Archimedes' principle states that the upward
buoyant force that is exerted on a body immersed
in a fluid, whether fully or partially submerged,
is equal to the weight of the fluid that the
body displaces and acts in the upward direction
at the center of mass of the displaced fluid.
Archimedes' principle is a law of physics
fundamental to fluid mechanics. It was formulated
by Archimedes of Syracuse.
== Explanation ==
In On Floating Bodies, Archimedes suggested
that (c. 250 BC):
Any object, wholly or partially immersed in
a fluid, is buoyed up by a force equal to
the weight of the fluid displaced by the object.
Practically, Archimedes' principle allows
the buoyancy of an object partially or fully
immersed in a fluid to be calculated. The
downward force on the object is simply its
weight. The upward, or buoyant, force on the
object is that stated by Archimedes' principle,
above. Thus, the net force on the object is
the difference between the magnitudes of the
buoyant force and its weight. If this net
force is positive, the object rises; if negative,
the object sinks; and if zero, the object
is neutrally buoyant - that is, it remains
in place without either rising or sinking.
In simple words, Archimedes' principle states
that, when a body is partially or completely
immersed in a fluid, it experiences an apparent
loss in weight that is equal to the weight
of the fluid displaced by the immersed part
of the body.
== Formula ==
Consider a cuboid immersed in a fluid, with
one (hence two: top and bottom) of its sides
orthogonal to the direction of gravity (assumed
constant across the cube's stretch). The fluid
will exert a normal force on each face, but
only the normal forces on top and bottom will
contribute to buoyancy. The pressure difference
between the bottom and the top face is directly
proportional to the height (difference in
depth of submersion). Multiplying the pressure
difference by the area of a face gives a net
force on the cuboid – the buoyancy, equaling
in size the weight of the fluid displaced
by the cuboid. By summing up sufficiently
many arbitrarily small cuboids this reasoning
may be extended to irregular shapes, and so,
whatever the shape of the submerged body,
the buoyant force is equal to the weight of
the displaced fluid.
weight of displaced fluid
=
weight of object in vacuum
−
weight of object in fluid
{\displaystyle {\text{ weight of displaced
fluid}}={\text{weight of object in vacuum}}-{\text{weight
of object in fluid}}\,}
The weight of the displaced fluid is directly
proportional to the volume of the displaced
fluid (if the surrounding fluid is of uniform
density). The weight of the object in the
fluid is reduced, because of the force acting
on it, which is called upthrust. In simple
terms, the principle states that the buoyant
force (Fb) on an object is equal to the weight
of the fluid displaced by the object, or the
density (ρ) of the fluid multiplied by the
submerged volume (V) times the gravity (g)
or Fb = ρ x g x V. Thus, among completely
submerged objects with equal masses, objects
with greater volume have greater buoyancy.
Suppose a rock's weight is measured as 10
newtons when suspended by a string in a vacuum
with gravity acting on it. Suppose that, when
the rock is lowered into water, it displaces
water of weight 3 newtons. The force it then
exerts on the string from which it hangs would
be 10 newtons minus the 3 newtons of buoyant
force: 10 − 3 = 7 newtons. Buoyancy reduces
the apparent weight of objects that have sunk
completely to the sea floor. It is generally
easier to lift an object up through the water
than it is to pull it out of the water.
For a fully submerged object, Archimedes'
principle can be reformulated as follows:
apparent immersed weight
=
weight of object
−
weight of displaced fluid
{\displaystyle {\text{apparent immersed weight}}={\text{weight
of object}}-{\text{weight of displaced fluid}}\,}
then inserted into the quotient of weights,
which has been expanded by the mutual volume
density of object
density of fluid
=
weight
weight of displaced fluid
{\displaystyle {\frac {\text{density of object}}{\text{density
of fluid}}}={\frac {\text{weight}}{\text{weight
of displaced fluid}}}}
yields the formula below. The density of the
immersed object relative to the density of
the fluid can easily be calculated without
measuring any volume is
density of object
density of fluid
=
weight
weight
−
apparent immersed weight
.
{\displaystyle {\frac {\text{density of object}}{\text{density
of fluid}}}={\frac {\text{weight}}{{\text{weight}}-{\text{apparent
immersed weight}}}}.\,}
(This formula is used for example in describing
the measuring principle of a dasymeter and
of hydrostatic weighing.)
Example: If you drop wood into water, buoyancy
will keep it afloat.
Example: A helium balloon in a moving car.
When increasing speed or driving in a curve,
the air moves in the opposite direction to
the car's acceleration. However, due to buoyancy,
the balloon is pushed "out of the way" by
the air, and will actually drift in the same
direction as the car's acceleration.
When an object is immersed in a liquid, the
liquid exerts an upward force, which is known
as the buoyant force, that is proportional
to the weight of the displaced liquid. The
sum force acting on the object, then, is equal
to the difference between the weight of the
object ('down' force) and the weight of displaced
liquid ('up' force). Equilibrium, or neutral
buoyancy, is achieved when these two weights
(and thus forces) are equal.
== Refinements ==
Archimedes' principle does not consider the
surface tension (capillarity) acting on the
body. Moreover, Archimedes' principle has
been found to break down in complex fluids.
== Principle of flotation ==
Archimedes' principle shows the buoyant force
and displacement of fluid. However, the concept
of Archimedes' principle can be applied when
considering why objects float. Proposition
5 of Archimedes' treatise On Floating Bodies
states that:
Any floating object displaces its own weight
of fluid.
In other words, for an object floating on
a liquid surface (like a boat) or floating
submerged in a fluid (like a submarine in
water or dirigible in air) the weight of the
displaced liquid equals the weight of the
object. Thus, only in the special case of
floating does the buoyant force acting on
an object equal the objects weight. Consider
a 1-ton block of solid iron. As iron is nearly
eight times as dense as water, it displaces
only 1/8 ton of water when submerged, which
is not enough to keep it afloat. Suppose the
same iron block is reshaped into a bowl. It
still weighs 1 ton, but when it is put in
water, it displaces a greater volume of water
than when it was a block. The deeper the iron
bowl is immersed, the more water it displaces,
and the greater the buoyant force acting on
it. When the buoyant force equals 1 ton, it
will sink no farther.
When any boat displaces a weight of water
equal to its own weight, it floats. This is
often called the "principle of flotation":
A floating object displaces a weight of fluid
equal to its own weight. Every ship, submarine,
and dirigible must be designed to displace
a weight of fluid at least equal to its own
weight. A 10,000-ton ship's hull must be built
wide enough, long enough and deep enough to
displace 10,000 tons of water and still have
some hull above the water to prevent it from
sinking. It needs extra hull to fight waves
that would otherwise fill it and, by increasing
its mass, cause it to submerge. The same is
true for vessels in air: a dirigible that
weighs 100 tons needs to displace 100 tons
of air. If it displaces more, it rises; if
it displaces less, it falls. If the dirigible
displaces exactly its weight, it hovers at
a constant altitude.
While they are related to it, the principle
of flotation and the concept that a submerged
object displaces a volume of fluid equal to
its own volume are not Archimedes' principle.
Archimedes' principle, as stated above, equates
the buoyant force to the weight of the fluid
displaced.
One common point of confusion regarding Archimedes'
principle is the meaning of displaced volume.
Common demonstrations involve measuring the
rise in water level when an object floats
on the surface in order to calculate the displaced
water. This measurement approach fails with
a buoyant submerged object because the rise
in the water level is directly related to
the volume of the object and not the mass
(except if the effective density of the object
equals exactly the fluid density).
== See also ==
Eureka, reportedly exclaimed by Archimedes
upon discovery that the volume of displaced
fluid is equal to the volume of the submerged
object. Archimedes also used this principle
to help Hiero II with his problem with a golden
crown.(note that this idea is not Archimedes'
principle
