Pressure increases w/ increasing depth in fluid. At depth in fluid w/ density , pressure is: , where is atmospheric pressure at surface & is strength of gravity.
Pascal’s Principle: pressure applied to enclosed fluid is transmitted undiminished to every location in fluid as well as container walls.
Circles on box surface are ports where either pressure can be measured, or more pressure can be added.
Location indicates a piston where pressure is applied to the box.
Other pressures at ports include contribns due to depths (, or ) w/in box.
A component not mentioned in diagram of fluid-filled box is contribution from ambient (outside) pressure of local atmosphere.
Common application of Pascal’s Principle is in transmitting force using liquid-filled steel tubes (hydraulics).
Wood cube w/ edges long, having mass & sides of area , is held completely immersed in water w/ bottom face at depth ( ).
Common experience holds that cube
buoyed up. Diagram of vector
forces due to fluid pressure that impinge on four vertical
sides shows why cube is not pushed laterally by
Force due to pressure on bottom surface of cube is . Force on top surface is . Net force on cube due to water pressure is (an upward push)!
Archimedes of Syracuse (ca. 225 B.C.E.) experienced an
Eureka! moment while taking his bath:
volume for objects of quite irregular shapes can be accurately
determined by measuring water displaced by
Following scientific principle was not direct translation of Archimedes’ insight; rather mathematical rendering adapted for technologies giving accurate weight. Technologies for est. not accurate in 200 B.C.E.
Archimedes’ Principle: object immersed wholly or partially in fluid experiences upward buoyant force w/ magnitude equal to weight of fluid volume displaced by object: .
When avg. density for immersed portion of object is less than that of fluid: , object will float. Archimedes’ focus was on avg. density, not .
Note that it is average density of object that makes a difference. Objects made of solid steel sink, but steel ship floats because its interior is air.
Specific Gravity: dimensionless ratio of density for object to density for pure water at and (denominator is ).
Archimedes’ patron was the King of Syracuse: a man who jealously guarded his gold.
A contract was issued for one kilogram of the king’s gold to be fashioned by a smith into an elegant crown.
The crown was delivered, but the king suspected that some of his gold had been diverted toward enrichment of the smith.
Archimedes thought about this, but at first couldn’t discern how one kg crown could be distinguished from one kg pure gold.
His bath-time insight led to unfortunate
streaking incident through streets of Syracuse.
He applied insight indirectly: if crown less dense than gold sample, it occupies more volume, & feels greater buoyancy.
Archimedes’ Princ. provides method for measuring either avg. density or volume (for immersed part) of object. Net buoyancy is .
Buoyancy is also weight for water amount that has same volume as immersed portion of object: .
For the above reason, volume of object can be found:
Object’s avg. density would be mass divided by volume. Mass is typically found by observing weight in air: . Thus
Example: Solid ball has weight . When submerged in water, its apparent weight is . Find density of ball.
Buoyancy of immersed ball must satisfy , so .
Archimedes’ Principle states . We know values & , & so can solve equation to get .
Mass of ball must be
. This enables us to calculate
Note that general formula for this sort of calculation is . This avoids explicit measure of object’s volume. (Weights are downward, i.e. negative, forces in this context.)
Example: Given object w/ string (to hang it), scale & container of water:
(a). find object volume w/out directly measuring water displaced,
(b). find object density, & account for air buoyancy.
Note that estimate is not strictly valid because is affected by a small amount of air buoyancy.
Assume that there is a true value
for weight of
object, but that
; then set
net buoyancy to be
The formula for volume (including effect of air buoyancy) is now . This answers part (a).
Using formulas that incorporated air buoyancy affected only fifth decimal of density estimate (volume estimate was affected in third decimal).
New formulas mainly useful when accurate local values for , and are available.
Notable complications for these calculations arise whenever object is partially immersed. Then buoyancy is water-equivalent weight only of immersed part.
Example: A closed hard cylinder of unknown material floats in water with long axis held vertically. Top end-face is parallel with calm surface of water.
The water-line is at a location that is up the length of the cylinder. The cylinder is only partly immersed. Compute the density of the cylinder in .
Since cylinder floats, its weight is negative of buoyant force .
Buoyancy applies only for immersed part of cylinder, and so .
Weight is simply , and therefore .
Solving the equation for the density of the cylinder, we get:
Pressure at ocean depths is very important for deep submersibles.
Ocean depth can be measured using a number where (maximum depth).
Pressure at depth depends upon density of the water at various lesser depths, but those densities also depend upon pressure!
Simplistic model for pressure at ocean depths is , where is density of water at std. pressure, is acceleration due to gravity and is std. atmospheric pressure.
Simplistic model disregards the dual-dependency conundrum.
Pressure at depth is just accumulated weight of water above that depth.
Higher pressure at depth makes deeper water more dense than shallow water.
Higher density of deep water in turn contributes to higher pressure at depth.
Response of water to higher pressures is modeled via bulk compression modulus (measured in units of pressure, ):
where applied pressure is stress, relative compression is strain and (roughly 21,700 atmospheres) is the volumetric modulus.
Pulling apart the volume change as and rewriting the fraction gives:
Next consider column of water having height and bottom area that is sliced into slabs of thickness .
A slab of water with mass that near surface has density , when subjected to pressure at depth exhibits density:
Pressure experienced by the slab of water (among a total of slabs) near middle of its height can be represented via .
Total pressure at depth might then be estimated:
Limiting as increases w/out bound, earlier estimate becomes integral:
Integral makes precise the way in which pressure function
depends upon itself.
More convenient representation states the issue as first order d.e.:
Note ocean depth is positive: where (max. depth), , & .
makes this d.e. an
initial value problem.
One quadrature giving single arbitrary constant suffices to get closed formula:
in which the arbitrary constant .
The function is a segment of the lower arm of a square root function.
This function attains its maximum at ; an ocean depth that does not occur on our planet.
Both and the simplistic approximation are increasing functions of .
The is positive (concave up for every ), whereas .
Relative error in simplistic model is . That is, simplistic model is only about two percent too low at .
For most terrestrial purposes, function is already a very close approximation.
Further generalizations of problem (for eg. d.e. w/ varying ) can use as beginning approximation.