Pressure and DensityFluid Pressure

Archimedes PrincipleBuoyant Force

Pascal’s Principle

Table 4-4, p. 145

3

Pressure in Fluids

Fluid: a matter which can flow (gases or liquids).

2. Fluids exerts pressure in all directions.

Pressure is the same on everydirection in a fluid at a givendepth.

Note that the fluid is not in motion!

3. The force due to fluid pressure alwaysacts perpendicular to any surface it isin contact with.

1. Fluids may have different density.

How we can calculate pressure producing by any fluid?

Examples of fluids: air, water, alcohol,mercury.

Consider a column of a fluid. What is thepressure at the bottom of the column?

mg Fweight G ==h

b

a

DVm =

hg Dab

(abh)gDA

DVgA

FP G ====

DVgFweight G ==

hgDPP gauge ==

1. The weight of a fluid:

4

2. The pressure created by a fluid at the bottom of the column:

Pgauge is so-called “gauge” pressure.

Gauge

gravity todueon accelerati - volume,- density,- mass, - gVDm

gravity todueon accelerati - height,- density,- Dpressure, - ghPabhV =

Gauge pressure depends on density of fluid and depth!

Fluid PressureLab 5. Pressure & density.

Fig. 4-26, p. 149 Table 4-3, p. 139

Measurement of Atmospheric Pressure: Mercury Barometer

7

The glass tube is completely filled withmercury and then inverted into the bowl ofmercury.

360013D:Hg ofDensity mkg=

Why use such a dense liquid?

gDPh =

atmwater m.h 13410 ==

Airpressure

A column of mercury 76 cm = 760 mm exertsthe same pressure as the atmosphere:

hgDP =

atmmNmsmmkghDgP 1/10013.1)76.0)(/8.9)(/106.13( 25233 =!=!==

Atmospheric PressureAtmospheric pressure is defined as:

1 atm = 76 cm Hg = 1.01 105 Pa

How high would water rise in a closedinverted column, at 1 atm?

The pressure at the base of the water column,!gh, is 1 atm: 1.01 105 Pa = (103kg/m3)(9.8m/s2)(h);thus h=10.3 m

8

Fig. 4-27, p. 150

A balloon that was partially inflated near the sea level expanded as the experimenters climbed the mountain.

9

Fig. 4-28, p. 151

Gauge PressureWe have a narrow pipe with water of 20 m height and water tower with 20 m ofwater height (see below). Pressure gauge placed at the bottom shows:

hgPgauge !=

1) Larger pressure for the water tower

2) The same pressure

3) Larger pressure for the pipe with water

4) Depends on the radius of the pipe

Pressure gauge11

m20

water

Gauge pressure:

towerwaterpipe

Fig. 9.15

(FB=!Vg, Buoyant force=Weight of fluid displaced)

Archimides’ principle and Buoyant force.

The pressure acting on a bottom of the suspended metal block is greater than that on the top due to increase of pressure with depth

FB=(P2-P1)A = !fg(h2-h1)A = !fgV

P2

P1

!f is the density of the fluid

12

Archimedes Principle and Buoyancy

1. The buoyant force on the object floating in a fluid is equal to the weight of the object.

13

gmFweightF OGB ===

Static equilibrium:

Some objects placed in fluid can float on the surface or at some depth.The reason: buoyant force acts on them.BF

2. The buoyant force on the object immersed in a fluid is equal to the weight of the submerged fluid.

Buoyant force = Weight of water

Fig. 4-35, p. 155

Buoyant Force

objectW

objectWobjectbouyant WF =

Fbouyant <Wobject=== gmWF waterdisplacedwaterdisplacedbouyant

Wobject = mobjectg = (!objectVobject )g

C

B

Water

!B

!C

!A < !B < !CbouyantF

bouyantFobjectW

bouyantFA!A

= (!waterVdisplaced water )g = (!waterVsubmerged object )g

14

Fig. 4-32, p. 153

BUOYANT FORCE, ARCHIMEDES PRINCIPLE

Density of water is approx 1g/mlDensity of ice is approx 0.93g/mlDensity of 100% pure ethanol is 0.79g/ml

Will an ice cube float higher in water or in alcohol?

Thus ice cubes would not float at all in 100% pure ethanol.

Weight of ice =W = mg = !Vg W = (0.93g / cm3)(1cm3)10m / s2 ) = 9.3N

FBouyant = !FluidVg

Water : FBouyant = !FluidVg = (1g / cm3)(1cm3)(10m / s2 ) = 10N

Alcohol : FBouyant = !FluidVg = (0.79g / cm3)(1cm3)(10m / s2 ) = 7.9N

FBouyant >W

FBouyant <W

Ice cubes floats in water

Buoyant Force

15

Problem 1:A 1.5-kg block of wood is floating in water. What is the magnitude of thebuoyant force acting on the block?

Problem 2:A floating boat displaces 3 m3 of water (Dw = 1000 kg/m3).

N.)sm.kg)(.(gmthe objectweight of F sOB 7148951 ====

1. What is the mass of the water displaced by the boat?

2. What is the buoyant force acting on the boat?

N)sm

.kg)((gmced waterthe displaweight of F swB 40029893000 ====

kg)m)(mkg(VD m

VmD www

w

ww 300031000 3

3 ====

3. What is the weight of the boat? weight = 29 400 N

Fig. 4-40, p. 160 Fig. 4-38a, p. 158

The zeppelin LZ-129 Hindenburg was oneof the largest aircraft ever built: 245 mlong, 41 m in diameter, 211 890 m3 of gas.Designed to use helium, forced to usehydrogen due to US military embargo.

Zeppelin Hindenburg: Buoyant Force in Air

16

LZ-129

Boeing 747

Launched in 1936, crashedin 1937 in Lakehurst, NJ

The buoyant force occurs due to difference in density of air and helium or hydrogen

3kg/m1.2 :Air =air! 3kg/m0.18 :Helium =He! 3kg/m0.089 :Hydrogen2=H!

Nsmmkg/mVgHair 0153072)/8.9)(890211]()089.02.1[()( :carryCan 232 =!=! ""

Pascal’s Principle

Pascal’s principle is widelyapplied in hydraulic systems.

Pascal’s principle:pressure applied to an enclosed fluid is transmitted undiminished to all

parts of the fluid and to the walls of the container.

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piston

fluid

)areaforce

(pressure ==AFP

force

cylinder

In a fluid, the applied force creates a pressure that is transmitted everywherethroughout the fluid.

Pressure in fluids:

Hydraulic systems consists of two or more pistons:

ll rA 2!=

Use Pascal’s Principle in Hydraulic Systems

sA

s

ss AFP =

piston Small

lF

18

2rA !=sF

piston Large

lA

l

ll AFP =

ls PP =s

lsl A

AFF =

l

l

s

s

AF

AF =

l

sls A

AFF =

in

outinout A

AFF =