In physics, buoyancy () or upthrust, is an
upward force exerted by a fluid that opposes
the weight of an immersed object. In a column
of fluid, pressure increases with depth as
a result of the weight of the overlying fluid.
Thus the pressure at the bottom of a column
of fluid is greater than at the top of the
column. Similarly, the pressure at the bottom
of an object submerged in a fluid is greater
than at the top of the object. The pressure
difference results in a net upward force on
the object. The magnitude of the force is
proportional to the pressure difference, and
(as explained by Archimedes' principle) is
equivalent to the weight of the fluid that
would otherwise occupy the volume of the object,
i.e. the displaced fluid.
For this reason, an object whose average density
is greater than that of the fluid in which
it is submerged tends to sink. If the object
is less dense than the liquid, the force can
keep the object afloat. This can occur only
in a non-inertial reference frame, which either
has a gravitational field or is accelerating
due to a force other than gravity defining
a "downward" direction.The center of buoyancy
of an object is the centroid of the displaced
volume of fluid.
== Archimedes' principle ==
Archimedes' principle is named after Archimedes
of Syracuse, who first discovered this law
in 212 B.C. For objects, floating and sunken,
and in gases as well as liquids (i.e. a fluid),
Archimedes' principle may be stated thus in
terms of forces:
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
— with the clarifications that for a sunken
object the volume of displaced fluid is the
volume of the object, and for a floating object
on a liquid, the weight of the displaced liquid
is the weight of the object.
More tersely: buoyancy = weight of displaced
fluid.
Archimedes' principle does not consider the
surface tension (capillarity) acting on the
body, but this additional force modifies only
the amount of fluid displaced and the spatial
distribution of the displacement, so the principle
that buoyancy = weight of displaced fluid
remains valid.
The weight of the displaced fluid is directly
proportional to the volume of the displaced
fluid (if the surrounding fluid is of uniform
density). In simple terms, the principle states
that the buoyancy force on an object is equal
to the weight of the fluid displaced by the
object, or the density of the fluid multiplied
by the submerged volume times the gravitational
acceleration, g. Thus, among completely submerged
objects with equal masses, objects with greater
volume have greater buoyancy. This is also
known as upthrust.
Suppose a rock's weight is measured as 10
newtons when suspended by a string in a vacuum
with gravity acting upon 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 buoyancy
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.
Assuming Archimedes' principle to be reformulated
as follows,
apparent immersed weight
=
weight
−
weight of displaced fluid
{\displaystyle {\text{apparent immersed weight}}={\text{weight}}-{\text{weight
of displaced fluid}}\,}
then inserted into the quotient of weights,
which has been expanded by the mutual volume
density
density of fluid
=
weight
weight of displaced fluid
,
{\displaystyle {\frac {\text{density}}{\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 volumes.:
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.
During a period of increasing speed, the air
mass inside the car moves in the direction
opposite to the car's acceleration (i.e.,
towards the rear). The balloon is also pulled
this way. However, because the balloon is
buoyant relative to the air, it ends up being
pushed "out of the way", and will actually
drift in the same direction as the car's acceleration
(i.e., forward). If the car slows down, the
same balloon will begin to drift backward.
For the same reason, as the car goes round
a curve, the balloon will drift towards the
inside of the curve.
== Forces and equilibrium ==
The equation to calculate the pressure inside
a fluid in equilibrium is:
f
+
div
σ
=
0
{\displaystyle \mathbf {f} +\operatorname
{div} \,\sigma =0}
where f is the force density exerted by some
outer field on the fluid, and σ is the Cauchy
stress tensor. In this case the stress tensor
is proportional to the identity tensor:
σ
i
j
=
−
p
δ
i
j
.
{\displaystyle \sigma _{ij}=-p\delta _{ij}.\,}
Here δij is the Kronecker delta. Using this
the above equation becomes:
f
=
∇
p
.
{\displaystyle \mathbf {f} =\nabla p.\,}
Assuming the outer force field is conservative,
that is it can be written as the negative
gradient of some scalar valued function:
f
=
−
∇
Φ
.
{\displaystyle \mathbf {f} =-\nabla \Phi .\,}
Then:
∇
(
p
+
Φ
)
=
0
⟹
p
+
Φ
=
constant
.
{\displaystyle \nabla (p+\Phi )=0\Longrightarrow
p+\Phi ={\text{constant}}.\,}
Therefore, the shape of the open surface of
a fluid equals the equipotential plane of
the applied outer conservative force field.
Let the z-axis point downward. In this case
the field is gravity, so Φ = −ρfgz where
g is the gravitational acceleration, ρf is
the mass density of the fluid. Taking the
pressure as zero at the surface, where z is
zero, the constant will be zero, so the pressure
inside the fluid, when it is subject to gravity,
is
p
=
ρ
f
g
z
.
{\displaystyle p=\rho _{f}gz.\,}
So pressure increases with depth below the
surface of a liquid, as z denotes the distance
from the surface of the liquid into it. Any
object with a non-zero vertical depth will
have different pressures on its top and bottom,
with the pressure on the bottom being greater.
This difference in pressure causes the upward
buoyancy force.
The buoyancy force exerted on a body can now
be calculated easily, since the internal pressure
of the fluid is known. The force exerted on
the body can be calculated by integrating
the stress tensor over the surface of the
body which is in contact with the fluid:
B
=
∮
⁡
σ
d
A
.
{\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf
{A} .}
The surface integral can be transformed into
a volume integral with the help of the Gauss
theorem:
B
=
∫
div
⁡
σ
d
V
=
−
∫
f
d
V
=
−
ρ
f
g
∫
d
V
=
−
ρ
f
g
V
{\displaystyle \mathbf {B} =\int \operatorname
{div} \sigma \,dV=-\int \mathbf {f} \,dV=-\rho
_{f}\mathbf {g} \int \,dV=-\rho _{f}\mathbf
{g} V}
where V is the measure of the volume in contact
with the fluid, that is the volume of the
submerged part of the body, since the fluid
doesn't exert force on the part of the body
which is outside of it.
The magnitude of buoyancy force may be appreciated
a bit more from the following argument. Consider
any object of arbitrary shape and volume V
surrounded by a liquid. The force the liquid
exerts on an object within the liquid is equal
to the weight of the liquid with a volume
equal to that of the object. This force is
applied in a direction opposite to gravitational
force, that is of magnitude:
B
=
ρ
f
V
disp
g
,
{\displaystyle B=\rho _{f}V_{\text{disp}}\,g,\,}
where ρf is the density of the fluid, Vdisp
is the volume of the displaced body of liquid,
and g is the gravitational acceleration at
the location in question.
If this volume of liquid is replaced by a
solid body of exactly the same shape, the
force the liquid exerts on it must be exactly
the same as above. In other words, the "buoyancy
force" on a submerged body is directed in
the opposite direction to gravity and is equal
in magnitude to
B
=
ρ
f
V
g
.
{\displaystyle B=\rho _{f}Vg.\,}
The net force on the object must be zero if
it is to be a situation of fluid statics such
that Archimedes principle is applicable, and
is thus the sum of the buoyancy force and
the object's weight
F
net
=
0
=
m
g
−
ρ
f
V
disp
g
{\displaystyle F_{\text{net}}=0=mg-\rho _{f}V_{\text{disp}}g\,}
If the buoyancy of an (unrestrained and unpowered)
object exceeds its weight, it tends to rise.
An object whose weight exceeds its buoyancy
tends to sink. Calculation of the upwards
force on a submerged object during its accelerating
period cannot be done by the Archimedes principle
alone; it is necessary to consider dynamics
of an object involving buoyancy. Once it fully
sinks to the floor of the fluid or rises to
the surface and settles, Archimedes principle
can be applied alone. For a floating object,
only the submerged volume displaces water.
For a sunken object, the entire volume displaces
water, and there will be an additional force
of reaction from the solid floor.
In order for Archimedes' principle to be used
alone, the object in question must be in equilibrium
(the sum of the forces on the object must
be zero), therefore;
m
g
=
ρ
f
V
disp
g
,
{\displaystyle mg=\rho _{f}V_{\text{disp}}g,\,}
and therefore
m
=
ρ
f
V
disp
.
{\displaystyle m=\rho _{f}V_{\text{disp}}.\,}
showing that the depth to which a floating
object will sink, and the volume of fluid
it will displace, is independent of the gravitational
field regardless of geographic location.
(Note: If the fluid in question is seawater,
it will not have the same density (ρ) at
every location. For this reason, a ship may
display a Plimsoll line.)It can be the case
that forces other than just buoyancy and gravity
come into play. This is the case if the object
is restrained or if the object sinks to the
solid floor. An object which tends to float
requires a tension restraint force T in order
to remain fully submerged. An object which
tends to sink will eventually have a normal
force of constraint N exerted upon it by the
solid floor. The constraint force can be tension
in a spring scale measuring its weight in
the fluid, and is how apparent weight is defined.
If the object would otherwise float, the tension
to restrain it fully submerged is:
T
=
ρ
f
V
g
−
m
g
.
{\displaystyle T=\rho _{f}Vg-mg.\,}
When a sinking object settles on the solid
floor, it experiences a normal force of:
N
=
m
g
−
ρ
f
V
g
.
{\displaystyle N=mg-\rho _{f}Vg.\,}
Another possible formula for calculating buoyancy
of an object is by finding the apparent weight
of that particular object in the air (calculated
in Newtons), and apparent weight of that object
in the water (in Newtons). To find the force
of buoyancy acting on the object when in air,
using this particular information, this formula
applies:
Buoyancy force = weight of object in empty
space − weight of object immersed in fluidThe
final result would be measured in Newtons.
Air's density is very small compared to most
solids and liquids. For this reason, the weight
of an object in air is approximately the same
as its true weight in a vacuum. The buoyancy
of air is neglected for most objects during
a measurement in air because the error is
usually insignificant (typically less than
0.1% except for objects of very low average
density such as a balloon or light foam).
=== Simplified model ===
A simplified explanation for the integration
of the pressure over the contact area may
be stated as follows:
Consider a cube immersed in a fluid with the
upper surface horizontal.
The sides are identical in area, and have
the same depth distribution, therefore they
also have the same pressure distribution,
and consequently the same total force resulting
from hydrostatic pressure, exerted perpendicular
to the plane of the surface of each side.
There are two pairs of opposing sides, therefore
the resultant horizontal forces balance in
both orthogonal directions, and the resultant
force is zero.
The upward force on the cube is the pressure
on the bottom surface integrated over its
area. The surface is at constant depth, so
the pressure is constant. Therefore, the integral
of the pressure over the area of the horizontal
bottom surface of the cube is the hydrostatic
pressure at that depth multiplied by the area
of the bottom surface.
Similarly, the downward force on the cube
is the pressure on the top surface integrated
over its area. The surface is at constant
depth, so the pressure is constant. Therefore,
the integral of the pressure over the area
of the horizontal top surface of the cube
is the hydrostatic pressure at that depth
multiplied by the area of the top surface.
As this is a cube, the top and bottom surfaces
are identical in shape and area, and the pressure
difference between the top and bottom of the
cube is directly proportional to the depth
difference, and the resultant force difference
is exactly equal to the weight of the fluid
that would occupy the volume of the cube in
its absence.
This means that the resultant upward force
on the cube is equal to the weight of the
fluid that would fit into the volume of the
cube, and the downward force on the cube is
its weight, in the absence of external forces.
This analogy is valid for variations in the
size of the cube.
If two cubes are placed alongside each other
with a face of each in contact, the pressures
and resultant forces on the sides or parts
thereof in contact are balanced and may be
disregarded, as the contact surfaces are equal
in shape, size and pressure distribution,
therefore the buoyancy of two cubes in contact
is the sum of the buoyancies of each cube.
This analogy can be extended to an arbitrary
number of cubes.
An object of any shape can be approximated
as a group of cubes in contact with each other,
and as the size of the cube is decreased,
the precision of the approximation increases.
The limiting case for infinitely small cubes
is the exact equivalence.
Angled surfaces do not nullify the analogy
as the resultant force can be split into orthogonal
components and each dealt with in the same
way.
=== Static stability ===
A floating object is stable if it tends to
restore itself to an equilibrium position
after a small displacement. For example, floating
objects will generally have vertical stability,
as if the object is pushed down slightly,
this will create a greater buoyancy force,
which, unbalanced by the weight force, will
push the object back up.
Rotational stability is of great importance
to floating vessels. Given a small angular
displacement, the vessel may return to its
original position (stable), move away from
its original position (unstable), or remain
where it is (neutral).
Rotational stability depends on the relative
lines of action of forces on an object. The
upward buoyancy force on an object acts through
the center of buoyancy, being the centroid
of the displaced volume of fluid. The weight
force on the object acts through its center
of gravity. A buoyant object will be stable
if the center of gravity is beneath the center
of buoyancy because any angular displacement
will then produce a 'righting moment'.
The stability of a buoyant object at the surface
is more complex, and it may remain stable
even if the centre of gravity is above the
centre of buoyancy, provided that when disturbed
from the equilibrium position, the centre
of buoyancy moves further to the same side
that the centre of gravity moves, thus providing
a positive righting moment. If this occurs,
the floating object is said to have a positive
metacentric height. This situation is typically
valid for a range of heel angles, beyond which
the centre of buoyancy does not move enough
to provide a positive righting moment, and
the object becomes unstable. It is possible
to shift from positive to negative or vice
versa more than once during a heeling disturbance,
and many shapes are stable in more than one
position.
== Fluids and objects ==
The atmosphere's density depends upon altitude.
As an airship rises in the atmosphere, its
buoyancy decreases as the density of the surrounding
air decreases. In contrast, as a submarine
expels water from its buoyancy tanks, it rises
because its volume is constant (the volume
of water it displaces if it is fully submerged)
while its mass is decreased.
=== Compressible objects ===
As a floating object rises or falls, the forces
external to it change and, as all objects
are compressible to some extent or another,
so does the object's volume. Buoyancy depends
on volume and so an object's buoyancy reduces
if it is compressed and increases if it expands.
If an object at equilibrium has a compressibility
less than that of the surrounding fluid, the
object's equilibrium is stable and it remains
at rest. If, however, its compressibility
is greater, its equilibrium is then unstable,
and it rises and expands on the slightest
upward perturbation, or falls and compresses
on the slightest downward perturbation.
==== Submarines ====
Submarines rise and dive by filling large
ballast tanks with seawater. To dive, the
tanks are opened to allow air to exhaust out
the top of the tanks, while the water flows
in from the bottom. Once the weight has been
balanced so the overall density of the submarine
is equal to the water around it, it has neutral
buoyancy and will remain at that depth. Most
military submarines operate with a slightly
negative buoyancy and maintain depth by using
the "lift" of the stabilizers with forward
motion.
==== Balloons ====
The height to which a balloon rises tends
to be stable. As a balloon rises it tends
to increase in volume with reducing atmospheric
pressure, but the balloon itself does not
expand as much as the air on which it rides.
The average density of the balloon decreases
less than that of the surrounding air. The
weight of the displaced air is reduced. A
rising balloon stops rising when it and the
displaced air are equal in weight. Similarly,
a sinking balloon tends to stop sinking.
==== Divers ====
Underwater divers are a common example of
the problem of unstable buoyancy due to compressibility.
The diver typically wears an exposure suit
which relies on gas-filled spaces for insulation,
and may also wear a buoyancy compensator,
which is a variable volume buoyancy bag which
is inflated to increase buoyancy and deflated
to decrease buoyancy. The desired condition
is usually neutral buoyancy when the diver
is swimming in mid-water, and this condition
is unstable, so the diver is constantly making
fine adjustments by control of lung volume,
and has to adjust the contents of the buoyancy
compensator if the depth varies.
== Density ==
If the weight of an object is less than the
weight of the displaced fluid when fully submerged,
then the object has an average density that
is less than the fluid and when fully submerged
will experience a buoyancy force greater than
its own weight. If the fluid has a surface,
such as water in a lake or the sea, the object
will float and settle at a level where it
displaces the same weight of fluid as the
weight of the object. If the object is immersed
in the fluid, such as a submerged submarine
or air in a balloon, it will tend to rise.
If the object has exactly the same density
as the fluid, then its buoyancy equals its
weight. It will remain submerged in the fluid,
but it will neither sink nor float, although
a disturbance in either direction will cause
it to drift away from its position.
An object with a higher average density than
the fluid will never experience more buoyancy
than weight and it will sink.
A ship will float even though it may be made
of steel (which is much denser than water),
because it encloses a volume of air (which
is much less dense than water), and the resulting
shape has an average density less than that
of the water.
== See also
