Theoretical mechanics - Hoskins

In a first course in Theoretical Mechanics the primary object to be gained by the student is a thorough grasp of fundamental mechanical principles. In most cases it is impossible to go beyond this object in the time available for the course. In the preparation of this text-book, the aim has been to present the fundamental principles in as clear and simple a manner as possible, and to enforce them by a sufficient number of illustrative examples. The examples are for the most part simple applications of the theory presented in the text, many numerical exercises being included. The solution of exercises involving numerical data forms a part of the work of the student of which the importance should be emphasized. In the desire to cover as much ground as possible, such work is too apt to be neglected.

The mathematical training required for using the book is that usually implied by an elementary knowledge of Differential and Integral Calculus. The most essential portions of Part I (Statics) may, however, be read without such knowledge.

As regards arrangement, the aim has been to observe the order of difficulty of the different topics treated, so far as this was possible without doing violence to the logical relations. This is one reason for placing Statics first, instead of treating it in its strictly logical position as a special case of Kinetics. Part II may, however, be read before Part I, if any teacher should prefer this order.

As to scope, the primary object has been to meet the needs of students of engineering in American universities and technical colleges. More has, however, been included than it will usually be possible or desirable to cover in an elementary course. The precise limits of the course must be determined by the judgment of the teacher, taking due account of the relative importance of the various parts of the subject and of the limitations of time. It is hoped that the arrangement of the book is such as to facilitate whatever abridgment the teacher may find desirable.

In case it is necessary to cut the course down to the narrowest possible limits, the following suggestions as to the most essential parts to retain may be of service.

If this book is in any degree successful in meeting the needs of students of engineering, it is hoped that it may be of service also to those pursuing the subject for its intrinsic scientific interest or as a preparation for the study of mathematical physics. The opinion is sometimes expressed that the needs of these different classes of students require essentially different methods of treating the subject. This view, so far as it refers to the fundamental parts of an elementary course, is not shared by the author of this text-book. For all students, the matter of first importance is the clear understanding of fundamental general principles and the ability to apply them.

Motions take place in accordance with definite laws. This statement means that the motions of bodies depend in an invariable way upon certain definite conditions. The fundamental object of the science of Mechanics is to investigate these conditions and to formulate the laws in accordance with which they determine the motion.

In its present form the science of Mechanics rests upon a few fundamental laws of great generality. These laws are generalizations from experience, and in the presentation of the principles of the science they are taken as postulates, propositions not deducible from anything more fundamental. So far as these postulates and the principles deduced from them give a true account of the motions of natural bodies, they constitute a natural science.

These fundamental laws, however, involve certain conceptions of matter and of motion which are ideal. This is necessarily the case. The motion of a body can be completely specified only by describing the motion of every ultimate portion of which it is composed; yet the ultimate structure of matter is wholly unknown. The bodies to which the laws of motion apply are therefore defined in an ideal way. Moreover, the very conception of motion involves the abstract notions of Geometry, with the notion of time added. Because of this ideal character of the laws, the science based upon them is properly called Theoretical Mechanics.

2. Material Bodies. A body is any definite portion of matter. No attempt need be made to define matter or to enumerate its properties. A sufficiently definite preliminary notion is supplied by ordinary experience.

The characteristics of material bodies which are of importance in a study of their motions are the following : (1) Everybody has a definite volume and a definite figure; (2) everybody possesses a definite mass; (3) bodies exert forces upon one another.

3. Mass. The mass of a body is often briefly defined as its "quantity of matter." These words, however, convey no definite idea of the meaning of mass as a factor in the determination of motion. A satisfactory definition of mass cannot be given in advance of a discussion of the fundamental laws of motion.

We conceive of mass as the one invariable characteristic of matter. Every individual portion of matter is regarded as possessing a definite mass whose value is uninfluenced by changes of position or by physical or chemical transformations. The volume and shape of a body may change, the forces it exerts upon other bodies and those which they exert upon it are different under different circumstances; but its mass is regarded as an absolute constant.

4. Force. A force is an action exerted by one body upon another, tending to change the state of motion of the body acted upon.

A force may be conceived as a push or a pull, acting upon a definite portion of a body. Such a push or pull always tends to change the motion of the body; but this tendency may be counteracted in whole or in part by the action of other forces.

Mechanics is often called the science of motion and force, because of the importance of force in the development of the laws of the science.

Force, like mass, is a quantity whose significance cannot be satisfactorily explained except by a full discussion of the fundamental laws of motion.

5. Particle. A body may be conceived to be divided into very small parts, each of which may be called a particle. Ideally, there is no limit to this process of subdivision. For the purposes of mathematical analysis it is often conceived to be carried so far that the linear dimensions of the particles become vanishingly small.

These particles may be conceived in either of two ways, corresponding to two different conceptions of the structure of matter.

(1) It may be assumed that matter occupies space continuously. By this it is to be understood that every portion of matter whose mass is finite occupies a finite volume.

(2) It may be assumed that any definite portion of matter consists of particles, each of which possesses finite mass but occupies no finite volume.

In the analysis of the motion of a particle, it is regarded as a geometrical point endowed with mass. A body whose linear dimensions are small in comparison with the range of its motion is often regarded as a particle.

6. Rigid Body. A rigid body may be defined as a body whose particles do not change their distances from one another.

An important part of the science of Theoretical Mechanics deals with bodies which are assumed to satisfy this definition.

Actual solid bodies undergo appreciable changes of shape and size; and if sufficiently small portions could be observed, they would doubtless be found to be in rapid motion, thus departing very far from the condition specified in the definition of rigidity. But disregarding the motions of ultimate particles and considering only the motion of a body as a whole, the theory of the motion of an ideal rigid body describes with great accuracy the motion of an actual solid body whose shape remains nearly constant.

7. Position and Motion. In the foregoing definitions and explanations it has been assumed that position and motion need no definitions. It is, in fact, doubtful whether any definitions can be given which convey clearer notions than the words themselves.

By the position of a particle is meant its relation in space to some body taken as a standard of reference.

A particle is in motion when its position is changing. A body is in motion when the particles composing it are in motion.

What shall be chosen as the body of reference in specifying the position and motion of a particle is a matter of arbitrary choice. Position and motion are thus not absolute but relative. The motion of a particle with reference to one body may be very different from its motion with reference to another. (See Arts. 267, 268.)

8. Kinds of Quantity. The science of Mechanics deals with quantities of four fundamental kinds: time, space, mass, force. In the development of the principles of the science other quantities are introduced which are derived from these but involve no other elementary conceptions.*

9. Divisions of the Subject. The general subject of Mechanics may be subdivided in various ways.

First, the basis of the subdivision may be the nature of the bodies dealt with. On this basis there are the following divisions:

(a) Mechanics of a Particle and of systems of particles in general.

(b) Mechanics of Solid Bodies, including (1) rigid and (2) non-rigid solids.

(c) Mechanics of Fluids, including (1) liquids and (2) gases.

The present work deals mainly with particles and with rigid solids.

Second, the basis of subdivision may be the fundamental kinds of quantity involved. On this basis the whole subject, and each of the above divisions, may be divided as follows:

(a) Kinematics, treating of motion, without reference to the causes producing or influencing it. Under this there are (1) Pure Kinematics, treating of motion apart from the idea of mass, and (2) Mass-Kinematics, treating of motion and mass.

(b) Dynamics, treating of forces and of their influence upon the motions of bodies. Dynamics, the science of force, is again subdivided into Statics and Kinetics.

Statics treats of the conditions of equivalence of systems of forces, and especially of the conditions under which forces are balanced so that they do not affect the motions of the bodies acted upon. Kinetics treats of the laws in accordance with which the motions of bodies are influenced by the forces acting upon them and by their masses.

Strictly speaking, Kinetics includes Statics; for from a knowledge of the effects of forces upon the motions of bodies may be derived all principles relating to the equivalence of different systems of forces and to the conditions under which forces are balanced. Statics and Kinetics are, however, often treated as coordinate branches of Dynamics; first, because Statics, although a special case, is a case of great importance, and second, because the principles of Statics can be developed to a large extent independently of those of Kinetics.

The arrangement of subjects in this book does not strictly follow either of the above classifications, but is designed to meet the needs of beginners by presenting the different topics somewhat in the order of their difficulty. The arrangement adopted is as follows:

Introductory chapter, treating of certain principles which have application in various branches of the general subject of Mechanics.

Part I. Statics; the discussion of the subject being limited mainly to systems of forces acting in the same plane upon a particle or a rigid body.

Part II. Motion of a Particle; including Kinematics and Kinetics, but limited mainly to the case of motion in a plane.

Part III. Motion of Systems of Particles and of Rigid Bodies; including Kinematics and Kinetics, but dealing mainly with motion in a plane.

The remaining portion of this introductory chapter is devoted to certain principles which, while not strictly included in the subject of Mechanics, are of fundamental importance in the development of the science.


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