Mechanics 1
By Adolphe Ratiarison
African Virtual university Université Virtuelle Africaine Universidade Virtual Africana
Mechanics 1
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I. Mechanics 1 ______________________________________________ 3 II. Prerequisite Course or Knowledge _____________________________ 3 III. Time ____________________________________________________ 3 IV. Materials _________________________________________________ 3 V. Module Rationale __________________________________________ 4 VI. Content __________________________________________________ 4 6.1 Overview ___________________________________________ 4 6.2 Graphic Organizer _____________________________________ 6 VII. General Objectives _________________________________________ 8 VIII. Specific Learning Objective(s) _________________________________ 9 IX. Learning Activities _________________________________________ 11 X. Key Concepts ____________________________________________ 16 XI. Required Readings ________________________________________ 17 XII. Essentiel Resources _______________________________________ 25 XIII. Useful links ______________________________________________ 35 XIV. Learning Activities _________________________________________ 49 XV. Synthesis of the Module ___________________________________ 218 XVI. Summative Evaluation _____________________________________ 220 XVII. References _____________________________________________ 227 XVIII. Grades and results of student’s evaluation ____________________ 230 XIX. Main Author of the Module ________________________________ 231 XX. File Structure ___________________________________________ 232
Table of conTenTs
I. Mechanics 1
Pr Adolphe RATIARISON
II. Prerequisites / Previous knowledge needed
To follow this module, the students should possess knowledge of the following concepts:
• Space, time, translation, rotation, reference system, change of coordinates
• cartesian coordinates
• cylindrical coodinates
• spherical coordinates
• polar coordinates
• vector field paths
• velocity fields
• differentiation
• integration
III. Times
120 hours
IV. Materials
• Internet access
• Computer with CDROM
• Adequate software (MS office)
• Television
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V. Module Rationale
This module is part of a training program for teachers. It helps to lay the basic skills acquired in the secondary school teaching system.
It addresses the movement of objects that are crucial in the physical universe.
The description of these movements is the essential work of physicists in the development of science since Galileo and Aristotle.
This module helps the learner to better understand the laws governing move- ment.
VI. content
6.1 Brief summary
This module of Mechanics 1 addresses aspects experienced in daily life and in our environment including:
• Physical quantities and vector operators;
• Kinematics of a material point in one dimension and two dimensions:
- Research of parametric equations and trajectories of a moving object - Calculation of velocity and acceleration vectors in different coordinate systems
- The composition law of velocities and accelerations
• Static solids (forces acting on a system)
• The dynamics of material points using Newton’s laws
• The concepts of Work, Energy, Power, Mechanical theorem of kinetic energy and the conservation of mechanical energy.
This module comprises of 4 units : Unit 1 : (15 hours).
The measurable quantities of physics.
• classification and measurement, and errors in measurement,
• vectors,
• scalars,
• vector operations.
Unit 2 : (30 hours).
Kinematics of a material point :
• One-dimensional movement,
• Two and three-dimensional movement Unit 3 : (30 hours).
• Statics of solids Unit 4 : (45 hours).
• Composition law of movement
• Dynamics of a material point
• Work, energy, and mechanical power
• Oscillators
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6.2 Graphic organizer
5
Mechanics 1 Module
General physics
Scalars and Vectors
Vector Operations
Kinematics of a material point
Straight movement
Curvilinear movement
Coordinate systems
Uniform movement
Varied uniform movement
Sinusoidal movement
Circular movement
Cycloidal movement
Helical movement
Cylindrical coordinates
Spherical coordinates
Solid equilibrium
Dynamics of a material point
Torques Center of gravity
Equilibrium conditions
Equilibrium stability
Galilean reference
Non-Galilean reference
Composition of movements
Newton’s 3 laws
Kinetic moment
Work-Energy-Power
Oscillators
6
GeneralitiesOf kinematics
Parametric equations
Components of speed
Components of acceleration
Intrinsic components of acceleration Trajectory equation
Components of velocity and acceleration in different coordinate systems
x= x(t); y=y(t) f(x,y)=Cte
Work Energy
Power
Work of a constant force
Kinetic energy theorem Work of conservative forces
Work of non-conservative forces
Mechanical energy theorem G énéralités sur la
cinématique du point
Equatio ns para métr iques
Composante s de la vit esse
Composante s de l’ accélération
Composante s intr ins èques de l’ accélération Equatio n de la traj ectoire
Composante s de la vit esse et de
l’ac céléra tion dan s diffé r ents syst èmes de coordonn ées
x= x(t); y=y(t) f(x,y)=Cte
) t ( y );
t (
x• •
) t ( y ), t (
x• ••
•
R
;v dt
dv 2
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VII. General objectives
The student should be able to:
1. learn about
- Physical quantities,
- Concepts of energy and work - Relation between energy and work 2. learn and apply the following concepts :
- One-dimensional movement
- Two and three-dimensional movement - Newton’s 3 laws
- Kinetic energy
- Working in a group to find solutions to an exercise
3. Students should be able to set, carry out experiments and analyse expe- rimental data to establish relations between physical quantities
VIII. specific objectives related to learning activities
Units Specific objectives
Unit 1: (15 H)
Measurable quantities of physics.
- Their classification and nature, - Sources of error in measurement, - Vectors,
- Scalars, - Vector operators
Specific learning objectives - Define a physical quantity - Give examples of physical quantities - Give units of physical quantities
Specific objectives of theoretical knowledge - Distinguish a vector quantity from a scalar quantity
Unit 2 : (30 H)
Kinematics of a material point : - One-dimensional movement, - Two or three-dimensional movement
Learning objectives- Define the trajectory of a moving object - Define average velocity
- Recall the components of acceleration vectors in a (o, x, y, z) coordinate system
Specific objectives of theoretical knowledge - Write the parametric equations of movement.
- Calculate average velocity of a moving object.
- Calculate the instantaneous velocity of a moving object.
- Calculate average acceleration of a moving object.
- Calculate the instantaneous acceleration of a moving object
- Integrate the instantaneous velocity - Integrate the instantaneous acceleration - Trace the trajectory of a moving object - Caclulate the intrinsic (local) components of the acceleration
Specific objectives of group work - Accomplish an exercise with group members
Unit 3 : (30 hours) Statics of solids
Specific objectives of theoretical knowledge - Document forces acting on a body
- Determine equilibrium conditions of a solid : - in rectangular translation
- in rotation
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Unit 4 : (45 hours)
- The law of movement composition- - Dynamics of material points – - Work, energy, and mechanical power – - Oscillators
Specific learning objectives - Newton’s 3 laws
- Apply Newton’s laws to solve problems Specific learning objectives - Apply the kinetic energy theory
Specific objectives of theoretical knowledge - Calculate the work done by a constant force - Calculate the work done by a varying force - Calculate gravitational potential energy - Calculate the kinetic energy of a moving object - Calculate the mechanical energy of a system - Apply the kinetic energy theorem
- Apply the conservation of mechanical energy theorem to a system
Specific learning objective (Optional educational material)
Be able to use two forms of evaluation Note when evaluations will take place
IX. Teaching and learning activities
9. Preliminary evaluation
Title of preliminary evaluation: MECHANICS TEST 1
Justification : This test aims to evaluate knowledge that will be essential to understand this module.
QUESTIONS
1. Movement means the ………of a moving object from one place to another.
2. We call the trajectory of a moving object the set of points that a moving object follows when the time t…………..
3. A vector is a straight line : a. True
b. False
4. Velocity is a scalar : a. True
b. False
5. Accelerating a car causes the scalar product of its velocity and its acceleration to be positive:
a. True b. False
6. We apply a force to :
a. produce movement of an object b. modify the movement of an object c. deform an object
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Check the correct answer(s)
7. You are in a car approaching a curve with velocity v. When the car 7-1. turns to the left, your body is thrown to the left :
a. true b. false
7-2. turns to the right, your body is thrown to the left : a. true
b. false
8. When a driver of a car moving with a speed ,v, on a level road, brakes sud- denly,
a. passenger in the car is thrown to a. the front b. the back
c. the left d. the right
Check the correct answer(s).
9. Two vectors which are equal in intensity have between them an angle a. R de- notes the resultant of two vectors and U the common module of these two vectors.
Associate the correct answers by matching the letter and number a. a = 90 ° 1. R=0
b. a = 0 ° 2. R=U
2
c. a = 180 ° 3. R=2U d. a = 45° 4. R=U3
10. A brick of mass M placed on a smooth table is : a. subject only to the action of its own weight b. not exposed to any force
c. exposed only to the reaction of the table
d. exposed to its own weight and the reaction of the table
Choose the correct answer.
11. The product of force and distance is : a. a vector
b. a scalar
12. Kinetic energy and work have the same units a. True
b. False
13. Power is the integral of energy a. True
b. False
14. Gravity is a force a. True
b. False 15. Mass is a scalar
a. True b. False Correct Answers
1. Movement means the displacement of a moving object object from one place to another. Movement is essentially related to displacement.
2. We call the trajectory of a moving object the set of points that a moving ob- ject follows when the time, t varies. Good answer, in fact displacement is related to time.
3.
a. Read the question carefully before answering
b. Very good. A vector is always oriented and has a measure which is not the case of a ray.
4.
a. We are talking about a velocity vector, therefore it cannot be a scalar.
b. Very good, velocity is a vector.
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5. True. For an accelerated movement, the scalar product of the acceleration and the velocity is positive
6.
a. Correct. A force can displace objects.
b. Correct. A force can also deviate the movement of an object.
c. Correct. In fact, if we wanted to break an object, we could apply a force to it.
7.
7-1.
a. Watch out, think carefully.
b. Correct. You understand that we are ultimately thrown in the opposite direc- tion.
7-2.
a. Correct. We are thrown in the opposite direction.
8.
a. Correct. We are thrown in the opposite direction.
b. Incorrect c. Incorrect d. Incorrect
9. a2 ; b3 ; c1. are the correct answers, and you know how to calculate the result of two forces.
For all other answers (a1, a3, b2, b1, c2, c3, d1, d2, d3, d4, a4, b4, c4,) : Think before answering.
10.
a. Incorrect.
b. Think about it, it is impossible
c. A solid in equilibrium is subjected to at least 2 forces d. Correct. You are recalling action and reaction.
11.
a. Watch out. Work is a scalar product of two vectors, but is not itself a vector.
b. Correct. The work of a force is a scalar.
12.
a. Correct. Work is a form of energy.
b. Incorrect.
13
a. Watch out. Energy and power do not have the same units.
b. Correct. The integration of energy cannot provide power.
14.
a. Correct. Gravity is indeed a force.
b. Incorrect. Think of the units.
15.
a. Correct. Unlike gravity, mass is a scalar.
b. Reread your lesson.
Comments of students following Evaluation test in mechanics 1 (100-200 words)
You took the pre-evaluation test.
• If you have correctly answered all question, you have a grade of A +. You will not have difficulty following this module.
• If you answered 75% of the test correctly, you have an A grade. You will have no difficulty following this module.
• If you answered 60% of the test correctly, you have a B grade. You could very well succeed in this module, but you put in some extra effort.
• If you answered between 45% and 50% of the test correctly, you have a C grade. You must take some supplemental courses.
• If you answered under 45% of the test correct, you have the note D. You should strive to learn everything, while still following the module as it is taught.
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X. Key concepts (glossary)
1. Acceleration: Change in velocity per unit of time
2. Freefall: Movement of an object subject only to its own weight (all resistant forces are neglected)
3. Kinetic Energy: Energy possessed by a body in motion 4. Potential energy: Energy stored by a body due to its position
5. Force: any means capable of producing a motion, to modify or deform an object.
6. Movement: Movement of a moving object object from one point to an- other
7. Mechanical power: work of a force per unit time 8. Referential: Object of reference
9. Mechanical work: Energy provided by a force when its point of applica- tion moves.
10. Speed: Change in the position of a moving object object per unit of time
XI. Required readings
UNIT 1 :
Measurable physical quantities.
- Their classification and measure,
- Different sources of error of measurement, - Vector quantities,
- Scalar quantities, - Vector operators
There are four required readings for Unit 1. They are grouped in Appendix 1.
Reading #1
Complete references :
RATIARISON Adolphe (2006). Grandeurs physiques – Mesures-Incertitudes- opérations vectorielles. Madagascar. Université d’Antanarivo.
The first two parts of this document are drawn from the following sites:
http://www.bipm.fr/fr/si/si_brochure/chapter1/1-2.html http://www.cegep-ste-foy.qc.ca/freesite/index.php?id=3113 http://www.ulb.ac.be/cours/psycho/content/cognum/calcul.html
Summary : The value of a physical quantity is usually expressed as the prod- uct of a number by a unit. For a particular quantity, we can use many different units. Among these units, we distinguish those of the International System (SI) based on seven base quantities.
The measurement of a physical quantity can be done directly, such as the length with the meter, the voltage with a voltmeter, or indirectly such as a sur- face area obtained by the product of the length by width.
Finally, the various operations on vectors are detailed.
Justification:
- Any physicist must know the units of measure because we cannot add two different sizes without expressing in the same unit.
- The vector addition is not only part of the composition of forces, but it is also vitally important in the composition of movement that we will see later.
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Reading #2
Complete references :
http://tanopah.jo.free.fr/seconde/Vct2.html Addition, opposing, and subtraction of vectors.
Summary : This course and nearly all elements and programming within, were designed and made by Jerome ONILLON. It is listed by the Irish tavern.
Addition and subtraction of two vectors are well detailed. It highlights the properties of vector addition as: commutativity, associativity, existence of neu- tral elements without forgetting the Chasles relationship
Justification: This completes the reading #1 The parallelogram rule used for addition and subtraction of vectors is well explained.
Reading #3
Complete references :
http://formation.etud.u-psud.fr/pcsm/physique/outils_nancy/apprendre/ chapitre2/
partie2/Title1res.htm Vectors. Vector addition.
Summary : Vector addition is an internal composition law and has the fol- lowing properties :
• Associativity
• Commutativity
• Neutral element
• Symmetric element
Hence, we can talk about subtracting a vector from another, and the Chasles relationship.
The multplication of a vector by a scalar is an external composition law.and has the following properties:
•
Distributivity with respect to vector addition :
•
Distributivity with respect to scalar addition :
•
Associativity :
•
Neutral element :
These properties are followed by :
• The determination of the position of a point M on a segment AB,
• The linear combination of two vectors.
Justification : Starting from the linear combination of several vectors, we can define the centroid of several points affected by weights ai
Reading #4
Complete references :
http://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_vectorielle Vector geometry.
Summary : We develop addition and subtraction of vectors, and multipying a vector by a scalar. Properties of a scalar product, vector product, mixed pro- duct and double vector product.
Justification : Allows readers to gain an in-depth understanding of vector operations.
Unité 2 : Kinematics of a material point 1D, 2D, and 3D movement
There are 3 required readings in Unit 2. They are grouped in Appendix 2.
Reading #5
Complete references :
RATIARISON, A. (2006). Cinématique du point. Mouvement à 1D, 2D ou 3D. Madagascar. Université d’Antananarivo. Cours inédit
Summary : The generality of the kinematic point concerns the definition of referentials, tracking a moving object in space, the curvilinear abscissa, the velocity vectors and acceleration vectors.
This manual will then examine the rectilinear uniform motion and uniformly varied motion.
When considering curvilinear movement, we emphasize the intrinsic compo- nents of acceleration, circular motion, cycloidal and spiral motion.
Finally, the different coordinate systems and components of velocity and ac- celeration vectors in these coordinate systems are considered.
Justification: Before we study the dynamics of a material point, we must have the kinematics of the point. To do this we need to know the topics listed above.
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Reading #6
Complete references :
http://abcsite.free.fr/physique/meca/me_ch3.html Kinematics of a point
Summary : This reading completes the previous calculations of components of speed and acceleration vectors in different coordinate systems. Polar coor- dinates and semi-polar coordinates are also still taught.
In this reading we encounter the so-called hodograph.
The different diagrams are clearly legible.
Justification: This course is easy to read, and can offer significant help to students.
Reading #7
Complete references :
http://www.chez.com/mecasite/Mecanique/cinematsol.htm Kinematics of a point.
Summary : This reading reinforces our knowledge of movement, the average velocity, average acceleration, instantaneous velocity and instantaneous ac- celeration. The rotational motion and uniform circular motion varied uniformly are also highly developed.
Justification: In addition to the two previous readings, it completes the course of the kinematics of the point.
UNIT 3 : Equilibrium of a solid on a horizontal plane
In Unit 3, there are 3 required readings, which are grouped in Appendix 3.
Reading #8
Complete reference :
RATIARISON, A. (2006). Equilibre d’un solide sur un plan – Faculté des Sciences- Université d’Antananarivo –MADAGASCAR, Cours inédit Summary : This reading is mainly concerned with the equilibrium of a solid on a plane. A solid can slide or rotate on a plane if it is not in equilibrium. To introduce the equilibrium of a solid, we speak of torque, which is a system of free vectors. This system of free vectors is reduced to the resultant forces and resultant moment of all forces applied to the system considered. The equili- brium condition is defined by a torque of zero, meaning a general zero result and zero moment.
Justification: In the module only zero resultant forces have been defined, but to broaden the knowledge of students we must also mention the zero moment of the forces applied to the system in question.
Unit 3 :
Reading #9
Complete reference :
Statique du solide taken from « http://fr.wikipedia.org/wiki/Statique_du_solide » A Wikipedia article.
Solid Statics
Summary : The possible movements, sometimes called degrees of freedom are of two kinds: translations (3 main directions) and rotation (around the three directions). While the translations may not be caused by forces, rotations are generated by moments of these forces, or other pairs of force. When the equi- librium point requires that the establishment of 3 algebraic relations (equation of vector forces in 3 dimensions), while that of the solid demands the consid- eration of 3 additional equations (moments vector equation). The fundamental principle of statics can then be considered:
1. the theorem of the resultant (sum of forces is zero).
2. the theorem of the moment (sum of moments is zero).
Justification: The study of equilibrium of a solid always requires the con- sideration of these 2 theorems, even if in some simple cases of mechanics of a point, they seem to be resolved with one of the 2 parts. Generally, it is not possible to treat the two aspects separately (forces and moments): it is actually a complex 6-dimensional problem.
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Unit 3 :
Reading #10
Complete reference :
http://www.ac-poitiers.fr/cmrp/cpge/docs/Coursdemodelisationetdestatique.doc Solid statics
Summary : The mechanical action is anything likely to maintain a body at rest, create or modify a motion to deform a body, and manifests itself in two forms:
- The translational motion due to the resultant forces applied to the solid - The rotation due to the resultant moment of these forces
Before stating the Fundamental Principle of Statics (FPS), the author speaks of the modeling contact activities:
- Contact with a fluid on a solid, - The contact of two solids.
Justification: One of the characteristics of this reading is the mechanical ac- tions applied to a balance:
- The mechanical action at a distance (gravity, electromagnetic, electrostatic, ...)
- The mechanical action of contact (pressure, contact, ...) This reading is very beneficial for students.
Unit 4 :
Composition of movements Dynamics of material points-
Work, energy, and mechanical power – Oscillators
Reading #11
Complete reference :
RATIARISON, A. (2006). Composition de mouvement, Dynamique du point matériel, Travail – Energie - Puissance, Oscillateurs– Faculté des Sciences- Université d’Antananarivo –MADAGASCAR, Cours inédit
Topics on dynamics and oscillators were taken from : http://abcsite.free.fr/index.html
Summary : This unit begins by addressing the law of movement composition and Newton’s 3 laws, with their practical applications.
It continues by discussing evidence on Coriolis inertial forces.
It highlights the definition and calculation of work produced by the conserva- tive forces and that produced by non-conservative forces.
It establishes the theorem of kinetic energy and the theorem of mechanical energy.
It concludes with the study of damped harmonic oscillators.
Justification: To get a general idea of absolute motion and relative motion, the course begins with the generalization of different velocities and accelerations of the three Newton laws, and theorems of mechanical and kinetic energy on the basis of the dynamic point.
Reading #12
Complete reference :
Papanicola Robert, http://www.sciences-indus-cpge.apinc.org/IMG/pdf/ CIN2_
DERIVATION_VECTORIELLE.pdf Vectorial derivation.
Summary : This course of vector derivation leads to the composition law of motion. It therefore complements the course of Ratiarison Adolphe. For the concept of the composition of three rotations, the author brings the three Euler angles, namely precession, nutation and proper rotation.
Justification:The three Euler angles are not on the agenda because in prac- tice this concerns the kinematics of sound. It is therefore not essentially that a student spends considerable time on it. The composition of rotations is well developed in the course of Ratiarison.
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Reading #13
Complete reference :
http://abcsite.free.fr/physique/meca/me_ch3.html Dynamics of a material point
Work, energy, and power Oscillators
Summary : A complete course on the dynamics of a material point. It follows the site : http://abcsite.free.fr/physique/meca/me_ch3.html that was already cited in the kinematics portion.
Justification : A course that is easy to read.
Reading #14
DIOUF, S. (2004). L’Evaluation des apprentissages. Sénégal. Université Cheikh Anta DIOP de Dakar. FASTEF (ex ENS)
Summary: This text is recommended to respond to an optional formal evalua- tion of educational nature. It contains different parts including:
Evaluation that deals with various issues relating to the assessment
The different forms of evaluation where it is also about the roles and moments of assessment
Strategies for collecting information. In this section you will find ways to cor- rect the issues relating to objective and subjective correction.
It also includes the steps of building a subject of examination and the charac- teristics of the evaluation.
Justification: Reading this text enables students to answer questions correctly during a formative assessment of educational nature. All answers to the assess- ments are contained in this text.
XII. essential resources
Resource #1
Complete reference :
RATIARISON, A. (2006), Cours de mécanique générale 1 Faculté des Sciences -Université d’Antananarivo - Madagascar
Summary : This mechanics course is taught in the first year of university in the Faculté des Sciences, at the Ecole Normale and l’Ecole Supérieure Poly- technique at l’Université d’Antananarivo.
It discusses the vector operators, and the kinematic point in a Galilean refer- ence. The laws of composition of movements are dealt with in the kinematics portion. Central acceleration movements are also developed.
Justification: This is useful for practical exercises from a distance.
Resource #2
Complete reference:
PEREZ, J. P. (1997). Mécanique – Fondements et applications. Université Paul Sabatier Toulouse – France. Edition MASSON, 120 bd St Germain 75 280 Paris Cedex 06
Summary : It is a comprehensive manual for students from first to third year.
The mechanics of a material point, the dynamics of solids, the movement of central acceleration, oscillators, analytical mechanics, and fluid mechanics are treated.
Justification: Students will always need this book during their studies, be- cause apart from the topics of kinematics and mechanics of a solid, there are also fluid mechanics topics and several applications that are treated..
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Resource #3
Complete reference :
http ://www.hazelwood.k12.mo.us/grichert/sciweb/applets.htlm
Summary : This site is a compilation of links leading to other internet sites containing physics-based simulations.
Justification : Students will always have a need for this manual during their studies, as many physics topics are developed.
Resource #4
Complete reference:
CAZIN, M. (1995), Cours de mécanique générale et industrielle– Gautier Villars – tome 1, NY 1003 -1995
Summary : A complete course, but slightly difficult to read. However, it contains many applications and is very useful in further studies.
Justification : The student will find many applications to mechanics.
Resource #5
Complete reference:
The Free High School Science Texts: A Textbook for High - School Students Studying Physics.- FHSST Authors1 - December 9, 2005 -http://savannah.
nongnu.org/projects/fhsst
Summary : Many physics topics are in this manual, including mechanics, electricity, optics, and electromagnetism.
The student can learn all about physics by consulting this site.
Justification: Students will always find a use for this book, since the physics topics are very well developed.
Resource #6
Complete reference:
http://www.google.ca/search?client=firefox_a&rls=org.mozilla%3Aen-US%Aofficial-
&hl=en&q=c3%A9equilibre+d%27un+solide+sur+plan&meta
Summary : This site contains practical exercises of equilibrium of a solid on a place, subject to many forces.
Justification: Students can do practical exercises on the site.
Resource #7
Complete reference:
http://www.chimix.com/an5/prem5/hotp5/force1htlm
Summary : This site contains practical exercises of equilibrium of a solid on a place, subject to many forces. Visualisation of free fall and apparent motion.
Justification: Students can do practical exercises on the site.
Resource #8
Complete reference:
http://fr.wikipedia.org/wiki/statique-du-solide
Summary : This page summarizes static equilibrium of a solid.
Justification : Students can do practical exercises on the site.
Resource #9
Complete reference:
http://formation.edu-psud.fr/pcsm/physique/outils_nancy/apprendre/Chapitre2/Title- 1res.htm
Summary : Developed in this site:
- the vector sum and its properties (associativity, commutativity, identity element, element symmetrical), the difference of two vectors, the rela- tionship Chasles
- the product of a vector by a scalar and its properties (distributivity over vector addition, distributivity with respect to the addition of scalar, asso- ciativity, and the existence of the neutral element).
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Applications of vector addition that are found:
- The position of a point on a line, - The linear combination of 2 vectors - Coplanar vectors
- The centroid of n weighted points
- Isobarycentre of 3 points, not aligned (center of gravity of a triangle) - Isobarycentre of 4 points in space (center of gravity of a tetrahedron) Justification: This document is a valuable resource that can complete the course and teach that the research of the center of gravity is a vector addition.
Resource #10
Complete reference:
http://formation.edu-psud.fr/pcsm/physique/outils_nancy/apprendre/Chapitre3/partie3 / Title1res.htm
Summary : This document geometrically represents complex numbers. Addi- tion and subtraction of complex numbers relates to addition and subtraction of vectors.
Justification: This site demonstrates that addition and subtraction of vectors has other applications in the field of science.
Resource #11
Complete reference:
http://msch2.microsoft.com/fr-fr/library/system.windows.forms.paddings.op_addition.
aspx
http://msch2.microsoft.com/fr-fr/library/system.windows.forms.paddings.op_methods.
aspx
http://mathexel.site.voila.fr/index.htlm
Summary : These documents execute the addition and subtraction of vectors on the computer screen.
They show different operations using the « Padding » method.
Justification: The student is, with the aid of an instructor, familiarized with vector operations.
Resource #12
Complete reference:
http://www.sciences.univ-nantes.fr/physique/perso/corticol/bibliohtml/lissajou_
j.html
http://www.sciences.univ-nantes.fr/physique/perso/corticol/bibliohtml/cycloi_j.html http://www.sciences.univ-nantes.fr/physique/perso/corticol/bibliohtml/pndhgs_
j.html
Summary: The first document sets out a Lissajous curve corresponding to the addition of two perpendicular sinusoidal movements of their phase.
A cycloid is the trajectory of a fixed point of a circle when it rolls without slip- ping on a line. The second document shows how to trace a cycloid as a func- tion of the travel speed of the circle.
The third document highlights the oscillation of a cycloidal pendulum, which is called the Huygens pendulum.
Justification: The visualization of these phenomena provides a very specific vision for the student. The equations of Lissajous curves are very complex un- less they are reduced to that of the ellipse. Thus, visualization of these curves is very helpful to students. Similarly, an abstract design and a cycloid cycloidal pendulum is difficult.
Resource #13
Complete reference :
http://electronics.free.fr/school/article.phys3?id_article=9#5
Summary : This site belongs to a young Moroccan born February 5, 1988 who is passionate about philosophy, mathematics and computing. That’s why he created this site which is a medium of exchange of knowledge and experience.
This article is a summary of the kinematics of a particle in a Galilean refer- ence. It defines:
- The position vector - The velocity vector - The acceleration vector
- The average acceleration vector - The average velocity
- Cartesian coordinates of the acceleration vector
- Coordinates of the acceleration vector in the Frenet reference It develops some specific movements:
- The rectilinear motion (uniform and uniformly varying) - The circular motion (uniform and uniformly varying)
Justification: We have already developed all of the topics, but it is interesting to see what others are doing.
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Resource #14
Complete reference:
http://www.chez.com/Mecanique/cinematipts.htm
Summary : After defining the kinematic characteristics (velocity and accelera- tion) the author speaks of:
- Uniform rectilinear movements and uniformly varying
- Movements of rotation, the normal components and tangential acceleration - Movements of uniform rotation and uniformly varying
Justification: We have already developed all of the topics, but it is interesting to see what others are doing.
Resource #15
Complete reference :
http://www.chez.com/Mecanique/dynamiqu.htm Summary : This site contains :
Fundamental principle of the dynamics of solids under rectilinear translation.
D’Alembert ;s principle ;
Rotational movement with respect to a fixed axis.
Justification: We have already developed all of the topics, but it is interesting to see what others are doing.
Resource #16
Complete reference :
http://www.chez.com/Mecanique/energeti.htm
Summary : This site summarizes the mechanical energy of a point. The work of a force, the work of a couple, the gravitational potential energy, elastic energy of a spring, kinetic energy of a solid translation, the kinetic energy of a solid rotation, the average power, the power developed by a force, and the concept of performance are shown.
Justification : This completes our course.
Resource #17
Complete reference :
http://www.sciences.univ-nantes.fr/physique/perso/gtulloue/aquadiff.html
Summary : The purpose of this site is to illustrate the solution of linear dif- ferential equations of first and second order frequently encountered in physics.
The solution itself is developed in mathematics courses and will not be detailed here. However, there is a summary of results and examples.
Illustrations and animations found here include:
- The presentation of the harmonic oscillator.
- The horizontal linear oscillator - The elastic pendulum
- The weighted pendulum - The tension of a pendulum wire - The period of the pendulum weight - The Botafumeiro
- The cycloidal pendulum
Justification : This is useful for practical exercises from a distance.
Resource # 18 Complete reference
http://www.n-vandewiele.com/TDMeca2.pdf
Summary : Seven corrected exercises concerning the composition of move- ment and changing reference states.
Justification: Since there are not many exercises concerning the compostion of movement, the student should familiarize themselves with this topic from other resources as well.
Resource # 19
Complete reference :
http://www.ens-lyon.fr/Infosciences/Climats/Dynam-atmo/Cours-Coriolis Summary : The purpose of this site is to supplement knowledge of the Coriolis force. The author tries to introduce the concepts gradually so that the Coriolis force is understandable with minimal prior knowledge.
• This site consists of seven paragraphs:
• The first shows the existence of forces of inertia and the Coriolis force.
• The second gives us the mathematical expression of the Coriolis force.
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• The third shows some manifestations of the Coriolis force.
• The fourth examines the movement of air masses moving on the ground to see how clear the Coriolis force is.
• The fifth gives us additional information.
• The sixth offers a simple experiment to demonstrate the deviation of trajectories in a rotating frame
• The seventh, a conclusion, examines a planet invented by Antoine de Saint Exupéry in “Le Petit Prince”
Justification: This site is not only dedicated to the mathematical formulation of the Coriolis force, but describes the various manifestations of the Coriolis force in everyday life. It is useful, necessary and even essential that teachers and students read this site.
Resource #20
Complete reference :
http://www.ucd.ma/fs/modules/meca1/um1./modules3/cin2.htm
http://perso.orange.fr/rmchs/physique_05/cours_physique/cours_mecach5_cinemati- que.pdf
Summary : A course on different coordinate systems and on the laws of com- position of velocities and accelerations.
Justification: We have already developed all of the topics, but it is interesting to see what others are doing.
Resource #21
Complete reference :
http://www.keepschool.com/cours-fiche-les_systèmes_oscillants Summary :
This site serves as a supplement to courses on oscillators. It talks about:
- classification of experimental oscillators (experimental properties and characteristic properties of oscillators)
- free mechanical oscillators (simple pendulum, horizontal elastic pendu- lum)
- mechanical oscillators forced (torsion pendulum in forced oscillations, resonance phenomena)
Justification:
The course of this site is given a simple way. It can help the student understand oscillators.
Resource #22
Complete reference:
http://www.logitheque.com/fiche.asp?I=18755
Summary : Static is an educational software used in physical science at the high school level dealing with the static point of solid material subjected to one or more forces (or forces couples) to show the conditions of equilibrium solids.
STATIC revolves around six themes related to statics - Equilibrium point;
- Equilibrium of a solid rotation about an axis;
- Equilibrium of a solid subjected to couples of forces;
- Equilibrium of a rod rotating about an axis;
- Static solid on an inclined plane with or without friction forces;
- Static floaters (Archimedes principle).
Justification: The learner will deepen their knowledge on the equilibrium of a solid rotation about an axis and of static floaters.
Resource #23
Complete reference :
http://www.univ-lemans.fr/enseignements/physique/01/statique.htm
Summary : A brief summary of the equilibrium and dynamics of points in a Galilean reference.
Justification : This module summarizes the different forces that can act on a point or an object.
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Resource #24
Complete reference :
http://www.univ-lemans.fr/enseignements/physique/02/meca/couplage3.html Summary : We have here a simulation of a coupling of 3 mechanic oscilla- tors, neglecting friction. The independent pulsations of the oscillators are w12
= K/M1, w22 = K/M2,. w32 = K/M3.
Each mass is subjected to the restoring force of 2 springs attached to it.
The movement equations are :
If the 3 masses are equal, the solution to the system is :
To treat all cases in the program, this system of coupled differential equations is solved numerically using the Runge-Kutta order of 4.
By assumption, the initial velocity of the two masses is always zero.
One can see that for any initial conditions the solution is usually a complex aspect. It is a linear combination of the three proper modes.
It is of the form: Xi = Ai.cos (wpt) + Bi.cos (wqt) + Ci.cos (wrt) (i = 1, 2, 3) The values of the constants Ai, Bi and Ci are a function of initial conditions.
Justification: The simulation work helps students to understand physical phe- nomena.
XIII. Useful links
Useful link #1
Title : Parabolic trajectory
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/tir_para_
j.html
Screen capture :
Justification :
We can see the envelope trajectory (parabola of safety) if we vary the direction and intensity of the initial velocity. The learner can calculate the equation of the parabola of safety.
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Useful link #2
Title :Trajectory and free fall
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/tirchu_
j.html
Screen capture :
Justification :
This animation shows the meeting point of two moving object objects. The learner can calculate the z component of the meeting point of the projectile and the other object, and the time at which they will meet.
Useful link #3
Title : Movement of the moon and a solar planet.
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/
terlun_j.html Screen capture :
Justification :
The animation shows the movement of the moon and a solar planet. The trajec- tories are presented in two different references.
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Useful link #4
Title :Longitudal oscillations of a spring
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/prlong _j.html
Screen capture :
Justification :
One can observe the sinusoidal signal propagation along the spring and the displacement of a localized point M of the spring.
Useful link #5
Title : Multiple pendulums
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/pendmu _j.html
Screen capture :
Justification :
The animation here shows the conservation of mechanical energy. You can choose the number of pendulums N released without initial velocity and com- mon initial angle.
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Useful link # 6
Title :Coupled oscillators
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/osc- cpl_j.html
Screen capture :
Justification :
We have here the coupling of three springs of respective stiffness k, k0 and K.
You can vary the stiffness k0 in the second spring and we can have the proper modes of periods T1 and T2.
Useful link #7
Title :Uniform rotational movement
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/manege_
j.html.
Screen capture :
Justification :
The student may vary the period of rotation of the carousel, the pathways of rays Ra and Rb, and calculate the centrifugal force of inertia applied to A and B.
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Useful link #8
Title :Trajectory of a grenade
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/gre- nad_j.html
Screen capture :
Justification :
In this site we find another reference that is the centroid reference. We there- fore study two different references (the fixed reference and the centroid refer- ence).
Useful link #9
Title : :Oscillation with solid friction
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/frtsol_
j.html
Screen capture
Justification :
By changing the parameters b, x0 and T0, we visualize the range of equilib- rium and location of extremum. The student can then write the equation of motion.
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Useful link #10
Title :Trajectory of the sun in the galaxy
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/galaxi_
j.html
Screen capture:
Justification :
The trajectory of the sun is not perfectly circular because it describes a circle of radius R in the plane Oxy galactic center O, but it varies more along the axis Oz, perpendicular on both sides of this plane.
Useful link #11 Title : The Big Wheel
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/gdroue_
j.html
Screen capture :
Justification:
Study of circular movement in a vertical plane.
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Useful link #12
Title :Epicycle of Ptoémée
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/epi- clc_j.html
Screen capture :
Justification :
The Earth T and another planet P demonstrate a circular motion around the sun. It shows the relative motion of P relative to T.
Useful link #13
Title :Trajectory of a dog running after its master
URL : http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/chien_
j.html
Screen capture :
Justification :
By varying the velocity of the dog and its handler, it displays the path of the dog and the student can find the equation of the trajectory based on these ve- locities.
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Useful link #14
Title :Trajectory of 4 flies
http://www.sciences.univ-nantes.fr/ physique/perso/cortial/bibliohtm/4mouche_j.html Screen capture :
Justification :
Before finding the distance traveled by each fly, the student tries to find the equation of the trajectory of each fly.
XIV. learning activities
Learning activity 1
Activity title
• Vector quantities
• Addition and subtraction of vectors
• Vector operations Specific learning objectives The learner must be able to:
• Properly represent a physical quantity
• Recall some units of physical quantities
• Add vectors
• Subtract two vectors
• Perform scalar products
• Recall the physical meaning of the scalar product
• Perform vector products
• Recall the physical meaning of vector products
• Calculate the double vector product
• Calculate the mixed product Activity summary
The main aim of this module is the dynamics of material points subjected to various forces that can be represented by vectors. Thus, this activity is to famil- iarize oneself with the common vector operators. As the movement of particles can be movements of translation and / or rotation, we can not confine ourselves to only addition and subtraction of vectors, but we must also investigate other operations of vectors, such as the vector product.
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Key concepts
General result: vector sum of several vectors
Scalar product: Operation vector giving the projection of a vector on the sup- port another to a constant.
Vector product: Operation vector whose norm is equal to the surface gener- ated by the two vectors and the vector product is directly perpendicular to both vectors.
Mixed product: Operation vector containing both the scalar product and vector product, scalar representing the volume generated by the 3 vectors.
Double vector product: Twice the vector product.
Appropriate readings (IN APPENDIX 1)
1° RATIARISON, A. (2006). Grandeurs physiques – Mesures-Incertitudes- opérations vectorielles.Madagascar. Université d’Antanarivo
2° ht tp://tanopah.jo.free.fr/seconde/Vct2.html Addition and subtraction of vectors
3° http://formation.etud.u-psud.fr/pcsm/physique/outils_nancy/apprendre/
chapitre2/partie2/Title1res.htm Vectors. Addition of vectors.
4° http://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_vectorielle Vector geometry.
Wikipedia article.
Appropriate resources
ANSERMET J.-P. (Version 2004-2005), La mécanique rationnelle – Formation de base des Sciences et des ingénieurs – Institut de Physique des nanos- tructures- Ecole Polytechnique Fédérale de Lausanne de Lausanne– PHB – Ecublens, 1015 Lausanne
TIPLER, P. A. (1995) , Physics for Scientists and Engineers – New York, NY 1003. Worth Publishers
PEREZ, J. P. (1997), Mécanique: Fondements et applications, MASSON The Free High School Sciences: A Textbook for High School Students Studing
physics – FHSSt Authors- December 9, 2005 from http://savannah.nongnu.
org/projects/fhsst
http://www.google.ca/search?client=firefox_a&rls=org.mozilla%3Aen-US%Aofficial-
&hl=en&q=c3%A9equilibre+d%27un+solide+sur+plan&meta
Useful links
http://www.infoline.ru/g23/5495/Physics/English/waves.htlm http://www.infoline.ru/g23/5495/index.htlm
http://formation.edu-psud.fr/pcsm/physique/outils_nancy/apprendre/Chapi- tre2/Title1res.htm
http://formation.edu-psud.fr/pcsm/physique/outils_nancy/apprendre/Chapi- tre3/partie3 /Title1res.htm
http://msch2.microsoft.com/fr-fr/library/system.windows.forms.paddings.
op_addition.aspx
http://msch2.microsoft.com/fr-fr/library/system.windows.forms.paddings.
op_methods.aspx
http://mathexel.site.voila.fr/index.htlm
Detailed activity description
In this activity, there are twenty independent questions on vector calculus.
They can check whether students master the various operations on vectors.
The students will discuss on an online chat to have the same understanding of the different parts of the course: Physical - Measurement-Uncertainty-vector operations. For each exercise, students will be organized in groups for collab- orative work.
These 20 questions and corrected exercises include:
- Units of physical quantities.
- The practical meaning of vector operations,
- Addition and subtraction vector treated graphically and analytically;
- The vector product;
- Double vector product;
- The combination product.
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Evaluation of Activity 1
Twenty questions and exercises
Exercise 1
What does following physical quantity signify ? : M=5,25 ± 0,02 Kg
Exercise 2
Write the following expression correctly : D= 15,83379 ± 0,173 m
Exercise 3
Consider m as a physical quantity defined by the following equality : m=m1-m2-m3
Write Dm as a function of ∆m1, ∆m2 et ∆m3
Exercise 4
What does the relative uncertainty
∆a
a
.represent.Give the precision of the measure if we have: m= 25,4 ± 0,2 Kg
Exercise 5
How do we add two vectors : - If they are parallel - If they are not parallel
Exercise 6
What is the scalar product of a vector with any unit vector?
Exercise 7
What does the modulus of the vector product of two vectors represent?
Exercise 8
When is the vector product equal to zero ?
Exercise 9
What does the mixed product of three vectors represent geometrically ?
Exercise 10
The mixed product of three vectors is invariant under permutation of these three vectors. Write the mixed product in different forms
(
V1,V2,V3)
Exercise 11. Vector division.
Consider the equality :
a ∧ x = b
(1) where a and b are 2 given vectors and x the unknown vector. Our goal is to solve this equation in the following scenarios :a) What is the solution x if a=0 and b=0 b) What is the solution x si
a ≠ 0
and b=0 c) What is the solution x if a=0 and b≠ 0d) We suppose
a ≠ 0
and b≠0 and a is not perpendicular to be) We suppose the general case:
a ≠ 0
and b≠ 0 and a perpendicular to b• We suppose that the equation in question has a particular solution x0 . We can thus writea∧ x0 =b (2). By subtracting term by term in equations (1) and (2), write the general form of the solution x of equation (1) knowing a particular solutionx0
• We are looking for a particular solution x0 of equation (2). For this we multiply vectorally the left of equation (2). What is the particularity of
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x0 to achieve it in an easier fashion. Write x0.
• Write the solution x of equation (1).
• We represent x by the vector OM . What is the complete M portion of the solution of the equation a∧OM =b.
Exercise 12
Consider 2 vectors V and 1 V , of respective modulus 5 m/s and 3 m/s , 2
that have an angle of a=30° between then. Calculate
( V +
1V
2)
2 .Find the modulus of the vector sum
(
V +1 V2)
. Trace the vector sum.Exercise 13
A boat is crossing a river at a speed of 6 km / h. The velocity of the incom- ing flow perpendicular to the boat is 3 km / h (these velocities are measured from the reference ground, say, by an observer located on a shoreline). In what direction the boat is headed?
Exercise 14
Prove that the diagonals of a parallelogram intersect in the middle.
Exercise 15
Prove that the diagonals of a rhombus are perpendicular.
Exercise 16
Determine a unit vector perpendicular to the plane formed by the vectors A = i
2 − 6
j− 3
k and B = i + j k4 3 −
Exercise 17
Show that the sine law holds in a triangle.
Exercise 18
Consider two vectors . Show that
ur A
∧ B ur
2+ A ur
.B ur
( )
2= A ur
2B ur
2Exercise 19
How is it that a person moving along the x axis with velocity U must tilt their umbrella to protect most of the rain that falls parallel to the y-axis with veloc- ity V.
Exercise 20
Consider 3 non-aligned points and a point O as an origin We let OA=a, , OB=b,. OC=c
Show that :
a r
∧ b r + b r
∧ c r + c r
∧ a r
is a perpendicular vector in the plane ABC.
Corrected exercises Evaluation of activity 1
1°) M=5,25 ± 0,02 Kg signifies that 5,23 Kg < M < 5,27 Kg
2°) A more correct way of writing it would be D= 15,8 ± 0,2 m
3°) m=m1-m2-m3 ⇒∆m = ∆m1+ ∆m2 + ∆m3
4°) The relative uncertainty
∆a
a
represents the precision of the measurement.If we have : m= 25,4 ± 0,2 Kg, the precision of the measurement of m is
∆m m = 0,2
25,4 = 0,8%
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5) If the two vectors are parallel, we draw the second vector on the line of action of the first vector.
If the two vectors are not parallel, we use the parallelogram rule.
6°) The scalar product of a vector with a given unit vector is an orthogonal pro- jection of the first vector on the unit vector.
7°) The modulus of the vector product of two vectors is the area of the paral- lelogram generated by the two vectors.
8°) The vector product is zero only if : - one of the vectors is zero
- the two vectors are parallel.
9°) The mixed product of three vectors represents the volume of the parallelepiped defined by the three vectors.
10°)
V ur
1• V uru
2∧ V uru
3( ) = V ( ur
1,V uru
2,V uru
3) = V uru
2• V ( uru
3∧ V ur
1) = V uru
3• V ( ur
1∧ V uru
2)
11°)
Vector division a)
a r
∧ x r
= b r
➙0 r
= 0 r
➙ indetermination : all vectors x of the vectorial space is a solution to (1)
b)
a r
∧ x r
= 0 r
➙
x r
= la r
, the vectors x parallel to the vector a are solutions to (1) c) Equation (1) is written 0=b, which is impossible : There is no vector x satisfying equation (1).
d) no solution since
a r
∧ x r
= b r
implies that
a r
⊥ b r
e)a r
∧ x r
= b r
(1)•
a r ∧ x uru
0= b r
(2) (1)-(2) ➙a
r ∧ (x r
− x uru
0) = 0 r
➙
x r = x uru
0+ la r
•
a r
∧ a r
∧ x uru
0( ) = a r ∧ b r
➙( ) a r .x uru
0a r − a
2x uru
0= a r ∧ b r
. . If x0 is perpendicular to a, we can easily havex
0uru = 0 r
➙
uru x
0= − a r
∧ b r a
2 •x r
= − a r
∧ b r a
2+ la r
b
a
a2
a ∧b
−
M
H
( )
The group of points M, at the extremities of vectors x belong to the segment
(∆) passing by the point H, such tat
