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Unit Seven: Solids and Fluids

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Title: Unit Seven: Solids and Fluids


1
Unit SevenSolids and Fluids
GE253 Physics
  • John Elberfeld
  • JElberfeld_at_itt-tech.edu
  • 518 872 2082

2
Schedule
  • Unit 1 Measurements and Problem Solving
  • Unit 2 Kinematics
  • Unit 3 Motion in Two Dimensions
  • Unit 4 Force and Motion
  • Unit 5 Work and Energy
  • Unit 6 Linear Momentum and Collisions
  • Unit 7 Solids and Fluids
  • Unit 8 Temperature and Kinetic Theory
  • Unit 9 Sound
  • Unit 10 Reflection and Refraction of Light
  • Unit 11 Final

3
Chapter 7 Objectives
  • Distinguish between stress and strain, and use
    elastic moduli to compute dimensional changes.
  • Explain the pressure-depth relationship and state
    Pascal's principle and describe how it is used in
    practical applications.
  • Relate the buoyant force and Archimedes'
    principle and tell whether an object will float
    in a fluid, on the basis of relative densities.
  • Identify the simplifications used in describing
    ideal fluid flow and use the continuity equation
    and Bernoulli's equation to explain common
    effects of ideal fluid flow.
  • Describe the source of surface tension and its
    effect and discuss fluid viscosity.

4
Reading Assignment
  • Read and study College Physics, by Wilson and
    Buffa, Chapter 7, pages 219 to 250
  • Be prepared for a quiz on this material

5
Written Assignments
  • Do the homework on the handout.
  • You must show all your work, and carry through
    the units in all calculations
  • Use the proper number of significant figures and,
    when reasonable, scientific notation

6
Introduction
  • Until now, we have beendealing with ideal
    objects
  • Now we will look at objects that can stretch,
    bend, break, flow, and compress when forces are
    applied

7
Introduction
  • In this week, we will study real, macroscopic
    objects, such as wires, blocks, and containers of
    fluids.
  • We will keep the concept of force and see what
    happens to macroscopic objects when forces act on
    them and see what forces macroscopic objects
    exert.

8
Stress and Strain
  • The stress of taking physics may cause a strain
    on your brain
  • But in physics, these terms have different and
    precise meanings.
  • We will learn to use these terms as we study the
    effects of forces on solids and fluids in this
    unit, not to accelerate objects, but to deform
    them.

9
Stress and Strain
  • Stress (force/area) causes strain ( deformation
    of some type)

10
Stress and Strain
  • In physics, stress is related to the force
    applied to an object
  • Strain is related to how the object is deformed
    as a result.
  • Both also depend on the size of the object.

11
Stress
  • The diagram shows a wire or rod of length Lo and
    area A.
  • When you apply a force F on the wire, it
    stretches by a distance .
  • When you apply a force F in the opposite
    direction on a rod, the rod is compressed by a
    distance .
  • The greater the force F, the greater is the
    amount of stretching or compressing.

A
F
L0
12
Stress
  • It is clear that a thick wire is harder to
    stretch than a thin one.
  • For this reason, you can define the concept of
    stress as
  • Stress F / A
  • The units of stress are N/m2.

13
Strain
  • The longer the wire or rod, the greater is the
    amount of stretching. For this reason, you can
    define strain as
  • Strain ?L / L
  • The vertical lines stand for absolute value,
    which means the same strain results from positive
    ?L due to stretching or negative ? L due to
    compressing.

14
Strain
  • A strain of 0.05 on a wire means that the wire
    has been stretched to a length 5 greater than
    its original length.

15
Youngs Modulus
  • NOTE This graph shows Youngs Modulus how much
    a wire will stretch if you apply different forces
    on it.

YoungsModulusRegion
16
Graph
  • Engineers need to know the force need to stretch
    material a certain distance
  • What force cause the bridge to sag into the
    waves?
  • Graphs show strain on the X axis and stress on
    the Y axis

17
Youngs Modulus
  • Youngs Modulus works in the straight line part
    of the graph only
  • The ratio of stress to strain in a material is
    known as Youngs modulus
  • Y (F/A) / ( ?L / L)
  • Where, F is force A is cross sectional area
    perpendicular to the applied force and L is
    length of the object being stressed.
  • NOTE Youngs Modulus is an ELASTIC Modulus
    because materials bounce back to their original
    shape in that part of the graph.

18
Youngs Modulus
  • The bigger the modulus, the stronger the
    material, and the more force it takes to cause a
    specific deformation.
  • Because material bounce back to their original
    shapes, moduli (plural) are elastic
  • The units of Youngs modulus are the same as
    those of stress, N/m2.
  • Youngs modulus for steel is 20 x 1010 N/m2 and
    for bone is 1.5 x 1010 N/m2 for example.

19
Practice
  • The femur (upper leg bone) is the longest and
    strongest bone in the body.
  • Let us assume that a typical femur is circular
    and has a radius of 2.0 cm.
  • How much force is required to extend the bone by
    0.010?

20
Calculations
  • Y (F/A) / ( ?L / L)
  • From the chart Ybone 1.5 x 1010N/m2
  • R 2cm(1m/100cm) .02m
  • A ?R2 ?(.02m)2 1.26 x 10-3m2
  • ?L / L .01 .01/100 1 x 10-4
  • Y ( ?L / L) (F/A)
  • F A Y ( ?L / L)
  • F 1.26 x 10-3m2 1.5 x 1010N/m2 1 x 10-4
  • F 1.89 x 103 N (about 425 pounds)

21
Practice
  • A mass of 16 kg is suspended from a steel wire of
    0.10-cm-diameter.
  • By what percentage does the length of the wire
    increase?
  • Youngs modulus for steel is 20 x 1010 N/m2.

22
Calculation
  • Y (F/A) / ( ?L / L)
  • F W mg 16kg 9.8m/s2 157 N
  • R .1cm(1m/100cm)/2 .0005m
  • A ? R2 ? (.0005m)2 7.85 x 10-7m2
  • ( ?L / L) (F/A) / Y (F/ ? R2 ) / Y
  • (?L/L) (157N/ 7.85 x 10-7m2) / 20x1010N/m2
  • (?L/L) .001 0.1
  • NOTE The change in length is inversely
    proportional to the SQUARE of the radius!

23
Shear Modulus
  • To distort a rectangular solid, apply a force on
    one of its faces in a direction parallel to the
    face.
  • Simultaneously, you must also apply a force on
    the opposite face in the opposite direction.
  • The diagram on the screen shows how you can
    distort an object by applying force.

24
Shear
  • The diagram on the screen shows that the amount
    of distortion can be measured by the new angle Ø
    , which the faces make.
  • Shearing stress is defined as F/A, where F is the
    tangential force and A is the area of the surface
    that the force acts on.
  • Shearing strain is the angle Ø .
  • Similar to linear distortions, shearing strain
    is directly proportional to shearing stress.
  • The constant of proportionality is called the
    shear modulus (S)
  • S (F/A) / F

25
Shear
  • Shear modulus has the same units as Youngs
    modulus, N/m2.
  • For many substances, shear modulus is
    approximately one-third of the Youngs modulus.

26
Bulk Modulus
  • You can also distort a rectangular solid by
    applying forces perpendicular to its surfaces.
  • This stress causes the solid to become smaller
    and is known as volume stress or pressure.
  • Volume stress is defined as F/A, where F is the
    force perpendicular to the surfaces, and A is the
    area of the surface.

27
Bulk Modulus
  • Volume strain is the change in volume divided by
    the original volume.
  • Similar to shearing stress and stress, strain is
    directly proportional to volume stress.
  • The constant of proportionality is called bulk
    modulus (B) and is defined as
  • B (F/A) / ( ?V / V)

28
Bulk Modulus
  • Solids are usually surrounded by air that exerts
    a compressing force on their surface.
  • Pressure Force/Area
  • Therefore, when you apply a force F, you increase
    this force, or more precisely, the volume stress.
  • Bulk modulus is usually written as
  • B (F/A) / ( ?V / V)
  • B ?p / ( ?V / V)
  • ?p is the increase in pressure above normal air
    pressure

29
Table
1.0 x 10 9 4.5 x 109 26 x 109 2.2 x 109
30
Compare Bulk Modulus
  • NOTE
  • B ?p / ( ?V / V)
  • For a gas, it takes the smallest pressure to
    create a change in volume, so gasses have the
    smallest bulk modulus
  • Solids require a big pressure to have a change in
    volume, so they have the biggest bulk modulus
  • Bigger modulus implies a stronger material

31
Practice
  • By how much must you change the pressure on a
    liter of water to compress it by 0.10?

32
Calculation
  • B ?p / ( ?V / V)
  • B 2.2 x 109N/m2
  • ( ?V / V) .1 (1/100) .001
  • ?p B ( ?V / V)
  • ?p 2.2 x 109 N/m2 .001 2.2 x 106 N/m2
  • Because there is always air pressure, this is the
    INCREASE in pressure to cause the change in
    volume

33
Pressure in a Fluid
  • The molecules in a solid are tightly bound.
  • In a liquid or a gas, however, the molecules are
    in motion and free to move.
  • The diagram shows a cubical container filled with
    gas.
  • The molecules of the gas are represented by the
    red dots.
  • The blue arrows show the direction they are
    moving in.

34
Pressure
  • When a gas molecule with momentum p hits the wall
    and bounces back, it exerts an impulse equal to
    2p on the wall.
  • To calculate the force exerted by a single
    molecule, you need to know how long the collision
    lasted.
  • The effect of a single molecule is very small,
    but the effect of a room full, like in a tornado,
    can be huge.

35
Measuring Pressure
  • The diagram shows a device that measures the
    pressure of a gas.
  • Because gas molecules can move freely, the
    pressure of a gas will be the same no matter
    where you place the gauge.
  • Furthermore, any changes in pressure applied to
    the container will be transmitted throughout the
    gas volume by the moving molecules.

36
Pascals Principle
  • Because liquid molecules are also free to move,
    the same principles also apply to liquids and
    thus all fluids.
  • Pascals principle states this effect
  • Pressure applied to an enclosed fluid is
    transmitted undiminished to every point in the
    fluid and to the walls of the container.

37
Atmospheric Pressure
  • When a gas is in a container, it exerts uniform
    pressure throughout the container.
  • In the case of atmosphere, however, the air stays
    at the Earths surface because of the force of
    gravity.
  • Air pressure is greatest at sea-level and
    decreases as the altitude increases.
  • The pressure of our atmosphere at sea-level is
  • This amount of pressure is defined as 1
    atmosphere (atm).

38
Density
  • A liquid is much more dense than a gas.
  • As a result, the effect of gravity on a liquid is
    much greater than that on a gas in the same sized
    container on the earth.
  • In ordinary sized containers, a gas exerts
    uniform pressure throughout the container it is
    in, but a liquid does not.
  • To refresh your memory, density is the mass of an
    object or system of particles divided by the
    volume it occupies.
  • ? m / V

39
Depth and Pressure in a Liquid
  • The diagram explains the relationship between
    pressure and depth in a liquid.
  • It shows a container of water with an imaginary
    rectangular column of water.
  • The column of water has a surface area A, and a
    weight mg.
  • Hence, a person holding the column of water
    experiences a force mg, and a pressure mg/A,
    which is the pressure of the water at the bottom
    of the container.

40
Demonstration
41
Pressure
  • You can derive the equation for pressure using
    the relationship between mass and density
  • ? m/V gt m ? V
  • V A h gt m ? Ah
  • p F/A mg/A ? Ah g / A
  • p ? g h
  • The total pressure of the liquid at the bottom of
    the container is
  • p p0 ? g h where p0 normal air pressure

42
Density
43
Practice
  • What is the total pressure exerted on the back of
    a scuba diver in a lake at a depth of 8.00 m?

44
Total Pressure
  • Pressing down on the divers back is all the water
    above him, plus all the AIR above the water.
  • pwater ? g h 1000kg/m3 x 9.8m/s2 x 8m
  • pwater 7.84 x 104 N/m2 7.84 x 104 Pa
  • pAir 1 Atmosphere 101kPa
  • Ptotal 7.84 x 104 Pa 101kPa
  • Ptotal 1.79 x 105 Pa

45
Barometers
  • A barometer is a device that measures atmospheric
    pressure.
  • To make a barometer at home, turn a glass upside
    down under water.
  • When you lift it straight up, it will remain
    filled with water.
  • Air pressure holdsthe water up

46
Barametric Pressure
  • At the surface of the Earth, the mercury in a
    barometer rises to a height of 760 mm.
  • This means atmospheric pressure is
  •  
  • The density of mercury is taken from the density
    table shown previously.
  • Here we have introduced another unit of pressure
    called the torr, named after a famous scientist,
    and representing the pressure corresponding to 1
    mm of mercury.

47
Pascals Principle
  • The pressure of a gas in a container is the same
    at every point in the gas and on the walls of the
    container.
  • Pressure in a liquid varies with depth
  • The diagram on the screen shows a piston that
    exerts a force F over an area A on a body of
    water.
  • This creates a pressure p that can be experienced
    throughout the fluid.

48
Pressure
  • Total pressure is the sum of the pressure from
    the weight of the fluid AND the added pressure
    from the piston

49
Hydraulic Lift
  • Two pistons are connected to a hydraulic lift.
  • When you press the input piston, the output
    piston rises.
  • The pressure on the input piston is the same as
    that on the output piston.
  • Keep in mind that the force is given by the
    pressure times the area. F P A
  • The output piston has a much larger area than the
    input piston.
  • Therefore, the output piston exerts a much larger
    force than the input piston, and you can easily
    lift an automobile. F P A

50
Lift a Car
  • p1 p2
  • F1/A1 F0/A0
  • F0 F1 (A0 /A1 )

51
Practice
  • The input and lift (output) pistons of a garage
    lift have diameters of 10 cm and 30 cm,
    respectively. The lift raises a car with a weight
    of 1.4 x 104 N.
  • (a) What is the force on the input piston?
  • (b) What is the pressure to the input piston?

52
Calculations
  • p1 p2
  • F1/A1 F0/A0
  • R1 10cm (1m/100cm)/2 .05m
  • R2 30cm (1m/100cm)/2 .15m
  • A1 ? R12 ?(.05m)2 7.85x10-3m2
  • A2 ? R22 ?(.15m)2 7.07x10-2m2
  • F1 F0(A1/A0)
  • F1 1.4 x 104 N (7.85x10-3m2/ 7.07x10-2m2)
  • F1 1,550N
  • p1 F1/A1 1,554N / 7.85x10-3m2 2.0x105N/m2
  • p2 F2/A2 1.4 x 104 N / 7.07x10-2m2
  • p2 2.0x105N/m2 checks!

53
Thought Experiment
  • If the output piston of a hydraulic lift has a
    very large area, a two-year-old can lift the
    Empire State Building.
  • Does this make sense?
  • Is energy conserved?

54
Results
  • Yes, it does make sense.
  • Simple machines, such as levers, transform small
    input forces into large output forces.
  • Energy is the ability to do work.
  • It is the product of force and distance.
  • Machines transform small forces applied over
    large distances to large forces applied over
    small distances.

55
Practice
  • A 60-kg athlete does a single-hand handstand.
  • If the area of the hand in contact with the floor
    is 100 cm2, what pressure is exerted on the
    floor?

56
Calculation
  • p F/A
  • A 100 cm2 (1m/100cm)2 1.00x10-2m2
  • p mg/A 60kg 9.8m/s2/ 1.00x10-2m2
  • p 5.88x104N/m2

57
Practice
  • An oak barrel with a lid of area 0.20 m2 is
    filled with water. A long, thin tube of
    cross-sectional area 5.0 X 10-5 m2 is inserted
    into a hole at the center of the lid, and water
    is poured into the tube.
  • When the water reaches 12 m high, the barrel
    bursts.
  • What was the weight of the water in the tube?
  • What was the pressure of the water on the lid of
    the barrel?
  • What was the net force on the lid due to the
    water pressure?

58
Calculations
  • W m g ? V g ? Ah g
  • A 5.0 X 10-5 m2 , h 12 m
  • W1000kg/m3 5.0 X 10-5 m2 12 m 9.8m/s2
  • W 5.88 N
  • p F/A 5.88 N / 5.0 X 10-5 m2
  • p 1.18x105N/m2
  • OR
  • p ?gh 1000kg/m3 9.8m/s2 12m 1.18x105N/m2
  • Checks!

59
Calculations
  • p F / A
  • F p A
  • F 1.18x105N/m2 0.20 m2
  • F 2.36 x 104N (More than 5000 pounds!)

60
Archimedes
  • Have you ever yelled, 'Eureka!' when you figured
    out how to solve a challenging problem?
  • Well the story goes that when Archimedes figured
    out the principle named after him, he was so
    excited that he not only yelled, 'Eureka,' but
    also went running through town from the public
    bath naked!
  • You may not get that excited over Archimedes
    principle, but you will learn some useful and
    interesting things about fluid buoyancy and fluid
    flow in this lesson.

61
Archimedes' Principle
  • The upward force experienced by an object when it
    is immersed in a liquid or a gas is called
    buoyant force.
  • Buoyant force is equal in magnitude to the weight
    of the volume of fluid displaced.
  • This rule is called Archimedes' principle.
  • BF mg ? V g

62
Archimedes Principle
  • The diagrams on the screen illustrate Archimedes
    principle.
  • Part (a) shows buoyant force as the force that
    pushes up when you hold a piece of wood under
    water.
  • Part (b) shows buoyant force as the force that
    reduces the weight of an object weighing 10 N
    to 8 N.

63
Practice
  • A spherical helium balloon has a radius of 30 cm.
    What is the buoyant force acting on it in air?

64
Calculations
  • Find the volume if R 30 cm .3 m
  • V (4/3) ? R3 .113 m3
  • BF mg ? V g 1.29kg/m3 .113 m3 9.8m/s2
  • BF 1.43 N (about 1/3 of a pound)
  • If the balloon and the air in the balloon has
    weight less then 1.43 N, it will float if more,
    it will sink

65
Practice
  • The diagram shows a partially submerged iceberg
    at rest.
  • It is in equilibrium because its buoyant force is
    equal to the weight of the iceberg.
  • Suppose a uniform solid cube of material 10 cm on
    each side has a mass of 700 g. Will the cube
    float? If it will float, how much of the cube
    will be submerged?

66
Calculations
  • Suppose the cube were completely SUBMERGED!
  • LSide 10 cm .1m
  • V LWH (.1m)(.1m)(.1m) 10-3 m3
  • BF ? V g 1x103Kg/m3 10-3 m3 9.8m/s2
  • BF 9.8N
  • Weight of the object mg
  • W 700 g (1kg/1000g) 9.8m/s2 6.86 N
  • The buoyant force is greater than the weight!
  • The object will be pushed up to the surface and
    will rise above the surface until the weight of
    the fluid displaced just balances the weight of
    the object

67
Floating Object
  • V LWH (.1m)(.1m)(.1m) 10-3 m3
  • W 700 g (1kg/1000g) 9.8m/s2 6.86 N
  • BF 6.86 N ? V g 1x103Kg/m3 V 9.8m/s2
  • V 6.86 N / (1x103Kg/m3 9.8m/s2 )
  • V 7 x 10-4 m3
  • 7 x 10-4 m3 is the volume of the cube below the
    surface
  • The remainder
  • 10-3 m3 - 7 x 10-4 m3 3 x 10-4 m3 is above the
    surface
  • It is 70 submerged

68
Ideal Fluid Flow
  • To mathematically describe fluid flow like we
    have projectile motion, for example can be very
    difficult in real cases.
  • Consider the upper portion of the rising smoke
    it looks almost random!
  • In order to obtain a basic description of fluid
    flow, we will make some simplifications and
    consider ideal fluid flow, so the flow looks more
    like the lower part of the rising smoke stream.
  • The conditions for an ideal fluid are that its
    flow is

69
Ideal Fluid Flow
  • Steady The flow is not turbulent particles in
    the fluid do not collide, and their paths do not
    cross. The flow looks smooth.
  • Irrotational There is no rotational motion in
    small volumes of the fluid, so there are no
    whirlpools. A paddle wheel completely
          embedded in the stream does not rotate.
  • Nonviscous Viscosity is negligible. This means
    the fluid flows easily, without significant
    friction or resistance to the flow.
  • Incompressible The density is constant
    throughout the fluid.
  • When a fluid's flow has these four
    characteristics, we say it is an ideal fluid.
    Unless specified otherwise, we will consider
    these four conditions to be true in the fluids we
    consider.

70
Equation of Continuity
  • When a liquid flows through a tube, the amount of
    liquid entering the tube must equal the amount of
    liquid coming out of it.
  • If the cross-sectional area of the tube varies
    across its length, then the speed of the liquid
    must vary too.
  • The diagram on the screen shows what happens to
    the speed of water when the nozzle has a
    smaller cross-sectional area than the tube.

71
Equation of continuity
  • If the liquid is incompressible, the density of
    the liquid remains the same throughout the tube.
    If the liquid is incompressible, the density of
    the liquid remains the same throughout the tube.
  • Hence, as the above equation simplifies to
  • Av constant
  • Area x velocity constant
  • Volume/time constant
  • Equation of continuity is an equation that
    describes the fact that the amount of fluid
    entering a tube is equal to the amount of fluid
    leaving the tube.

72
Equation of Continuity
  • High cholesterol in the blood can cause fatty
    deposits called plaques to form on the walls of
    blood vessels.
  • Suppose plaque reduces the effective radius of an
    artery by 25.
  • Because the area is smaller, what must happen to
    the velocity?

73
Bernoullis Equation
  • Bernoulli's equation states that when a liquid is
    moving fast, its pressure is reduced.
  • The diagram below shows a tube of varying
    cross-sectional areas.
  • The pressure indicators show that the pressure is
    lower in the middle region, where the smaller
    cross sectional area results in the flow rate
    being greater.

74
Bernoulli's Equation
  • To observe Bernoulli's equation, conduct the
    following experiment.
  • Hold two corners of a sheet of paper with both
    hands.
  • Rotate your hands so that the paper takes the
    shape of the wing of an airplane.
  • Blow over the top of the paper.
  • You will observe that the sheet of paper rises.
  • It lifts up because air flows on one side of the
    paper, but is stationary on the other.
  • This is the principle behind the lifting of air
    planes the orientation and shape of the wing of
    an airplane causes the air flowing over it to
    flow faster than the air under it.

75
Wing Design
76
Surface Tension
  • When there is a boundary between a liquid and a
    gas, some remarkable phenomena occur.
  • The figures on the screen show two water/air
    phenomena and one soap solution/air boundary
    phenomena.

77
Surface Tension
  • 1) An insect is able to walk on water. 2) Water
    in air forms circular droplets.3) Soap solutions
    form bubbles.
  • In the phenomenon where bubbles are made, there
    are two surfaces the inside and the outside of
    the bubble.
  • Although the bubble is thin compared to a piece
    of paper, it is very thick compared to the size
    of a molecule.
  • The surface in an air-liquid boundary is only two
    or three molecules thick.

78
Surface Tension
  • The forces of attraction between the molecules of
    a liquid are lower than those of a solid.
  • All the same, they do exist because otherwise the
    molecules of a liquid would escape into the
    atmosphere.
  • In a liquid, a molecule, which is at a distance
    from the surface, is surrounded by other
    molecules.
  • A molecule on the surface, however, is not
    entirely surrounded by molecules.
  • The molecules on the surface are held together by
    a horizontal force between them, which is called
    surface tension.

79
Viscosity
  • You must have noticed that in January, molasses
    flows slower than water or kerosene.
  • This happens because of viscosity.
  • Viscosity is a fluids internal resistance to
    flow.
  • It can be measured with a device called a
    viscosimeter.
  • There are various types of viscosimeters.

80
Flow Rate
  • When fluid is flowing in a pipe, as a result of
    viscosity, the speed of the fluid near the
    surface of the pipe is less than that at the
    center.
  • The flow rate is the average volume of a fluid
    that flows beyond a given point during a time
    interval
  • The diagram on the screen shows that is not easy
    to calculate the average flow rate.
  • Flow rate has the unit of m3/s.

81
Summary
  • What are the elastic modulie?
  • What is the relationship between pressure and
    depth in a fluid?
  • What is Archimedes principle?
  • What are the essential principles of fluid flow?
  • What causes surface tension and what is
    viscosity?
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