Review: Pressure
Physics 201, Lecture 26
Today’s Topics
n
Fluid Mechanics (chapter 14)
n  Review: Pressure
n  Buoyancy, Archimedes’s Principle (14.4)
n  Fluid Dynamics, Bernoulli’s Equation (14.5,14.6)
n  Applications of Fluid Dynamics
q  Fluid exerts force on the objects it contact:
molecules constantly hit the surface  F =Σ(Δmv/Δt)
q  Force exerted by fluid distributes over contact surface
q  Pressure: P = Force / Area  P
≡ F/A
§  Unit: N/m2 ≡ Pascal (Pa)
§  Pressure depends on only the magnitude of F
§  Pressure also definable for solids when the contact surface is
regular
§  Be careful with “p”: momentum/power/pressure?
§  Atmospheric (room air) pressure: 1.01 x105 Pa
Review: Pascal’s Principle
Same level  same pressure
Regardless of shape, etc.
(in the same fluid)
q  At different level: P = P0 + ρgh
q  Compare PA , PB ?
Ø  Define a reference level CD : PC=PD
Δh
q  Pascal’s Principle:
Pressure is a contained fluid is transmitted to every point, in
every direction, regardless of the shape of the container.
Pressure is the same at same height/depth.
water
oil
Quick Quiz
A
B
C
D
Ø  Compare level AB to level CD:
PA=PC - ρHggΔh < PD – ρH2ogΔh =PB
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Exercise: Hydraulic Jack
q  Explain the working principle of the hydraulic press
§  Force ratio F1:F2=?
P2 = P1 →
F2 F1
=
A2 A1
→
F2 A2
=
F1 A1
>> 1
§  Displacement ratio Δx1:Δx2=?
→ Δx1 A1 = Δx 2 A2
The magnitude of buoyant force always
equals the weight of the fluid displaced
by the object:
Δx
A
→ 2 = 1
Δx1 A2
recall : W = F • Δx
W2 F2 Δx 2
=
= 1:1 !
W1 F1Δx1
Archimedes of Syracuse
( 287 – 212 BC)
B=Mdispg= ρfluidVdispg
§  Work ratio W1:W2 =?
€
q  All object immersed in fluid is subject to an
buoyant force (B) exerted by the fluid.
q  Archimedes Principle:
oil is not compressible
€
Buoyancy and Archimedes's Principle
One can “save” force,
but not work (energy)!
ü
ü
ü
ü
Buoyant force is always upwards
It is independent of shape
It is independent of the density of the object
It is caused by pressure difference over depth
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Recall: Density
Floating or Sinking
If ρobj > ρfluid
Fg = mobjg = ρobj Vobj g
B = ρfluid Vobj g < Fg
If ρobj < ρfluid
Fg = mobjg = ρobj Vobj g
B = ρfluid Vobj g > Fg
sinking !
floating !
(eventually, sunken at the bottom)
(eventually, floating at top)
Density = Mass/Volume
ρ ≡ m/V
§  Definable for solids and fluids
§  Basic Unit: kg/m3
§  Independent of shape
§ Solids and liquids:
ρ very weakly depends on
temperature and pressure
§  Gases:
ρ strongly depends on
temperature and pressure
§ Solid form is not necessarily
“heavier” than liquid form
Substance
Density ρ
(103 kg/m3)
Water
1.00
Ice
0.917
Mercury
13.6
Lead
11.3
Copper
8.92
Iron
7.86
Aluminum
2.70
Wood
0.550
Blood
1.06
Oil
0.92-0.98
Alcohol
0.82
Room Air
0.00129
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Exercise: Floating Iceberg
q  What is the portion of a floating iceberg that is under water?
( ρice =0.9, ρwater =1.0 )
Ø  Floating: Fg = B
Fg = ρice Vwhole_ice g
B =ρwater Vice_in_water g
à  ρice Vwhole_ice = ρwater Vice_in_water
à  Vice_in_water : Vwhole_ice = ρice : ρwater = 0.9 :1 = 90%
Quiz: Ice on a Cup of Water
q  Convince yourself that the only 10% of the ice is showing
above the water level.
q  Quiz: when all ice is melted, is the water level
higher, lower, or remains the same
Solution:
Before melting B= miceg
= ρwaterVdispg
à  Vdisp= mice/ρwater
After melting, mice becomes
Vice_melt ρwater of water.
à  Vice_melt =mice/ρwater =Vdisp
Follow up quiz: What about Ice on Oil (ρoil=0.94, ρIce=0.9)?
Exercise:
King Hiero II of Syracuse’s Crown
q  The legend says that Archimedes was once asked by the King
of Syracuse to tell whether a new crown he had just acquired
was made of pure gold. Here is what Archimedes did:
Ø  Weigh the crown in air, he got:
Win_air = 7.84 N = ρcrownVcrown
Ø  then weigh the crown in water and got:
Win_water = 6.84 N = Win_air - B
Ø  So the buoyancy B= 7.84-6.84 = 1.00N
= ρwaterVcrown
then he can obtain
ρcrown = ρwater 7.84/1.00 = 7.84 x103kg/m3
<<ρGold =19.3 x103kg/m3
So the king was indeed cheated!
Fluid Dynamics
q  Pascal’s Principle and Archimedes principle are for fluids that are
in equilibrium. What about fluid that flows?
Ø  The subject is called Fluid Dynamics (a course by itself!)
q  For this course, we are considering a simpler
(but still very useful ) model: ideal fluid flow.
flow of streamline
q  Ideal Fluid Flow:
§  The flow is of steady streamline (i.e. non-Turbulence)
§  The fluid is non-viscous (i.e. ignore all internal friction)
§  The fluid is incompressible. (i.e. density is constant)
§  The fluid is ir-rotational (i.e. zero angular momentum)
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Fluid Dynamics: Continuity
The amount of (ideal) fluid moving in at point 1
shall equal the amount moving out at point 2
Quiz: Water Flow From Faucet
q  Quiz: Why the water is getting narrower while going down?
q  Answer:
By gravity, water is accelerated while going down v2 > v1
Per continuity: A1v1=A2v2
 A2<A1 !
v1
v2
A1
A2
Continuity Equation
A1v1= A2v2 (=constant)
Fluid Dynamics: Bernoulli’s Equation
The net work done between point 1 and 2
shall equal change energy
Demo: Paint Sprayer
q  Explain why water can be sucked up (and sprayed out)
direct result of
energy conservation
Bernoulli’s Equation
P+ ½ ρv2 + ρgy=constant
q  Also: Ping-Pong balls in air tubes, Bernoulli ball etc.
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Demo: The Venturi Tube
Discussion: Airplane Wing
to be countered
by propeller/jets
q
Larger A, lower v
same height: P+ ½ ρv2 = constant
lower v  higher P
higher P  lower liquid height
Air flow faster
flying direction
Air flow slower
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