Chapter 5: FLUID MECHANICS

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley

Chapter 5:

FLUID MECHANICS


This lecture will help you understand:

• Density• Pressure• Buoyancy in a Liquid• Archimedes’ Principle• Pressure in a Gas• Atmospheric Pressure• Pascal’s Principle• Buoyancy in a Gas• Bernoulli’s Principle


Density

Density• Importantproperty of materials (solids, liquids,

gases)• Measure of compactness of how much mass an

object occupies• “lightness” or “heaviness” of materials of the

same size


Density

• Equation :

• Units of: – mass in grams or kilograms– volume in cm3 or m3

– density in kg/m3 or g/cm3

Example: The density of mercury is 13.6 g/cm3, so mercury has 13.6 times as much mass as an equal volume of water

(density 1 g/cm3).

density = massvolume


Density

Weight density• in equation form:

often expressed in pounds per cubic foot

example: density of salt water is 64 lb/ft3, more dense

than fresh water (density 62.4 lb/ft3)

weight density = weightvolume


Which of these has the greatest density?

A. 100 kg of lead

B. 100 kg of water

C. Both are the same

D. None of the above

DensityCHECK YOUR NEIGHBOR


Which of these has the greatest density?

A. 100 kg of lead

B. 100 kg of water

C. Both are the same


Explanation:

They have the same mass and weight, but different volumes. Any amount of lead is more dense than any amount of water.

DensityCHECK YOUR ANSWER


Pressure

• force per unit area that one object exerts on another

• equation:

• depends on area over which force is distributed

• units in lb/ft2, N/m2, or Pa (Pascals)

pressure = forcearea


PressureExample: The teacher between beds of nails is unharmed

because the applied force is spread over many nails. Combined surface area of the nails results in a tolerable pressure that does not puncture the skin.


Pressure in a Liquid• Force per unit area that a liquid exerts on

something• Depth dependent and not volume dependent

Example: Swim twice as deep and the pressure due to the weight of water above you is twice as much. (For total pressure, add to this the atmospheric pressure acting on the water surface.)


Pressure in a Liquid

Effects of water pressure• acts perpendicular to surfaces

of a container

• liquid spurts at right angles from a hole in the surface curving downward– The greater the depth, the greater

the exiting speed


Pressure in a Liquid

• Acts equally in all directions

Examples: • your ears feel the same amount of pressure under

water no matter how you tip your head• bottom of a boat is pushed upward by water pressure• pressure acts upward when pushing a beach ball

under water


Pressure in a Liquid• Independent of shape of container

whatever the shape of a container, pressure at any particular depth is the same

• Equation:

liquid pressure = weight density depth


Water Tower

• Force of gravity acting on the water in a tall tower produces pressure in pipes below that supply many homes with reliable water pressure.


Suppose water from a tall tower supplies a nearby home. If water faucets upstairs and downstairs are turned fully on, will more water per second flow from the downstairs or the upstairs faucet? Or will water flow in each be the same?

A. Downstairs.

B. Upstairs.

C. Same.

D. Not enough information in problem.

PressureCHECK YOUR NEIGHBOR


Suppose water from a tall tower supplies a nearby home. If water faucets upstairs and downstairs are turned fully on, will more water per second flow from the downstairs or the upstairs faucet? Or will water flow in each be the same?

A. Downstairs

B. Upstairs

C. Same

D. Not enough information in problem.

Explanation:

Water pressure depends on the depth below the free surface. Downstairs faucets are simply “deeper” and receive greater pressure, which means greater rate of water flow.

PressureCHECK YOUR ANSWER


Does a 3-meter deep lake or a 6-meter deep small pond exert more pressure on a dam?

A. The three-meter deep lake.

B. The six-meter deep small pond.

C. Same amount of pressure is exerted (atmospheric) so same force.

D. Not enough information given in the question.

PressureCHECK YOUR NEIGHBOR


PressureCHECK YOUR ANSWER

Does a 3-meter deep lake or a 6-meter deep small pond exert more pressure on a dam?

A. The three-meter deep lake.

B. The six-meter deep small pond.

C. Same amount of pressure is exerted (atmospheric) so same force.

D. Not enough information given in the question.


Buoyancy in a Liquid

Buoyancy • apparent loss of weight of a submerged object• amount equals the weight of water displaced


Archimedes’ Principle

Archimedes’ Principle• discovered by Greek scientist Archimedes• relates buoyancy to displaced liquid• states that an immersed body (completely or

partially) is buoyed up by a force equal to the weight of the fluid it displaces

• applies to gases and liquids


Archimedes’ PrincipleApparent weight of a submerged object• weight out of water – buoyant force

Example: if a 3-kg block submerged in water apparently “weighs” 1 kg, then the buoyant force or

weight of water displaced is 2 kg

(BF = wt out of water – apparent wt = 3 kg – 1 kg = 2 kg)



• Displacement rule: A completely submerged object always displaces a volume of liquid equal to its own volume.

Example: Place a stone in a container that is brim- full of water, and the amount of

water overflow equals the volume of the stone



• Buoyant force is equal to the weight of fluid displaced. It can also be understood by pressure differences.

• The greater pressure against the bottom of the box, minus the pressure on the top, results in an upward force—the buoyant force.



Buoyant Force• Buoyant force is equal to the

weight of fluid displaced. • Understood by pressure

differences greater pressure against the box – pressure on the top of box


On which of these blocks submerged in water is the buoyant force greatest?

A. 1 kg of lead.

B. 1 kg of aluminum.

C. 1 kg of uranium.

D. All the same.

Archimedes’ PrincipleCHECK YOUR NEIGHBOR


On which of these blocks submerged in water is the buoyant force greatest?

A. 1 kg of lead.

B. 1 kg of aluminum.

C. 1 kg of uranium.

D. All the same.

Explanation:

The largest block is the aluminum one. It displaces more water and therefore experiences the greatest buoyant force.

Archimedes’ PrincipleCHECK YOUR ANSWER


Archimedes’ PrincipleFlotation• Principle of flotation

– A floating object displaces a weight of fluid equal to its own weightExample: A solid iron 1-ton block may displace 1/8 ton of water

and sink. The same 1 ton of iron in a bowl shape displaces a greater volume of water—the

greater buoyant force allows it to float


The reason a person finds it easier to float in salt water, compared with fresh water, is that in salt water

A. the buoyant force is greater.

B. a person feels less heavy.

C. a smaller volume of water is displaced.

D. None of the above.



The reason a person finds it easier to float in salt water, compared with fresh water, is that in salt water

A. the buoyant force is greater.

B. a person feels less heavy.

C. a smaller volume of water is displaced.


Explanation:

A floating person has the same buoyant force whatever the density of water. A person floats higher because a smaller volume of the denser salt water is displaced.



On a boat ride, the skipper gives you a life preserver filled with lead pellets. When he sees the skeptical look on your face, he says that you’ll experience a greater buoyant force if you fall overboard than your friends who wear Styrofoam-filled preservers.

A. He apparently doesn’t know his physics.

B. He is correct.



On a boat ride, the skipper gives you a life preserver filled with lead pellets. When he sees the skeptical look on your face, he says that you’ll experience a greater buoyant force if you fall overboard than your friends who wear Styrofoam-filled preservers.

A. He apparently doesn’t know his physics.B. He is correct.

Explanation:He’s correct, but what he doesn’t tell you is you’ll drown! Your life preserver will submerge and displace more water than those of your friends who float at the surface. Although the buoyant force on you will be greater, the net force downward is greater still!



Pressure in a Gas

• Gas pressure is a measure of the amount of force per area that a gas exerts against containing walls.

• Here the force is exerted by the motion of molecules bouncing around.

• Temperature is a measure of the KE per molecules of the gas.


Pressure in a GasRelationship between pressure and density• Gas pressure is proportional to density

Example:

– Air pressure and air density inside an inflated tire are greater than the atmospheric pressure and density outside

– Twice as many molecules in the same volume air density doubled

– For molecules moving at the same speed (same temperature), collisions are doubled pressure doubled


Pressure in a Gas

Double density of air by• Doubling the amount of air• Decreasing the volume to half


Pressure in a Gas

Boyle’s Law• Relationship between pressure and volume for ideal

gases• An ideal gas is one in which intermolecular forces play

no role• States that pressure volume is a constant for a given

mass of confined gas regardless of changes in pressure or volume (with temperature remaining unchanged)

• pressure volume = constant means that P1V1 = P2V2


When you squeeze a party balloon to 0.8 its volume, the pressure in the balloon

A. is 0.8 its former pressure.

B. remains the same if you squeeze it slowly.

C. is 1.25 times greater.

D. is 8 times greater.

Pressure in a GasCHECK YOUR NEIGHBOR


When you squeeze a party balloon to 0.8 its volume, the pressure in the balloon

A. is 0.8 its former pressure.

B. remains the same if you squeeze it slowly.

C. is 1.25 times greater.

D. is 8 times greater.

Explanation:

Boyle’s law, sweet and simple: P(1.0 V) = 1.25 P(0.8 V).

Pressure in a GasCHECK YOUR ANSWER


Earth’s AtmosphereAtmosphere• ocean of air • exerts pressure

The Magdeburg-hemispheres demonstration in 1654 by Otto von Guericke showed the large magnitude of atmosphere’s pressure.


Atmospheric Pressure

Atmospheric pressure• Caused by weight of air• Varies from one locality to another• Not uniform• Measurements are used to predict

weather conditions


Atmospheric Pressure

• Pressure exerted against bodies immersed in the atmosphere result from the weight of air pressing from above

• At sea level is 101 kilopascals(101 kPa)

• Weight of air pressing down on 1 m2 at sea level ~ 100,000 N, so atmospheric pressure is ~ 105 N/m2


Atmospheric Pressure• Pressure at the bottom of a column of air reaching to the

top of the atmosphere is the same as the pressure at the bottom of a column of water 10.3 m high.

• Consequence: the highest the atmosphere can push water up into a vacuum pump is 10.3 m

• Mechanical pumps that don’t depend on atmospheric pressure don’t have the 10.3-m limit


Mechanical Pump

• When the piston is lifted, the intake valve opens and air moves in to fill the empty space.

• When the piston is moved downward, the outlet valve opens and the air is pushed out.


Barometers

Barometer• Device to measure atmospheric pressure• Also determines elevation

Aneroid barometer• Small portable instrument that measures

atmospheric pressure• Calibrated for altitude, then an altimeter


Atmospheric pressure is caused by the

A. density of Earth’s atmosphere.

B. weight of Earth’s atmosphere.

C. temperature of the atmosphere.

D. effect of the Sun’s energy on the atmosphere.

Atmospheric PressureCHECK YOUR NEIGHBOR


Atmospheric PressureCHECK YOUR ANSWER

Atmospheric pressure is caused by the

A. density of Earth’s atmosphere.

B. weight of Earth’s atmosphere.

C. temperature of the atmosphere.

D. effect of the Sun’s energy on the atmosphere.


Two people are drinking soda using straws. Do they suck the soda up? Could they drink a soda this way on the Moon?

Atmospheric PressureCHECK YOUR NEIGHBOR

A. Yes and yes.B. No, they suck the air out and the

atmospheric pressure pushes the soda up. Yes, they could do the same thing on the Moon.

C. No, they reduce air pressure in the straw and the atmospheric pressure pushes the soda up. No, they could not do the same thing on the Moon.

D. Yes. No, they could not do the same thing on the Moon.


Atmospheric PressureCHECK YOUR ANSWER

Two people are drinking soda using straws. Do they suck the soda up? Could they drink a soda this way on the moon?

A. Yes and yes.B. No, they suck the air out and the

atmospheric pressure pushes the soda up. Yes, they could do the same thing on the Moon.

C. No, they reduce air pressure in the straw and the atmospheric pressure pushes the soda up. No, they could not do the same thing on the Moon.

D. Yes. No, they could not do the same thing on the Moon. The Moon does not

have an atmosphere.


Pascal’s Principle

Pascal’s principle• Discovered by Blaise Pascal, a scientist and theologian

in the 17th century• States that a change in pressure at any point in an

enclosed fluid at rest is transmitted undiminished to all points in the fluid

• Applies to all fluids—gasesand liquids


Pascal’s Principle• Application in hydraulic press

Example: – Pressure applied to the left piston is transmitted to the

right piston

– A 10-kg load on small piston (left) lifts a load of 500 kg on large piston (right)


A 10-kg load on the left piston will support a 500-kg load on the right piston. How does the pressure of fluid against the lower part of the left piston compare with the pressure against the lower right piston?

A. More pressure on the left piston.

B. More pressure on the right piston.

C. Same pressure on each.

D. Same force on each.

Pascal’s PrincipleCHECK YOUR NEIGHBOR


Pascal’s PrincipleCHECK YOUR ANSWER

A 10-kg load on the left piston will support a 500-kg load on the right piston. How does the pressure of fluid against the lower part of the left piston compare with the pressure against the lower right piston?

A. More pressure on the left piston.

B. More pressure on the right piston.

C. Same pressure on each.

D. Same force on each.


Pascal’s Principle

• Since the pressure in the fluid is the same at both ends of the tube, one can cleverly change the force and area to mechanically multiply each.

• This principle underlies a lot!

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2

1

1

21

A

F

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F

PP

1P

2

2

A

F

1

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Pascal’s Principle• Application for gases and liquids

– seen in everyday hydraulic devices used in construction

– in auto lifts in service stations• increased air pressure produced by an air

compressor is transmitted through the air to the surface of oil in an

underground reservoir.The oil transmits thepressure to the piston,which lifts the auto.

(Here surface area of reservoir is irrelevant.)


In a hydraulic device, it is impossible for the

A. output piston to move farther than the input piston.

B. force output to exceed the force input.

C. output piston’s speed to exceed the input piston’s speed.

D. energy output to exceed energy input.

Pascal’s PrincipleCHECK YOUR NEIGHBOR


In a hydraulic device, it is impossible for the

A. output piston to move farther than the input piston.

B. force output to exceed the force input.

C. output piston’s speed to exceed the input piston’s speed.

D. energy output to exceed energy input.

Comment:

This illustrates the conservation of energy, a cornerstone of all of science.

Pascal’s PrincipleCHECK YOUR ANSWER


Buoyancy in a Gas• Archimedes’ principle

applies to fluids—liquids and gases alike.

• Force of air on bottom of balloon is greater than force on top.

• Net horizontal forces cancel, but not vertical ones, which supplies the buoyant force.

• And this buoyant force equals the weight of displaced air!


Is there a buoyant force acting on your classmates at this moment? Defend your answer.

A. No. If there were, they would float upward.

B. Yes, but it is insignificant compared with their weights.

C. Only in water, but not in air.

D. None of these.

Buoyant ForceCHECK YOUR NEIGHBOR


Is there a buoyant force acting on your classmates at this moment? Defend your answer.

A. No. If there were, they would float upward.

B. Yes, but it is insignificant compared with their weights.

C. Only in water, but not in air.

D. None of these.

Buoyant Force CHECK YOUR ANSWER


Fluid Flow

Continuous flow• Volume of fluid that flows past any cross-section

of a pipe in a given time is the same as that flowing past any other section of the pipe even if the pipe widens or narrows.

• Fluid speeds up when it flows from a wide to narrow pipe

• Motion of fluid follows imaginary streamlines


Bernoulli’s Principle

Bernoulli’s Principle• Discovered by Daniel Bernoulli, a 15th century

Swiss scientist• States that where the speed of a fluid increases,

internal pressure in the fluid decreases• Applies to a smooth, steady flow



Streamlines• Thin lines representing fluid motion• Closer together, flow speed is greater and pressure

within the fluid is less (note the larger bubbles!)• Wider, flow speed is less and pressure within the fluid is

greater (greater pressure squeezes bubbles smaller)



Laminar flow• Smooth steady flow of constant density fluid

Turbulent flow• Flow speed above a critical point becomes

chaotic


What happens to the internal water pressure in a narrowing pipe of moving water?

A. Pressure is higher.

B. Pressure remains unchanged.

C. Pressure is less.

D. None of these.

Bernoulli’s PrincipleCHECK YOUR NEIGHBOR


Bernoulli’s PrincipleCHECK YOUR ANSWER

What happens to the internal water pressure in a narrowing pipe of moving water?

A. Pressure is higher.

B. Pressure remains unchanged.

C. Pressure is less.

D. None of these.

Comment:

This reduction in pressure would be apparent if air bubbles were in the flowing water. Note their sizes increase in the narrow part, due to reduced pressure there!


Applications of Bernoulli

• Moving air gains speed above the roof of a house. This change in air velocity means reduced pressure on the roof.

• Therefore, air pressure inside the house is greater, which can raise the roof.


The pressure in a stream of water is reduced as the stream speeds up. How then can a stream of water from a fire hose actually knock a person off his or her feet?

A. It can’t, as Bernoulli’s principle illustrates.

B. The pressure due to water’s change in momentum can be much greater than the water’s internal pressure.

C. Bernoulli’s principle works only for laminar flow, which the stream is not.


Bernoulli ApplicationCHECK YOUR NEIGHBOR


Bernoulli ApplicationCHECK YOUR ANSWER

The pressure in a stream of water is reduced as the stream speeds up. How then can a stream of water from a fire hose actually knock a person off his or her feet?

A. It can’t, as Bernoulli’s principle illustrates.

B. The pressure due to water’s change in momentum can be much greater than the water’s internal pressure.

C. Bernoulli’s principle works only for laminar flow, which the stream is not.


Explanation:

There’s a basic distinction between the pressure within flowing water and the pressure it can exert when its momentum is changed. The pressure that knocks one off his or her feet is due to the change in the water’s momentum, not the pressure within the water.


Airplane wing

• The vertical vector represents the net upward force (lift) that results from more air pressure below the wing than above the wing.

• The horizontal vector represents the air drag force.


Air speeds up as it is blown across the top of the vertical tube. How does this affect the air pressure in the vertical tube, and what then occurs?

A. The air jet pulls liquid up the tube.

B. Liquid mysteriously rises in the tube.

C. Reduced air pressure in the tube (due to Bernoulli) lets atmospheric pressure on the liquid surface push liquid up into the tube where it joins the jet of air in a mist.

D. Liquid in the vessel somehow turns to mist.

Bernoulli ApplicationCHECK YOUR NEIGHBOR


Bernoulli ApplicationCHECK YOUR ANSWER

Air speeds up as it is blown across the top of the vertical tube. How does this affect the air pressure in the vertical tube, and what then occurs?

A. The air jet pulls liquid up the tube.

B. Liquid mysteriously rises in the tube.

C. Reduced air pressure in the tube (due to Bernoulli) lets atmospheric pressure on the liquid surface push liquid up into the tube where it joins the jet of air in a mist.

D. Liquid in the vessel somehow turns to mist.


Bernoulli Boats

• When the speed of water increases between boats, Bernoulli must be compensated for or else the boats collide!


Bernoulli Umbrella

• Why does Nellie Newton blame Bernoulli for her predicament?

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Chapter 5: FLUID MECHANICS