floating object. The buoyant force is equal to the
density of water, times the volume of displaced water, times
gravity.
Buoyant
force = Density of water * Volume of displaced water * g
Weight of displaced water =
Density of water * Volume of displaced water *
g
A floating iceberg is in a
state of equilibrium. This means the buoyant force is equal to
the weight of the iceberg.
Buoyant
force = Weight of iceberg
Density of
ice * volume of iceberg * g =
Density of
water * volume of displaced water * g
cancelling
g which exists on both sides leaves
Density of
ice * volume of iceberg = Density of water * volume of displaced
water
Divide
both sides by density of salt water and by volume of iceberg
gives
Density of
ice/density of water = volume displaced water / volume of
iceberg
Density of
ice = 917 kg/m^3
Density of
salt water = 1024 kg/m^3
Meaning 917/1024 = volume of
displaced water / volume of iceberg = 0.896 = 89.6 percent
The volume of water displaced
is the volume of the submerged portion of the iceberg, which is
89.6 % of the iceberg, with the remaining 10.4 percent above
water.