Physics Module 6
Properties of Matter
Prepared by Sisay Shewamare
African Virtual university
Université Virtuelle Africaine
Universidade Virtual Africana
African Virtual University Notice
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African Virtual University Table of Contents
I.
Properties of Matter_ ________________________________________ 5
II.
Prerequisite Course or Knowledge_ _____________________________ 5
III. Time_____________________________________________________ 5
IV. Materials__________________________________________________ 5
V.
Module Rationale_ __________________________________________ 5
VI. Content___________________________________________________ 6
6.1
6.2
6.3
Overview_____________________________________________ 6
Outline_ _____________________________________________ 6
Graphic organizer______________________________________ 7
VII. General Objective(s)_________________________________________ 8
VIII. Specific Learning Objectives___________________________________ 8
IX
Pre-assessment_ ___________________________________________ 9
X.
Key Concepts (Glossary)_____________________________________ 15
XI. Compulsory Readings_______________________________________ 17
XII. Compulsory Resources______________________________________ 18
XIII. Useful Links_ _____________________________________________ 20
XIV. Teaching and Learning Activities_______________________________ 23
XV. Synthesis Of The Module_ ___________________________________ 72
XVI. Summative Evaluation_______________________________________ 73
XVII.References_ ______________________________________________ 78
XVIII. Main Author of the Module __________________________________ 79
African Virtual University Foreword
This module has four major sections
The first one is the Introductory section that consists of five parts vis:
1.
Title:
2.
Pre-requisite Knowledge: In this section you are provided with infromation
regarding the specific pre-requisite knowledge and skills you require to start
the module. Carefully look into the requirements as this will help you to decide
whether you require some revision work or not.
3.
Time Required: It gives you the total time (in hours) you require to complete
the module. All self tests, activities and evaluations are to be finished in this
specified time.
4.
Materials Required: Here you will find the list of materials you require to
complete the module. Some of the materials are parts of the course package
you will receive in a CD-Rom or access through the internet. Materials recommended to conduct some experiments may be obtained from your host
institution (Partner institution of the AVU) or you may acquire, borrow or by
some other means.
5.
Module Rationale: In this section you will get the answer to questions like
“Why should I study this module as pre-service teacher trainee? What is its
relevance to my career?”
The second is the Content a section that consists of three parts:
6.
Overview: The content of the module is briefly presented. In this section you
will find a video file (Quicktime, .move) where the author of this module is
interviewed about this module. The paragraph overview of the module is
followed by an outline of the content including the approximate time requiered to complete each section. A graphic organization of the whole content is
presented next to the outline. All these three will assist you to picture how
content is organized in the module.
7.
General Objective(S): Clear, informative, concise and achievable objectives
are provided to give you what knowledge skills and attitudes you are expected
to attain after studying the module.
8.
Specific Learning Objectives (Instructional Objectives): Each of the specific objectives, stated in this section, are at the heart of a teaching learning
activity. Units, elements and themes of the module are meant to achieve the
specific objectives and any kind of assessment is based on the objectives
intended to be achieved. You are urged to pay maximum attention to the
specific objectives as they are vital to organize your effort in the study of the
module.
African Virtual University The third section is the bulk of the module. It is the section where you will
spend more time and is refered to as the Teaching Learning Activities. The
gist of the nine components is listed below:
9.
Pre-assessment: A set of questions, that will quantitatively evaluate your
level of preparedness to the specific objectives of this module, are presented
in this section. The preassessment questions help you to identify what you
know and what you need to know, so that your level of concern will be raised
and you can judge your level of mastery. Answer key is provided for the set
of questions and some pedagogical comments are provided at the end.
10. Key Concepts: This section contains short, concise definitions of terms used
in the module. It helps you with terms with which you might not be familiar
to the module.
11.
Compulsory Readings: A minimum of three compulsory reading materials
are provided. It is mandatory to read the documents.
12. Compulsory Resources: A minimum of two video, audio with an abstract in
text form is provided in this section.
13. Useful Links: A list of atleast ten websites is provided in this section. It will
help you to deal with the content in greater depth..
14. Teaching And
Learning Activities: This is the heart of of the module. You
need to follow the learning guidance in this section. Various types of activities are provided. Go through each activity. At times you my not necessarily
follow the order in which the activities are presented. It is very important to
note:
• formative
and summative evaluations are carried out thoroughly
compulsory readings and resources are done
• as many as possible useful links are visited
• feedback is given to tha author and communication is done
• all
Enjoy your work on this module.
African Virtual University I.
Properties of Matter
By Sisay Shewamare, Jimma University Ethiopia
II.
Pre-Requisite Course Or Knowledge
In order to study this module, you need to complete the modules on Mechanics
I, Mechanics II, and Electricity and Magnetism. This module also assumes you
have taken introductory course in Calculus.
III.
Time
The time you require to complete this module is 120hrs. For chapterwise
breakdown see section 6 of the module
IV.
•
•
•
•
•
V.
Materials
Internet Connection
Compulsory Readings And Compulsory Resources
(As Listed In Sections 11 & 12)
Standard Weights
Wires Made Of Different Substances
Software Package
Module Rationale
Science teaching in secondary schools is expected to enable learners to work in
scientific ways (apply scientific principles), stimulate their curiosity and deepen
their interest in the natural and physical world.
In this module you will study the behavior of solids when subjected to strains,
and the behavior of fluids in different contexts is studied. You will also understand the thermal and electrical conductivity (also known as transport properties)
of metals.
The study of mechanical, thermal and electronic properties of materials will not
only help you for advanced studies in solid state Physics and electronics physics,
but also will give you a cutting edge in teaching technological applications of the
Physical Sciences for your future students.
African Virtual University Fig: Which properties of Tungsten wire make it very convenient for the construction of
a bulb fillament?
VI.
Content
6.1 Overwiew
In this module you will study elastic and transport properties of materials like
elasticity, fluid flow, diffusion, osmosis, thermal and electrical conductivities
of a materials
At the beginning, activities leading you through the details of the effects of force
on various types of materials are presented. Then you will come across activities
that will enable you describe the properties of fluids and use these properties
to arrive at principles and laws such as Archimedes principle, Pascals law and
Bernoull’is equation.
The module includes properties like viscosity, diffusion, thermal properties
conductivity, expansion), Electrical conductivity of metals, semiconductors and
alloys. These properties are also known as transport properties.
6.2 Outline
Elasticity (30hours)
Load and strees;
strain
Stress Strain relationship:Hooke’s law
Compressibility, Elasticity and Plasticity
Young’s modulus
Poisson’s ratio
•
•
•
•
•
•
Fluids •
•
•
•
(45 hours)
Density
Pressure
Fluid at rest
Measuring pressue
African Virtual University •
•
•
•
•
Pascal’s Principle
Archimedes Principle
Equilibrium of floating object
Bernoulli’s equation
The flow of real fluid
Transport properties (45 hours)
•
•
•
•
•
Diffusion
Viscosity
Thermal conductivity
Thermal expansion
Electrical conductivity of metals, semiconductors and alloys.
6.3Graphic Organizer
A. Elasticity
Viscosity
Diffusion
conductivity
Expansion
Metals
Semiconductors
Alloys
Thermal Properties
C. Transport
Properties
Properties of
Matter
Electrical conductivity
B. Fluids
Stress.
Strain
Compressibility
Plasticity
Young's Modulus
Poson Ratio
Density
Pressue
Fluids at rest
Measuring Pressue
Pascal's Principle
Archimedes Principle
Equilibrium of floating objects
Equation ofContinuity
Bernoulli's Equation
The flow of real fluids
African Virtual University VII.
General Objective(s)
After completing this module you would be able to:
•
•
•
•
•
Explain the concept of elastic properties of materials
Describe the transport properties of materials
Appreciate the properties of fluids and apply the concepts to a range of
contexts.
Use thermal conductivity of matteials to solve porblems
Use Elcectrical conductivity of materials to solve problems.
VIII.Specific Learning Objectives (Instructional Objectives)
Content
Elasticity (35 hours)
• Load and strees;
• strain
• Stress Strain relationship:Hooke’s
law
• Compressibility, Elasticity and
Plasticity
• Young’s modulus
• Poisson’s ratio
Learning objectives
After Completing this section you
would be able to:
• Determine the effect of force on
materials
• Calculate Young’s modulus for a
range of materials
• Calculate Poisson’s ratio for a
given material
• Predict material properties
African Virtual University Fluids (45 hours)
•
•
•
•
•
•
•
•
•
Density
Pressure
Fluids at rest
Measuring pressue
Pascals Principle
Archimedes Principle
Equilibrium of floating object
Bernoulli’s equation
The flow of real fluids
• Describe basic properties of fluid
(density,pressure)
• Apply the properties of fluids
(Archimedes principle, Pascal’s
law)
• Evaluate fluid motion
(continuity,turbulance real fluids )
• Use Bernoulli’s equation
Transport properties (45 hours)
•
•
•
•
•
Diffusion
Viscosity
Thermal conductivity
Thermal expansion
Electrical conductivity of
metals, semiconductors and
alloys.
• Analyse particle motion in fluids
• Describe relative properties of
solids, liquids and gases
• Evaluate the effects of heat on
materials e.g. calculate thermal
expansion
• Calculate the effective concentration of mobile electrons in metals, alloys and semiconductors
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IX.
Pre-assessment
This pre assessment questions comprizes questions from the prerequisite
knowledge as well as question that assess your mastery level of the objectives
stated in this module. If your performance is more than 70% you can proceed to
work on this module.
However if your performance is less than 70% you need to revise some of your
school Physics. The depth of the revision work you need is proportional to how
far your performance is away from the required minimum
Answers to the questions are provided immediately after the questions.
How does air support an aircraft?.
9.1Questions
1. Figure 1 the weight of the liquid, density ρ , at x is kept constant while the
liquid flows out of the narrow tube at depth h below x. The velocity v of the
liquid from the narrow tube is
a) ; hρg
b) 2gh
c)
d)
2gh
gh
e)) . 2ghρ
x
h
v
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2. A hot air balloon moving upwards has a total weight of 200N and a volume
of 20m3. Assuming the air density of 1.2kgm-3, the net upward force on the
balloon in N is then about
a)
b)
c)
d)
e)
.24
36
40
176
240
3. When a stone of mass m at the end of a string is whirled in vertical circle at
constant speed
a)
b)
c)
d)
e)
The tension (force) in the string stays constant
The tension is least when the stone reaches the bottom of the circle
The tension in the string is always mg
the weight mg is always the centripetal force
the tension is greatest when the stone is at the bottom of the circle
4. At the olympic high-diving competition, a diver from the top board curves
her body in order to
a)
b)
c)
d)
e)
dive cleanly in to the water
spin more
increase her energy
spin more slowly
increase her speed
5. When streached beyound its elastic limit, a metal rod such as steel
a)
b)
c)
d)
becomes plastic
has no energy
obeys Hooke’s
becomes colder
6. Figure 2 shows three mass in a row. The force on the 1kg mass is zero if the
distance x in meters is
a)
b)
c)
d)
e)
2
3
9kg
4
5
6
1kg
x
15
Figure 2
4kg
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7. The time constant of the circuit shown in Figure 3 is 4s. The time constant of
the circuit shown in figure 4 is thus:
a)
b)
c)
d)
e)
8s
4s
2s
1s
0.5s
C
R
Figure 3
C
C
R
Figure 4
R
8. At what temperature are the reading from a Fahrenheit thermometer and
Celisius thermometer the same.
a)
b)
c)
d)
e)
-20
40
32
-40
72
9. Which of the following are semiconductor materials?
a) gallium arsenide
b) germanium c) silicon
d) all of the above
10. Why are semiconductors valuable in modern electronics ?
a)
b)
c)
d) use low power
reliable
fast switching
all of the above
11.Which electronic devices are primarily made from semiconductors ?
a) transistors
b) .resistors
c) capacitors
d) none of the above
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12.How does the conductivity in pure semiconductors vary with temperature?
a) conductivity increases as temperature goes down
b) conductivity increases as temperature goes up
c) conductivity does not change with temperature
13. What explains why semiconductors have different electrical properties from
metals?
a)
b)
c)
d)
more valence electrons
fewer valence electrons
band gap structure
no differences
14. Both _electrons _ and _holes_ are considered charge carriers.
15. A diode contains both _n-type_ and __p-type_ regions.
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9.2Answer Key
1) C
2) C
3) E
4) B
5) A
6) E
7) B
8) D
9) D
10) D
11) A
12) B
13) C
14) electron
hole
15) n-type p-type
9.3 Pedagogical Comment For Learners
The module is presented in such a way that you will find yourself in a variety of
activities like reading, going through worked examples, experimenting virtually
and in the real lab, online discussion with study group, solving problems etc.
This is possible partly by the package you receive with this module and via the
internet. Your effort to experience all compulsory materials and as many resources as possible has no substitute. Infact learning takes place with the learner’s
effort. Therefore you are advised to work all the problems provided and consult
the references suggested.
The concepts presented are best understood in experimental tests. It is a very
good idea if you keep in touch with the AVU partner University.
The last thing you have to do is evaluate yourself whether you have achieved the
expected learning outcomes mentioned at the begining of the module.
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X.
Key Concepts (Glossary)
ELASTICIY: Is the property of a material, or a substance, or a body of returning
to its original size and shape after distortion or deformation by a force. (Source
Wikipedia consulted on … )
STRESS: Is a force per unit area, measured in newtons per meter squared ( Nm -2 ).
Examples of a stress include a tension, a thrust, and a shearing force.
STRAIN: Is the ratio of the dimensional change produced to the original dimension. When a stress is applied to a body a strain is produced. The body can be
distoreted or deformed, depending upon its elesticy. It may be a ratio of lengths,
areas, or volumes.
YOUNG’S MODULUS: Is the modulus of elastticty of a wire or rod stretched
longitudinally, or of a rod compressed longitudinally. It is measured in N m −2
Force F
=
Area
A
Extension x
Strain=
=
Length
l
Stress =
Youngs Modulus = E =
Stress F l
=
Strain Ax
C O M P R E S S I B I L I T Y: I n t h e r m o d y n a m i c s a n d f l u i d m e chanics, compressibility is a measure of the relative volume change
of fluid or solid as a response to a pressure (or mean stress) change.
β=−
1 ∂V
V ∂P
where V is volume and P is pressure. The above statement is incomplete,
because for any object or system the magnitude of the compressibility depends
strongly on whether the process is adiabatic or isothermal.
PLASTICITY: Is the property of a material, or a substance, of being permanently
deformed by a force, without breaking.
POISSON RATIO: When a sample of material is stretched in one direction,
it tends to get thinner in the other two directions. Poisson’s ratio ( ν , µ), named
after Simeon Poisson, is a measure of this tendency. Poisson’s ratio is the ratio of
the relative contraction strain, or transverse strain (normal to the applied load),
divided by the relative extension strain (in the direction of the applied load). For
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a perfectly incompressible material, the Poisson’s ratio would be exactly 0.5.
Most practical engineering materials have ? between 0.0 and 0.5. Cork is close
to 0.0, most steels are around 0.3, and rubber is almost 0.5. Some materials,
mostly polymer foams, have a negative Poisson’s ratio; if these auxetic materials
are stretched in one direction, they become thicker in perpendicular directions.
Assuming that material is compressed along y axis
v yx =
εx
εy
v
where yx is the resulting Poisson’s ratio,
axial strain.
εx
is transverse strain, and
εy
is
PASCAL’s PRINCIPLE: A change pressure applied to an enclosed fluid is
transmitted undiminished to every point of the fluid and the walls of the containing vessel.
ARCHIMEDE’S PRINICIPLE: Any body completely or partially submerged
in a fluid is buoyed up by a force equal to the weight of the fluid displaced by
the body
BERNOULLI’S EQUATION: As a fluid moves through a pipe of varying
cross section and elevation, the pressure will change along the pipe.
VISCOSITY: Is resistance to the internal friction between molecules. Viscosity
can be measured by an instrument called a viscometer. One way to measure
relative viscosity of liquids is to use a 5 ml pipette and a stop watch. Draw up
precisely 5.00 ml of the liquid and begin the stop watch as the liquid leaves the
pipette. The longer it takes to empty the more viscous is the liquid. Some liquids
like water have a low viscosity where other liquids like honey have a high viscosity. Viscosity will be affected by the temperature. At higher temperatures the
viscosity decreases as the molecules take on more kinetic energy allowing them
to move past each other faster
DIFFUSION: Diffusion is the movement of particles from higher chemical
potential to lower chemical potential (chemical potential can in most cases of
diffusion be represented by a change in concentration). An electric charge is an
attribute of matter that produces a force
THERMAL CONDUCTIVITY: Thermal expansion of solids:or a body is a
consequence of the change in the average separation between its constituent
atoms or molecules.
ELECTRICAL CONDUCTIVITY: Is a measure of a material’s ability to
conduct an electric current when an electrical potential difference is appplied
across the conductor. Its movable charges flow, giving rise to an electric current.
The conductivity σ is defined as the ratio of the current density to the electric
field strength J = σ E ,
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XI. Compulsory Readings
Reading #1 Mechanical Properties
Complete reference : http://dmoz.org/Science/Physics/Fluid_Mechanics_and_
Dynamics/
Abstract : The links on the above mentioned page lead you to html materials on
topics of Bernoulli’s Principle Animation, Calculations and Equations of Fluid
Mechanics, Classical Fluid Mechanics Problem Solutions - Solutions to Classical
Fluid Flow & Momentum Transfer Problems, Fluid dynamics course material,
Fluid Mechanics, and many more that are directly relevant to this module.
Rationale: The Open Directory Project is the largest, most comprehensive
human-edited directory of the Web. It is constructed and maintained by a vast,
global community of volunteer editors.
Date consulted: October, 2006
Reading #2 Gases Liquids and Solids
Complete reference http://en.wikipedia.org/wiki/Elasticity_%28physics%29
Abstract : The topics discussed in this document include Contents Modeling
elasticity, Transitions to inelasticity
Rationale: This is one chapter of a free text book maintained by www.lightandmatter.com It is available in pdf and html formats. The pdf files can be downloaded chapter by chapter d potential; introduction to special relativity; Maxwell’s
equations, in both differential and integral form; and properties of dielectrics and
magnetic materials
Date consulted: September, 2006
Reading #3 Solid Mechanics
Complete reference :http://en.wikibooks.org/wiki/Solid_Mechanics#Stress
Abstract : Topics in this reading material follows the continuum mechanics
approach, where the material properties to be the same even when we consider
infinitesimal areas and volumes. The alternative approach is to build up material
properties from basic equations relating atomic forces and interactions, and extending it to larger sets of such entities (e.g., molecular dynamics).
Rationale: This is part of a book on solid mechanics and it is a good reading
material for this module.
Date consulted: Nov, 2006
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XII.
Compulsory Resources
Resource #1
Effect of Temprature and Volume on the number of Collisions
Source;Lon-CAPA
URL: http://lectureonline.cl.msu.edu/~mmp/kap10/cd283.htm.
Date Consulted: Nov 2006
Description: This Java applet helps you understand the effect of temperature
and volume on the number of collisions of the gas molecules with the walls. In
the applet, you can change the temperature and volume with the sliders on the
left side. You can also adjust the time for which the simulation runs. The applet
counts all collisions and displays the result after the run. By varying temperature and volume and keeping track of the number of collisions, you can get a
good feeling of what the main result of kinetic theory will be.
Resource #2
Virtual Experiment on the Ideal Gas Law
Source;Uoregon University
URL: http://jersey.uoregon.edu/vlab/Piston/index.html
Date Consulted: Nov 2006
Description: This Java applet helps you to do a series of virtual experiments,
you will control the action of a piston in a pressure chamber which is filled
with an ideal gas. The gas is defined by four states: Temperature; Volume or
density; Pressure and Molecular Weight
There are 3 possible experiments to do. In the third experiment, labelled Ideal
Gas Law, you can select from the Red, Blue or Yellow gas containers. Each
gas in those containers has a different molecular weight and hence each will
respond differently under changing pressure conditions..
Resource #3
Computer Calculation of Phase Diagrams
Source: video.google.com
Complete Reference: http://video.google.com/videoplay?docid=13979881767
80135580&q=Thermodynamics&hl=en
Rationale: Thermodynamic models of solutions can be used together with
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data to calculate phase diagrams. These diagrams reveal, for a given set of all
parameters (such as temperature, pressure, magnetic field), the phases which
are thermodynamically stable and in equilibrium, their volume fractions and
their chemical compositions.
This lecture covers the pragmatic methods implemented in commercial software for the estimation of multicomponent, multiphase equilibria.
The content should be generally useful to scientists. This is the fifth of seven
lectures on the thermodynamics of phase transformations
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XIII. Useful Links
Useful Link #1
Title: Buoyant Force in Liquids
URL: http://www.walter-fendt.de/ph11e/buoyforce.htm
Screen Capture:
Description: This Java applet shows a simple experiment concerning the
buoyancy in a liquid: A solid body hanging from a spring balance is dipped into
a liquid (by dragging the mouse!). In this case the measured force, which is equal
to the difference of weight and buoyant force, is reduced. You can change (within
certain limits) the preselected values of base area, height and densities by using
the appropriate text fields.
Rationale: This virtual experiment conforms with activity 2 of the module.
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Useful Link #2
Title: Water Pressure and depth.
URL: http://www.mste.uiuc.edu/murphy/PicnicCooler/default.html
Screen Capture:
Description: This applet was written by Lisa Denise Murphy at the University of
Illinois. Early drafts were written in 1999. The current version was last revised in
January of 2000. Permission is given for students and teachers to use this applet,
provided acknowledgement is made of the source.
Rationale: This virtual activity is of use for activity 2
Useful Link #3
Title: Solid Mechanics
URL: http://en.wikibooks.org/wiki/Solid_Mechanics
Screen Capture:
Description: This is a book on solid mechanics. .
Rationale: The contents of activity 1 and activity 3 are entertained in greater
detail
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Useful Link #4
Title:Viscosity
URL: http://www.spacegrant.hawaii.edu/class_acts/ViscosityTe.html
Screen Capture:
Description: This is advanced description of viscosity for more curious readers.
Useful Link #5
Title: Thermal Conductivity
URL: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thercond.html
Screen Capture:
Description: An excellent presentation with many relevant liniks.
Rationale: supplements activity 2
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XIV. Teaching And Learning Activities
Activity 1: Elasticity of Materials
You will require 30 hours to complete this activity. In this activity you are guided
with a series of readings, Multimedia clips, worked examples and self assessment
questions and problems. You are strongly advised to go through the activities
and consult all the compulsory materials and as many as possible among useful
links and references.
Specific Teaching and Learning Objectives
•
•
Analyse the effects of force on materials
Define different types of coefficient of elasticity
Summary of the Learning Activity
In this activity, you will define the concepts of load, strees and strain. You will
also derive the mathematical equations for the stress and strain. In addition you
will be able to solve different problems. The simplest cases of deformations are
those
i)
in which a wire, fixed at its upper end, is pulled down by a weight at lower
end, bringing about a change in its length.
ii) in which an equal compression is applied in all directions, so that there
is a change of volume but no change in shape.
iii) in which a system of forces may be applied to a body such that, although
there is no motion of the body as a whole, there is relative displacement
of its contiguous layers, causing a change in shape or “form” of the body
with no change in its volume. In all these cases the body is said to be
Strained or deformed
Key Concepts
Load: The term load, in the present context, implies the combination of external
forces (for example the weight of the body itself, together with those connected
with it; centrifuge forces in the case of rotating wheels and pulleys; forces due
to friction or forces due to unequal expansion and contraction on changes of
temperature etc.) acting on a body and its effect is to change the form or the
dimensions of the body.
Stress: The restoring or recovering force per unit area set inside the body is
called strees.
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Strain: The change produced in the dimensions of a body under a system of
forces or couples in equilibrium, is called strain, and is measured by the change
per unit length (linear strain), per unit volume, (volume strain), or the angular
deformation, (shear strain, or simply shear) according as the change takes place
in length, volume or shape of the body.
Linear Elasticity: (also known as elasticity of length ) Is a property possessed by
bodies that increase in length when a tensile force is applied to the. The applied
force causes equal and an opposite force called restoring or recovering force set
insite the body.
Poisson Ratio: The Poisson’s ratio is related to elastic moduli K, the bulk modulus; n as the shear modulus; and Y, Young’s modulus, by the following. The
elastic moduli are measures of stiffness. They are ratios of stress to strain. Stress
is force per unit area, with the direction of both the force and the area specified.
restoring or recovering force per unit area set inside the body is called strees.
Compressibility: The Bulk Modulus is sometimes referred to as compressibility;
so that, compressibility of a body is equal to
1
where k is its Bulk modulus.
k
it must thus be quite clear that whereas Bulk modulus is stress per unit strain,
compressibility represents strain per unit stress restoring or recovering force per
unit area set inside the body is called strees.
List of Relevant Readings
Reference
Nelkon & Parker (1995), Advanced Level Physics, 7th ed, CBS Publishers & Ditributer, 11, Daryaganji New Delhi (110002) India. ISBN
81-239-0400-2.
Rationale: This reading assumes high school physics background of
the reader it suits this module
Reference
Flower B.H., Mendoz E (1970), Properties of Matter. John Wiley &
Son Ltd, ISBN 0471 26498 9R McCliment (1984). Phusics, Harcourt
Brace Jovanovich, Publishers, San Diogo .
Rationale: This reading provide easy sources of information. The
contents have been treated in lucid manner with adequate mathematical
support.
Reference
Grant Mathur D.S. (1985), Elements of Properties of Matter, Shaym
Lal Charitable Trust, Ram Nagar, New Delhi 110055. 284-360
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List of Relevant Resources
Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html
Reference: http://en.wikipedia.org/wiki/Young’s_modulus Summary: Young’s Modulus (E) (also known as the Young Modulus,
modulus of elasticity, elastic modulus or tensile modulus) is a measure
of the satiffness of a given material. It is defined as the ratio, for small
strains, of the rate of change of stress with strain
Reference: http://en.wikipedia.org/wiki/Elasticity_of_substitution
Summary: An important property of many structural materials is their
ability to regain their original shape after a load is removed. These
materials are called elastic.
List of Relevant Useful Links
Title: Elasticity
URL: http://en.wikipedia.org/wiki/Young’s_modulus
Abstract:- properties and mathematical equation is found
Title: work done in strain
URL: http://en.wikipedia.org/wiki/Young’s_modulus
Abstract: equation of work done
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Introduction to the Activity
All bodies can, more or less, be deformed by suitably applied force. The simplest
cases of deformation that you can do are the following
1. In which a wire, fixed at its upper end, is pulled down by a weight at lower
end, bringing about a change in its length
L
A(c r oss-section)
ΔL
(a)
F (Loa d attached)
Figure 1
System of forces and deformations defining elastic modulus of linear tension
2. In which an equal compression is applied in all directions, so that there is a
change of volume but no change in shape.
F
V
F
ΔV
F
F
(b)
Figure 2
System of forces and deformations defining elastic modulus of a change in volume
3. A system of forces may be applied to a body such that, although there is no
motion of the body as a whole, there is relative displacement of its contiguous
layers, causing a change in shape or “form” of the body with no change in its
volume
F
B ΔL B’
Lθ
A
C’
D
(c)
Figure 3 System of forces and deformations defining elastic modulus due to tangential forces
producing an angle of shear
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Detailed Description of the Activity
(Main Theoretical Elements)
* Insure clear learning guidance and variety of learning activities are provided
throughout the acitvity.
Elasticity
In all the above cases the body is said to be strained or deformed. When the
deforming forces are removed the body tends to recover its original condition.
For example, the wire, in the Figure 1, tends to come back to its original length
when the force due to the suspended weight is removed from it, or, a compressed
volume of air or gas throws back the piston when it recovers its original volume.
This property of a material body to regain its original condition, on the removal of
the deforming forces, is called elasticity. Bodies, which can recover, completely
their original condition, on the removal of the deforming forces, are said to be
perfectly elastic. On the other hand, bodies, which do not show any tendency to
recover their original condition are said to be plastic.
Linear elasticity,
Linear elasticity also known as elasticity of length, is a property possessed by
bodies that increase in length or breadth or width when a tensile force is applied
to them normally in those directions.
Young’s Modulus
When the deforming force is applied as shown in the Figure 1 to the body only
along in particular direction, the change per unit length in that direction is called
longitudinal, linear or elongation strain,
l
and the force applied per unit area
L
of cross–section is called longitudinal or linear stress
Y=
F
. Young’s modulus
a
F .L
L dF
. For uniform change Y = .
. For non uniform change Where a is the
a.l
a dl
cross sectional area of the rod, L is the length of the rod, F is the Load .
Stress: Is the tensile force per unit area and is denoted by σ.
σ
Young’s modulus, E = =
ε
F
A
e
l
=
FL
for a uniforn change.
eA
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L
dF
For non uniform change E = •
where A is the cross-sectional area of
A dl
the rod, l is the length of the rod and, F, is the load.
Bulk modulus.
Here, the force is applied normally and uniformly as shown in the Figure 2 to
the whole surface of the body; so that, while there is a change of volume, there
is no change of shape The force applied per unit area, (or pressure), gives the
Stress =
F
v
and the change per unit volume, the strain=
their ratio giving
A
V
F
F .V
V
the Bulk Modulus for the body. k = a =
=P
v
a.v
v
V
Modulus of Rigidity.
In this case, while there is a change in the shape of the body, there is no change in
this volume as shown in the Figure 4 Tangential force F is applied in the direction
shown point B shifts to B’, D to D’, i.e. the lines joining the two faces turn through
an angle θ .the face ABCD is then said to be sheared through an angle θ this angle
θ (in radians), through which a line originally perpendicular to the fixed face is
turned, gives the strain or the shear strain, or the angle of shear, as it is often called
as can be seen, θ =
BB ' l
= , where l is the displacement BB’ and L, the length
AB L
of the side AB or the height of the cube. In otherwords, θ =relative displacement
of plane AB’D’C distance from the fixed plane ABCD. Tangential stress is equal
to the force F divided by the area of the face BDdb( area=a),i.e. equal to
F
. The
a
ratio of the tangintial stress to the shear strain gives the coefficient of rigidity of
F
F
F .L
a
the material of the body denoted by n=
= a =
If the shearing strain
l
θ
a.l
L
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dF
is not proportional to the shear stress applied, we have n=
A
a
dθ
C
Figure 4 Module regidity
Work done in a strain
In order to deform a body, work must be done by the applied force. The energy
so spent is strored up in the body and is called the energy of strain. When the
applied forces are removed the stress disappears and the energy of strain appears
as heat.
Let us consider the work done during the three cases of strain.
Elongation strain-(stretch of a wire)
Then work done
W=
∫ F .dl
Now, Young’s modulus for the material of the wire, i.e.
E =
F .L
a.l
where
L- is the original length
l - the increase in length
a- cross sectional area
F- the force applied
Then the force applied
F =
E .a.l
L
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The work done during the stretch from 0 up to l
l
w=
∫
0
E .a
ldl
L
l
=
E .a
ldl
L ∫0
=
E .a l 2
L 2
1 ⎛ E .a.l ⎞
E .a.l
.l
But
F
=
L ⎟⎠
L
= ⎜
2⎝
Hence w =
=
1
Fl
2
1
(stretching force x stretch)
2
Work done per unit volume =
Volume Strain
1
l
F.
2 L.a
=
1F l
.
2a L
=
1
stress x strain
2
Let σ be the stress applied. Then, over an area a the force applied is σ.a, and
therefore, work done for a small movement dx, in the direction of σ, is equal to
σ.a.dx. Now, a.dx is equal to dv, the small change produced in volume. Thus,
work done for a change dv is equal to σ dv.
And, therefore total work done for the whole change in volume, from 0 to V, is
given by
V
W = ∫ σ dV
0
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K = σ.
V
V
; so that σ = K .
v
v
Where V is the original volume and
K is the Bulk modulus
v
and w =
k.v
∫V
dv
0
v
k
= ∫ VdV
V 0
=
1 k.v
.v
2 V
=
1
σv
2
=
1
stress x change in volume
2
1 v
2 V
Work done per unit volume = σ
1
stress x strain
2
=
Shearing Strain
Consider a cube of edge L,(Fig.(1)), with its lower foce DC fixed, and let F be
the tangential force applied to its upper face in the plane of AB, so that the face
ABCD is distorted into the position A’B’CD or sheared through an angle θ.
Let the displacement AA’ be equal to BB’= l . Then, work done during a small
displacement d l is equal to F.d l . And, therefore work done for the whole of the
displacement, from 0 to l is given by
l
w = ∫ F .dl
Now
0
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n=
F
, F = n.a.θ and a = L2 ,
a.θ
also θ =
l
L
where L is the length of each edge of the cube so that
l
F = n.L2 . = n.L.l
L
Work done during the whole stretch from 0 to l , i.e
l
w = ∫ n.L.l.dl
0
=
1
1
1
n.L.l 2 = F .l = tangential force x displacement
2
2
2
Work done per unit volume =
=
1 F .l 1 F l 1 F
=
=
.θ
2 L3 2 L2 L 2 a
1
stress x strain.
2
Thus, we see that in any kind of strain, work done per unit volume is equal to
1
stress x strain
2
Dimensions.
The strain of a wire has no dimenssion
The dimenssion of stress= ML−1T −2
The SI unit of modulus of elasticity is the Pascal
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Task: 1.1 Experiment on stretching of steel wire by different loads
Objectives
The learners will be able to demonstrat different types of deformation
The learners will be able to calculate of the ratio of linear stress to linear
strain
• The learners will be able to draw the relation between the stress versus
the strain
Problem
•
•
The following problem is helping to find the strength of the material as well as
it helps to answer the objectives
Hypothesis
Formulate an hypothesis about the relation ship between the load and the cross
sectional area of the steel wire (stress), the length of the steel to the extension of
the steel (strain), calculate the Young’s modulus.
Equipment
Two long thin steel wires
Rigid support
Different weight
One the wires carries a vernier scale
Procedure
1) Arrange the steel wires, the load, the vernier scale as shown
2) Put different loads at the place of w
B
P
Q
v
M
P and Q are steel wires
V vernier scale
Tensile force on Q
A
w
Figure 5
Experimental arrangement for stretching of steel wire by different loads
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3. P,Q are two long thin steel wires suspended beside each other from a rigid
support B
4. The wire P is kept taut by a weight A attached to its end and carries a scale
M graduated in millimeters
5. The wire Q carries a vernier scale v alongside the scale M
6. V measures the small extension e, or change in length of Q, when the load
w is increased, and this in turn increases the force F in the wire.
Questions
1. What do you observe
2. Calculate the stress
3. Calculate the strain
4. Plot the graph of the stress versus strain
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Task: 1.2 Experiment to exercise mathematical equations
Objectives
The learners will be able to derive the mathematical equations for solving problems on coefficient of elasticity
Problem
Derive the mathematical quation on elasticity for the following constants.
i)Young’s modulus (E)
ii)Bulk modulus(k)
iii)Bulk regidity(n)
Advise
If you have derived the mathematical equations that is very nice. If not please
check what is done in derivation
Formative evaluation 1
stress
strain
Figure 6
Graph of stress against strain
Problem 1
In this activity you are expected to show on for the graph of stress vs strain the
following
a) elastic range b) elastic limit
c) plastic range
Answer
a) red b) broken line Problem 2
Mention factors affecting Elasticity
c) red region
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Answer
Effect of hammering-rolling and a annealingEffect of impuritiesEffect of change
of temperature
Problems 3
Show that
a) A small and uniform strain on volume V is equivalent to three linear strain
each of magnitude v/3, in any three perpendicular?
Answer
Imagine a unit cube to be compressed equally and uniformly on all sides, so that
length of each edge decreases by a length l and its volume by a small amount v.
Then, clearly volume strain in the cube =
edge of the cube
v
= v , and linear strain a long each
V
l
=l
L
(
)
Since length of each edge of the cube now becomes L − l the new volume of
(
the cube becomes L − l
)
3
Decrease in volume of the cube, i.e
(
v=V − L −l
)
3
After calculating and negelecting the higher order of you can find
v = 3l
Then l =
v
3
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Formative evaluation
Show that
The bulk modulus for a gas
i)
at constant temperature (i.e. under isothermal conditions) is equal to its
pressure
ii) when temperature is not constant, (i.e. when the conditions are adiabatic)
it is equal to γ times its pressure, where γ =
Cp
Cv
Answer
Let p be the pressure and V, the volume of a gas , and let it be compressed by increasing the pressure (p+dp), so that the volume is reduced by dv, and becomes (V-dv)
then stress =
dF
= pressure applied =dp
dA
volume strain =
changeinvolume
originalvolume
bulk modulus for the gas, i.e. K = −
dP
V
dV
i) If the gas is compressed isothermally, its temperature remains constant, therefore
PV = const
P=
const
V
dp = −
const
dV
V2
Vdp = −
const
= BulkModulus = K
V
const
=K
V
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const
=p
V
Then
K = p bulk constant equal to the pressure
Answer
ii) if the gas compressed adiabatically
pV γ = const , γ =
Cp
Cv
p = CV − γ
Differentiating p with respect to V gives
dp = −γ V − γ − 1 dVconst
V
dp
γ
= γ const
dV V
Where −V
dp
= kBulk , const = pV γ
dV
γ
pV γ
γ
V
k = γ p Bulk constant
k=
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Activity 2: Fluids
You will require 45 hours to complete this activity. In this activity you are
guided with a series of readings, Multimedia clips, worked examples and self
assessment questions. You are strongly advised to go through the activities and
consult all the compulsory materials and as many as possible among useful links
and references.
Specific Teaching and Learning Objectives
•
•
•
•
Describe basic properties of fluid (density,pressure)
Apply the properties of fluids (archimedes principle)
Explain fluid motion (continuity, turbulance, real fluid)
Use Bernnoulli’s Equation
Summary of the Learning Activity
In this activity the learners will describe the pressure in fluids at rest, explain the
effects of the buoyant force on a submerged object and the distribution of fluid
in a closed conteiner.
The pressure P, in a fluid is the force per unit area that the fluid exerts on any
surface. The pressure in a fluid varies with depth(h) according to the expression
p = pa + ρ gh where Pa is atmospheric pressure (1.01x105N/m2) and ρ is the
density of the fluid,
You will state also Pascal’s law and Archimedes’s principle.
Fluid dynamics (fluid in motion) can be understood by assuming that the fluid is
non viscous and incompressible and that the fluid motion is a steady flow with no
turbulence.Using these assumptions,the flow rate through the pipe is a constant
That is A1V1=A2V2 .The sum of, kinetic energy per unit volume, and potential
energy per unit volume has the same value at all points along a streamline. That
is,
1
p + ρ v2 + ρ gy = constant
2
Bernoulli's equation
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Key Concepts
Pascal’s Principle: A change pressure applied to an enclosed fluid is transmitted
undiminished to every point of the fluid and the walls of the containing vessel
Archimeds’ Principle: Any body completely or partially submerged in a fluid is
buoyed up by a force equal to the weight of the fluid displaced by the body.:
Stream Line: Is the path taken by a fluid particle under steady flow is called a
stream. line
Bernoulli’s Equation: This equation gives an expresssion that deals with the
sum of the pressure, kinetic energy per unit volume, and potential energy per unit
volume has the same value at all points along a streamline
Introduction to the Activity
The knowledge of the existence of electrostatic charge goes back at least as far
as the …
Detailed Description of the Activity (Main Theoretical Elements)
States of matter
Matter is normally classified as being in one of its states, solid, liquid or gaseous. Often, this classification is extended to include a fourth state referred to
as plasma.
The fourth state of matter can occur when matter is heated to very high temperatures. Under this condition, one or more electrons surrounding each atom are
freed from the nucleus. The resulting substance is a collection of free electrically
charged particles: the negatively charged electrons and the positively charged
ions. Such an ionized gas with equal amounts of positive and negative charges
is called plasma.
Density and Pressure
•
ρ=
•
The density of a substance is defined as its mass per unit volume.
m
v
Specific gravity of a substance is defined as the ratio of its density to the
density of water at 4oc, which is 1x103kg/m3
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If F is the magnitude of the normal force on the piston and A is the area
of the piston, then the pressure, P, of the liquid at the level to which
the device has been submerged is defined as the ratio of force to area.
P=
F
A
∆F dF
P = lim
=
∆ A → 0 ∆A
dA
The unit of pressure in the SI system is Pascal (Pa)
1Pa = 1
N
m2
Variation of pressure with depth
Consider a fluid at rest in a container shown in the Figure 2.1 below
Figure 1
Variation of pressure with depth in a fluid the volume element is at rest, and the force
on it.
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We first note that all points at the same depth have the same pressure.
Consider the fluid contained with in an imaginary cylinder of cross-sectional
area A and height dy. The upward force on the bottom of the cylinder is PA and
the down ward force on the top is (P+dP) A. The weight of the cylinder, whose
volume is dv, is given by dW = ρgdV = ρgAdy , where ρ is the density of
the fluid. Since the cylinder is in equilibrium, the force must add to zero, and
so we get
∑F
y
(
)
= PA − P + dP A − ρgAdy
dP
= −ρg
dy
From this result, we see that an increase in elevation (positive by) corresponds to
a decrease is pressure (negative dp). If p1 and p2 are the pressure at the elevations
y1 and y2 above the reference level, and If the density is uniform, then integrating
P2
y2
∫ dP = − ∫ ρgdy
P1
y1
P2 - P1 = - ρg( y2 − y1 )
If the vessel is open at the top, then the Pressure at the depth h can be obtained.
Taking atmospheric pressure to be Pa = P2, and noting that the depth h = Y2
– Y1,
We find that:
P = Pa + ρgh
The absolute pressure P at a depth h below the surface of a liquid open to the
atmosphere is greater than atmospheric pressure by an amount ρgh .
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P2 = Pa
h
y2
• p1 =p
y1
Figure 2
The Pressure P at a depth h below the surface of a liquid open to the atmosphere
is given by P = Pa + ρgh
This result also verifies
(i) The pressure is the same at all points having the same elevation.
(ii) The pressure is not affected by the shape of the vessel.
Pascal’s principle
A change pressure applied to an enclosed fluid is transmitted undiminished to
every point of the fluid and the walls of the containing vessel.
Figure 3
A hydraulic press
P1 = P2
⇒
F1 F 2
=
A1 A2
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Pressure Measurements
One simple device for measuring pressure is the open-tube monometer shown
below.
Figure 4
The open – tube manometer
One end of a U – shaped tube containing a liquid is open to the atmosphere, and
the other end is connected to a system of unknown pressure P. The pressure at
point B equals P = Pa + ρgh where ρ is the density of fluid. But the pressure
at B equals the pressure at A.
PA = PB
P = Pa + ρgh The pressure P is called the absolute pressure while
P–Pa is called the gauge pressure.
Buoyant Forces and Archimedes’ Principle
Archimedes’ Principle can be stated as follows:
Any body completely or partially submerged in a fluid is buoyed up by a force
equal to the weight of the fluid displaced by the body.
In other words the magnitude of the buoyant force is equal to the weight of the
fluid displaced by the object.
W
B
B = W = ρ f Vg = mg where V is the volume of cube and ρ f is density of fluid,
m mass of water, W is the weight of fluid displaced.
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Case 1: A totally submerged object
when an object is totally submerged in a fluid of density ρ f , the upward buoyant
force is given by B = ρ f V0 g , Where V0 is the volume of the object. If the object
has a density ρ0 , its weight is equal to W = mg= ρ0V0 g , and the net force on it is
B – W = ( ρ f − ρ0 ) V0 g . Hence the density of the object is less than the density
of the fluid, the unsupported object will accelerate upward. If the density of the
object is greater than the density of the fluid, the unsupported object will sink.
Case II: A floating object
Consider an object in static equilibrium floating on a fluid; that is one which is
partially submerged. In this case, the upward buoyant force is balanced by the
downward weight of the object. If vf is the volume of the fluid displaced by the
object, then the buoyant force has a magnitude given by B = ρ f Vg . Since the weight of the object is W = mg = ρ0V0 g, and W = B, we see that ρ f Vg = ρ0V0 g , or
ρ0 V
=
ρ f V0
Fluid Dynamics
When fluid is in motion, its flow can be one of two main types of flow.
(i) steady flow which a flow where each particle of the fluid flows a smooth
path, and the paths of different particles do not cross each other.
(ii) a non-steady or turbulent which is an irregular flow characterized by small
whirl pool-like region.
Stream Lines
The path taken by a fluid particle under steady flow
is called a stream line. A particle at P flows one of
thesestreamlines, and its velocity V is tangent to the
streamline at each point along its path.
V
.
P
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The Equation Of Continuity
Consider a fluid flowing through a pipe of non – uniform cross – sectional
area.
V2
A2
∆x 2
V1
A1
∆x1
The particles in the fluid move along the streamlines in steady flow. At all points
the velocity of the particles is tangent to the stream line along which it moves.
In a small time interval Δt, the fluid at the bottom end of the pipe moves a distance Δx1 = v1 Δt. If A1 is the cross-sectional area in this region, then the mass
contained in the shaded region is Δm1= ρ1 A1 Δx1 = ρ1 A1 v1Δt. Similarly, the
fluid moves through the upper end of the pipe in the time Δt has a mass Δm2 =
ρ2 A2v2Δt. However, since mass is conserved and because the flow is steady, the
mass that crosses A1 in a time ∆t must equal the mass that crosses A2 in the time
∆t. Therefore ∆m1=∆m1, or
ρ1 A1V1 = ρ2 A2V 2
This is equation of continuity
A1V1 = A2V 2
The product of the area and the fluid speed at all points along the pipe is a
constant.
Bernoulli’s Equation
As a fluid moves through a pipe of varying cross section and elevation, the pressure will change along the pipe.
We shall assume that the fluid is incompressible and nonviscous and that it flows
V2
in an irrotational and steady manner.
P2 A2
V1
P1 A1
y1
∆x1
∆x 2
y2
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Consider the flow through a non-uniform pipe in a time Δt. Therefore the force on
the lower end of the fluid is P1A1 where P1 is the pressure at point 1. The work
done by this force is W1=F1Δx1=P1A1Δx1= P1ΔV, where ΔV is the volume of the
lower shaded region. In similar manner, the work done on the fluid at the upper
end in the time Δt is given W2=F2Δx2=-P2A2 Δx2= -P2 ΔV. This work is negative
since the fluid force opposes the displacement. Thus the network done by these
forces in the time Δt is w = (P1-P2) ΔV part of this work goes into changing the
kinetic energy of the fluid, and part into changing the gravitational potential
energy. If Δm is the mass passing through the pipe in the time Δt, then the change
in its kinetic energy is
∆k =
1
1
∆m v2 2 − ∆m v1 2
2
2
( )
( )
The change in its potential energy is
∆u = ∆mgy2 − ∆mgy1
We can apply the work energy theorem in the form w=Δk+Δu to its volume of fluid to give
( P
1
- P
2
)
Δ V =
1
1
∆m v2 2 − ∆m v1 2 + ∆mgy2 − ∆mgy1
2
2
( )
( )
∆m
If we divide each term by ΔV, and recall that ρ =
the above expression
∆V
reduces to
(P1-P2) =
1
1
ρv2 2 − ρv1 2 + ρgy2 − ρgy1
2
2
Rearranging terms we get
P1+ 1 ρv 2 + ρgy1 = P2+
2
1
1
ρv2 2 + ρgy2
2
This is Bernoulli’s equation as applied to a non-viscous, incompressible fluid in
steady flow. It is often expended as
P+
1
ρv 2 + ρgy constant
2
Bernoulli’s equation says that the sum of the pressure, (p), the kinetic energy per
unit volume (
1 2
ρυ ), and potential energy per unit volume ( ρgy ) has the same
2
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value at all points along a stream line.
When the fluid at rest v1=v2=0 and the above equation becomes
(
)
P1 − P2 = ρg y2 − y1 = ρgh
Which agrees with Bernoulli’s equation
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Learning Activities
Task 2.1. Calculation of speed in fluid flow
(a) A water hose 2cm in diameter is used to fill a 20 litre bucket. If it takes
1min to fill the bucket, what is the speed v at which the water leaves the
hose?
(b) If the diameter of the hose is reduced to 1cm, what will the speed of the
water be as it leaves the hose, assuming the same flow rate?
Task 2.2. Using Archimedes principle to compare densities
(a) A plastic sphere floats in water with 0.5 of its volume submerged. This
same sphere floats in oil with 0.4 of its volume submerged. Determine
the ratio of densities of the oil and the sphere.
(b) A cube of wood whose one of the sides is 20cm has a density of 0.65x103
floats on water.
i. what is the distance from the top of the cube to the water level?
ii. how much lead weight has to be placed on top of the cube so that its top is just level with the water?
Task 2.3. Using fluid dynamics equations to solve problems
1. Determine the absolute pressure at the bottom of a lake that is 30m
deep.
2. A swimming pool has dimensions 30m X 10m and a flat bottom. When
the pool is filled to a depth of 2m with fresh water, what is the total force
due to the water on the bottom? On each end? On each side?
3. The spring of the pressure gauge has a force constant of 1000N/m, and
the piston has a diameter of 2cm. Find the depth in water for which the
spring is compressed by 0.5cm?
Task 2.4 Using fluid dynamics equations to solve
The open vertical tube in the figure shown below contains two fluids of densities
ρ1 And ρ2 , which do not mix. Show that the pressure at the depth h1 +h2 is given
by the expression P = Pa+ ρ1 gh1 + ρ2 gh2
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Formative Evaluation 2
1. The rate of flow of water through a horizontal pipe is 2m3/min. Determine
the velocity of flow at a point where the diameter of the pipe is
(a) 10cm
(b) 5cm
2. What is the hydrostatic force on the back of Grand Coulee Dam if the water
in the eservoir is 150m deep and width of the dam is 1200m?
3. Calculate the buoyant force on a solid object made of copper and having a
volume of 0.2m3 if it is submerged in water. What is the result if the object is
made of steel?
4. In air an object weighs 15N. When immersed in water, the same object weighs
12N. When immersed in another liquid, it weights13N. Find
a. The density of the object and
b. The density of the other liquid
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Activity 3: Transport Properties
You will require 25 hours to complete this activity. In this activity you are
guided with a series of readings, Multimedia clips, worked examples and self
assessment questions.. You are strongly advices to go through the activities and
consult all the compulsory materials and as many as possible among useful links
and references.
Specific Teaching and Learning Objectives
•
•
•
•
Analyse particle motion in fluids
Describe relative properties of solids, liquids and gasses
Discuss the effects of heat on materials – e.g. calculate thermal expansion
Calculate the effective concentration of mobile electrons in metals, alloys
and semiconductors
Summary of the Learning Activity
In this unit you will learn the transport properties of gases (molecules) in a system by considering that diffusion, viscocity and heat conduction as a transport
process. In addition you will in detailed describtion of conduction and thermal
expansion of metals using mathematical approach. The transportaion of electron
is discussed in terms of the effective concentration of mobile electrons in metals,
alloys and and semiconductors
Key Concepts
Diffusion: Is the movement of particles from higher chemical potential to lower
chemical potential (chemical potential can in most cases of diffusion be represented by a change in concentration).An electric charge is an attribute of matter
that produces a force.
Osmosis: If two solutions of different concentration are separated by a semipermeable membrane which is permeable to the smaller solvent molecules but
not to the larger solute molecules, then the solvent will tend to diffusion across
the membrane from the less concentrated to the more concentrated solution this
process is called osmosis.
Electron diffusion: resulting in electric
Heat Conduction: The conduction of heat is also a process of diffusion in which
random thermal energy is transferred from a hotter region to a colder one without
bulk movement of the molecules themselves.
Viscous motion: of fluids can be far more complicated than diffusion or heat
conduction and we will be forced to consider only the steady state equation.
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Thermal expansion of solids or a body: Is a consequence of the change in the
average separation between its constituent atoms or molecules
Electrical conductivity: Is the ability of different types of matter to conduct an
electric current
Semiconductors: are materials whose conductivity is between that of conductors
(generally metals) and that of nonconductors or insulators.
Alloy: Is a metal composed of more than one element
Key terms
•
•
•
•
•
•
•
•
Momentum diffusion
Brownian motion
Diffusion equation
Fick’s law of diffusion
Heat flow
Osmosis
Osmotic pressure
Transport phenomena
List of Relevant Readings
Reference:- Viscosity
Abstract: Viscosity is the resistance or the internal friction between molecules.
Viscosity can be measured by an instrument called a viscometer. Some liquids
like water have a low viscosity whereas other liquids like honey have a high
viscosity. Viscosity will be affected by the temperature. At higher temperatures
the viscosity decreases as the molecules take on more kinetic energy allowing
them to move past each other faster
List of Relevant Resources
Reference:- http://video.google.com/videoplay?docid=-4559185597114887
235&q=electric+charge&hl=en
Summary: This resource is video show on electric charges
Reference: - …http://en.wikipedia.org/wiki/Electrical_conductivity
Summary:- To analyse the conductivity of materials exposed to alternating
electric field
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Introduction to the Activity
Diffusion is the transport of a material or chemical by molecular motion. If molecules of a chemical are present in an apparently motionless fluid, they will exhibit
microscopic erratic motions due to being randomly struck by other molecules in
the fluid. Individual particles or molecules will follow paths sometimes known
as “random walks.”
In such processes, a chemical initially concentrated in one area will disperse. That
is, there will be a net transport of that chemical from regions of high concentration
to regions of low concentration.
An analogous form of diffusion is called conduction. In this case, heat is the
“chemical” that is transported by molecular motion. As in chemical diffusion,
heat migrates from regions of high heat to regions of low heat. The mathematics
describing both conduction and diffusion is the same.
Figure 1
Consider two containers of gas A and B separated by a partition. The molecules of both
gases are in constant motion and make numerous collisions with the partition
Detailed Description of The Activity (Main Theoretical Elements)
Gases Liquids and Solids
As a useful, though not complete, classification it can be said that matter exists
in three states, as gas, liquid or solids. This statement is justified by the fact that
there exist many substances which can undergo sharp, easily identifiable, reproducible and reversible transitions from one state to the other. Water is the classical
example: its freezing and melting, boiling and condensation have been contemplated since the time of the ancient Greek scientists. There are obvious contrast
between the properties of ice, water and steam or water vapour which make their
description as solid, liquid and gas quite unambiguous. Similarly, most metals
are solid, they melt under well defined conditions of temperature and pressure to
form liquids and boil at higher temperatures to produce gases.
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If all substances possessed such clear demarcations, it would be easy to define
the different states of matter. But there are very many substances like glasses or
glues which one normally thinks of as being solid but which do not melt at sharply
defined temperatures; when heated they gradually become plastic, till they become
recognizably liquid. Other solids such as wood or stone are inhomogeneous and
it is difficult to describe their structure in detail.
Prosperities and structures of gases
Gases have low densities they are highly compressible over wide ranges of
volume, they have no rigidity and low viscosities. The molecules are usually
a large distance apart compared with their diameter and there is no regularity
in their arrangement in space. Given the positions of two or three molecules, it
is not possible to predict where a further one will be found with any precision.
The molecules are distributed at random throughout the whole volume. The low
density can be readily understood in terms of the comparatively small number of
molecules per unit volume. The high compressibility follows from the fact that
the average distance between molecules can be altered over wide limits. The
molecules can move long distances without encountering one another, so there
is little resistance to motion of any kind, which is the basis of the explanation of
the low viscosity.
Properties and structure of liquids
Liquids have much higher densities than gases and their compressibility is low.
They have no rigidity but their viscosity is greater than that of ordinary gases.
The molecules are packed quite closely together and each molecule is bonded to
a number of neighbors but still the pattern as a whole is a disordered one. The
molecules are moving with just the same order of velocity as in a gas at the same
temperature, though the motion is now partly in the form of rapid vibrations and
partly translational.
Properties and structure of solids
Solids have practically the same densities and compressibilities as liquids. In
addition they are rigid; under the action of small forces they do not easily change
their shape.
An important property of those solids which have a well-defined melting – point
is that they are close packed, and the arrangement is highly regular. Substances
which do not melt sharply but show a gradual transition to the liquid when heated
are said to be amorphous and show no trace of regularity of external shape.
In crystalline solids, the molecules are arranged in regular three dimensional
patterns or lattices, If the crystal has been carefully prepared, the regular arran-
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gement persists over distances of several thousand molecules in any direction
before there is an irregularity, but if it has been subjected to strains or distortions
the regular arrangement may be perfect and uninterrupted only over much shorter average distances. In metals the ions are closely packed together, so that the
distance between the centre of an ion and that of one of its nearest neighbours
is equal to the diameter of one ion, or something close to it. In other crystals,
the packing together of the molecules may be relatively open, but even in light
solids such as ice the distance between the centers of any molecule and its near
neighbors in only twice the diameter of a molecule. In solids, the molecules are
again moving with the same order of magnitude of velocity as in gases or liquids,
but the motion is confined to vibrations about their mean positions.
Transport Processes
So far we have learned the properties of solids, liquids and gases which are in
equilibrium. In this activity we will deal with systems which are nearly but not
quite in equilibrium in which the density (or the temperature or the average momentum) of the molecules varies from place to place. Under these circumstances
there is a tendency for the non-uniformities to die away through the movement
the transport of molecules down the gradient of concentration (or of their mean
energy down the temperature gradient or their mean momentum down the velocity gradient).
Diffusion
Diffusion is the movement of molecules from a region where the concentration is
high to one were it is lower, so as to reduce concentration gradients. This process
can take place in solids, liquids and gases (though this part you will be mostly
concerned with gases). Diffusion is quite independent of any bulk movements
such as winds or convection currents or other kinds of disturbance brought about
by differences of density or pressure or temperature (although in practice these
often mask effects are due to diffusion).
One gas can diffuse through another when both densities are equal. For example,
carbon monoxide and nitrogen both have the same molecular weight, 28, so that
there is no tendency for one or other gas to rise or fall because of density differences: yet they diffuse through each other. Diffusion can also take place when a
layer of the denser of two fluids is initially below a layer of the lighter so that the
diffusion has to take place against gravity. Thus, if a layer of nitrogen is below a
layer of hydrogen, a heavy stratum below a light one, then after a time it is possible
to detect some hydrogen at the bottom and some nitrogen at the top, and after a
very long time both layers will be practically uniform in concentration.
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Diffusion coefficients of gases α and β can be measured with a suitable geometrical arrangement of two vessels with different initial concentrations together
with some method of measuring those concentrations such as a chemical method
or mass spectroscopy, for example. If the rates of change of concentration with
time are plotted, the diffusion coefficient can be deduced; the equations describing
the process are given in diffusion equation.
t= 1/4D
concentration
concentrati
on
T= 1/2D
T=1/D
Figure 2
Concentration as a function of x for different values of time t
The diffusion equation
We will begin by taking a macroscopic view of the phenomenon, that is, we will
write down equations which involve such variables as concentrations or fluxes but
will not specifically mention individual molecules. We define the concentration
α as the number of molecules n per unit volume. Let us consider the simple case
where n varies with one coordinate only the x-axis. In Figure 1. the concentration
at all points in the plane x is n, at (x+dx) it is (n+dn). Then diffusion takes place
down the concentration gradient, from high to low concentration; we are assuming
that bulk disturbances are absent. We next define the flux J of particles as the
number of particles on average crossing unit area per second in the direction of
increasing x. Notice that both concentration and flux can be measured in moles
instead of numbers of molecules: this is equivalent to dividing all through our
equations by Avogadro’s number N.
n
n+dn
J
X
X+dX
X
Figure 3
Coordinates used in the definition of diffusion
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In general, the flux J may change with position x and may also change with time t.
In other words, J may be a function of x and t so we write it as J (x,t). Of course,
there are circumstances where J may be the same for all x, or where it is constant
with time, but the most general situation is that j depends on both.
It is an experimental fact that, at any instant that flux at any position x is proportional to the concentration gradient there:
( )
∂n
or
∂x
( )
∂n
∂x
J x,t α −
J x,t = − D
………………………3.1
where D is called the diffusion coefficient. This is known as Fick’s law.
By itself, Eq. (3.1) is adequate to describe ‘steady-state’ conditions where currents
and concentrations do not change with time so that the flux can be written J(x).
For example, if a tube of length l cm with constant cross-sectional area A cm2 has
molecules continually introduced at one end and extracted at the other end at the
same rate, the concentration gradient becomes -∆n/ l , where ∆n is the difference of
concentration between the two ends. The number of particles crossing any plane
in the tube per second is then –DA∆n/ l and this does not change with time.
Consider, however, the much more general situation where initially a certain
distribution of concentration is set up and then subsequently the molecules diffuse so as to try to reach a uniform concentration. Concentrations are, therefore,
changing with time and particles must be accumulating in the region between
x0 and (x0+dx) or moving from it. Therefore, the number crossing area A of the
plane x0 is not equal to that crossing the same area at (x0+dx). The flux entering
this volume is
⎛ ∂n ⎞
Jx0 = - D ⎜ ⎟
⎝ ∂x ⎠ x = x
0
The flux leaving the slice can be written Jx0+dx where
⎛ ∂J ⎞
dx + ...
Jx0+dx = Jx0+ ⎜⎝ ∂x ⎟⎠
and we can neglect higher terms. The rate of movement of molecules from the
slice is equal to the difference between the two values of AJ, and also equal to
the volume of the slice, A dx, times the rate of decrease of n:
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-
∂J
∂n
Adx = Ad
∂x
∂t
∂J
∂n
=−
………………………… (3.2)
∂x
∂t
That is
Combining this with equation (3.1) and eliminating J:
∂n
∂⎛
∂n ⎞
∂2 n
= − ⎜ − D ⎟ = D 2 ………………………………………. (3.3)
∂t
∂x ⎝
∂x ⎠
∂x
if we assume that D is constant independent of the concentration. This is called
the diffusion equation, and since n depends on x and t it could be written n(x,t).
If the process takes place in 3 dimensions, J is a vector whose components are
(Jx,Jy,Jz) and the above equations become
⎛ ∂n
∂n
∂n ⎞
J = iJ x + jJ y + kJ z =−D ⎜ i + j + k ⎟ = − D grad n
∂y
∂z ⎠
⎝ ∂x
−
∂n ∂J x ∂J ∂J
=
+
+ = div J
∂t ∂x ∂y ∂z
Where i,j and k unit vectors parallel to x,y and z. Eliminating J:
⎛ ∂2 n ∂2 n ∂2 n⎞
∂n
= − div(− D grad n) = D∇ 2 n= D ⎜ 2 + 2 + 2 ⎟
∂t
∂y
∂z ⎠
⎝ ∂x
Thus we have a system of three equations. (3.1) is an experimental law linking
the flux at any point with the concentration gradient there. (3.2) is the continuity
equation expressing the fact that molecules cannot disappear, and (3.3) combines
these two equations. Eq. (3.1) is adequate for steady-state conditions, where
conditions do not vary with time; but for the general case (3.3) may be used.
These are typical of transport equations with the provision that for energy and
momentum diffusion, the coefficients in the three equations are not all identical
as they are here.
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Heat conduction
Heat can be transferred by conduction, convection or radiation. The process of
transferring heat through a body is called thermal conduction. The physical property known as thermal conductivity is a measure of how efficient the material
will conduct heat through it. The thermal conductivity of a substance is defined
as the amount of heat transfer per unit area per unit time per unit temperature
gradient through a body. Mathematically, thermal conductivity can be treated in
a very similar way to diffusion leading to very similar types of mathematical
functions. Thermal conductivity is very important when designing for thermal
insulation, thermal isolation, efficient heat transfer and cooling systems
The conduction of heat is also a process of diffusion in which random thermal
energy is transferred from a hotter region to a colder one without bulk movement
of the molecules themselves. In a hot region of a solid body, they have extra
kinetic energy. By a collision process, this energy is shared with and transferred
to neighbouring molecules, so that the heat diffuses through the body though the
molecules themselves do not migrate. The macroscopic equations describing
conduction in one dimension x are, firstly, the experimental law for the heat
flux
Q = −k
∂T
∂x
…………………..(3.4)
(where Q is the heat flux across unit area, measured in W cm-2, k is the thermal
conductivity and T is the temperature) and, secondly, the continuity equation
∂Q
∂T
= − Cp
∂x
∂t
……………………..(3.5)
which expresses the conservation of energy in the form that the heat which is
absorbed by a slice of a body goes into raising its temperature. C is the specific
heat per unit mass, ρ the density so that C ρ is the specific heat per unit volume.
Combining these two equations to eliminate Q:
∂T ⎛ k ⎞ ∂ 2 T
=
∂t ⎜⎝ Cp ⎟⎠ ∂x 2
………………(3.6)
⎛ k ⎞
⎟ is called the thermal diffusivity by analogy with Eq. (3.3). E1.
⎝ Cρ ⎠
where ⎜
(3.4) by itself is adequate for steady-state conditions, as when for example heat
is fed into one end of a bar and extracted at the other and all temperatures are
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constant with time, and T can be calculated as a function of x alone. But when
conditions are not steady, and T varies with time as well as position, Eq. (3.6)
describes the situation.
Viscosity
For completeness, a third simple transport process the diffusion of momentum
by viscous forces will be mentioned here, briefly. Viscous motion of fluids can
be far more complicated than diffusion or heat conduction and we will be forced
to consider only the steady state equation.
Z
Moving plate
Y
Ux
X
Stationary plate
Figure 4
Coordinates used in the definition of viscosity.
Consider a gas or liquid confined between two parallel plates (Fig 4). Let the
lower plate be stationary and the upper plate be moving in the direction shown,
which we will call the x-direction. Molecules of fluid very near the plate will
be dragged along with it and have a drift velocity, Ux parallel to x, superposed
on their thermal velocity. We will assume that Ux is much less than the mean
thermal speed or the speed of sound. Molecules of fluid near the stationary
plate will, however, remain more or less with zero drift velocity.
Eventually a regime will be set up in which there is a continuous velocity gradient
across the fluid from bottom to top. In this state, molecules will be continuously
diffusing across the space between the plates and taking their drift momentum
with them. Considering an area of a plane parallel to the xy plane in the fluid,
molecules which diffuse across from above to below will carry more drift momentum than those which diffuse from underneath to the top. In other words,
the more rapidly moving layer tends to drag a more slowly moving layer with it,
because of this diffusion of momentum.
In macroscopic terms, a shearing stress (force per unit area) is necessary to maintain this state of motion. The experimental law is
Pxz = η
∂U x
∂z
……………..(3.7)
where Pxz is the force per unit area in the x direction due to a gradient of Ux in
the z-direction and η is called the coefficient of viscosity. Provided the direction
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of the force is clearly understood, it is not necessary to include a minus sign, as
this depends on the convention for the choice of axes.
We started by considering a fluid in Figure4, but Eq. (3.7) can be applied to solids
because the right-hand side can be written
dθ
, where θ is an angle of shear. It
dt
is difficult to imagine a solid subjected to a shear which goes on increasing with
time, but it is quite common for solids to be sheared to and for in an oscillatory
fashion. Forces are then required to provide the accelerations, but in any case
the viscosity gives rise to the dissipation of energy and the production of heat. It
is usual to refer to this as due to the internal friction of solids.
It is implied in Figure 4. that
∂U x
is constant and that Ux increases proportio
∂z
nally with z. This is so if the coefficient η is a constant. For many liquids this
holds, but there are notable exceptions when η varies with the velocity gradient
or rate of shear so that the velocity profile is not linear
When we come to write down equations representing the motion of a fluid while
it is not in a steady state but accelerating, we meet a situation which is much more
complicated than the diffusion or heat conduction cases. For one thing, there are
always mass-acceleration terms which have no analogue in the other phenomena.
For another, a kind of regime may be set up where the flow is not streamline as
illustrated in Figure 4 but turbulent, and vortices or eddies are present which add
an element of randomness to the flow pattern. We can, however, usefully adopt a
mathematical representation of the simple situation of Fig 4. We can imagine the
liquid divided into layers, each one sliding over the one underneath it on imaginary
rollers like long axle rods parallel to the y-axis. These rollers are not there in any
real sense, but they can lead one to define a quantity called the vorticity which is
always present in a flowing fluid even when no macroscopic vortices are present.
(In a simple case like Fig. 4 the vorticity degenerates into the velocity gradient.)
Now in the general case of an accelerating fluid with non-uniform velocity it is
the vorticity which diffuses throughout the fluid, though the equation it obeys is
not of a simple form
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Task 3.1 Measurement of the viscosity of gases
In his classic experiments to measure the viscosity of gases at low pressures,
Maxwell used a torsion apparatus in which a number of circular glass discs
were arranged to swing in between fixed ones (Fig 5). He found the damping
coefficient of the oscillations. If we neglect the energy loss in the torsion wire
itself and assume that the discs would go on swinging for a very long time if all
the gas were removed, we can calculate the damping as follows.
Figure 5
Principle of the apparatus for measurement of viscosity by the damping of torsional
oscillation.
Consider one surface of one plate, and select an annulus ring between radii r and
(r+dr). Then (assuming streamline flow) the force on this annulus ring, whose
area is 2 π dr, is
dF=
( ) (2πrdr ) η rω
d
………3.8
Where the linear velocity is r ω , ω being the angular velocity, and d is the spacing
between adjacent moving and stationary surfaces. The contribution to the couple
is the couple is the radius times the force:
dG=
2πηϖ 3
r dr d
………..3.9
and the total couple is
G=
2πηϖ
d
∫
a
0
r 3 dr =
πηϖ 4
a
2d ………..3.10
Where a is the radius of the disc. If there are n discs, each with two surfaces,
there are 2n such contributions.
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t Law
Solutions Of The Diffusion Equation: The
Consider figure3.3 coordinates used in the definition of diffusion the length is
along the x-axis and the ends are at x=0 and x =. On the face x=0,N0 molecules
are initially all concentrated in a thin layer and are subsequently allowed to diffuse
into the material. We will denote the number at time t which are within a slice
between x and (x+dx) by n (x,t) A dx. Then the appropriate solution of Eq.(3.3)
shows that the concentration.
( )
N0
n x,t =
(
A π Dt
−
)
1
e
x
2
……………. 3.11
4 Dt
2
We can, therefore calculate the mean net distance traveled by a molecule at any time t.
()
x t =
A
xn x,t dx
N 0 ∫0
∞
( )
we find x =
2
π
( )
Dt
1
2
We find the mean net distance traveled is proportional to the square root of the
time. This is perhaps an unexpected result: one is used to traveling twice the
distance when the time is doubled, but for the random process of diffusion this is
not so. Of course, some molecules go much further than this, other less far, and it
is the mean which we have calculated. Stated differently , our results shows that
to diffuse a mean distance. X, the time required is proportional to x2 . This is an
important characteristic of the diffusion process.
Thermal Expansions of solids and liquids.
Most solids expand as their temperature increases. The thermal expansion of solids
or a body is a consequence of the change in the average separation between its
constituent atoms or molecules. Suppose the linear dimension of the body along
some direction is l at some temperature. The length increases by an amount ∆l
for a change in temperature ∆T
l0
∆l
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Then ∆l α ∆T
∆l = l ∆T
∆l = α l ∆T
Where α is coefficient of linear expansions of solids?
The linear dimension of the body also change with temperature, it follows that area and volume of a body also change with temperature.
l
ω
λ
∆V =
βV0 ∆T
β = 3α
β is the coefficient of volume expansion
β = 3α for isotopic solid where the coefficient of linear expansion is the same
in all direction.
For a side of volume l , ω , λ
+ ∆V = (l + ∆l )(ω + ∆ω ) (λ + ∆λ) = (l + 2∆T )(ω +αωT ) (λ +αλ∆T )
= l λ ω (1 +α∆T )
= l λ ω (1 +α∆T ) 3 = l λ ω ( + 3 α ∆T ) + 3(α∆T )2 + (α∆T )3
= V (1 + 3α∆T ) + 3( α ∆T )2 + (α∆T )2
(1 +α∆T )
(1 +α∆T )
Comparing (α∆T )3 << α∆T
α∆T 2
<< α∆T
Then we neglect α∆T 3 and α∆T 2
V+ ∆V =[ 1+3 α∆T + 3
( α∆T 2 ) + ( α∆T 3 )]
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∆V = [V 3 α∆T + 3
(α∆T )2
+ (α∆T )3 ]
∆A= V 3α∆T
3 α =
∆V
V ∆T
For a flat plate
∆A= 2 A∆T
2 α δ
∆A= δ A∆T
Electrical conductivity
Electrical conductivity is the ability of different types of matter to conduct an
electric current. The electrical conductivity of a material is defined as the ratio of
the current per unit cross-sectional area to the electric field producing the current.
Electrical conductivity is an intrinsic property of a substance, dependent on the
temperature and chemical composition, but not on the amount or shape.
Electrical conductivity is the inverse quantity to electrical resistivity. For any
object conducting electricity, one can define the resistance in ohms as the ratio of
the electrical potential difference applied to the object to current passing through
it in amperes. For a cylindrical sample of known length and cross-sectional area,
the resistivity is obtained by dividing the measured resistance by the length and
then multiplying by the area.
The conductivity (σ) of a material is determined by taking the reciprocal of the
measured electrical resistance (R) to the flow of electricity in a length (L) of
material divided by the cross-sectional area (A). σ =
Conductivity is temperature dependent. σ T ' =
where
1⎛ L⎞
.
R ⎜⎝ A ⎟⎠
σT
1 + α(T − T ')
σT′ is the electrical conductivity at a common temperature, T′
σT is the electrical conductivity at a measured temperature, T
σ is the temperature compensation slope of the material,
T is the measured temperature,
T′ is the common temperature
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Metals generally have very high electrical conductivity. The electrical conductivity of copper at room temperature, for instance, is over 70 million siemens per
meter. On an atomic level this high conductivity reflects the unique character
of the metallic bond in which pairs of electrons are shared not between pairs of
atoms, but among all the atoms in the metal, and are thus free to move over large
distances. Many metals undergo a transition at low temperatures to a superconducting state, in which the resistance disappears entirely and the conductivity
becomes infinite. The superconduction process involves a coupling of electron
motion with the vibration of the atomic nuclei and inner-shell electrons, to allow
net current flow without energy loss.
Electrical conductivity in the liquid state is generally due to the presence of ions.
Substances that give rise to ionic conduction when dissolved are called electrolytes. The conductivity of one molar electrolyte is of the order of 0.01 siemens per
meter, far less than that of a metal, but still very much larger than that of typical
insulators. Sodium chloride (common table salt), composed of sodium ions and
chloride ions, is a very poor conductor in the solid state. If it is dissolved in water,
however, it becomes a good ionic conductor. Likewise, if it is melted, it becomes
a good conductor. Substances such as hydrogen chloride or acetic acid are nonconductors in the pure state but give rise to ions and thus electrical conductivity
when dissolved in water. In modern electrochemistry, substances of the sodium
chloride type, which are actually composed of ions, are termed true electrolytes,
while those that require a solvent for ion formation, like hydrogen chloride, are
termed potential electrolytes.
The unit of electrical conductivity in the International System of Units (SI)
system is the siemens per meter, where the siemens is the reciprocal of the
ohm, the unit of electrical resistance, represented by the Greek capital letter
omega ( Ω ). An older name for the siemens is the mho, which, of course, is
ohm spelled backwards (which was written as an inverted Greek omega).
Semiconductors are materials which have a conductivity between conductors
(generally metals) and nonconductors or insulators (such as most ceramics). Semiconductors can be pure elements, such as sillicon or germanium, or compounds
such as gallium arsenide or cadmium selenide. In a process called doping, small
amounts of impurities are added to pure semiconductors causing large changes
in the conductivity of the material.
Metals and alloys
An alloy is a metal composed of more than one element. Engineering alloys
include the cast-irons and steels, aluminum alloys, magnesium alloys, titanium
alloys, nickel alloys, zinc alloys and copper alloys. For example, brass is an alloy
of copper and zinc. This versatile construction material has several characteristics,
or properties, that we consider metallic:
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(1) It is strong and can be readily formed into practical shapes.
(2) Its extensive, permanent deformability, or ductility, is an important asset in
permitting small amounts of yielding to sudden and severe loads. Many Californians have been able to observe moderate earthquake activity that leaves
windows (of relatively brittle glass) cracked while steel support framing still
functions normally.
(3) A freshly cut steel surface has a characteristic metallic luster, and
(4) a steel bar shares a fundamental characteristic with other metals: it is a good
conductor of electrical current. Although structural steel is a special common
example of metals for engineering, a little thought produces numerous others
[such as gold, platinum, lead and tin].
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Learning Activities
Task 3.1. The mean distance travelled by a molecule at any time t.
Calculate the mean distance travelled by a molecule at any time t
N0
( )
n x,t =
∞
(
A π Dt
2
use e−α x dx =
∫
0
−
)
1
e
x
2
4 Dt
diffusion equation
2
1 π
2 α
solution
x=
2
π
1
(Dt) 2
Task 3.2: Derive the surface and volume expansion coefficients
a) For volume expasion show that
∆V =
βV ∆T
β = 3α
b) For a flat plate show that
∆A= δ A∆T
δ = 2α
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Task 3.1 Problem
1. Consider a composite structure shown on below. Conductivities of
the layer are: k1 = k3 = 10 W/mK, k3 = 16 W/mK, and k4 = 46 W/mK. The
convection coefficient on the right side of the composite is 30 W/m 2K.
Calculate the total resistance and the heat flow through the composite.
2. An aluminum tube is 3m long at 200C. What is its length at 1000C.
3. A metal rod made of some alloy is to be used as a thermometer. At 00C its
length is 40cm, and at 1000C its length is 40.06cm.
a. What is the linear expansion coefficient of the alloy?
b. What is the temperature when its length is 39.975cm?
4. At 200C, an aluminum ring has an inner diameter of 5cm, and a brass rod has
a diameter of 5.05cm.
a. To what temperature must the aluminum ring be heated so that it will just slip
over the brass rod?
b. To what temperature must both be heated so the aluminum ring will slip off
the brass rod? Would this work?
∆V
5. Calculate the fractional change in the volume (
) of an aluminum bar that
V
0
undergoes a change in temperature of 30 C
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Solution
1. First, draw the thermal circuit for the composite. The circuit must span between
the two known temperatures; that is, T1 and T∞.
Next, the thermal resistances corresponding to each layer are calculated:
Similarly, R2 = 0.09, R3 = 0.15, and R4 = 0.36
To find the total resistance, an equivalent resistance for layers 1, 2, and 3 is found
first. These three layers are combined in series:
The equivalent resistor R1,2,3 is in parallel with R4:
Finally, R1,2,3,4 is in series with R5. The total resistance of the circuit is:
Total thermal resistance Rtotal = R1,2,3,4 + R5 = 0.46
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The heat transfer through the composite is:
= 173.9 W. ← heat flow through the composite
Formative Evaluation 3
1.What is the properties of semiconductor
a) it is an in sulators b) it is con ductors C it is material which has a conductivity between conductors (generally metals) and nonconductors or insulators
2. The hollw cylinder as shown in the figure has the length L and inner and
outer radii a and b. It is made of a material with resistivity ρ . A potential
difference is set up between the inner and outer surface of the cylinder so
that current flow radially through the cylinder. What is yhe resistance to this
radial current flow
b
a
Solution
dR =
R=
R=
ρdr
2πrL
ρ
2π L
b
∫
a
dr
r
ρ
b
ln
2π L a
3. Derive the diffusion equation in 1D
4. State the properties of solid ,liquid and gas
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XV. Synthesis Of The Module
Electricity and magnetism I
needs your expertise.
Expected Solutions To Some Problems Set
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XVI. Summative Evaluation
Summative evaluation
1. Determine Youn’s modulus, Bulk modulus and Poisson’s ratio and derive
a relation between them
2. Asteel wire 2mm in diameter is just stretched between two fixed points at
a temperature of 200C. Determine its tension when the temperature falls
to 100C. (coefficient of linear expansion of steel is 0.000011 and Young’s
modulus for steel is 2.1x1012dynes per sq.cm)
Solution
let the length of wire be Lcm
then, on a fall in temperature, from 200C to 100C, its length will decrease by an
amount
∆L = Lα∆T
= L90.000011)(10)
∆L = (L )11x10−5
-the strain produced in it =
∆L
L
=(L)(11x10-5)/L
=11x10-5
-strees =T/ π r2
=T/ π (0.1)2
Young’s moduolus (Y) =
stress
strain
=7.3x106dyne.
3. Define stress, strain and Young’s modulus.
4. A copper wire 3 meters long of Young’s modulus 2.5x1011dyne/cm2 has a
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diameter of 1mm. If a weight of 10kg is attached to one end what extension is
produced? If poisson’s ratio is 0.26, what lateral compression is produced?
Solution
Original length of the wire (L)=3m
Young’s modulus for the wire (Y)=12.5x1011dynes/cm2
Radius of the wire (r)= ½ mm
Its area of cross section= π r2
Force applied (F)= 10kgmwt.= 981x104 dyne
From the relation
Y=
F .L
, then
a.l
l=
Poisson’s ratio, δ =
0.26=
F .L
= 0.2997cm
a.Y
lateralstrain
longitudinalstrain
lateralstrain
l
L
Lateral strain = 0.26x l
L
= 2.6x10-4
This, therefore, gives the value of lateral strain, i.e, d/D, where d is the decrease
in diameter
(d/D) = 2.6x10-4
d = D(2.6x10-4) = 2.6x10-5cm is lateral compression
5. Establish an expression for the workdone in streching a wire through 1cm,
assuming Hooke’s law to hold. Find the work done in joules in stretching a
wire of cross-section 1sq.mm and length 2meters through 0.1mm, if young’s
modulus for the materials of the wire is 2x1012dynes/cm2
Solution
Work done =(1/2) stretching x the stretch
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= (1/2) F. l
= ½ .(Y.a)/L . l
=5x10-4 joule
6. Show that the bulk modulus k, Young’s moduous E and the Poisson’s ratio
δ are connected together by the relation k =
E
3 1 − 2δ
(
)
Solution
We have k =
1
then
3 α − 2β
(
Therefore k =
)
Y
3 1 − 2δ
(
1
⎛
β⎞
3α ⎜ 1 − 2 ⎟
α⎠
⎝
where E =
1
β
and δ =
α
α
)
7. Show that the rigidity n, and young’s modulus E are connected by te relation
n=
1
where δ is the poisson ratio
2 1+ δ
(
)
Solution
we have n =
n=
But Y =
1
α
1
2 α+β
(
)
1
⎛
β⎞
2α ⎜ 1 + ⎟
⎝ α⎠
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δ=
β
α
Therefore n =
1
2 1+ δ
(
)
8. Water flows along a horizontal pipe, whose cross- section is not costant.
The pressure is in cm/sec. Find the pressure at a point where the velocity is
65cm/s.
Solution
p1=1cm=1 x 13.6 x 981 dynes/cm2
V1= 35cm/s, V2 = 65 cm/s, ρ = 1 gm/cm3
P2 =?
Appling Bernoullis relation
P1 – P2 =
=
1
1
ρV1 2 − ρV 2 2
2
2
1
ρ V1 2 − V 2 2
2
(
)
P2= 0.89cm of mercury
9. Define the coefficient of viscosity. Give examples of some viscous substances.
How would you determine the coefficient of a liquid?
10.State
a) the law of fluid pressure
b) The principle of Archimedes
11. A string supports a solid iron object of mass 180gm totally immersed in a
liquid of density 800kg m-3. Calculate the tension in the string if the
density of iron is 8000kg/m3
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Solution
The tension in the string = weight of an object in the air – the weight of liquid
displaced
T= Mg-mg
where m=(.18/8000) x 800 =18gm
=(0.18 x 10 - 0.018 x 10 )
=(1.8 - .18 )
=1.62N
12. At 200C, an aluminum ring has an inner diameter of 5cm, and a brass rod has
a diameter of 5.05cm.
a) To what temperature must the aluminum ring be heated so that it will just
slip over the brass rod?
To what temperature must both be heated so the aluminum ring will slip off the
brass rod? Would this work?
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XVII. References
Finn, C. B.P (1993). Thermal Physics , Chapman & Hall, London.
Raymond A. Serway (1992). PHYSICS for Scientists & Engineers. Updated
Version.
Kleppner & Kolenkow An introduction to mechanics.
Douglas D. C. Giancoli Physics for Scientists and Engineers. Vol. 2. Prentice
Hall.
Sears, Zemansky and Young, College Physics, 5th ed.
Sena L.A. (1988) Collection of Questions and Problems in Physics, Mir Publishers Moscow.
Nelkon & Parker (1995), Advanced Level Physics, 7th ed, CBS Publishers
& Ditributer, 11, Daryaganji New Delhi (110002) India. ISBN 81-2390400-2.
Godman A, and Payne E.M.F, (1981) Longman Dictionary of Scientific Usage.
Second Impression, ISBN 0 582 52587 X, Commonwealth Printing press
Ltd, Hong Kong.
Siegel R. and Howell J. R., (1992) Thermal Radiation Heat Transfer, 3rd ed.,
Hemisphere Publishing Corp., Washington, DC.
Kittel C. and Kroemer H., (1980) Thermal Physics, 2nd ed., W. H. Freeman
and Co., San Francisco, CA.
Zemansky M. W. and Dittman R. H., (1981) Heat and Thermodynamics, 6th
ed., McGraw Hill Book Co..
Halliday D., Resnick R., and Walker J. (1997), Fundamentals of Physics, 5th
ed., John Wiley and Sons
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XIX. Main Author of the Module
About the author of this module:
Sisay Shewamare
Department of physics, Jimma University,
Ethiopia, East Africa.
P.O.Box (personal), (Institutional)
E-mail : sisayshewa20@yahoo.com
Tel: +251-91-7804396
Brief Biography: My name is Sisay Shewamare I am living in Ethiopia I am
working in Jimma university department of physics. You are always welcome to
communicate with the author regarding any question, opinion, suggestions, etc
about this module.
Thank you