Fluid MechanicsFluid Mechanics

Chapter 8Chapter 8

Fluids and Buoyant Fluids and Buoyant Force Force

Section 1Section 1

Defining a FluidDefining a Fluid

�� AA fluidfluid is a nonsolid state of matter in which the is a nonsolid state of matter in which the atoms or molecules are free to move past each atoms or molecules are free to move past each other, as in a gas or a liquid.other, as in a gas or a liquid.

�� Both liquids and gases are considered fluids Both liquids and gases are considered fluids because they can flow and change shape.because they can flow and change shape.

�� Liquids have a definite volume; gases do not.Liquids have a definite volume; gases do not.

Density and Buoyant ForceDensity and Buoyant Force

�� The concentration of matter of an object is The concentration of matter of an object is called the called the mass densitymass density..

�� Mass density is measured as the mass per Mass density is measured as the mass per unit volume of a substance.unit volume of a substance.

ρ = mV

mass density = massvolume

Densities of Common SubstancesDensities of Common Substances

Density and Buoyant ForceDensity and Buoyant Force

�� The The buoyant forcebuoyant force is the upward force is the upward force exerted by a liquid on an object immersed exerted by a liquid on an object immersed in or floating on the liquid.in or floating on the liquid.

�� Buoyant forces can keep objects afloat.Buoyant forces can keep objects afloat.

Buoyant Force and ArchimedesBuoyant Force and Archimedes’’PrinciplePrinciple

�� The Brick, when added will cause the water to The Brick, when added will cause the water to be displaced and fill the smaller container.be displaced and fill the smaller container.

�� What will the volume be inside the smaller What will the volume be inside the smaller container?container?

�� The same volume as the brick!The same volume as the brick!

Buoyant Force and ArchimedesBuoyant Force and Archimedes’’PrinciplePrinciple

�� ArchimedesArchimedes’’ principle describes the magnitude principle describes the magnitude of a buoyant force.of a buoyant force.

�� ArchimedesArchimedes’’ principle: principle: Any object completely or Any object completely or partially submerged in a fluid experiences an partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the upward buoyant force equal in magnitude to the weight of the fluid displaced by the object.weight of the fluid displaced by the object.

FFBB = = FFgg (displaced fluid)(displaced fluid) = = mmffggmagnitude of buoyant force = weight of fluid displacedmagnitude of buoyant force = weight of fluid displaced

Buoyant ForceBuoyant Force

�� The raft and cargo The raft and cargo are floating are floating because their because their weight and weight and buoyant force are buoyant force are balanced.balanced.

Buoyant ForceBuoyant Force�� Now imagine a small hole Now imagine a small hole

is put in the raft.is put in the raft.�� The raft and cargo sink The raft and cargo sink

because their density is because their density is greater than the density of greater than the density of the water.the water.

�� As the volume of the raft As the volume of the raft decreases, the volume of decreases, the volume of the water displaced by the the water displaced by the raft and cargo also raft and cargo also decreases, as does the decreases, as does the magnitude of the buoyant magnitude of the buoyant force.force.

Buoyant ForceBuoyant Force

�� For a floating object, the buoyant force equals the For a floating object, the buoyant force equals the objectobject’’s weight.s weight.

�� The apparent weight of a submerged object The apparent weight of a submerged object depends on the density of the object.depends on the density of the object.

�� For an object with density For an object with density ρρOO submerged in a fluid submerged in a fluid of density of density ρρff, the buoyant force , the buoyant force FFBB obeys the obeys the following ratio: following ratio:

Fg (object)

FB

=ρO

ρf

ExampleExample�� A bargain hunter purchases A bargain hunter purchases

a a ““goldgold”” crown at a flea crown at a flea market. After she gets market. After she gets home, she hangs the crown home, she hangs the crown from a scale and finds its from a scale and finds its weight to be 7.84 N. She weight to be 7.84 N. She then weighs the crown while then weighs the crown while it is immersed in water, and it is immersed in water, and the scale reads 6.86 N. Is the scale reads 6.86 N. Is the crown made of pure the crown made of pure gold? Explain.gold? Explain.

SolutionSolution

ρρ

=

=

– apparent weightg B

g O

B f

F F

FF

( )

ρ ρ

=

=

– apparent weightB g

gO f

B

F F

FF

�� Choose your equations:Choose your equations:

�� Rearrange your equations:Rearrange your equations:

SolutionSolution

�� Plug and Chug:Plug and Chug:

�� From the table in your book, the density From the table in your book, the density of gold is 19.3 of gold is 19.3 ×× 101033 kg/mkg/m33. .

�� Because 8.0 Because 8.0 ×× 101033 kg/mkg/m33 < 19.3 < 19.3 ×× 101033

kg/mkg/m33, the crown cannot be pure gold., the crown cannot be pure gold.

( )ρ ρ

ρ

=

= = ×

= ×

3 3

3 3

7.84 N – 6.86 N = 0.98 N

7.84 N1.00 10 kg/m

0.98 N

8.0 10 kg/m

B

gO f

B

O

F

FF

Your Turn IYour Turn I

�� A piece of metal weighs 50.0 N in air and 36.0 N A piece of metal weighs 50.0 N in air and 36.0 N in water and 41.0 N in an unknown liquid. Find in water and 41.0 N in an unknown liquid. Find the densities of the following:the densities of the following:�� The metalThe metal�� The unknown liquidThe unknown liquid

�� A 2.8 kg rectangular air mattress is 2.00 m long A 2.8 kg rectangular air mattress is 2.00 m long and 0.500 m wide and 0.100 m thick. What and 0.500 m wide and 0.100 m thick. What mass can it support in water before sinking?mass can it support in water before sinking?

�� A ferry boat is 4.0 m wide and 6.0 m long. When A ferry boat is 4.0 m wide and 6.0 m long. When a truck pulls onto it, the boat sinks 4.00 cm in the a truck pulls onto it, the boat sinks 4.00 cm in the water. What is the weight of the truck?water. What is the weight of the truck?

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Fluid PressureFluid Pressure

Section 2Section 2

Pressure Pressure

�� Deep sea divers wear atmospheric diving Deep sea divers wear atmospheric diving suits to resist the forces exerted by the suits to resist the forces exerted by the water in the depths of the ocean.water in the depths of the ocean.

�� You experience this pressure when you You experience this pressure when you dive to the bottom of a pool, drive up a dive to the bottom of a pool, drive up a mountain, or fly in a plane.mountain, or fly in a plane.

PressurePressure

�� Pressure Pressure is the magnitude of the force on a is the magnitude of the force on a surface per unit area.surface per unit area.

�� PascalPascal’’s principle states that pressure applied to s principle states that pressure applied to a fluid in a closed container is transmitted a fluid in a closed container is transmitted equally to every point of the fluid and to the equally to every point of the fluid and to the walls of the container.walls of the container.

P = FA

pressure = forcearea

PressurePressure

�� The SI unit for pressure is the The SI unit for pressure is the pascalpascal, Pa., Pa.�� It is equal to 1 N/mIt is equal to 1 N/m22..�� The pressure at sea level is about 1.01 x The pressure at sea level is about 1.01 x

101055 Pa.Pa.�� This gives us another unit for pressure, the This gives us another unit for pressure, the

atmosphere, where 1 atmosphere, where 1 atmatm = 1.01 x 10= 1.01 x 1055 PaPa

PascalPascal’’s Principles Principle

�� When you pump a bike tire, you apply When you pump a bike tire, you apply force on the pump that in turn exerts a force on the pump that in turn exerts a force on the air inside the tire. force on the air inside the tire.

�� The air responds by pushing not only on The air responds by pushing not only on the pump but also against the walls of the the pump but also against the walls of the tire.tire.

�� As a result, the pressure increases by an As a result, the pressure increases by an equal amount throughout the tire.equal amount throughout the tire.

PascalPascal’’s Principles Principle�� A hydraulic lift uses A hydraulic lift uses

Pascal's principle.Pascal's principle.�� A small force is applied A small force is applied

(F(F11) to a small piston of ) to a small piston of area (Aarea (A11) and cause a ) and cause a pressure increase on the pressure increase on the fluid.fluid.

�� This increase in pressure This increase in pressure ((PPincinc) is transmitted to the ) is transmitted to the larger piston of area (Alarger piston of area (A22) ) and the fluid exerts a and the fluid exerts a force (Fforce (F22) on this piston.) on this piston.

F1

F2

A1

A2

2

2

1

1

AF

AF

Pinc ==

1

212 A

AFF =

ExampleExample

�� The small piston of a hydraulic lift has an The small piston of a hydraulic lift has an area of 0.20 marea of 0.20 m22. A car weighing 1.20 x 10. A car weighing 1.20 x 1044

N sits on a rack mounted on the large N sits on a rack mounted on the large piston. The large piston has an area of piston. The large piston has an area of 0.90 m0.90 m22. How much force must be applied . How much force must be applied to the small piston to support the car?to the small piston to support the car?

SolutionSolution

�� Plug and Chug:Plug and Chug:�� FF11 = (1.20 x 10= (1.20 x 1044 N) (0.20 mN) (0.20 m22 / 0.90 m/ 0.90 m22))�� FF1 1 = 2.7 x 10= 2.7 x 1033 NN

2

2

1

1

AF

AF =

2

121 A

AFF =

Your Turn IIYour Turn II�� In a car lift, compressed air exerts a force on a In a car lift, compressed air exerts a force on a

piston with a radius of 5.00 cm. This pressure is piston with a radius of 5.00 cm. This pressure is transmitted to a second piston with a radius of transmitted to a second piston with a radius of 15.0 cm.15.0 cm.�� How large of a force must the air exert to lift a 1.33 x How large of a force must the air exert to lift a 1.33 x

101044 N car?N car?�� A person rides up a lift to a mountain top, but the A person rides up a lift to a mountain top, but the

personperson’’s ears fail to s ears fail to ““poppop””. The radius of each . The radius of each ear drum is 0.40 cm. The pressure of the ear drum is 0.40 cm. The pressure of the atmosphere drops from 10.10 x 10atmosphere drops from 10.10 x 1055 Pa at the Pa at the bottom to 0.998 x 10bottom to 0.998 x 1055 Pa at the top.Pa at the top.�� What is the pressure difference between the inner and What is the pressure difference between the inner and

outer ear at the top of the mountain?outer ear at the top of the mountain?�� What is the magnitude of the net force on each What is the magnitude of the net force on each

eardrum?eardrum?

PressurePressure

�� Pressure varies with depth in a fluid.Pressure varies with depth in a fluid.

�� The pressure in a fluid increases with The pressure in a fluid increases with depth.depth.

( )

ρ= +

× ×

0

absolute pressure = atmospheric pressure +

density free-fall acceleration depth

P P gh

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Fluids in MotionFluids in Motion

Section 3Section 3

Fluid FlowFluid Flow�� Moving fluids can exhibit Moving fluids can exhibit laminarlaminar (smooth) (smooth)

flow or flow or turbulentturbulent (irregular) flow.(irregular) flow.

Laminar Flow Turbulent Flow

Fluid FlowFluid Flow

�� An An ideal fluidideal fluid is a fluid that has no internal is a fluid that has no internal friction or viscosity and is incompressible.friction or viscosity and is incompressible.

�� The ideal fluid model simplifies fluidThe ideal fluid model simplifies fluid--flow flow analysisanalysis

Fluid FlowFluid Flow

�� No real fluid has all the properties of an No real fluid has all the properties of an ideal fluid, it helps to explain the properties ideal fluid, it helps to explain the properties of real fluids.of real fluids.

�� Viscosity refers to the amount of internal Viscosity refers to the amount of internal friction within a fluid. High viscosity equals friction within a fluid. High viscosity equals a slow flow.a slow flow.

�� Steady flow is when the pressure, Steady flow is when the pressure, viscosity, and density at each point in the viscosity, and density at each point in the fluid are constant.fluid are constant.

Principles of Fluid FlowPrinciples of Fluid Flow

��The continuity equation results from The continuity equation results from conservation of mass.conservation of mass.

��Continuity equation:Continuity equation:AA11vv11 = = AA22vv22

Area Area ×× speed in region 1 = area speed in region 1 = area ×× speed in region 2speed in region 2

Principles of Fluid FlowPrinciples of Fluid Flow

�� The speed of fluid flow The speed of fluid flow depends on crossdepends on cross--sectional area.sectional area.

�� BernoulliBernoulli’’s principle s principle states that the pressure states that the pressure in a fluid decreases as in a fluid decreases as the fluidthe fluid’’s velocity s velocity increases.increases.

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