The forces at work in buoyancy
In
physics,
buoyancy ( ) is
the upward force that keeps things afloat. The net upward buoyancy
force is equal to the magnitude of the weight of fluid displaced by
the body. This force enables the object to float or at least seem
lighter.
Archimedes' principle
Archimedes' principle is named after Archimedes of Syracuse, who first
discovered this law. Archimedes' principle may be stated
thus:
Archimedes' principle does not consider the
surface tension (capillarity) acting on the
body.
The weight of the displaced fluid is directly proportional to the
volume of the displaced fluid (if the surrounding fluid is of
uniform density). Thus, among completely submerged objects with
equal masses, objects with greater volume have greater
buoyancy.
Suppose a rock's weight is measured as 10
newton when suspended by a string in a
vacuum. Suppose that when the rock is lowered
by the string 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.
Assuming Archimedes' principle to be reformulated as follows,
- \mbox{apparent immersed weight} = \mbox{weight} - \mbox{weight
of displaced fluid}\,
then inserted into the quotient of weights, which has been expanded
by the mutual volume
- \frac { \mbox{density}} { \mbox {density of fluid} } = \frac {
\mbox{weight}} { \mbox {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:
- \frac { \mbox {density of object}} { \mbox {density of fluid} }
= \frac { \mbox {weight}} { \mbox {weight} - \mbox {apparent
immersed weight}}\,
(This formula is used for example in describing the measuring
principle of a
dasymeter and of
hydrostatic weighing.)
Ex. If you drop wood into water buoyancy will keep it afloat.
Forces and equilibrium
This is the equation to calculate the pressure inside a fluid in
equilibrium. The corresponding equilibrium equation is:
- \mathbf{f}+\operatorname{div} \sigma=0
where
f is the force density exerted by some outer
field on the fluid, and
σ is the
stress tensor. In this case the stress tensor
is proportional to the identity tensor:
- \sigma_{ij}=-p\delta_{ij}.\,
Here \delta_{ij}\, is the
Kronecker
delta. Using this the above equation becomes:
- \mathbf{f}=\nabla p.\,
Assume the outer force field is conservative, that is it can be
written as the negative gradient of some scalar valued
function:
- \mathbf{f}=-\nabla\Phi.\,
Then we have:
- \nabla(p+\Phi)=0 \Longrightarrow p+\Phi =
\text{constant}.\,
Hence 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 our case we have
gravity, so Φ = −
ρgz where
g is the
gravitational acceleration,
ρ is the mass density of the
fluid. Let the constant be zero, that is the pressure zero where
z is zero. So the pressure inside the fluid, when it is
subject to gravity, is
- p=\rho g z.\,
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 forces.
The buoyant force exerted on a body can now be calculated easily,
since we know the internal pressure of the fluid. We know that the
force exerted on the body can be calculated by integrating the
stress tensor over the surface of the body:
- \mathbf{F}=\oint \sigma \, d\mathbf{A}
The surface integral can be transformed into a volume integral with
the help of the
Gauss–Ostrogradsky
theorem:
- \mathbf{F}=\int \operatorname{div}\sigma \, dV = -\int
\mathbf{f}\, dV = -\rho \mathbf{g} \int\,dV=-\rho \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 buoyant 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:
- \rho V_\text{disp}\, g, \,
where
ρ is the
density of the
liquid,
V disp is the volume of the displaced body of
liquid, and
g is the
gravitational acceleration at the
location in question.
If we replace this volume of liquid by a solid body of the exact
same shape, the force the liquid exerts on it must be exactly the
same as above. In other words the "buoyant force" on a submerged
body is directed in the opposite direction to gravity and is equal
in magnitude to
- \rho V g. \,
The net force on the object is thus the sum of the buoyant force
and the object's weight
- F_\text{net} = m g - \rho V 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.
Commonly, the object in question is floating in equilibrium and the
sum of the forces on the object is zero, therefore;
- mg = \rho V g, \,
and therefore
- m = \rho V. \,
showing that the depth to which a floating object will sink (its
"
buoyancy") is independent of the variation of the
gravitational
acceleration at various locations on the surface of the Earth.
- (Note: If the liquid 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 is common to define a
buoyant mass
m_{b} that represents the effective
mass of the object with respect to gravity
m_b = m_\mathrm{o} \cdot \left( 1 -
\frac{\rho_\mathrm{f}}{\rho_\mathrm{o}} \right)\,
where m_{\mathrm{o}}\, is the true (vacuum) mass of the object,
whereas ρ
_{o} and ρ
_{f} are the average densities
of the object and the surrounding fluid, respectively. Thus, if the
two densities are equal, ρ
_{o} = ρ
_{f}, the object
appears to be weightless. If the fluid density is greater than the
average density of the object, the object floats; if less, the
object sinks.
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 buoyant
force, which, unbalanced against 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 buoyant force on an object acts
through the
centre of buoyancy,
being the centroid of the displaced volume of fluid. The weight
force on the object acts through its
center of gravity. An object will be stable
if an angular displacement moves the line of action of these forces
to set up a 'righting
moment'. See
also
Angle of loll.
Compressive fluids
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. As a
submarine expels water from its buoyancy
tanks (by pumping them full of air) it rises because its volume is
constant (the volume of water it displaces if it is fully
submerged) as its weight 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 rise and dive by filling large 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. Normally, precautions are taken to ensure
that no air has been left in the tanks. If air were left in the
tanks and the submarine were to descend even slightly, the
increased pressure of the water would compress the remaining air in
the tanks, reducing its volume. Since buoyancy is a function of
volume, this would cause a decrease in buoyancy, and the submarine
would continue to descend.
The height of a balloon tends to be stable. As a balloon rises it
tends to increase in volume with reducing atmospheric pressure, but
the balloon's cargo does not expand. The average density of the
balloon decreases less, therefore, than that of the surrounding
air. The balloon's buoyancy decreases because the weight of the
displaced air is reduced. A rising balloon tends to stop rising.
Similarly, a sinking balloon tends to stop sinking.
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 has a buoyancy that is greater than
its own weight. If the fluid has a surface, such as water in a lake
or the sea, the object will float 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.An object with a higher average density than the fluid has
less 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
References
External links