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Pressure and Pressure Archimedes Principle Buoyant ’s Principle

Table 4-4, p. 145

Pressure in Fluid Pressure Consider a column of a fluid. What is the Lab 5. Pressure & density. Fluid: a which can flow ( or ). pressure at the bottom of the column?

Pressure is the same on every 1. The of a fluid: direction in a fluid at a given Examples of fluids: air, water, alcohol, . depth. Fweight G == mg m = DV G == DVgFweight - - mass, Dm - density,V - , g - acceleration due to h 1. Fluids may have different density. 2. The pressure created by a fluid at the bottom of the column: 2. Fluids exerts pressure in all directions. F DVg D(abh)g P G === = Dhg 3. The force due to fluid pressure always A A ab gauge == hgDPP b acts perpendicular to any surface it is = abhV in contact with. a Gauge P - pressure, D - density, - height, gh - acceleration due to gravity

Note that the fluid is not in motion! Pgauge is so-called “gauge” pressure.

How we can calculate pressure producing by any fluid? Gauge pressure depends on density of fluid and depth! 3 4

Fig. 4-26, p. 149 Table 4-3, p. 139 Measurement of : Mercury

The glass tube is completely filled with Atmospheric Pressure mercury and then inverted into the bowl of mercury. A column of mercury 76 cm = 760 mm exerts Atmospheric pressure is defined as: the same pressure as the : 1 atm = 76 cm Hg = 1.01 105 Pa = hgDP Air kg pressure Density of Hg = 60013D: m3 How high would water rise in a closed Why use such a dense ? P inverted column, at 1 atm? h = gD The pressure at the base of the water column, !gh, is 1 atm: 1.01 105 Pa = (103kg/m3)(9.8m/s2)(h); P Dg h == ! 3 kg /106.13( 3 )( /8.9 2 )( msmm = ! 25 =1/10013.1)76.0 atmmN thus h=10.3 m = .h 3410 m water =1 atm 7 8

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

9 Fig. 4-27, p. 150

Gauge Pressure

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

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 pipe water 20 m tower

Gauge pressure:

gauge = !hgP

Pressure gauge 11 Fig. 4-28, p. 151 Archimedes Principle and Archimides’ principle and Buoyant force. Some objects placed in fluid can float on the surface or at some depth.

(FB=!Vg, Buoyant force=Weight of fluid displaced) The reason: buoyant force F B acts on them.

Fig. 9.15 Static equilibrium:

FB = weight = FG = mO g

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

1. The buoyant force on the object floating in a fluid is equal to the P2 ! is the density of the fluid f weight of the object.

2. The buoyant force on the object immersed in a fluid is equal to the The pressure acting on a bottom of the suspended metal block weight of the submerged fluid. is greater than that on the top due to increase of pressure with depth 12 13

Buoyant force = Weight of water Buoyant Force F A bouyant B Fbouyant !A < !B < !C !A !B F W bouyant object C Wobject Water !C

Fbouyant = Wobject Wobject

Wobject = mobject g = (!objectVobject )g

Fbouyant < Wobject Fbouyant = Wdisplaced water = mdisplaced water g =

= (!waterVdisplaced water )g = (!waterVsubmerged object )g 14 Fig. 4-35, p. 155

BUOYANT FORCE, ARCHIMEDES PRINCIPLE

Fig. 4-32, p. 153 Buoyant Force Will an ice cube float higher in water or in alcohol? Problem 1: Density of water is approx 1g/ml A 1.5-kg block of wood is floating in water. What is the magnitude of the Density of ice is approx 0.93g/ml buoyant force acting on the block? m Density of 100% pure ethanol is 0.79g/ml F = weight of the object = m g = (1.5 kg)(9.8 ) = 14.7 N B O s s Problem 2: 3 3 2 Weight of ice = W = mg = !Vg W = (0.93g / cm )(1cm )10m / s ) = 9.3N 3 3 A floating boat displaces 3 m of water (Dw = 1000 kg/m ). F = ! Vg Bouyant Fluid 1. What is the mass of the water displaced by the boat?

3 3 2 mw kg 3 Water : FBouyant = !FluidVg = (1g / cm )(1cm )(10m / s ) = 10N FBouyant > W Dw = mw = DwVw = (1000 3 )(3 m ) = 3000 kg Vw m Ice cubes floats in water 2. What is the buoyant force acting on the boat?

3 3 2 m Alcohol : FBouyant = !FluidVg = (0.79g / cm )(1cm )(10m / s ) = 7.9N F < W FB = weight of the displaced water = mw g = ( 3000 kg)(9.8 s ) = 29 400 N Bouyant s Thus ice cubes would not float at all in 100% pure ethanol. 3. What is the weight of the boat? weight = 29 400 N 15

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

Zeppelin Hindenburg: Buoyant Force in Air Pascal’s Principle

F force The zeppelin LZ-129 Hindenburg was one Launched in 1936, crashed Pressure in fluids: P = (pressure = ) of the largest aircraft ever built: 245 m in 1937 in Lakehurst, NJ A 3 long, 41 m in diameter, 211 890 m of . In a fluid, the applied force creates a pressure that is transmitted everywhere Designed to use helium, forced to use hydrogen due to US military embargo. throughout the fluid. Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished to all LZ-129 parts of the fluid and to the walls of the container. force Boeing 747 piston fluid The buoyant force occurs due to difference in and helium or hydrogen Pascal’s principle is widely applied in hydraulic systems. Air : 1.2 kg/m3 Helium : 0.18 kg/m3 Hydrogen : 0.089 kg/m3 cylinder !air = !He = !H 2 =

2 3 2 Can carry : ("air ! "H )Vg = [(1.2 ! 0.089) kg/m ](211 890 m )(9.8 m / s ) = 2 307 015 N 16 17 Use Pascal’s Principle in Hydraulic Systems

Hydraulic systems consists of two or more pistons: Small piston Large piston A = ! r 2 F s F Al l A s 2 Al = !r l

Fs Fl Ps = Pl = As Al

F A F A F = s l F = l s l A s A F F s l P = P s = l s l F A As Al in out Fout = Ain 18