Density

LIFE IN A FLUID
Dr Julia Bryant
Module outline
Fluid statics
• What is a fluid?
Density !
Pressure!
• Fluid pressure and depth
Pascal’s principle
• Buoyancy
Archimedes’ principle
Fluid dynamics!
• Reynolds number
• Equation of continuity!
• Bernoulli’s principle
• Viscosity and Turbulent flow
• Poiseuille’s equation
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http://www.physics.usyd.edu.au/teach_res/jp/fluids/wfluids.htm
Static fluids
Density
What is a fluid?
Definition (" Greek letter rho)
• A gas or a liquid that takes the shape of a container
• A fluid can flow.
• A fluid is a collection of randomly arranged molecules
held together by weak cohesive forces and the forces
exerted by the walls of the container.
"
density of object!
m
V
mass of object
volume of object"
solid
gas
liquid
!=
m
V
Units
SI unit
kg.m-3
Other units
g.mL-1
1 mL = 1 cm3 = (10-2)3 m3 = 10-6 m3
1 g = 10-3 kg
1 g.mL-1 = 103 kg.m-3
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Density values (kg.m-3)
Interstellar space
10-20
Best laboratory vacuum
10-17
Air (20 °C and 1 atm)
1.21
Styrofoam
100
cork
Ice
240
917
Water (20 °C and 1 atm)
998
SaltWater
1024
Whole blood
1060
Earth (average)
~5500
Iron
7900
Mercury
13 600
Osmium (most dense element)
22 500
Lead
DEMO
11 340
Uranium nucleus
1017
Neutron star (core)
1018
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Static fluids
Why does a cork float?
Why does ice float?
Pressure
P
Average density of an object < density of
fluid => object floats
Good Beer
p=
F
A
F
A
force at right angles to surface
area of contact over which force acts
F
1006.8
A
Units
Aerogel – the
least dense
solid on earth
1.5 kg.m-3
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pressure
p
Pa
pascal
force
F
N
newton 1 N = 1 kg.m.s-2
1 Pa = 1 N.m-2
area
A
m2
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How large is atmospheric pressure?
Atmospheric Pressure patm
Rough calculation
p=F/A
Pressure at Earth’s surface due to
weight of atmosphere
but F=ma
p=mg/A
but ! = m/V
m=!V
101.3 kPa
"air = 1.21 kg.m-3
1013 hPa (hecto means 102)
g = 9.8 m.s-2
p=!Vg/A
1013 mb
h ~ 10 km
p=!Ahg/A
1.013x105 Pa
1.013 b
(millibar)
(bar)
p=!hg
760 mmHg
p = (1.21)(104)(9.8) Pa
14.7 lb.in2 (14.7 psi)
p = 1.19 x105 Pa
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What is the force that crushes a steel
beer can when the air is pumped out
of it?
Atmospheric Pressure patm
Pressure on your head ~ 1 atm
Force to collapse can F = p A
Force = pA
=1.01 x
105
N/m2
x # x (10
Assume atmospheric pressure acts only on outside
surface of can
p = patm = 1.013 x 105 Pa
cm)2
~ 3200 N = 0.33 ton
= 23.5 cases beer (stubbies)
Consider a cylinder of radius R = 0.035 m and height
h = 0.10 m
A = 2(! R2) + 2! R h
Why don’t you collapse?
Are people with big
heads under more
pressure?
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F = (1.013 x 105){(2!)(0.035)2 + (2!)(0.035)(0.10)} N
F = 3007.4 N
The atmospheric pressure of Jupiter is 90
times that of earth, that would crush a car
absolutely flat!
DEMO
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How big is this force?
Consider a full keg of beer which weighs 72.8kg =>
F=mg = 72.8x9.8 = 713 N =>force equivalent to the
weight of > 4 full kegs.
?
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What does it mean when we are told to inflate a tyre to 100
kPa?
Answer
Pressure gauges measure the pressure above and below
atmospheric (or barometric) pressure.
Need to distinguish between a gauge pressure and
absolute pressure.
What does it mean when we are told to
inflate a tyre to 100 kPa?
200
100
300
0
400
patm = 101 kPa
pgauge = 100 kPa
pabsolute = (101 + 100) kPa = 201 kPa
pabsolute = pgauge + patm
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?
How does a gas or liquid exert a pressure on its container walls?
Answer
Impact of a molecule on the wall of the
container exerts a force on the wall and
the wall exerts a force on the molecule.
Many impacts occur each second and
the total average force per unit area is
called the pressure.
How does a gas or liquid exert
a pressure on its container
walls?!
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What weight can the Magdeburg plates support?
- what is the force on the bottom plate?
patm = 1x105 Pa
R = 24 cm
p1
F1
A = # R2
p=F/A
p0
Say F1=0.9F0
F=pA
F0 = (1x105)(#)(0.24)2 N
mg
F0
F0 = 18,095 N
$F=0
F1 + mg = F0
In 1964 Otto van Guericke, mayor of Magdeburg, placed two
35.5cm hemispheres together and pumped out the air; 16 horses
could not pull them apart.
mg = F0 - 0.9 F0 = 0.1 F0
mg=0.1 x 18,095 = 1809.5 N
DEMO15
Max m= 1809.5/9.8 = 185 kg
= 13 cases beer!
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m = 70 kg
FG = 700 N
A = 10 mm x 10 mm
A = (100)(10-3)(10-3) m2
A = 10-4 m2
p=F/A
p = 700 / 10-4 Pa
p = 700 x 104 Pa
pressure !!!
p = 7 x 106 Pa
What pressure is applied to the ground if a person stood on one heel?
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pressure !!!
p / patm = 7x106 / 1x105 atm = 70 atm
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Why is it better is use a sharp knife?
Why is it better is use a sharp knife?
p=
F
A
A
p
F
force at right angles to surface
area of contact
over
which
force acts
Stress A
is uni-directional
(you
press
downward
with
the knife). "
Pressure is omni-directional (you pump air into a
tyre and it inflates in all directions). !
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F
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A
FLUID PRESSURE
Pressure increases with depth in a fluid
p
• Provided that an object immersed in a
fluid is not moving, the force on each
small part of the contact surface is
perpendicular to the surface so the
interaction can be described completely in
terms of pressure.
Mass = m
= " dV
= " A dy
Fluid element is in equilibrium so net forces are zero.
• Fluid pressure always acts perpendicular
to any surface irrespective of how the
surface is oriented.
Horizontal force is due to pressure only, and p is the same on both
sides.
force
p
p + dp
area
• If there is relative motion between the
fluid and solid object, the force has
components parallel to the surface and
the interaction is not simply described by
the pressure.
dy
pressure
• Pressure has no intrinsic direction of its
own (pressure is scalar)
pA - (p +dp)A - mg = 0
mg
m= " V
pA - pA - dpA - "Vg = 0
A
y
p
-dp A - g " A dy = 0
p
dp = - " g
dy
As y increases, p decreases
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Snorkelling
Pressure increases with depth in a fluid
Snorkelling
In a static fluid, with uniform density ",
Pressure at depth h = pressure acting on surface
+ pressure due to height of liquid
patm
ph = p0 + F
A
F = weight of column liquid of cross sectional area A
F = mg
%p = ph – patm
p0 pressure acting on surface
200 mm
m="V
= " Ah
F = ma = " Ahg = "gh
A
A
ph = p0 + "gh
If the pressure at at the surface
increases then the pressure at depth
h increases by the same amount.
ph = patm + " g h
h
ph
A
ph
Weight of
column
of liquid
F
%p = " g h
%p = (1030)(9.8)(0.2) Pa
%p = 2x103 Pa
if h > 200 mm
Liquid – uniform density !
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%p too large, lungs collapse
How to burst open a barrel of beer.
You dive deep down into the water.
Does it matter whether it is in the ocean or a swimming pool?
!
If a barrel is full of beer and a long narrow pipe is inserted in
the top, no matter how strong the barrel, it can be burst by
adding a high enough column of beer, even though the total
weight of beer in the tube is small compared to that of the beer
in the barrel.
The pressure within a uniform stationary
fluid is the same at all points in the
same horizontal plane.
ph = p0 + "gh
h
Pascal’s experiment:
The radius of the lid was 20 cm, and the height of the water
in the tube was 12m when the barrel burst.
The pressure exerted by a static fluid depends only upon the depth of
the fluid, the density of the fluid, and the acceleration of gravity
(a) What was the force of the fluid pushing out on the top of
the barrel?
F=PA P= "gh so F= "gh #R2
Recal Fatm
F = (1x103x9.8x12x # x 0.22) N = 14,800 N
ph = p0 + ! g h
~3200 N
Static pressure does not depend upon mass or surface area of liquid
and the shape of container due to pressure exerted by walls.
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(b) If the tube had an inner radius of 3mm, what mass of
water caused the pressure that burst the barrel?
M= "h #R2 = 1x103x12 x 9x10-6 Kg = 0.34 Kg = 340g
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Example problem:
The pressure on the surface of a lake is atmospheric pressure, Pat.
(a) At what depth is the pressure twice atmospheric pressure?
(b) If the lake was full of mercury, at what depth is the pressure 2Pat?
A scuba diver's lungs can rupture
if he doesn't breathe while surfacing?
(a) p = pat + "gh
When p=2pat,
Why?
2pat = pat + "gh
pat = "gh
h = pat /"g
= 1.01 x 105/(1x 103 x 9.8) = 10.3m
(b) hHg = pat /"Hgg
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= 1.01 x 105/(13645 x 9.8) = 75cm
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Another example problem:
And another example problem:
Blood flows into the aorta through a circular opening of
radius 0.9cm.
If the blood pressure is 120 torr, how much force must be
exerted by the heart?
A 1500kg car rests on four tires, each of which is inflated to a gauge
pressure of 200kPa. What is the area of contact of each tire with
the road if the four tires support the weight equally?
F = pgaugeA
1 torr = 133.322Pa = 133.322 N.m-2
A=F/pgauge
Fcar=1500 x 9.8
A = 1500 x 9.8/(4 x 2 x 105 ) = 1.8 x 10-2 m2 =184 cm2
120 torr =120 x 133.322 N.m-2
F = pA
F = 120 x 133.322 x # x (9x10-3 )2 N
= 4.07 N
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Why do your ears feel different when you dive into deep water?
Why does atmospheric pressure change as you go up a mountain?
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