Peter Guthrie Tait.

A treatise on dynamics of a particle, with numerous examples online

. (page 1 of 23)
Online LibraryPeter Guthrie TaitA treatise on dynamics of a particle, with numerous examples → online text (page 1 of 23)
Font size
QR-code for this ebook


ill



.•: I f












^



GIFT OF
Daughters of
Frederick Slate




A



A TREATISE



ON



DYNAMICS OF A PAETICLE.



A TREATISE ON



DYNAMICS OF A PARTICLE,



WITH NUMEROUS EXAMPLES,



BY



PETER GUTHRIE TAIT, M,A.



rOBMERLY FELLOW OF ST PETER'S COLLEGE, CAJIBBIDGE,
PKOFESSOB OF NATURAL PHBLOSOPHT Ei THE UNIVEBSITr OF EDIXBCBGH,

AND THE LATE

WILLIAM JOHN STEELE, B.A.,

FEI.LOW OF ST PETER'S COLLEGE, CAMBBFDGE.



FOURTH EDITION.

Carefully revised.



MACMILLAN AND CO.

i8;8

[All R' (jilts reserved.]



,^\



4^,



-■V



Cambrt>cfe :

PMKTED BY C. J. CLAY, M.A,

AT THE UNIYEKSITY PBESS.






PREFACE



To the first edition of this work, published ia 185G, the
following was prefixed : —

"In the present Treatise will be found all the ordinary
propositions, connected with the Dynamics of particles, which
can be conveniently deduced without the use of D'Alembert's
Principle.

" Its publication has been delayed by many unforeseen
occurrences ; more especially by the early and lamented death
of Mr Steele, whose portion of the work was left uncompleted,
and whose assistance in its final arrangement and revision
would have been invaluable. The principal portions due to
him are the greater part of Chapters III., V. and VIII.
together with a few pages of Chapter I.

"Considerable use has been made of Pratt's Mechanical
Fhilosophij : indeed a large portion of Chapter XI. is reprinted
verbatim from that work.

" Throufrhout the book will be found a number of illus-
trative examples introduced into the text, and for the most
part completely worked out ; others with occasional solutions
or hints to assist the student are appended to each Chapter.
For by far the greater portion of these the Cambridge
Senate-House and College Examination Papers have been
applied to."



994594



Tl PREFACE.

To this was added, in the second edition, published in
1865 :—

" I am glad of the opportunity, presented by the call for
a second edition, to make reparation for many of the faults
of the first. Numerous trivial errors, and a few of a more
serious character, have now been corrected ; many sections
and several new examples have been added ; and the whole
of the second Chapter has been rewritten, upon the basis of
the corresponding portion of Thomson and Tait's Natural
Philosophy/ which, though as yet unpublished, was printed off
nearly two 3^ears ago.

"When I wrote that Chapter, in 1855, I had not read
Newton's admirable introduction to the Principia; and I
endeavoured to make the best of the information I had then
acquired from English and French treatises on Mechanics.
These five pages, faulty and even erroneous as I have since
seen them to be, cost me almost as much labour and thought
as the utterly disproportionate remainder of my contributions-
to the volume. And I cannot but ascribe this result, in part
at least, to the vicious system of the present day, which
ignores Newton's Third Law of Motion, though constantly
assuming it (tacitly) as an axiom ; and erects Statics upon
a separate basis from Kinetics, thereby necessitating several
additional Physical Axioms, the splitting of Newton's Second
Law into two, and the introduction of a so-called Statical
measure of Force.

" To be enabled to preserve the title of the work, I have
added (apropos of the Second Law of Motion) a few hints
about Statics of a particle.

" The examples are, for the most part, reprinted verhatim
from the papers in which they were set ; in a few the lan-
guage has been altered, or the theorem involved has been
generalized ; several, however, have defied all attempts at



PREFACE. Vll

improvement, and now stand in their unintelligibility as a
warning, to the Candidate for Mathematical Honours, of the
ordeal he may have to pass through.

" To several important theorems more than one demon-
stration has been appended : with the object of exhibiting the
use of the various processes by applying them to the de-
duction of results of real value, instead of to the solution of
' Problems ' of unquestionable absurdity.

" Various friends to whom I have applied for suggestions
as to any important changes which they might think desirable
in this second edition, and especially I. TODHUNTER, Esq.,
of St John's, have replied that they had none to offer, as
they liked the book well enough in its original form. This
has prevented me from attempting a thorough alteration of
style which I had contemplated, viz. to cease breaking up
^the subject into detached propositions — specially fitted for
writing out' I retain my o^vn opinion, however, that
this is not the form in which such a treatise ought to be
written ; although there can be no doubt that it offers certain
advantages to the student whose sole object in reading is to
pass an examination.

" The treatise is intended to be merely an analytical one :
for the full discussion and experimental demonstration of the
elementary principles on which the analysis is founded, the
reader must be referred to works on Natural Philosophy ; of
which, so far as mere Abstract Dyu amies is concerned, we
have a most admirable example in the Principia. For the
general application of modern theories to the whole range of
physical phenomena, the reader is referred to the forthcoming
work on Natural Philosophy by Professor W. Thomson and
myself, in which the subject will be developed from the grand
basis of Conservation of Energy.



VIU PREFACE.

" I have been dissuaded from introducing into this work
the Newtonian notation for Fluxions. It is true that in
Kinetics of a particle it is not very greatly superior to the
ordinary notation of differential coefficients : though, when
the general equations of motion of a system have to be treated,
in the beautiful manner invented by Lagrange, a partial use
of it is absolutely necessary. Newton's idea of Fluxions was
purely Kinematical; and, in fact, the fundamental ideas of
the Differential Calculus are essentially involved in the most
elementary considerations regarding velocity. It is also to be
observed, that, whenever we wiite f (x) for the differential
coefficient oi f[x), we are really employing the principal fea-
ture of Newton's notation, though in a form somewhat more
expressive than his.

" It is possible that in this edition a few of the objection-
able terms or methods, which the first edition contained, may
have remained undetected — but I hope that in every essential
respect the volume will be found to be an improvement on
its predecessor.

" I am encouraged in this hope by the fact that the sheets
in passing through the press have been read by J. Stirling,
Esq., of Trinity, to whose care and knowledge I am indebted
for many valuable suggestions."

To the above I have now little to add save this — that the
work of weeding and improvement, combined with cautious
and I hope judicious addition, has been carried still farther.

Some of the newly added Examples bear witness to the
great improvement which has recently taken place in the
Tripos Examination papers. For this the University has
mainly to thank the combination of genuine scientific know-
ledge with extraordinary originality presented by Prof. James
Clerk Maxwell, of Trinity, formerly of St Peter's.



PREFACE. IX

The Third Edition, published in 1871, was indebted to
\V. D. NiVEX, Esq., of Trinity, for a careful re\'ision of the
proofs, for the selection of numerous additional Examples,
and for a few foot-notes indicating the processes now most
commonly employed in certain parts of the analysis.

The present edition has been thoroughly revised by
Prof. Greenhill, who has not only at great labour verified
and (where necessary) corrected the Examples, but has
endeavoured to adapt the book to the present requirements
of the Tripos by the free introduction of Elliptic Functions,
&c. which, in my Cambridge days, were under the ban of the
Board of Mathematical Studies.

My attention has been called to the fact that several
sections of this book, in which some novelties appear, have
been translated almost lettei' for letter and transferred, with-
out the slightest allusion to their source, to the pages of a
German work. Several other books have ob%-iously been
similarly treated. It is well that this should be kno^^n, as
the English authors might otherwise come to be supposed
to have adopted these passages siinpliciter from the Geniian.

P. GUTHEIE TAIT.



College, Edinburgh.
April, 1878.



CONTENTS.



PAGES

Preface ... ... v — ix

Chapter I. Kinematics ... ... 1 — 34

Division of the subject, §§ 1—3.

Velocity, §§ 4—7.

Composition and Eesolution of Velocities, §§ 8 — 11.

Acceleration, §§ 12 — 19.

Hodograpli, § 20.

Moment of Velocity, §§ 21 — 24.""

Motion of a point deduced from the given acceleration, § 25.

Kelative Velocity and Acceleration, §§ 26 — 36.

Angular Velocity and Acceleration, §§ 37—40.

Velocity and Acceleration relative to Moving Axes, §§ 41— 43,

Examples 34—41

Chapter II. Laws of Motion. 42 — 58

Definitions of Mass, Density, Particle, Force, Momentum,
Vis Viva, Kinetic Energy, Measm-e of Force, Compo-
nent of Force, &c. &c. 44 — 57.

Definition, and Properties, of Center of Inertia, § 58.

Definition of Moment of Momentum, § 59.

Definition of Work done by a force, and consequences of tbo
definition, §§ 60, 61.

Definition of Potential Energy, § 62.

Newton's Laws of Motion, with their consequences — as
Measure of Time, Parallelogram of Forces, Conserva-
tion of Momentum and of Moment of Momentum, &c.
§§ 63-72.



J.



Xll CONTENTS.

PAGES

Scholium to tlie Third Law, with its interpretation.

D'Alembert's principle, Horse-power, Conservation of

Energy in Ordinary Mechanics, §§ 73 — 75.
Conservation of Energy, Impossibility of Perpetual Motion,

Joule's experimental results, §§ 76 — 78*.

Chapter III. Eectilixear Motion 59 — 79

Constant Force, §§ 79 — 87.

Force varying according to different powers of the distance,
§§ 88—105.

Examples 79—85

Chapter IY. Parabolic Motion ... ... ... 86 — 108

Projectile in vacuo, §§ 106 — 119.

Projectile in vacuo when the changes in the direction and
magnitude of gravity are considered, §§ 120, 121.

Force constant indirection, but notinmagnitude,§§ 1 22 — 129.

Newton's investigation of the motion of a luminous cor-
puscle, § 130.

Examples ... ... ... ... 108 — 112

Chapter V. Central Orbits 113 — 144

General Equations, §§ 131, 132.

Attraction proportional to the distance, § 133.

Polar Form of General Equations, and consequences, §§ 134

—144.
Properties of Apses, §§ 145 — 148.
Orbits under the Law of Gravitation, §§ 149 — 158.
Elliptic motion; definitions and immediate deductions,

§§ 159—162.
Kepler's Problem, §§ 163 — 167.
Lambert's Theorem, § 168.

Examples 144 — 166

Chapter YI. Constrained Motion ... ... 1G7 — 221

Preliminary remarks on Constraint, § 169.
Motion on Smooth Plane Curve, Cycloidal and Common
Pendulum, &c., Direct Problem, §§ 170 — 179.



CONTENTS. Xlll

PAGES

Inverse Problems — Bracliistoclirone, &c., §§ i8o — 186.

Motion on Smooth Surface, §§ 187 — 189.

Particular Case — Spherical Pendulum, §§ 190, 191.

Double Pendulum, § 192.

Effect of the Earth's rotation on simple pendulum, §§ 193 —

195-
Constraint by String attached to a moving Point, §§ 196—

198.
Constraint by Smooth Tube in motion, §§ 199— 203.
Constraint by Eough Curve, §§ 204, 205.
Constraint by Eough Surface, ? 306.

Examples ... ... ... ... ... ... 221 — 25.6



Chapter YIL Motion in a Eesistixg Medium ...237 — 250

General Statement of the Problem, § 207.
J Kectilinear Motion with various appUed forces and various
laws of resistance— Terminal Telocity, &c., §§ 208 —

212.

Curvilinear Motion, under various laws of resistance and
various forces. Approximate determination of path of
projectile with low trajectory, §§ 212 — 217.

Equation of Central Orbit in resisting medium, §§218, 219.

Examples 251 — 25S



Chapter YIII. General Theorems 259—308

Constraint perpendicular to dii-ection of motion, §§ 220,

221.
All central forces have a potential, § 222.
Conservation of Energy, and Equipotential Surfaces, §§

223, 224.
Inverse Problem as to conservative forces, § 225.
Deductions from Conservation of Energy, §§ 226—229.
Least, or Stationary, Action, §§ 230 — 237.
Varying Action, §§ 238 — 243.
The principle applied to the investigation of a planetary

orbit, §§ 244—248.
Application to Cotes' spirals, § 249.
Lagrange's Equations in Generalized Co-ordinates, §§ 250,

251.



XIV CONTEXTS.



Application to Bracliistochrones, § 252.

The bracbistochrone wlien the force is central, § 253.

The bracbistochrone normal to a series of isochronous sur-
faces, § 254.

Connection between the forces under which curves may be
described as free paths or bracbistochrones, §§ 255 —
260.

Motion about moving center, §§ 261, 262.

Hodographs, §§ 263 — 270.

Case of resisted motion in an equiangular spiral, § 271.

Examples 309 — 318



Chapter IX. Impact 319 — 335



Preliminary Eemarks, Coefficient of Eestitution, § 272.

Dh-ect Imjsact of Spheres, §§ 273, 274.

Impact of Sphere on Fixed Surface, § 275.

Impact of Smooth Spheres generally, Apparent Loss of

Energ}', §§ 276, 277.
Impulsive Tension in Chain, §§ 278 — 283.
Continuous Series of Indefinitely Small Impacts, §§ 284 —

286.
Disturbed Planet, § 287.

Examples ... ... 335-350



Chapter X. Motion of Two or More Particles 351 — 370

I. Free Motion. General Equations, §§ 289, 290.
Conservation of Momentum, § 291 ; of Moment of Momen-
tum, § 292 ; of Energy, § 293.

Particular Case of Two Particles, only, §§ 294—297.

II. Constrained Motion. Conditions of Constraint, § 298.
Two particles, in space, connected by inextensible string,

§299.
String constrained by pulley, § 300.
Chain slipping over pulley, § 301.
Complex pendulum, §§ 302, 303.
Limits of the treatise, § 304.

Examples ... 37o— 378#

General Examples 378 — 395



CONTENTS. XV

PAGES

Appendix 39C— 407

A. On the iutcgi-ation of the equations of motion about

a center of attraction 396

B. Motion on a cycloid 400

C. Brachistochrone, for gravity 401

C\. , for any forces 403

C\. General property of free path and brachistochrone for

any force whose direction is constant ih'ul.

D. Of two curves, convex upwards, joining two points in

a vertical plane, the inner is described in less time

than the outer ... ... ... ... 405

E. Inverse problem— To find the equation of the con-

straining curve when the time of descent, to the
lowest point, through any arc, is given as a function
of the vertical height fallen through 406



EREATA.



Page 85, line 22, for cos am sffl' — (modulus k) read en ( J jjJ



at
aK

Page 141, Article 165. The development of it in terms of t is

w = ni + 2S - Jm i^^^) si^ ^'^ "*»
w=i m

where J„j (ne) = - / cos m{t- e sin i) di

is Bessel's function of the n*^ order.

For the development of r and 6 in terms of t, the coefficients being
Bessel's functions, see Todhunter's Treatise on Legendre's, Laplace's, and
Bessel's Functions.

^ Page 152, Example 13. The result may be written more simply

^ ae

r = adn — .
c

^ ' Page 195, line 14. Add "p being the radius of curvature of the normal
section of the sm-face through the tangent to the path."

^ Page 198, Une 15. For z^ read z.

^ Pages 200— 202. For tai + A; read iai + ^.

- Page 202, Ime 12. For tt ^^^/ read - ^^,^ .



Page 202, line 13. For k read if.
^ Page 333, last line. For ^ read



DYNAMICS OF A PAKTICLE



CHAPTER I.

KINEMATICS. . .

1. Dijnamics is the Science which investigates the action
of Force; and naturally divides itself into two parts as
follows.

2. Force is recognized as acting in two w^ays : in Statics
so as to compel rest or to prevent cliange of motion, and in
Kinetics so as to produce or to change motion.

3. In Kinetics it is not mere motion which is investi-
gated, but the relation o^ forces to motion. The circumstances
of mere motion, considered without reference to the bodies
moved, or to the forces producing the motion, or to the forces
called into action by the motion, constitute the subject of a
branch of Pure Mathematics, which is called Kinematics.
To this, as a necessary introduction, we devote the present
chapter.

4. The rate of motion (or the rate of change o^ position)
of a point is called its Velocity. It is greater or less as the
space passed over in a given time is greater or less : and it
may be constant, i.e. the same at every instant ; or it may be
variable.

Constant velocity is measured by the space passed over in
unit of time, and is, in general, expressed in feet per second ;
if very great, as in the case of light, it may be measured in
miles per second. It is to be observed, that Time is here
used in the abstract sense of a uniformly-increasing quantit}-

T. D. 1



2 KINEMATICS.

— wliat in the differential calculus is called an independent
variable. Its ph3^sical definition is given in Chap. II.

5. Thus, a point moving uniformly with the velocity v
describes a space of v feet each second, and therefore vt feet
in t seconds, t being any number whatever. Putting s fur
the space described in t seconds, we have

s = I't.

Ifence with ii'nii velocity a point describes unit of space in
unit of time.

' 'G. It is well to observe that since, by our formula, we
have generally

s

'=1'

and since nothing has been said as to the magnitudes of s
and t, we may take these as small as we choose. Thus we
get the same result tvhether lue derive v from the sjmce described
m a million seconds, or from that described in a millionth of a
second. This idea is very useful, as it will give confidence
in results when a variable velocity has to be measured, and
we find ourselves obliged to approximate to its value by
considering the space described in an interval so short, that
during its lapse the velocity does not sensibly alter in value.

7. Velocity is said to be variable when the moving point
does not describe equal spaces in equal times. The velocity
at any instant is then measured by the space which woidd
have been described in a unit of time, if the point had moved
on uniformly for that interval with the velocity which it had
at the instant contemplated. This is a most important, and
in fact a fundamental, conception, which the student must
thoroughly realize before he can usefully proceed farther. It
lies at the root of all the correct methods ever devised for the
purpose of measuring the rate at which change, of any kind,
is going on.

Let V be the velocity of the point at the time t, measured
from a fixed epoch, s the space described by it during that
time, and s + ^s the space described during a greater interval



KINEMATICS. 6

t + St. Suppose t\ to be the greatest, and v^ the least, velo-
city with which the point moves during the time Bt ; then
v^St, i\^U would be the spaces whicli a point would describe
in that interval, moving uniformly with these velocities
respectively. But the actual velocity of the point is not
greater than v^, and not less than v^, therefore as regards the
actual space described,

hs is not greater than i\U, and not less than i\pty

Ss
'' Bt '^ '^'

however small Et may be. But, as St continually diminishes,
i\ and ^2 tend continually to, and ultimately become each
equal to, V, Therefore, proceeding to the limit,

ds

dt=''

If V be negative in this expression, it indicates that s
diminishes as t increases ; the positive case, w^hich we have
taken as the standard one, referring to that in which s and t
increase together. It follows that, if a velocity in one direc-
tion be considered positive, in the opposite direction it must
be considered negative; and consequently the sign of the
velocity indicates the direction of motion.

This is, of course, on the supposition that the velocity
alters continuously, and not by jerks. It would require an
infinite force to produce in an infinitely short time such a
change of velocity in a material particle. Hence as we are
preparing for physical applications only, such cases may be
excluded for the present. They will be treated in the chapter
on luipact.

8. It will be easily seen that the idea of velocity ex-
plained above is equally applicable whether the point be
considered as moving in a straight, or in a curved, line. In
the latter case, however, the direction of motion continually
changes ; and it will be necessary to know at every instant
the direction, as well as the magnitude, of the point's velo-
city. This is usually, and in general most conveniently, done

1—2



4 KINEMATICS.

by considering tlie velocities of the j^oint parallel to tlie
three co-ordinate axes respectively. For, if the co-ordinates of
the point be represented by x, y, z, the rates of increase of
these, or the velocities parallel to the corresponding axes, will
by reasoning analogous to the above be

dx dy dz
It' di' di'

Denoting by v the whole velocity of the point, we have

and, if a, y8, 7 be the angles which the direction of motion
makes with the axes,



cos a = ^7- = -7
ds as



d_.n
dx dt

dt



dx
di

dy
dt

dz

— = v cos 7 = v^.

at '



or — = -y cos a = v^, suppose.

Cut



Similarly, ~^~ = v cos

dt



Hence, ^ Jll^ are to be found from the whole velo-
dt dt dt
city V, by resolving as it is called ; I e. by multiplying by
the direction-cosines of the direction of motion. They are
called the Component Velocities of the point :^ and, with refer-
ence to them, V is called the Besidtant Velocity.

9. It follows from the above, that, if a point be moving
in any direction, we may suppose its velocity to be the result-
ant of three coexistent velocities in any three directions at



KINEMATICS. O

riglit angles to each other ; or, more generally, in any threo
directions not coplanar. But the rectangular resolution is
the simplest and best except in some very special applications.

Let v^, Vy, V, be the rectangular components of the velo-
city V of a moving point, then the resolved part of v aloDg
a line inclined at riiglat angles \, /x, v to the axes will be

V_^ cos \ + Vy cos fJb + V, COS V.

For, let a, /3, 7 be the angles which the direction of the
point's motion makes with the axes, 6 the angle between
I his direction and the given line. Then since

cos 6 = cos a cos A, + cos /S cos /x + cos 7 cos Vy

the resolved part of v along that line is



V cos



a cos \ -f cos /3 cos yu, + cos 7 cos v\



i\ COS X + Vy cos jjb + i\ cos V,



10. These propositions are virtually equivalent to the
following obvious geometrical construction : —

To compound any two velocities as OA, OB in the figure;
where OA, for instance, represents in magnitude and direc-
tion the space w^hich w^ould be described in one second by
a point moving with the first of the given velocities— and



similarly OB for the second ; from A draw J. C parallel and
ecpial to OB. Join 06': — then 0(7 is the resultant velocity
in magnitude and direction. For the motions parallel to OA
and OB arc independent.

00 is evidently the diagonal of the parallelogram two of
whose sides are OA, OB.



6 KINEMATICS.

Hence the resultant of any two velocities as OA, AC, in
the figure is a velocity represented by the third side, OC, of
the triangle OA C.

Hence if a point have, simultaneously, velocities repre-
sented by OA, AG, and CO, the sides of a triangle taken, in
the same order, it is at rest.

Hence the resultant of velocities represented by the sides
of any closed polygon whatever, whether in one plane or not,
taken all in the same order, is zero.

Hence also the resultant of velocities represented by all
the sides of a polygon but one, taken in order, is represented
by that one taken in the opposite direction.

When there are two velocities or three velocities in two
or in three rectangular directions, the resultant is the square
root of the sum of their squares — and the cosines of the in-
clination of its direction to the given directions are the ratios
of the components to the resultant.

[Newton's Method of Fluxions was devised simply to
express this and other fundamental conceptions in Kinematics.
To him s, x, y, i, or (as we now somewhat less conveniently write

them) -t7 , -TL y Zj^ > 'jj ' ^^'® simply the velocity of the moving


1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Online LibraryPeter Guthrie TaitA treatise on dynamics of a particle, with numerous examples → online text (page 1 of 23)