Lecture 20

Static fluids

Today’s Topics:

• Density• Pressure• Pressure, Depth and Pressure Gauges• Pascal’s Principle• Archimede’s Principle

Solids and Fluids

• Solids– Maintain their shape…. Generally don’t flow– Limited compressibility– Resistant to shear

• Fluids– Can Flow …. Adapt to the shape of their container– Little resistance to shear

• Liquids …. Limited compressibility• Gases …. Larger compressibility

– Fills the container

Mass DensityThe mass density of a substance is the mass of a substance divided by its volume:

Vm

=r

SI Unit of Mass Density: kg/m3 (g/cm3)

Ice has a lower density than liquid water:

Lakes (water) freeze from the top down!

Example – Mass DensityThe average density of the material in intergalactic space is approximately 2.5 × 10–27 kg/m3. What is the volume of a gold sample, ρ = 19,300 kg/m3, that has the same mass as 8.0 × 1024 m3 of intergalactic space?

33

33

3

m kg/m kg

So,kg

)m kg/m :is space ticintergalac of m of mass The

6

2427

24

100.1300,1902.0

02.0100.8)(105.2(

100.8

-

-

´===

=´´==

´

=Þ=

r

r

rr

MV

MVM

MVVM

gold

Pressure

AFP =

SI Unit of Pressure: 1 N/m2 = 1Pa

Pascal

Pressure is uniform in all directions!

Gauge pressure vs. absolute pressure

Atmospheric Pressure at Sea Level: 1.013 x 105 Pa = 1 atmosphere

Arises from the weight of the air (fluid) above us

Pressure and Depth012 =--=å mgAPAPFy

mgAPAP += 12

rVm =

VgAPAP r+= 12

AhV =

AhgAPAP r+= 12

hgPP r+= 12

Example: PressureA gas sample is confined within a chamber that has a movable piston. A small load is placed on the piston; and the system is allowed to reach equilibrium. If the total weight of the piston and load is 70.0 N and the piston has an area of 5.0 x 10−4 m2, what is the pressure exerted on the piston by the gas? Note: atmospheric pressure is 1.013 x 105 Pa.

load

piston

gas sample

The pressure exerted by the gas on the piston will be equal to the pressure exerted by the piston on the gas!

Pa

N/m m

N

22

5

54

104.2

10013.1100.50.70

´=

´+´

=

+=

-

+ pressurecAtmospheriA

WP loadPiston

Don’t forget atmospheric pressure!

Example – Pressure and DepthA column of water of height 70.0 cm supports a column of an unknown liquid as suggested in the figure (not drawn to scale). Assume that both liquids are at rest and that the density of water is 1.0 x 103 kg/m3. Determine the density of the unknown liquid.

70.0 cm

27.0 cm

A

33

kg/m cm

cm) kg/m

:height)that at point any(or point AAt

33

1060.20.27

0.70)(100.1(´=

´==

=+=+

=

unknown

waterwaterunknown

unknownunknownwaterwater

unknownunknownwaterwater

unknownwater

hh

hhpressurecAtmospherighpressurecAtmospherigh

PP

rr

rrrr

Pressure Gauges (Barometer)

ghPP r+= 12

ghPatm r=

( )( )( )

mm 760m 760.0

sm80.9mkg1013.6Pa 1001.1

233

5

==

´´

==g

Ph atm

r

P2(A) = P2(B)

Pascal’s PrincipleAny change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls.

( )m 012 gPP r+=

1

1

2

2

AF

AF

=

÷÷ø

öççè

æ=

1

212 AAFF

Are we getting something for nothing?

Let’s calculate the work done on the input and output sides:

( )APPAPAPFB 1212 -=-=

ghPP r=- 12

ghAFB r=hAV =

! gVFB

fluiddisplaced

of mass

r=

Buoyant Force

Archimede’s PrincipleAny fluid applies a buoyant force to an object that is partiallyor completely immersed in it; the magnitude of the buoyantforce equals the weight of the fluid that the object displaces:

! !fluid displaced

ofWeight fluid

forcebuoyant of Magnitude

WFB =

Example -- Archimede’s PrincipleA small sculpture made of brass (rbrass = 8470 kg/m3) is believed to have a secret central cavity. The weight of the sculpture in air is 15.76 N. When it is submerged in water, the weight is 13.86 N. What is the volume of the secret cavity?