© 2013 Pearson Education, Inc. Chapter 15 Fluids and Elasticity Chapter Goal: To understand...

© 2013 Pearson Education, Inc.

Chapter 15 Fluids and Elasticity

Chapter Goal: To understand macroscopic systems that flow or deform.

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Gases

A gas is a system in which each molecule moves through space as a free particle until, on occasion, it collides with another molecule or with the wall of the container.

Gases are fluids; they flow, and exert pressure.

Gases are compressible; the volume of a gas is easily increased or decreased.

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Liquids

Liquids are fluids; they flow, and exert pressure. Liquids are incompressible; the volume of a liquid

is not easily increased or decreased.Slide 15-20


Volume

An important parameter of a macroscopic system is its volume V.

The S.I. unit of volume is m3.

Some unit conversions:

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1 m3 = 1000 L 1L = 1000 cm3

1m3 = 106 cm3


Density

The ratio of an object’s or material’s mass to its volume is called the mass density, or sometimes simply “the density.”

The SI units of mass density are kg/m3.

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A piece of glass is broken into two pieces of different size. How do their densities compare?

A. 1 > 3 > 2.B. 1 = 3 = 2.C. 1 < 3 < 2.

QuickCheck 15.1

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A piece of glass is broken into two pieces of different size. How do their densities compare?

A. 1 > 3 > 2.B. 1 = 3 = 2.C. 1 < 3 < 2.

QuickCheck 15.1

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Density characterizes the substance itself, not particular pieces of the substance.

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Densities of Various Fluids


Example 15.1 Weighing the Air

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Pressure

What is “pressure”? Consider a container of

water with 3 small holes drilled in it.

Pressure pushes the water sideways, out of the holes.

In this liquid, it seems the pressure is larger at greater depths.

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A fluid in a container presses with an outward force against the walls of that container.

The pressure is defined as the ratio of the force to the area on which the force is exerted.

The SI units of pressure are N/m2, also defined as the pascal, where 1 pascal = 1 Pa = 1 N/m2.

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Pressure


Learning About Pressure

FACTS about PRESSURE


There are two contributions to the pressure in a container of fluid:

1. A gravitational contribution, due to gravity pulling down on the liquid or gas.

2. A thermal contribution, due to the collisions of freely moving gas molecules within the walls, which depends on gas temperature.

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Pressure


Atmospheric Pressure

The global average sea-level pressure is 101,300 Pa. Consequently we define the standard atmosphere as

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The reason is that a fluid exerts pressure forces in all directions.

The air underneath your arm exerts an upward force of the same magnitude, so the net force is close to zero.

If you hold out your arm, which has a surface area of about 200 cm3, the atmospheric pressure on the top of your arm is 2000 N, or about 450 pounds.

How can you even lift your arm?

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Pressure


Example 15.2 A Suction Cup

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The shaded cylinder of liquid in the figure, like the rest of the liquid, is in static equilibrium with .

Balancing the forces in the free-body diagram:

The volume of the cylinder is V = Ad and its mass is m = Ad.

Solving for pressure:

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Pressure in Liquids

net = 0


Example 15.3 The Pressure on a Submarine

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Liquids in Hydrostatic Equilibrium

No! A connected liquid in hydrostatic equilibrium rises to

the same height in all open regions of the container.Slide 15-40


What can you say about the pressures at points 1 and 2?

A. p1 > p2.B. p1 = p2.C. p1 < p2.

QuickCheck 15.2

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What can you say about the pressures at points 1 and 2?

A. p1 > p2.B. p1 = p2.C. p1 < p2.

QuickCheck 15.2

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Hydrostatic pressure is the same at all points on a horizontal line through a connected fluid.


Liquids in Hydrostatic Equilibrium

No! The pressure is the same at all points on a horizontal line

through a connected liquid in hydrostatic equilibrium.Slide 15-43


An iceberg floats in a shallow sea. What can you say about the pressures at points 1 and 2?

A. p1 > p2.B. p1 = p2.C. p1 < p2.

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QuickCheck 15.3


An iceberg floats in a shallow sea. What can you say about the pressures at points 1 and 2?

A. p1 > p2.B. p1 = p2.C. p1 < p2.

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QuickCheck 15.3



Example 15.4 Pressure in a Closed Tube

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What can you say about the pressures at points 1, 2, and 3?

A. p1 = p2 = p3.B. p1 = p2 > p3.C. p3 > p1 = p2.D. p3 > p1 > p2.E. p1 = p3 > p2.

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QuickCheck 15.4


What can you say about the pressures at points 1, 2, and 3?

A. p1 = p2 = p3.B. p1 = p2 > p3.C. p3 > p1 = p2.D. p3 > p1 > p2.E. p1 = p3 > p2.

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QuickCheck 15.4



Gauge Pressure

Many pressure gauges, such as tire gauges and the gauges on air tanks, measure not the actual or absolute pressure p but what is called gauge pressure pg.

where 1 atm = 101.3 kPa.

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A tire-pressure gauge reads the gauge pressure pg, not the absolute pressure p.


Example 15.5 An Underwater Pressure Gauge

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Barometers

Figure (a) shows a glass tube, sealed at the bottom and filled with liquid.

We seal the top end, invert the tube, place it in an open container of the same liquid, and remove the seal.

This device, shown in figure (b), is a barometer.

We measure the height h of the liquid in the tube.

Since p1 = p2:

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Pressure Units


The Hydraulic Lift

Consider a hydraulic lift, such as the one that lifts your car at the repair shop.

The system is in static equilibrium if:

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Force-multiplying factor If h is small, negligible


Suppose we need to lift the car higher.

If piston 1 is pushed down a distance d1, the car is lifted higher by a distance d2:

The Hydraulic Lift

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Work is done on the liquid by the small force; work is done by the liquid when it lifts the heavy weight.

What about PE grav of the liquid!!!


Example 15.7 Lifting a Car

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Example 15.7 Lifting a Car


Buoyancy

Consider a cylinder submerged in a liquid.

The pressure in the liquid increases with depth.

Both cylinder ends have equal area, so Fup > Fdown.

The pressure in the liquid exerts a net upward force on the cylinder:

Fnet = Fup – Fdown. This is the buoyant force.

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The buoyant force on an object is the same as the buoyant force on the fluid it displaces.

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Buoyancy


When an object (or portion of an object) is immersed in a fluid, it displaces fluid.

The displaced fluid’s volume equals the volume of the portion of the object that is immersed in the fluid.

Suppose the fluid has density f and the object displaces volume Vf of fluid.

Archimedes’ principle in equation form is:

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Buoyancy


A heavy lead block and a light aluminum block of equal sizes are both submerged in water. Upon which is the buoyant force greater?

A. On the lead block.B. On the aluminum block.C. They both experience the same buoyant force.

QuickCheck 15.5

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A heavy lead block and a light aluminum block of equal sizes are both submerged in water. Upon which is the buoyant force greater?

A. On the lead block.B. On the aluminum block.C. They both experience the same buoyant force.

QuickCheck 15.5

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Same size both displace the same volume and weight of water.


Example 15.8 Holding a Block of Wood Underwater

VISUALIZE


Example 15.8 Holding a Block of Wood Underwater


Tactics: Finding Whether an Object Floats or Sinks

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Two blocks are of identical size. One is made of lead and sits on the bottom of a pond; the other is of wood and floats on top. Upon which is the buoyant force greater?

A. On the lead block.B. On the wood block.C. They both experience the same buoyant force.

QuickCheck 15.6

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Two blocks are of identical size. One is made of lead and sits on the bottom of a pond; the other is of wood and floats on top. Upon which is the buoyant force greater?

A. On the lead block.B. On the wood block.C. They both experience the same buoyant force.

QuickCheck 15.6

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The fully submerged lead block displaces more much water than the wood block.


A Floating Object

The volume of fluid displaced by a floating object of uniform density is:

The volume of the displaced fluid is less than the volume of the uniform-density object:

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Most icebergs break off glaciers and are fresh-water ice with a density of 917 kg/m3.

The density of seawater is 1030 kg/m3.

89% of the volume of an iceberg is underwater!

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A Floating Object


Which floating block is most dense?

A. Block a.B. Block b.C. Block c.D. Blocks a and b are tied.E. Blocks b and c are tied.

QuickCheck 15.7

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Which floating block is most dense?

A. Block a.B. Block b.C. Block c.D. Blocks a and b are tied.E. Blocks b and c are tied.

QuickCheck 15.7

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Blocks a, b, and c are all the same size. Which experiences the largest buoyant force?

A. Block a.B. Block b.C. Block c.D. All have the same

buoyant force.E. Blocks a and c have the

same buoyant force, but the buoyant force on block b is different.

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Blocks a, b, and c are all the same size. Which experiences the largest buoyant force?

A. Block a.B. Block b.C. Block c.D. All have the same

buoyant force.E. Blocks a and c have the

same buoyant force, but the buoyant force on block b is different.

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QuickCheck 15.8


Boats

The figure shows a basic model of a boat.

The boat will float if the weight of the displaced water equals the weight of the boat.

The minimum height of the sides is:

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The ideal-fluid model provides a good description of fluid flow in many situations.

This model consists of three assumptions:

1. The fluid is incompressible; it is more like a liquid than a gas.

2. The fluid is nonviscous; it is more like water than syrup.

3. The flow is steady; it is more like laminar flow than turbulent flow.

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Fluid Dynamics


The rising smoke in the photograph begins as laminar flow, but at some point undergoes a transition to turbulent flow.

The ideal-fluid model can be applied to the laminar flow, but not to the turbulent flow.

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Fluid Dynamics


The laminar airflow around a car in a wind tunnel is made visible with smoke.

Each smoke trail represents a streamline.

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Fluid Dynamics


Comparing two points in a flow tube of cross section A1 and A2, we may use the equation of continuity:

where v1 and v2 are the fluid speeds at the two points. This is because the volume flow rate Q, in m3/s, is constant:

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Fluid Dynamics


Water flows from left to right through this pipe. What can you say about the speed of the water at points 1 and 2?

A. v1 > v2.B. v1 = v2.C. v1 < v2.

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QuickCheck 15.9


Water flows from left to right through this pipe. What can you say about the speed of the water at points 1 and 2?

A. v1 > v2.B. v1 = v2.C. v1 < v2.

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QuickCheck 15.9

Continuity: v1A1 = v2A2


Bernoulli’s Equation


The energy equation for fluid in a flow tube is:

An alternative form of Bernoulli’s equation is:

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Try this simple experiment!

What happened? As the air speed

above the strip of paper increases, the pressure has to decrease to keep the quantity p + ½rv2 + rgy constant.

The air pressure above the strip is less than the air pressure below the strip, resulting in a net upward force on the paper.

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Gas flows from left to right through this pipe, whose interior is hidden. At which point does the pipe have the smallest inner diameter?

A. Point a.B. Point b.C. Point c.D. The diameter doesn’t change.E. Not enough information to tell.

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QuickCheck 15.10


Gas flows from left to right through this pipe, whose interior is hidden. At which point does the pipe have the smallest inner diameter?

A. Point a.B. Point b.C. Point c.D. The diameter doesn’t change.E. Not enough information to tell.

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QuickCheck 15.10

Smallest pressure fastest speed smallest diameter


Example 15.11 An Irrigation System

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Venturi tubes measure gas speeds in environments as different as chemistry laboratories, wind tunnels, and jet engines.

The gas-flow speed can be determined from the liquid height h.

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Venturi Tubes


Suppose you clamp one end of a solid rod while using a strong machine to pull on the other end, as shown in figure (a).

Figure (b) shows graphically the amount of force needed to stretch the rod.

In the elastic region:

where k is the slope of the graph.

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Elasticity


F/A is proportional to L/L. We can write the proportionality as:

The proportionality constant Y is called Young’s modulus.

The quantity F/A is called the tensile stress.

The quantity L/L, the fractional increase in length, is called strain:

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Elasticity


Concrete is a widely used building material because it is relatively inexpensive and, with its large Young’s modulus, it has tremendous compressional strength.

Elastic Properties of Various Materials


Example 15.13 Stretching a Wire

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Bars A and B are attached to a wall on the left and pulled with equal forces to the right. Bar B, with twice the radius, is stretched half as far as bar A. Which has the larger value of Young’s modulus Y?

A. YA > YB.B. YA = YB.C. YA < YB.D. Not enough information to tell.

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QuickCheck 15.11


Bars A and B are attached to a wall on the left and pulled with equal forces to the right. Bar B, with twice the radius, is stretched half as far as bar A. Which has the larger value of Young’s modulus Y?

A. YA > YB.B. YA = YB.C. YA < YB.D. Not enough information to tell.

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QuickCheck 15.11

FA

YLL

Area of B increases by 4. If YB = YA, stretch would be only L/4. Stretch of L/2 means B is “softer” than A.


A volume stress applied to an object compresses its volume slightly.

The volume strain is defined as V/V, and is negative when the volume decreases.

Volume stress is the same as the pressure.

where B is called the bulk modulus. The negative sign in the equation ensures that the pressure is a positive number.

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Volume Stress and the Bulk Modulus

© 2013 Pearson Education, Inc. Chapter 15 Fluids and Elasticity Chapter Goal: To understand...

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