gh
B m
V
F
P
A
P liq Vdisp g
Av const.
12
P v const.
2
13-0
Fluid - Statics
- density
=
M
V
kg
m3
F
force
Pressure: P = A = area
units:
N
m2
= Pa
lb
lb
or
or 2
2
ft
in
Atmospheric pressure
1 Atm = 14.7
lb
i n2
1 Atm = 1.01 x 105 Pa
13-1
sample
13-1a
Pascals Principle
F1
PA1
F2
P
Pressure is transmitted
through fluid without loss
PA2
Area, A2
F1(Total) = F1 - P A1
Piston
at rest :
F2(Total) = F2 - P A2
0 = F2 - PA2
0 = F1 - PA1
F1
A1
=P=
F2
A2
F2 =
A2
F1
A1
13-2
Applications of Pascals Principle
AL
Fs = ( A ) Fs
s
Fs
As
=P=
FL
AL
This factor can be made
to be 1,000 or more.
- Hydraulic Press
- Hydraulic Jack
13-3
Pressure in a Liquid (or Fluid)
Pressure Force per unit Area
Any arbitrary
cylinder
vacuum
V=Ah
V
Ft=0 top
mg= (Ah) g
F(total) = -Ft-mg+ Fh
fluid at
rest
vac
0 = -0- Ahg + PhA
Ph = hg
mg
h
PATM
V=Ah
A
Fh= PhA
Pressure due to weight of water on top
If Po added at top (instead of vacuum) it is transmitted to bottom.
Ph = Po + hg
13-4
Note: h=0 at surface and all hs are + #s
13-5
Torricelli Barometer
13-5
13-6
Atmospheric Pressure
A barometer compares the pressure due to the atmosphere to
the pressure due to a column of fluid, typically mercury. The
mercury column has a vacuum above it, so the only pressure is
due to the mercury itself.
13-6a
The Siphon
13-7
Physics of the Flush Toilet
Water from the flush tank empties
abruptly into toilet bowl, filling the
siphon tube to a level lower than
the bottom of the bowl.
The siphon action is set up and the
bowl empties. Then, air enters the
tube, which stops the siphon.
Picture is from www.howstuffworks.com
Finally, the remaining water from the flush tank refills the
bowl and the flush valve closes. The flush tank continues
to fill until the filler float closes the filler valve.
13-8
Applications
The blood pressure in a vein is about
P = 2.4 x 103 Pa
How high must a Saline or
Pharmaceutical fluid to flow into a vein?
g hmin = 2.4x103 Pa
hmin
2.4x103
= (9.8) (103 kg/m3)
= 0.24 m
= 24 cm ( ~9.5 inch)
What is the difference in blood pressure
between the feet and heart?
hH to F ~ 1.2 m
hH to F
1 mm Hg = 133.3 Pa
Blood = 1.05 x 103 kg/m3
13-9
Measurement of blood pressure with
finger tip monitor is a function of finger height
13-9a
Archimedes Principle
buoyant force
mass of water displaced
by V= volume object (submerged)
density of water (medium)
13-10
Archimedes Principal
fluid
Gravity force G
object
Gravity force, G, bigger sinks
Buoyant force, B, bigger floats
Buoyant force B Total force of fluid on object
density of object greater than density of fluid - sinks
density of object less than density of fluid - floats
increasing density of object
13-10a
Net on force object= buoyant force + gravitational force
Neglecting friction how fast will object sink or rise?
13-10b
Cormorant
wings not
water proof
low
buoyancy
Ducks
water proof wings
trap air
Mallard
high
buoyancy
13-10c
Canvas back
Archimedes
King Hiero of Syracuse had a goldsmith make a crown out of an amount of gold.
Q. to Archimedes, did the the goldsmith cheat, mixing some silver inside the crown?
Archimedes- thought of in bathtub Eureka!
Weight in
a Liquid
=
Weight
(in Air)
-
Weight of the
Water displaced
Any solid body submerged
in a liquid loses the weight
of the liquid displaced by it.
Mass/weight of crown was ok but density was too low
mixed in lighter (not Au) element!
The crown & gold nugget
balance in air, but not in
the water.
13-11
13-12
Flow in Incompressible Fluids
Crossectional area
A1
A2
v1
v2
P1
P2
Pressure
Assumptions & Conditions
1) No frictional energy loss in fluid or with surfaces
2) Laminar flow of stream lines (i.e. no turbulence)
3) Incompressible fluid
13-13
H2O is
incompressible
Flow in Incompressible Fluids
Crossectional area
A1
A2
v1
v2
P2
F2
F1
P1
x2 = v2 t
x1 = v1 t
V - Volume
V1 = A1x1
V2 = A2x2
= A1v1t
= A2v2t
Conservation of mass/fluid volume: V1 = V2
(what goes in must come out)
A1v1t = A2v2t
13-14
A1v1 = A2v2
Flow in Incompressible Fluids
Crossectional area
A1
A2
v1
v2
P2
F2
F1
P1
x 2 = v 2 t
x 1 = v 1 t
Work Energy Theorem
Work Done, W
F1 = P1A1
F2 = P2A2
W = +F1x1 - F2x2
W=
= P1A1x1 - P2A2x2
V1
but V1 = V2
W = K - change in Kinetic Energy
V2
W = (P1-P2) V
1
2
m2v22 -
1
m1v12
2
(P1 - P2) V = Vv22 - V v12
2
2
Bernoullis Equation
P + v2 = constant
2
13-15
Pressure
must decrease
Velocity increases
Bernoulli
Faster flowing air lower pressure
P+
1
2
v2 = constant
if velocity increases,
pressure must decrease
13-15a
13-16
13-16
Bernoullis Equation + pot. energy
P1 +
1
2
v12 = P2 +
1
2
y2
P2
v
y1
v22
P1
If y stays the same
If v stays the same
P2 + g y2 = P1 + g y1
v
0
If both height and velocity change
12
P + v + gy constant
2
12
12
P1 + v1 + gy1 P2 + v 2 + gy2
2
2
13-17
The horizontal velocity of water emerging from holes in the
side of a container is given by
12
P + v + gy constant
2
When it comes out P=PATM
v = 2gh
PATM
y=0
PATM
12
PATM + v2 + g (h2 ) y=-h1
2
h3
12
v2 gh2
2
v2 2 gh2
2
h1
h2
v2 2 gh2
13-17a
Surface Tension
(extra nor required)
A molecule in the center experiences
forces from all directions, while one
on the surface have a net force inward.
Ink drop
Untreated film
Contact angle
Ink drop
Treated film
Contact angle
Image from Peter Prentice - PolyTech Consultants
13-18-extra
Drop of
liquid