Table Of ContentELEMENTS OF
Theoretical Mechanics
for Electronic Engineers
by
FRANZ BULTOT
DOCTEIIR EN SCIENCES
MEMBER IT THE
ACAD~MIE ROYALE DES SCIENCES D'OIITRE-MER
CHARGY DR COURS AT THE
INSTITUT NATIONAL DE RADIO]LECTRICIT)
BRIIXELLES
Translated from the French by
ELIZABETH S. KNOWLSON
PERGAMON PRESS
OXFORD • LONDON • EDINBURGSRH • NEW YORK
PARIS • FRANKFURT
Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.1
Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1
Pergamon Press Inc., 122 East 55th St., New York 22, N.Y.
Pergamon Press GmbH, Kaiserstrasse 75, Frankfurt-am-Main
Copyright ® 1965
Pergamon Press Ltd.
First English edition 1965
Library of Congress Catalog Card No. 63-16857
FOREWORD TO THE ENGLISH EDITION
IT is a welcome change to find that this volume is more in the style of a set
of lectures than a text-book. Instead of the usual lists of unworked exer-
cises, there are worked examples and many notes on the application of most
sections of the theory to modern electrical developments. These are of
considerable assistance in convincing practically-minded students that the
mathematical work really is useful and therefore worth following.
The sections on oscillatory motions and their counterparts in electrical
circuits and radio are developed particularly well. The use of vector not-
ations throughout is commendable and a good introduction is given to the
differential operators of vector field theory.
The book should be of considerable interest and value to students taking
a Higher National Diploma or a Diploma in Technology or those in the
final year of the Higher National Certificate in Electrical Engineering. A
student working on his own could follow most of the work having a mathe-
matical knowledge of roughly first year Higher National Certificate standard.
J. E. CAFFALL
Department of Engineering
Oxford College of Technology
d
FOREWORD
THEORETICAL mechanics is rightly considered to be one of the fundamental
branches of instruction which are essential in training an engineer.
Even through in French technical literature there are already many
treatises and courses on theoretical mechanics — and some of these are
excellent — it must be admitted that they are all very much alike.
M. F. BuLTOT has sought to produce something new. Although the general
laws of mechanics naturally remain the same in his hands as in those of
anyone else, he has aimed at giving a specific direction to his work in writing
a course on theoretical mechanics for future electronic engineers. He has
chosen his numerous examples and applications — without which theore-
tical mechanics would be merely an abstract, arid science — from those which
have either a direct or an indirect bearing on electronics.
M. Γ. BuLTOT, who for many years has been a brilliant and authoritative
teacher in a well-known College of Advanced Technology, has had the
opportunity of putting his work to the test. Indeed, this work came into
being as a result of what he considered electronics required from theoretical
mechanics. Its contents have already been taught for several years before
being presented for publication so that the pedagogical qualities of the work
are to be as highly recommended as its fundamental soundness.
Anxious, however, not to be too narrow in his approach, M. F. BULTOT
has been wise enough to widen his horizon and thus to ensure for his book
the interest not only of future electronic engineers but also of all future
engineers and technologists training to an advanced level.
I am convinced that this work, which fills a gap in French scientific
literature, will be greatly esteemed by the ever-increasing number of those
who are interested in bringing our Higher Technical Education right up to
date.
Louis DE BBUYNE
Ingenieur Civil A.I.G.
Inspecteur de TEnseignement Technique
6
PREFACE
Trns work has been written for use by students of higher technical col-
leges, and especially for those specialising in electronics.
In writing this course on theoretical mechanics we have been guided by
the twofold desire to include in it only those notions which are generally
used in the specialised branches of electronics, and to show, on every
occasion, the link between mechanics and electricity. That is why we have
inserted in this volume, in addition to a number of classical exercises, a
good many applications to problems in electronics. In doing this it has
been our intention to prevent the student from being confronted with the
often awkward problem of the interpenetration of subjects, and to show
the future electronic engineer the extent to which a course on theoretical
mechanics is of interest to him. There is, furthermore, in fine, an index of
the electronic topics of which the mechanical principles are set out in this
book.
Containing also a chapter devoted to statics, the contents of this work
meet the usual requirements for the curricula of the special colleges for
technical engineers.
In the first half of the book only elementary notions of mathematics
are required (except for some applications which are marked with an
asterisk and which can, incidentally, be left out to begin with). This course
on mechanics can, therefore, be easily followed even if it is taught con-
currently with a course on calculus, as is generally the case in advanced
technical teaching.
It should be noted that the calculations, even where simple, have always
been worked out in detail in order to make the work easier for the student,
or even for someone working on his own who may find this course of interest.
We have pleasure, finally, in expressing our deep gratitude to M. L. De
Bruyne, Inspecteur de L'Enseignement Technique, who has been kind
enough to offer many helpful and discerning remarks on the work and our
thanks to Mr. Caffall for writing the Foreword to the English Edition.
FRANZ BuLTOT
7
1
VECTORS-VECTOR SYSTEMS-
VECTOR FUNCTIONS
1.1. VECTOR
A. DEFINITIONS
A vector is a directed line segment (Fig. 1.1). The initial point A is
called the origin and the terminal point B the head of the vector A B.
The line to which the segment belongs is called the vector support.
The vector AB is characterised by:
(1) Its magnitude, which is the length of the segment A B.
(2) Its alignment in space, which is the alignment of its support.
(3) Its sense, which is that going from the origin A to the head B.
A
Fm. 1.1
A vector quantity is a quantity which possesses direction. Force, velocity
and acceleration, for example, are vector quantities because they possess
direction. On the other hand, a quantity which possesses no direction is
called a scalar quantity. Temperature, electric charge of a conductor, volume
of a body, for example, are scalar quantities.
The magnitude of the vector P is denoted by 0 P l or by P.
B. FREE VECTORS AND BOUND VECTORS
If the origin of the vector is not localized the vector is called free; if to
the contrary, it is called applied or bound. Two parallel bound vectors
of the same length and sense are termed equal or equivalent. In the first
three paragraphs j- we shall consider only free vectors.
t Except at the point u of 1.1.
9
10 ELEMENTS OF THEORETICAL MECHANICS
C. SUM OR RESULTANT OF VECTORS
The sum of two vectors P and P is a vector extending from the origin
1 2
of P to the head of P , provided that the head of P is taken for the origin
1 2 1
of P (Fig. 1.2). The sum of several vectors is obtained in the same way
2
(Fig. 1.3).
P
2
R
Fta. 1.2 Fm. 1.3
It can be seen immediately that (Fig. 1.4)
Pl { P2 = P2 -{- P1;
P2
R + R=R+R
FIG. 1.4 FIG. 1.5
the addition of vectors is therefore commutative. It can also be easily verified
that (Fig. 1.5)
Ri { R2 + R3 = (R1 + R2) + U3;
the addition of vectors is therefore associative (or distributive if one reads
the preceding relation from right to left).
D. PRODUCT AND QUOTIENT OF A VECTOR BY A SCALAR
The product (the quotient) of a vector P by a scalar m is a vector whose
magnitude is equal to the product (quotient) of the magnitude of P by the
VECTORS-VECTOR SYSTEM:S-VECTOR FUNCTIONS 11
scalar m, whose alignment is that of P and whose sense is identical with
or opposite to that of P according to whether m is positive or negative
(Fig. 1.6).
FIG. 1.6
It is agreed to denote by - P the vector of opposite sense to P (but
having the same magnitude and alignment).
E. DIFFERENCE OF TWO VECTORS
The subtraction of a vector P is the addition of the vector of opposite
sense, -P (Fig. 1.7).
FIG. 1.7
F. BASE
Three vectors having unit magnitude and not all parallel to the same
planeform a base.Whentheyare mutuallyperpendicular, the baseis called
orthogonal (we shall subsequently always adopt an orthogonal base). We
shall denote these vectors byt
Any vector P can always be considered as the sum of three vectors
Px, Py, P, taken with the same directions as the base vectors. The vectors
pe» P11'Pzare the resolutes of the vector P.
t Certainauthorsuse l, j, k.
12
ELEMENTS OF THEORETICAL MECHANICS
If X, Y, Z denote the magnitudes of the vectors R , P , P it follows that
5 z
Px = C 1, P, = U 1,, Pz = Z 1;
hence
R =C1c -}- Y1y +Z 1z.
The scalars X, Y, Z are the coordinate components of the vector P.
Note that X, Y, Z are the projections, on the three axes, of the seg-
ment OA, the magnitude of the vector P. They are also the Cartesian co-
ordinates of the head A of the vector P when the latter has its origin in O
(Fig. 1.8).
G. MAGNITIIDE OF A VECTOR
When the base is orthogonal, it can be seen (Pythagoras's theorem) that
the magnitude of P + P, is equal to 1c2 -+ U2 and that the magnitude
x
of (R + P ) + P is equal to j/(C2 -{- U2) -{- Z2 (Fig. 1.8). The magni-
11 z
tude of the vector P is therefore given by the formula
I r 1c2 + y2 + Z2 (1.2)
and the cosines of the angles a, b, y between the base vectors and the
vector P by
cos a = RI I, cos b= RI I, cos y = RI. (1.3)
These are known as the direction cosines of the vector P. It can be seen
immediately that
cost a ± cos2 -}- cos2 y = C2 ± U2 + Z2 (1.4)
ß IRI2
VECTORS—VECTOR SYSTEMS—VECTOR FUNCTIONS 13
H. ANALYTICAL EXPRESSIONS OF THE EQUALITY AND OF THE
ADDITION OF VECTORS
If the bound vectors P of components C , U , Z and P of components
1 1 l 1 2
C2, U2, Z2 are equal, the projections of the magnitudes 1R11 and 1R21 are
necessarily equal and we have
U1 = U2, (1.5)
Z1 = Z2.
If
R3 = R1 + R2,
we have
C315, -{- U31u -{- Z3 1 = C11,,, -{- U1 1u -I- Z11z -}- C21,, -¤- U2 1u + Z212
= (C1-I- C2) 1c -~- ( U1 -i-- U2) 1u + (Z1 -~- Z2) 1z ;
hence, by virtue of (1.5)
C3 — C1+C2,
U3 = U + U2 , (1.6)
1
I Z3 = Z1 + Z2 •
It also follows that:
(1) the vector obtained by multiplying a vector by a scalar has for com-
ponents the products by the scalar of the components of the vector;
(2) two vectors of opposite sense have components of equal magnitudes
but opposite sign.
Exercises
(1) Add the vectors 131(2, 2, —6), R2(1, 1, 1), R3(0, 0, 3) and R4(-2, —1, 0).
R=1,+21,-21.
1 z
(2) Calculate the magnitude of the vector R found in 1.
‚P1 = 1/I + 4-F 4= 3.
(3) Multiply this vector P by the scalar —3.
R'=-3R= -31 -61 +61,.
x 1
1.2. SCALAR PRODUCT OF TWO VECTORS
The scalar product R1 . P2 of the vectors R1 and P2 is a scalar equal to the
product of the magnitudes of the two vectors and the cosine of the angle formed
by the two vectors (Fig. 1.9)
R1 • R2 = 1R11 1R21 cos O. (1.7)
