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Chapter 9 Fluids

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Objectives for Today Hydrostatic Pressure; P = gh Hydrostatic Pressure; P = gh Buoyancy; Archimedes’ Principle Buoyancy; Archimedes’ Principle F buoyancy = g(Volume displaced) F buoyancy = g(Volume displaced) Pascal’s Equation Pascal’s Equation P=F/A = f/a P=F/A = f/a Continuity Equation Continuity Equation A 1 V 1 =A 2 V 2 A 1 V 1 =A 2 V 2 Bernoulli’s Equation Bernoulli’s Equation P +1/2 v 2 + gh = constant P +1/2 v 2 + gh = constant

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Density The density of a substance of uniform composition is defined as its mass per unit volume: The density of a substance of uniform composition is defined as its mass per unit volume: Units are kg/m 3 (SI) or g/cm 3 (cgs) Units are kg/m 3 (SI) or g/cm 3 (cgs) 1 g/cm 3 = 1000 kg/m 3 1 g/cm 3 = 1000 kg/m 3

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Pressure The force exerted by a fluid on a submerged object at any point if perpendicular to the surface of the object The force exerted by a fluid on a submerged object at any point if perpendicular to the surface of the object

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Variation of Pressure with Depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium All points at the same depth must be at the same pressure All points at the same depth must be at the same pressure Otherwise, the fluid would not be in equilibrium (Think weather) Otherwise, the fluid would not be in equilibrium (Think weather)

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Pressure and Depth Examine the darker region, assumed to be a fluid Examine the darker region, assumed to be a fluid It has a cross- sectional area A It has a cross- sectional area A Extends to a depth h below the surface Extends to a depth h below the surface Three external forces act on the region Three external forces act on the region

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Pressure and Depth equation P o is normal atmospheric pressure P o is normal atmospheric pressure = kPa = kPa = 14.7 lb/in 2 = 14.7 lb/in 2 The pressure does not depend upon the shape of the container The pressure does not depend upon the shape of the container

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Pressure Units One atmosphere (1 atm) = One atmosphere (1 atm) = 760 mm of mercury 760 mm of mercury kPa kPa 14.7 lb/in lb/in 2

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Pressure Calculation Hoover Dam Hoover Dam Average Head Average Head meters of water meters of water Max Pressure; Max Pressure; ??? ??? Worksheet #1

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Pressure Calculation P = Po + gh P = Po + gh h=158.4 meters h=158.4 meters = 1000 kg/m 3 = 1000 kg/m 3 Pressure: Pressure: Po + gh = 101.3KPa x 9.8 x Pa Po + gh = 101.3KPa x 9.8 x Pa = KPa + 1,553,300 Pa = KPa + 1,553,300 Pa = 1655 KPa = 1655 KPa

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Why Black and White?

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Power turbines

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Downstream

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Video Clip

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Archimedes' Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object.

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Buoyant Force The upward force is called the buoyant force The upward force is called the buoyant force The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object

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Archimedes’ Principle: Totally Submerged Object The upward buoyant force is B=ρ fluid V obj g The upward buoyant force is B=ρ fluid V obj g The downward gravitational force is w=mg=ρ obj V obj g The downward gravitational force is w=mg=ρ obj V obj g The net force is B-w=(ρ fluid -ρ obj )gV obj The net force is B-w=(ρ fluid -ρ obj )gV obj

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Totally Submerged Object The object is less dense than the fluid The object is less dense than the fluid The object experiences a net upward force The object experiences a net upward force

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Totally Submerged Object The object is more dense than the fluid The object is more dense than the fluid The net force is downward The net force is downward The object accelerates downward The object accelerates downward

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Archimedes’ Principle: Floating Object F buoyancy = g(Volume displaced) F buoyancy = g(Volume displaced) The object is in static equilibrium. The object is in static equilibrium. The upward buoyant force is balanced by the downward force of gravity. The upward buoyant force is balanced by the downward force of gravity. Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level. Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level.

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Buoyancy in action Ship displacement 810 million N! 332 meters long How many cubic meters are displaced? Worksheet #2

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Got milk? Ship weighs 810 x 10 6 N = B Ship weighs 810 x 10 6 N = B Density of water = 1000 kg/m 3 Density of water = 1000 kg/m 3 Volume of water displaced is Volume of water displaced is B=(810 x 10 6 )=V disp x (1000 x 9.8) B=(810 x 10 6 )=V disp x (1000 x 9.8) V disp = cubic meters or V disp = cubic meters or 22 million gallons! 22 million gallons! B= fluid gV disp V disp =W ship / water g

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Pascal’s Principle A change in pressure applied to an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container. A change in pressure applied to an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container.

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Pascal’s Principle The hydraulic press is an important application of Pascal’s Principle The hydraulic press is an important application of Pascal’s Principle Also used in hydraulic brakes, forklifts, car lifts, etc. Also used in hydraulic brakes, forklifts, car lifts, etc.

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Application Worksheet #3a

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Fluids in Motion: Streamline Flow Streamline flow Streamline flow every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier also called laminar flow also called laminar flow Streamline is the path Streamline is the path different streamlines cannot cross each other different streamlines cannot cross each other the streamline at any point coincides with the direction of fluid velocity at that point the streamline at any point coincides with the direction of fluid velocity at that point

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Characteristics of an Ideal Fluid The fluid is nonviscous The fluid is nonviscous There is no internal friction between adjacent layers There is no internal friction between adjacent layers The fluid is incompressible The fluid is incompressible Its density is constant Its density is constant The fluid is steady The fluid is steady Its velocity, density and pressure do not change in time Its velocity, density and pressure do not change in time The fluid moves without turbulence The fluid moves without turbulence No eddy currents are present No eddy currents are present

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Equation of Continuity A 1 v 1 = A 2 v 2 A 1 v 1 = A 2 v 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant The product of the cross-sectional area of a pipe and the fluid speed is a constant Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter Av is called the flow rate – what are its units? Av is called the flow rate – what are its units?

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Application Worksheet #3b

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Bernoulli’s Equation Let’s take a minute to show how much you already know about this equation! Do a dimensional analysis -

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Bernoulli’s Equation What do the second and third terms look like? What happens we multiply by Volume?

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Conservation of energy States that the sum of the pressure, the kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline. States that the sum of the pressure, the kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline.

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Application Worksheet #4

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Applications of Bernoulli’s Principle: Venturi Meter Shows fluid flowing through a horizontal constricted pipe Shows fluid flowing through a horizontal constricted pipe Speed changes as diameter changes Speed changes as diameter changes Can be used to measure the speed of the fluid flow Can be used to measure the speed of the fluid flow Swiftly moving fluids exert less pressure than do slowly moving fluids Swiftly moving fluids exert less pressure than do slowly moving fluids

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Prairie Dogs Build burrows with two openings Build burrows with two openings One is even with ground, the other built up, why? One is even with ground, the other built up, why?

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Prairie Dogs He wants his family to have fresh air. He wants his family to have fresh air. Apply Bernoulli’s Eq’n to a breeze over both holes. Apply Bernoulli’s Eq’n to a breeze over both holes. Breeze

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Prairie Dogs How will the pressures over each hole compare? How will the pressures over each hole compare? What will this do the air in the tunnel? What will this do the air in the tunnel? Breeze

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Questions? Hydrostatic Pressure; P = gh Hydrostatic Pressure; P = gh Buoyancy; Archimedes’ Principle Buoyancy; Archimedes’ Principle F buoyancy = g(Volume displaced) F buoyancy = g(Volume displaced) Pascal; F/A=f/a Pascal; F/A=f/a Continuity Equation Continuity Equation A 1 V 1 =A 2 V 2 A 1 V 1 =A 2 V 2 Bernoulli’s Equation Bernoulli’s Equation P + 1/2 v 2 + gh = constant P + 1/2 v 2 + gh = constant

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Greek or Geek?

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Video Clip

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