Introduction
[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).
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...
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).
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.
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.
© Universitat Politècnica de Catalunya. All rights reserved