Pressure is a scalar quantity.
Positive pressure means
that adjacent fluid elements
are pushing each other
F dF
P= =
, F = PA
A dA
The pushing force acts
through the interface and is
proportional to the area of
the interface.
2x area => 2x force.
“Fluid exerts the same pressure
in all directions” means that the
force transmitted through an
interface near a given point does
not depend on the orientation of
the interface.
point of interest
Hydrostatic equilibrium
(no motion, no net forces)
Condition for hydrostatic
equilibrium – constant
pressure throughout the
fluid volume.
Variation of pressure
creates net force in the
direction of decreasing
pressure.
Hydrostatic equilibrium with
external forces: Gravity
Force from above
(pushing down)
Force from below
(pushing up)
PA
( P + dP) A
Gravitational force (pulling down)
dFg = mg = !gV = !gA dh
Balance (equilibrium) equation
PA + !gA dh = ( P + dP) A
!gA dh = AdP
dP
= !g
dh
Hydrostatic equilibrium with external forces:
Gravity
dP
= !g
dh
We have got a differential
equation…
Where do we go from here?
We need to integrate it. Can we?
If both ρ and g are constant, certainly yes.
P = !gh + P0
What is P0? And BTW, what is h?
P0 is the constant of integration – the value of pressure at h = 0.
It is natural to count h downwards from the fluid surface (h = depth).
Then P0 is the pressure at the surface – the atmospheric pressure.
If the atmospheric pressure changes, does the pressure at a given
depth change?
Yep!
Hydrostatic equilibrium with external forces:
Gravity
dP
= !g
dh
We have got a differential
equation…
Where do we go from here?
We need to integrate it. Can we?
If both ρ and g are constant, certainly yes.
P = !gh + P0
What is P0? And BTW, what is h?
Pascal’s law:
a pressure change anywhere in a fluid is felt
throughout the fluid.
Hydrostatic equilibrium with
external forces: Gravity
dP
= !g
dh
The density, ρ, is only constant in liquids.
Gasses are compressible, and one must
assume variable ρ(h).
Earth atmosphere:
" = " 0 exp(! h / h0 )
P = P0 exp(! h / h0 )
P = !gh + P0
Pressure at the surface of water or at 0 altitude
is the usual good candidate for P0
P0 = Patm
Ocean (or cup).
Atmosphere
P = Patm + ! w gh
P = Patm " ! air gh
Here h is the depth.
Pressure grows by about 1 atm
(105 Pa) every 10 m
Here h is the altitude.
Pressure drops by about 120 Pa
every 10 m
Drinking through a straw.
When you get a tooth extracted you are warned not to drink through
a straw. Why so?
The pressure of liquid at the top of the straw is
h
Ptop = Patm " !gh
Patm
For you to drink, the pressure inside your
mouth should drop below Ptop.
This implies a substantial negative pressure
difference, ρgh, between the blood vessels
in your gums and your mouth, which may
open the wound.
The total force acting on the
bottom of the vessel is
F = P! A
The most remarkable thing about the expression for
pressure is what it does not include.
The expression for hydrostatic pressure is easy to
see for the straight, unobstructed column, but not
obvious for more contorted geometries.
The force exerted on the bottom may be strikingly
different from the weight of the liquid!
P = !gh'
Hydraulic lift
A multiplication of
force can be achieved
by the application of
fluid pressure
according to Pascal's
principle, which for the
two pistons implies
P1 = P2
This allows the lifting of
a heavy load with a
small force, as in an auto
hydraulic lift…
but of course there can
be no multiplication of
work, so in an ideal case
with no frictional loss:
Win = Wout
Hydraulic lift is very similar to a lever….
“GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH”
Archimedes
A vacuum cleaner
Does the vacuum
suck the dust?
Strictly speaking
the pump of the
cleaner creates
lower pressure
inside it and the air
and dust are driven
into the hose by
the pressure of the
atmosphere.
evacuated
volume
A barometer
Pressure of the air, Patm,
drives mercury into the hollow
tube.
There is no air and no
pressure inside the tube.
h
Therefore Patm is
balanced by hydrostatic
pressure of the mercury.
Patm = ρgh
ρ = 13600 kg/m3
Ptop = Patm ! "gh = 0
- density of mercury
" Patm = !gh
A manometer – pressure gauge
Now there is a fluid (gas) under
pressure in the reservoir.
The difference between the
pressure inside the reservoir,
Pres, and Patm is now
balanced by hydrostatic pressure
of the mercury.
Pres " Patm = !gh
– gauge pressure
Pres = Patm + !gh
– absolute pressure
A manometer is measuring gauge pressure
Buoyant force
Archimede’s principle: The buoyant
force on an object is equal to the weight
of the fluid displaced by the object.
Fp = mw g = ! wVg
The buoyant force is applied to the center of gravity of the fluid
(center of the submerged volume of the body).
Archimedes principle:
The buoyant force on an object is equal to the weight of the
fluid displaced by the object, Vdisp ρfluid g.
Ship loaded with
50 ton of iron.
Ship empty.
Ship loaded with 50
ton of styrofoam.
Vsub ! water g = mship g
Volume of the submerged part of the ship (or any other floating object)
is equal to the mass of the ship divided by the density of water:
Vsub =
mship g
! water g
=
mship
! water
Equal Volumes Feel Equal Buoyant Forces
Suppose you had equal sized balls of cork, aluminum and
lead, with respective densities of 0.2, 2.7, and 11.3 times
the density of water. If the volume of each is 10 cubic
centimeters then their masses are 2, 27, and 113 gram.
Each would displace 10 grams of water, yielding apparent
masses of -8 (the cork would accelerate upward), 17 and 103
grams respectively (and weights of -0.08, 0.17 and 1.03 N).
Apparent mass can be defined as apparent weight divided by
the gravitational acceleration, g.
Center of gravity vs. center of buoyancy
Gravitational force is
applied at the center of
gravity.
Buoyancy force is
applied to the center of
buoyancy.
Center of gravity should
be below center of
buoyancy for stable
equilibrium.
Is that a necessary
condition of equilibrium?
There is something wrong with the picture on the left… What?
Center of buoyancy is the center of volume of the submerged part
of the boat. It cannot possibly be at or above the water-line!