Difference between revisions of "Introduction"

Latest revision as of 16:11, 12 July 2023

[math]\displaystyle{ \newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\Ss}{\textbf{S}} \newcommand{\deg}{^\textsf{o}} }[/math]

I.1 What is mechanics?

Mechanics is the branch of physics devoted to the study of the motion of material objects. As any other branch in science, it looks for answers to a set of questions. In the present case, these questions can be reduced to two fundamental ones:

How is that motion?
What does the motion depend on?

The answers to those questions are the two main chapters in this subject: kinematics and dynamics, respectively. In order to connect Newtonian mechanics to other branches in science, this course includes a third chapter: energetics.

Kinematics

Kinematics describes the motion of the objects without taking into account their material characteristics. Thus, their density and roughness are irrelevant in that description. In kinematics, we look for the mathematical expression of the different features of the motion under observation.

Dynamics

Dynamics studies the factors on which movement depends, and relates them mathematically with that motion. Some textbooks state that dynamics studies the “cause” of movement, but that is incorrect: when talking about “cause” and “effect”, the former should precede the latter. As we will see later on, all the variables appearing in the equations of Newtonian dynamics are simultaneous (that is, they are assessed at the same time instant). Thus, making a distinction between “cause” and “effect” is not possible.

Unlike kinematics, dynamics is not a description but an explanation of what is being observed. It is a theory, so it calls for experimental validation. Accurate experimental results may show discrepancies with the theoretical predictions. In that case, it is necessary to formulate a new theory (or correct the old one) consistent with those results.

Energetics

The original formulation of Newtonian dynamics is a vectorial one: its equations deal with vectorial variables. But there are also analytical (scalar) formulations for dynamics. The gateway to these formulations is energetics, which deals with different forms of energy and their transformations.

Energy is a concept that appears in all branches in science. Adding a chapter on energy to this course is providing a bridge to connect mechanics to all disciplines in science.

I.2 Models for material objects

A model is a simplified representation of reality. In mechanics, there are three models for material objects. From the simplest to the most complex one, those models are: particle, rigid body, deformable body and fluids.

Particle: the object is considered to be located at one point in space, thus it is not possible to talk about its dimensions and its orientation.

Rigid body: the object is modelled as a set of material points which keep constant their mutual distances.

Deformable body: the object is modelled as a set of material points which may mutually approach or separate.

Fluid: it is a set of material points whose mutual distances are never constant (liquids, gases).

A complex model is not better than a simple one (the best model for a cat is not a cat!). Good practice in science is to choose the simplest model that allows to answer the questions being asked. Hence, a model restricts the results that can be obtained (or restricts the questions that can be answered).

The particle is a model that can be used as far as the orientation and shape of the object are irrelevant for the problem under study. Thus, when investigating the duration of an Earth year, the Earth may be modelled as a particle (Figure I.1-A). When studying the transition day/night, the Earth orientation is relevant but not its deformation: it can be modelled as a rigid body (Figure I.1-B). However, if we want to study the tides, the Earth has to be modelled as a deformable body as both the orientation and the deformation are relevant (Figure I.1-C).

Figure I.1 Earth model according to the phenomenon under study

This course deals with the first two models. Deformable bodies and fluids are studied in mechanics of continuum media, and that is not included in the present course.

I.3 Limitations Newtonian mechanics

As any other theory, Newtonian mechanics has a limited range of application. It dates from the XVIIth century, hence it is associated with experiments on macroscopic objects with low speed (much lower than light speed) relative to the Earth. If we wanted to study microscopic objects or objects moving with high speed (close to the light speed), we would have to consider the two branches of mechanics that were developed at the beginning of the XXth century: quantum mechanics and relativistic mechanics, both excluded from this course.

Though we will not study microscopic or high-speed dynamics, this course contains examples with elements based on electromagnetic and thermodynamic phenomena (as motors). These two branches of science were developed in the XIXth century, and are beyond Newtonian mechanics. Nevertheless, we will treat them in a particular way in order to include them in some examples.

I.4 Reference frame

The motion of an object call for the existence of a space-time stage: a reference frame (or simply a frame). The definition of space and time goes beyond science, and it is still a question open to debate in philosophy. Here, we will simply give operative definitions and an effective mathematical model.

Time is a dimension of the universe that allows ordering the events according to a sequence (before, after, simultaneous) and comparing their duration (Figure I.2). In physics, what cannot be measured (not for lack of a measurement device but because of the impossibility to conceive that device!) does not exist. Accordingly, time may be defined as whatever is measured by a clock...

Figure I.2 Time allows ordering the events according to a before-after sequence

In Newtonian mechanics, time is an absolute dimension: it flows at the same rate for everybody, hence the sequence of events is the same for all observers. Two clocks that have been synchronized at a certain time will be synchronized forever regardless their relative motion. We may talk, then, of a unique universal clock.

Similarly, space may be defined as the dimension of the universe that allows ordering the objects according to three criteria: in front of/behind, right/left, above/under (Figure I.3). The fact that there are no more and no less than three criteria is related to the human perception of that dimension (some branches in physics consider up to 11 space dimensions).

Figure I.3 Space allows ordering the objects according to in front of/behind, right/left, above/under

Time and space are the essence of a reference frame, which is the stage where events take place and from which we observe the motion of material objects.

Let’s consider an object without dimensions (a particle). That particle moves relative to a reference frame if it is located at different points of its space at different times. Behind that statement there is the assumption that the points in the space of a reference are fixed to each other (their relative distances are constant). Thus, a good representation of a reference frame is a cloud of points that do not approach and do not separate from each other.

The concept of reference frame (set of points mutually fixed) is very close to that of rigid body (set of material points mutually fixed). For that reason, we usually associate a name of a material object to reference frames: we talk about the “ship reference frame”, the “plane reference frame” ... But one has to bear in mind that space and time are not material realities (or touchable realities!). Hence, the ship and the plane have to be replaced by an infinite set of points that do not approach or separate from the boundaries of those material objects: that cloud of points is a genuine representation of the reference frame. In general, the set of points fixed to the plane approaches or separates from that fixed to the ship: the ship and the plane are two different reference frames (though they share they absolute time of Newtonian mechanics, they do not share the space).

A more compact representation of a reference frame replaces the cloud of points by a trihedral representing the three dimensions of space (Figure I.4). The clock is not shown because it is the same for all reference frames.

Figure I.4 Two representations of a reference frame.

Though representing space through a trihedral suggests that there is a singular point (the intersection of the three axes of the trihedral), all points in a reference frame are equivalent. The concept of origin does not apply: there is no origin of the reference frame. The concept “origin” applies to a coordinate system (Cartesian, polar, cylindrical...), which may be used to define the position of points in space.

In some textbooks, reference frame and coordinate system are equivalent concepts. In this course, however, reference frame is a more abstract concept: it does not imply any particular coordinate system. Different coordinate systems may be used for a same reference frame (a same space).

A particular characteristic of space in Newtonian dynamics is that the distance between two points is an invariant (it is the same when measured form any reference frame). Hence, the size of an object is the same at every time instant in all reference frames.

Vector calculus >>>

@@ Line 3: / Line 3: @@
 \newcommand{\Ss}{\textbf{S}}
 \newcommand{\deg}{^\textsf{o}}</math>
-==I.1 Què és la mecànica?==
+==I.1 What is mechanics?==
-La mecànica és la part de la física que estudia el moviment dels objectes materials. Com a qualsevol branca de la ciència, el que busca és donar resposta a un conjunt de preguntes. En el cas que ens ocupa, aquestes preguntes es poden reduir a dues de fonamentals:
+Mechanics is the branch of physics devoted to the study of the motion of material objects. As any other branch in science, it looks for answers to a set of questions. In the present case, these questions can be reduced to two fundamental ones:
-:*Com és aquest moviment?
+:*How is that motion?
-:*De què depèn aquest moviment?
+:*What does the motion depend on?
-Les respostes a aquestes preguntes constitueixen els dos grans capítols d’aquesta disciplina: cinemàtica i dinàmica, respectivament.
+The answers to those questions are the two main chapters in this subject: kinematics and dynamics, respectively.
+In order to connect Newtonian mechanics to other branches in science, this course includes a third chapter: energetics.
-Per poder connectar la mecànica Newtoniana amb les altres branques de la ciència, aquest curs inclou un tercer capítol: l’energètica.
-'''Cinemàtica'''
+'''Kinematics'''
-:La cinemàtica descriu el moviment dels objectes sense fer atenció a les seves característiques materials. Així, si són molt o poc densos, rugosos o llisos, gruixuts o prims, res de tot això intervé en aquesta descripció. El que fa la cinemàtica és expressar matemàticament diferents aspectes del moviment que s’observa.
+:Kinematics describes the motion of the objects without taking into account their material characteristics. Thus, their density and roughness are irrelevant in that description. In kinematics, we look for the mathematical expression of the different features of the motion under observation.
-'''Dinàmica'''
+'''Dynamics'''
-:La dinàmica estudia els factors que influencien el moviment, i els relaciona amb aquest moviment. En alguns textos es diu que estudia el “per què” del moviment, però aquesta afirmació no és correcta: demanar-se per un “per què” és distingir entre causa i efecte. La primera hauria de precedir al segon. Però com es veurà més endavant, la formulació de la dinàmica Newtoniana és un conjunt d'equacions on totes les variables s’avaluen en el mateix instant de temps. Així, fer una distinció entre causa i efecte resulta impossible.
+:Dynamics studies the factors on which movement depends, and relates them mathematically with that motion. Some textbooks state that dynamics studies the “cause” of movement, but that is incorrect: when talking about “cause” and “effect”, the former should precede the latter. As we will see later on, all the variables appearing in the equations of Newtonian dynamics are simultaneous (that is, they are assessed at the same time instant). Thus, making a distinction between “cause” and “effect” is not possible.
-:A diferència de la cinemàtica, la dinàmica no és una descripció sinó una explicació del que s’observa. És, doncs, una teoria i, com a tal, necessita un recolzament experimental per ser validada. Els resultats d’experiments cada cop més precisos poden presentar discrepàncies amb les prediccions de la teoria, i demanar l’elaboració d’una de nova consistent amb aquests nous resultats.
+:Unlike kinematics, dynamics is not a description but an explanation of what is being observed. It is a theory, so it calls for experimental validation. Accurate experimental results may show discrepancies with the theoretical predictions. In that case, it is necessary to formulate a new theory (or correct the old one) consistent with those results.
-'''Energètica'''
-:En l’enfoc newtonià, la dinàmica és vectorial: les magnituds que apareixen en les formulacions són vectors. Però existeixen també formulacions analítiques (escalars) de la dinàmica. La porta d’entrada a aquestes formulacions és l’energètica, que tracta de les diferents formes d’energia i les seves transformacions.
+'''Energetics'''
-:Totes les branques de la ciència parlen d’energia. Afegir un capítol sobre energia a aquest curs és, doncs, proporcionar un pont que relaciona la mecànica amb les altres branques de la física.
+:The original formulation of Newtonian dynamics is a vectorial one: its equations deal with vectorial variables. But there are also analytical (scalar) formulations for dynamics. The gateway to these formulations is energetics, which deals with different forms of energy and their transformations.
+:Energy is a concept that appears in all branches in science. Adding a chapter on energy to this course is providing a bridge to connect mechanics to all disciplines in science.
@@ Line 32: / Line 33: @@
 -----------
-==I.2 Models d'objectes materials==
+==I.2 Models for material objects==
-Un model és una representació simplificada de la realitat. En mecànica, hi ha tres models per als objectes materials. De més senzill a més complex: partícula, sòlid rígid, sòlid deformable i fluid.
+A model is a simplified representation of reality. In mechanics, there are three models for material objects. From the simplest to the most complex one, those models are: particle, rigid body, deformable body and fluids.
-:* '''Partícula''': model d’objecte material que ocupa un sol punt a l’espai, de manera que no se li atribueixen dimensions ni pot ser orientat.
+:* '''Particle''': the object is considered to be located at one point in space, thus it is not possible to talk about its dimensions and its orientation.
-:* '''Sòlid rígid''': conjunt de punts materials fixos entre ells (és a dir, que mantenen les distàncies mútues constants).
+:* '''Rigid body''': the object is modelled as a set of material points which keep constant their mutual distances.
-:* '''Sòlid deformable''': conjunt de punts materials que poden no mantenir constants les distàncies entre ells.
+:* '''Deformable body''': the object is modelled as a set of material points which may mutually approach or separate.
-:*'''Fluid''': conjunt de punts materials que no mantenen mai constants les distàncies entre ells (líquids, gasos).
+:*'''Fluid''': it is a set of material points whose mutual distances are never constant (liquids, gases).
-Un model complex no és millor que un de senzill (el millor model de gat no és un gat!). La bona pràctica en ciència és triar el model més senzill possible que permet respondre les preguntes que es formulen. Per tant, un model limita els resultats que es poden obtenir (o restringeix les preguntes que es poden formular).
+A complex model is not better than a simple one (the best model for a cat is not a cat!). Good practice in science is to choose the simplest model that allows to answer the questions being asked. Hence, a model restricts the results that can be obtained (or restricts the questions that can be answered).
-El model partícula es pot fer servir sempre i quan l’orientació de l’objecte i la seva forma siguin irrellevants per al problema que s’estudia. Així, si es vol investigar la durada d’un any terrestre, la Terra es pot modelitzar com a partícula ('''Figura I.1-A'''). Si es vol estudiar l’alternança nit i dia, l’orientació de la Terra és rellevant però no la seva deformació: es pot modelitzar com a un sòlid rígid ('''Figura I.1-B'''). En canvi, si el fenomen a analitzar són les marees, cal un model de sòlid deformable de la Terra ja que tant l’orientació com la deformació són rellevants ('''Figura I.1-C''').
+The particle is a model that can be used as far as the orientation and shape of the object are irrelevant for the problem under study. Thus, when investigating the duration of an Earth year, the Earth may be modelled as a particle ('''Figure I.1-A'''). When studying the transition day/night, the Earth orientation is relevant but not its deformation: it can be modelled as a rigid body ('''Figure I.1-B'''). However, if we want to study the tides, the Earth has to be modelled as a deformable body as both the orientation and the deformation are relevant ('''Figure I.1-C''').
-[[File:I-1.jpg|thumb|center|400px|link=]]
+[[File:I-2-1-neut.png|thumb|center|400px|link=]]
-<center><small>'''Figura I.1''' Model de la Terra segons el fenomen que s’estudiï</small></center>
+<center><small>'''Figure I.1''' Earth model according to the phenomenon under study</small></center>
-Aquest curs de mecànica se centra en els dos primers models. L’estudi de sòlids deformables i fluids constitueix la mecànica dels medis continus, i queda exclosa del curs.
+This course deals with the first two models. Deformable bodies and fluids are studied in mechanics of continuum media, and that is not included in the present course.
@@ Line 57: / Line 58: @@
 -----------
-==I.3 Limitacions de la mecànica Newtoniana==
+==I.3 Limitations Newtonian mechanics==
-Com a qualsevol altra teoria, la mecànica Newtoniana té un marc de validesa limitat. Es tracta d’una teoria del segle XVII, i per tant associada a experiments d’escala macroscòpica sobre objectes amb moviments relatius a la Terra de baixa velocitat (molt inferior a la de la llum). Estendre la mecànica a l’àmbit microscòpic o d’altes velocitats és entrar en les dues branques de la mecànica que es van desenvolupar a començaments del segle XX: la mecànica quàntica i la mecànica relativista, que queden excloses d’aquest curs.
+As any other theory, Newtonian mechanics has a limited range of application. It dates from the XVIIth century, hence it is associated with experiments on macroscopic objects with low speed (much lower than light speed) relative to the Earth. If we wanted to study microscopic objects or objects moving with high speed (close to the light speed), we would have to consider the two branches of mechanics that were developed at the beginning of the XXth century: quantum mechanics and relativistic mechanics, both excluded from this course.
+Though we will not study microscopic or high-speed dynamics, this course contains examples with elements based on electromagnetic and thermodynamic phenomena (as motors). These two branches of science were developed in the XIXth century, and are beyond Newtonian mechanics. Nevertheless, we will treat them in a particular way in order to include them in some examples.
-Tot i deixar de banda aquestes dues situacions, aquest curs contempla exemples que inclouen elements basats en fenòmens electromagnètics i termodinàmics (com ara motors). Aquestes dues branques de la ciència es desenvolupen fortament a partir del segle XIX, i s’escapen del marc de la dinàmica Newtoniana. Tot i així, l’interès en incloure’ls en els exemples portarà a donar-los un tractament singular.
@@ Line 66: / Line 68: @@
 ------------------
-==I.4 Referència (o marc de referència)==
+==I.4 Reference frame==
-El moviment d’un objecte requereix l’existència d’un escenari espai-temporal: un '''marc de referència''' (o, simplement, una '''referència'''). La definició d’espai i temps és una qüestió que transcendeix la ciència i és objecte obert de debat en l’àmbit filosòfic. Aquí simplement donarem definicions operatives i una modelització matemàtica eficaç.
+The motion of an object call for the existence of a space-time stage: a '''reference frame''' (or simply a frame). The definition of space and time goes beyond science, and it is still a question open to debate in philosophy. Here, we will simply give operative definitions and an effective mathematical model.
-El temps és una dimensió de l’univers que permet ordenar de manera seqüencial els esdeveniments (abans de, després de, simultani amb) i comparar la seva durada ('''Figura I.2'''). En física, el que no es pot mesurar (no per manca d’aparell adequat sinó per impossibilitat de dissenyar-lo!) no existeix. Partint d’aquesta premissa, hi ha qui defineix el temps com allò que mesura un rellotge...
+Time is a dimension of the universe that allows ordering the events according to a sequence (before, after, simultaneous) and comparing their duration ('''Figure I.2'''). In physics, what cannot be measured (not for lack of a measurement device but because of the impossibility to conceive that device!) does not exist. Accordingly, time may be defined as ''whatever is measured by a clock...''
 [[File:I-4-1-cat,eng.png|thumb|center|400px|link=]]
-<center><small>'''Figura I.2''' El temps permet organitzar els esdeveniments segons seqüències abans-després</small></center>
+<center><small>'''Figure I.2''' Time allows ordering the events according to a before-after sequence</small></center>
-En mecànica Newtoniana, el '''temps''' es considera '''absolut''': flueix al mateix ritme per a tothom, i per tant la seqüència dels esdeveniments és la mateixa per a qualsevol observador. Dos rellotges sincronitzats en un cert instant seguiran sincronitzats per sempre més independentment de com es belluguin un respecte de l’altre. Es pot parlar, doncs, d’un únic rellotge universal.
+In Newtonian mechanics, '''time''' is an '''absolute''' dimension: it flows at the same rate for everybody, hence the sequence of events is the same for all observers. Two clocks that have been synchronized at a certain time will be synchronized forever regardless their relative motion. We may talk, then, of a unique universal clock.
-De manera anàloga, es pot definir l’'''espai''' com la dimensió de l’univers que permet ordenar els objectes segons tres criteris: més endavant o més enrere, més a la dreta o a l’esquerra més amunt o avall  ('''Figura I.3'''). Que siguin tres i no més o menys es relaciona amb la percepció humana d’aquesta dimensió (hi ha branques de la física que consideren fins a 11 dimensions espacials).
+Similarly, space may be defined as the dimension of the universe that allows ordering the objects according to three criteria: in front of/behind, right/left, above/under ('''Figure I.3'''). The fact that there are no more and no less than three criteria is related to the human perception of that dimension (some branches in physics consider up to 11 space dimensions).
 [[File:I-4-2-neut.png|thumb|center|400px|link=]]
-<center><small>'''Figura I.3''' L’espai permet organitzar els objectes segons davant-darrere, dreta-esquerra, a sobre-a sota</small></center>
+<center><small>'''Figure I.3''' Space allows ordering the objects according to in front of/behind, right/left, above/under</small></center>
-El temps i l’espai constitueixen la '''referència''' (o el marc de referència), que és l’escenari on tenen lloc i des d’on s’observen els moviments dels objectes materials.
+Time and space are the essence of a '''reference frame''', which is the stage where events take place and from which we observe the motion of material objects.
-Considerem el cas d’un objecte sense dimensions (modelitzat com a partícula). Aquesta partícula es mou respecte d’una referència si en instants de temps diferents es troba en punts diferents de l’espai d’aquesta referència. Això pressuposa que els punts de l‘espai d’una referència són fixos entre ells (mantenen les distàncies relatives constants). Una bona representació de referència, doncs, és un núvol de punts que ni s’apropen ni s’allunyen.
+Let’s consider an object without dimensions (a particle). That particle moves relative to a reference frame if it is located at different points of its space at different times. Behind that statement there is the assumption that the points in the space of a reference are fixed to each other (their relative distances are constant). Thus, a good representation of a reference frame is a cloud of points that do not approach and do not separate from each other.
-El concepte referència (conjunt de punts fixos entre ells) és molt proper al de sòlid rígid (conjunt de punts fixos materials entre ells). Per aquest motiu, usualment s’associa un nom d’objecte material a cada referència: es parla de “referència vaixell”, “referència avió”... Però no es pot oblidar que l’espai i el temps no són realitats materials (o palpables). Per tant, cal substituir mentalment el vaixell o l’avió per un conjunt (infinit!) de punts que ni s’apropen ni s’allunyen dels contorns d’aquests elements: aquest núvol de punts és una representació genuïna de la referència. El conjunt de punts fixos a l’avió s’apropa o allunya, en general, del núvol de punts fix al vaixell: es tracta de dues referències diferents (tot i compartir el temps absolut de la mecànica Newtoniana, no comparteixen l’espai).
-Una representació més compacta d’una referència consisteix en substituir el núvol de punts per un triedre representatiu de les tres dimensions espacials ('''Figura I.4'''). El rellotge es pot ometre perquè és el mateix per a totes les referències. L’espai, en canvi, no és únic.
+The concept of '''reference frame''' (set of points mutually fixed) is very close to that of rigid body (set of material points mutually fixed). For that reason, we usually associate a name of a material object to reference frames: we talk about the “ship reference frame”, the “plane reference frame” ... But one has to bear in mind that space and time are not material realities (or touchable realities!). Hence, the ship and the plane have to be replaced by an infinite set of points that do not approach or separate from the boundaries of those material objects: that cloud of points is a genuine representation of the reference frame. In general, the set of points fixed to the plane approaches or separates from that fixed to the ship: the ship and the plane are two different reference frames (though they share they absolute time of Newtonian mechanics, they do not share the space).
-[[File:I-4-3-cat.png|thumb|center|400px|link=]]
+A more compact representation of a reference frame replaces the cloud of points by a trihedral representing the three dimensions of space ('''Figure I.4'''). The clock is not shown because it is the same for all reference frames.
-<center><small>'''Figura I.4''' Dues representacions del concepte referència</small></center>
+[[File:I-4-3-eng.png|thumb|center|400px|link=]]
+<center><small>'''Figure I.4''' Two representations of a '''reference frame'''.</small></center>
-Tot i que la representació de l’espai mitjançant triedres sembla singularitzar un punt (la intersecció dels eixos del triedre), tots els punts d’una referència són equivalents. El concepte '''origen''' no li és aplicable: no existeix l’origen de la referència. Parlar d’origen és introduir un '''sistema de coordenades''' (cartesianes, polars, cilíndriques...) per definir la posició dels diversos punts de l’espai de la referència.
+Though representing space through a trihedral suggests that there is a singular point (the intersection of the three axes of the trihedral), all points in a reference frame are equivalent. The concept of '''origin''' does not apply: there is no ''origin of the reference frame''. The concept “origin” applies to a '''coordinate system''' (Cartesian, polar, cylindrical...), which may be used to define the position of points in space.
-En alguns textos, referència i sistema de coordenades es consideren equivalents. En aquest curs, però, el concepte referència és més abstracte: no implica cap sistema de coordenades concret. Per a una mateixa referència (un mateix espai), es poden fer servir molts sistemes de coordenades diferents.
+In some textbooks, reference frame and coordinate system are equivalent concepts. In this course, however, reference frame is a more abstract concept: it does not imply any particular coordinate system. Different coordinate systems may be used for a same reference frame (a same space).
-Una característica particular de l’espai en dinàmica Newtoniana és que la distància entre dos punts és un invariant (és la mateixa mesurada des de qualsevol referència). Per tant, la mida dels objectes (i la distància entre punts d'un sòlid rígid) és la mateixa en cada instant en totes les referències.
+A particular characteristic of space in Newtonian dynamics is that the distance between two points is an invariant (it is the same when measured form any reference frame). Hence, the size of an object is the same at every time instant in all reference frames.
@@ Line 113: / Line 115: @@
 <center>
-[[Vector calculus|vector calculus >>>]]
+[[Vector calculus|Vector calculus >>>]]
 </center>

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