Physics I Physics I Chap 15. Chap 15. Fluid Fluid Statics Statics Prof. WAN, Xin [email protected] http://zimp.zju.edu.cn/~xin wan/
Dec 29, 2015
Physics IPhysics I
Chap 15.Chap 15. Fluid StaticsFluid Statics
Prof. WAN, Xin
[email protected]://zimp.zju.edu.cn/~xinwan/
DensityDensity
Density (scalar): mass of a small element of material m divided by its volume V
– For infinitesimally small V:
= m / V
– For homogeneous material:
= m / V
Water BedWater Bed
The mattress of a water bed is 2.00 m long by 2.00 m wide and 30.0 cm deep. (a) Find the weight of the water in the mattress.
The density of water is 1 000 kg/m3, and so the mass of the water is
and its weight is
kgmmkgVM 333 102.1)3.022(/1000
NsmkgMg 423 1018.1/8.9102.1
About the weight of a compact car!
PressurePressure
Under static condition, force acts normal or perpendicular to the surface – vector.
Pressure: the magnitude of the normal force per unit surface area – scalar.
F p A
Fp
A
A
AF
Water Bed (Continued)Water Bed (Continued)
(b) Find the pressure exerted by the water on the floor when the bed rests in its normal position. Assume that the entire lower surface of the bed makes contact with the floor.
When the bed is in its normal position, the cross-sectional area is 4.00 m2
232
4
/1095.24
1018.1mN
m
NP
Common UnitsCommon Units
2
5
5
2
1 1 / [SI]
1 1.01325 10
1 10 1
1 760 760
1 14.7 / .
pascal N m
atmosphere Pa
bar Pa atm
atm mm of Hg torr
atm lb in
Origin of PressureOrigin of Pressure
Microscopic origin: collisions of molecules of the fluid with the surface– Action and reaction: Newton’s third law
liquid
V
freaction
Wall
Bulk ModulusBulk Modulus
Can density describe the resistance of a fluid to compression? Why?
For a fixed temperature, – Define:
bulk modulus
– Pressure change p – Fractional change in volume V/V
/
pB
V V
Bulk ModulusBulk Modulus
Solids & liquids– Large B, ~ 109 N/m2
– (Practically,) incompressible Gases
– Small B, ~ 105 N/m2
– Easily compressible
Pressure in a Fluid at RestPressure in a Fluid at Rest
( ) ( )
( ) 0
dm g Ady g gAdy
pA gAdy p dp A
yy+dy
A
pA
(p+dp)A
(dm)g
dpg
dy
Pressure in a Fluid at RestPressure in a Fluid at Rest
Assumption: , g independent of y
0 0
0 0
0 0 0
' '
( )
( )
p y
p y
dpg dp g dy
dy
p p g y y
p p g y y p gh
Mercury BarometerMercury Barometer
How to measure atmospheric pressure? E. Torricelli (1608-47)
For mercury, h = 760 mm.
How high will water rise?
No more than h = patm/g = 10.3 m
h patm
p=0
atmatm
PP gh h
g
Vacuum!
Atmospheric PressureAtmospheric Pressure
Gases are compressible. Thus, varies!
50 sea level
0 0
00
1.013 10
, for constant
p p Pa
ppV nRT T
p
dp pg g
dx p
dy
0 0
0 0
00
0 0
00 0
0
( / )0
''
'
ln ln ( )
p y
p y
g p h
gdp p dpg dy
dx p p p
gp p y y
p
p p e
dy
Atmospheric PressureAtmospheric Pressure
00 0
h p p gh
A.P. – an Improved VersionA.P. – an Improved Version
0 0 0
0 0
00
0 0 0
( )
( )
pV nRT
p T
p T
T T y y
Tdp pg g
dy p T y y
Temperature decreases 6oC for each 1,000 meters of elevation
A.P. – an Improved VersionA.P. – an Improved Version
0 0
0
0 0
0 0
0 0 0
0 0
0 0 0
00
( )
' '
' ( ' )
1
p y
p y
gT p
g T dydp
p p T y y
g Tdp dy
p p T y y
hp p T
0
000
gh
pp e
Q & A: On PressureQ & A: On Pressure
Q: Is it possible to stand on the roof of a four-story (12 m) building and drink, using a tube, from a glass on the ground? 12 m
A: Even if a person could completely remove all of the air from the tube, the height to which the outside air pressure pushes the water up the straw would not be high enough for him/her to drink the water, no matter how hard he/she sucks! Really?
Q & A on PressureQ & A on Pressure
12 m
Blaise PascalBlaise Pascal
Blaise Pascal (1623-62)
French mathematician & philosopher; founder of modern theory of probability; discovered Pascal's law of pressure and principle of the hydraulic press.
Pascal’s PrinciplePascal’s Principle
Pressure applied to an enclosed fluid is transmitted undimished to every portion of the fluid and to the walls of the containing vessel.
F
p
d
ext extp p gd p p
pext
Hydraulic LeverHydraulic Lever
/ / ,i i o o i i o oF A F A A d A d
Fi
mgAi Ao
Fo
di
do
/ /i i o o i oF A F A g d d
Hydraulic BrakeHydraulic Brake
Fred Duesenberg originated hydraulic brakes on his 1914 racing cars and Duesenberg was the first automotive marque to use the technology on a passenger car in 1921. In 1918 Malcolm Lougheed (who later changed the spelling of his name to Lockheed) developed a hydraulic brake system.
Archimedes of SyracuseArchimedes of Syracuse
Archimedes (circa 287-212 B.C.) Greek mathematician & inventorknown especially for work in mechanics; discovered principle of buoyancy; wrote on volumes of spheres and cylinders, value of , etc.
Archimedes’ PrincipleArchimedes’ Principle
A body wholly or partially immersed in a fluid is buoyed up by a force equal in magnitude to the weight of the fluid displaced by the body.
FB
mg
B fluid fluid
object
F Vg m g
mg Vg
Two CommentsTwo Comments
Buoyant force volume/density
Center of buoyancy (not fixed)Center of gravity
B fluid fluid
object
F Vg m g
mg Vg
Q&A: Galileo’s ScaleQ&A: Galileo’s Scale
Can you invent a scale that one can use to read the percentage of silver in the gold crown (let’s say, to help Archimedes to determine the exact percentage) without explicit calculations?