Physics IPhysics I

Chap 15.Chap 15. Fluid StaticsFluid Statics

Prof. WAN, Xin

xinwan@zju.edu.cnhttp://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

Temperature CorrectionsTemperature Corrections

0 50 kmT

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

Moorea, French PolynesiaMoorea, French Polynesia

SnorkelingSnorkeling

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 of SyracuseArchimedes of Syracuse

Give me a place to stand and I will move

the earth.

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?

EurekaEureka