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Chapter 2 Fluid Statics
Objective: to understand how pressure varies as we go up or down.
Why does pressure increase as we go down in water? __________
Aircraft Submarine
Blood Pressure
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Pascal’s Law
Figure 2.3 (p. 42)g (p )Notation for pressure variation in a fluid at rest with a free surface.
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Objective: to derive pressure variation with respect to height or depth
f( d )
f(x)f(x)
f(x+dx)
xxx x+dx
f(x+dx) = f(x) +
Fig. 2.2 (p.40)
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Figure 2.4 (p. 43)Fluid equilibrium in a container of arbitrary shape.q y p
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How do we lift a car at Pep Boys?
What is the principle of hydraulic jack?
Figure 2.5 (p. 45)Figure 2.5 (p. 45)Transmission of fluid pressure.
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Figure 2 6 (p 47)
2.4 Standard Atmosphere
Figure 2.6 (p. 47)Variation of temperature with altitude in the U.S. standard atmosphere.
What is the pressure outsidean aircraft cruising at normal altitude?
What is the benefit of flyingat that high altitude?
Less pressure Less air Less ______ Less fuel consumption more $
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Figure 2.6 (p. 47)Variation of temperature with altitude in the U.S.
d d h
What about an orbiting space station ?
standard atmosphere.
What is the benefit of flyingat the orbit?
No pressure No air No ______ No fuel consumption ??
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2.5 Measurements of Pressure
Gage Pressure and Absolute Pressure
1 atm = 101,000 Pa = 14.7 psia = 760 mmHg = 760 Torr = 29.92 in Hg
What is a vacuum pressure?A: Pressure below zero gage pressure.
Figure 2.7 (p. 48)Figure 2.7 (p. 48)Graphical representation of gage and absolute pressure.
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How do we measure the atmospheric pressure?
Who did it for the first time? Torricelli (1644)
P P ??
Invert
PB = Patm ??
What is PA?
Objective: to determine the height h.
Construct a force balance.
Figure 2.8 (p. 49)Mercury barometer.
Construct a force balance.
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2.6 Manometry
Objective: to determine gas pressure
P1 = P2 ?? gas1 2
P2 = P3 ??
gas
P3 = Patm + liquid
Figure 2.10 (p. 51)Simple U‐tube manometer.
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Figure E2.4 (p. 52)Example 2.4
Objective: to determine the pressure readingIn [psi]
Given: SG of Hg = 13.6Density of water = 1.94 slug/ft3
Gravity constant = 32.174 ft/s2
h1 = 36 inh2 = 6 inh3 = 9 in
P1 = P2 ??
P2 = Patm +
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Example 2.5
Flow meter using an orifice nozzle
How does it work?
Flow rate ~ pressure difference
Flow rate ~ (PA – PB)
Figure E2.5 (p. 53)
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2.6.3 Inclined‐tube manometer
Why do we use it?
What is the benefit?
Figure 2 12 (p 54)Figure 2.12 (p. 54)Inclined‐tube manometer
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Conventional Pressure gage: What is the principle? How does it work?
Figure 2.13 (p. 55)Figure 2.13 (p. 55)(a) Liquid‐filled Bourdon pressure gages for various pressure ranges. (b) Internal elements of Bourdon gages. The “C‐shaped” Bourdon tube is shown on the left, and the “coiled spring” Bourdon tube for high pressures of 1000 psi and above is shown on the right. (Photographs courtesy of Weiss Instruments, Inc.)
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Digital Pressure gage: What is the principle? How does it work?
Figure 2.14 (p. 56)Pressure transducer which combines a linear variable differential transformer (LVDT) with aPressure transducer which combines a linear variable differential transformer (LVDT) with a Bourdon gage. (From Ref. 4, used by permission.)
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Strain‐Gage Pressure Transducer: What is the principle? How does it work?
Figure 2.15b (p. 57)(b) h d f h d h h d d fl f h(b) Schematic diagram of the P23XL transducer with the dome removed. Deflection of the diaphragm due to pressure is measured with a silicon beam on which strain gages and an associated bridge circuit have been deposited.
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How to measure blood pressure?
Listen heart beat using a Stethescope
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2.8 Hydrostatic Force on a plane surface
What is the pressure at the bottom if h = 3 m?
Wh i h l f i h b ? B 4 2What is the total force acting on the bottom? Bottom area = 4 m2
Fi 2 16 ( 58)Figure 2.16 (p. 58)(a) Pressure distribution and resultant hydrostatic force on the bottom of an open tank. (b) Pressure distribution on the ends of an open tank.
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What happens if the bottom is a curved surface?
2 Liter Coke bottleTask: to estimate the force acting on the bottom
2 Liter Coke bottleHow?
PsinѲ
Ѳ
P
Hemisphere
Figure P2.94 (p. 89)
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2.8 Hydrostatic Force on a plane surface – Inclined surface
Figure 2 17 (p 58)Figure 2.17 (p. 58)Notation for hydrostatic force on an inclined plane surface of arbitrary shape.
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2.8 Hydrostatic Force on a plane surface – Inclined surface
Air
AirWater 3 m
Objective: to estimate the force acting on a square window (4 m x 4 m)
AirWater 3 m
7 m
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2.8 Hydrostatic Force on a plane surface – Inclined surface
Air
AirWater 3 m
Objective: to estimate the force acting on a circular window with a diameter of 4 m.
AirWater 3 m
7 m
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2.8 Moment created by Hydrostatic Force on a square window (4 m x 4 m)Air
AirWater 3 mObjective: to estimate the force F1
Hinge: What is zero at hinge?
7 m
Objective: to estimate the force F1required to keep the window closed.
F1
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2.8 Moment created by Hydrostatic Force on a square window (4 m x 4 m)
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HW 2.61 Given: Width = 4 ft; gate weight = 800 lbf
Objective: to determine tension in cable.
Figure P2 61 (p 85)Figure P2.61 (p. 85)
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2.11 Buoyancy
Why does a light object float?
Why does a heavy object sink?
Consider a force balance.Consider a force balance.
F = ma = ?
Wh t ki d f f d ? Figure 2.24 (p. 69)Buoyant force on submerged and floating bodies.
What kinds of forces do you see?
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2.11 Buoyancy
Derivation of buoyancy force.
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Example 2.10 BuoyancyGiven: Buoy diameter = 1.5 m; weight = 8.5 kN
Figure E2.10 (p. 71)Objective: to determine the tension in cable.
Consider a force balance.Consider a force balance.
F = ma = ?
Wh ki d f f d ?What kinds of forces do you see?
Volume of sphere = 3
61 dπ
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HW 2.105 BuoyancyFigure P2.105 (p. 90)
Consider a force balance.
F = ma = ?
What kinds of forces do you see?
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HW 2.102 Buoyancy – Inverted test tube partially filled with air floats in water
When we squeeze the plastic bottle
The test tube sinks. Why?
“The test tube sinks” means Buoyancy force decreases or increases?
Mathematically
C l i th it ti ?
Squeeze
Can we explain the situation?
Figure P2.102 (p. 90)
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2.12 Linear Motion: Carrying fuel in a container truck
Example 2.11 Stability problem
Is the truck accelerating or decelerating?
Direction of a truck
Figure E2.11 (p. 74)
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2.12 Linear Motion: Carrying fuel in a container truck
Derivation of Eq.(2.28) for an inclined surface (Accelerometer)
P = P(y z)P = P(y,z)
dP =
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Water to tame wind atop new skyscraper
A 300,000‐gallon, double‐chambered tank of water is going in near the top of the Comcast Center ‐ a creative solution by engineers to keep Philadelphia's tallest building from swaying too much in the wind.
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2.12.2 Rigid‐Body Rotation
Objective: to find a mathematical equation of the curved surface
DerivationP P( )P = P(r,z)Along the curved surface, what is constant? Is P constant?
dP = 0 ?
z
dP = 0 ?
d d2ωρρ radrdp
r == gdzdp ρ−=
gdzdrrdp ρωρ −= 2
Figure 2.30 (p. 75)Rigid‐body rotation of a liquid in a tank.
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Figure E2.12 (p. 76)