Experimental mechanics : a course of lectures delivered at the Royal College of Science for Ireland

EXPERIMENTAL MECHANICS

IE..TILE IS A PARABOLA.

EXPERIMENTAL MECHANICS

A COURSE OF LECTURES

DELIVERED AT THE ROYAL COLLEGE OF SCIENCE
FOR IRELAND

SIR ROBERT STAWELL BALL, LLD.,( F.R.S.

ASTRONOMER ROYAL OF IRELAND

FORMERLY PROFESSOR OF APPLIED MATHEMATICS AND MECHANISM IN TH!
ROYAL COLLEGE OF SCIENCE FOR IRELAND (SCIENCE AND ART DEPARTMENT)

WITH ILLUSTRATIONS
SECOND EDITION

Xondon

MACMILLAN AND CO.

AND NEW YORK
1888.

The Right of Translation and Reproduction is reserved

RICHARD CLAY AND SONS, LIMITED,
LONDON AND BUNGAY.

The First Edition -was printed in 1871.

PREFACE.

I HERE present the revised edition of a course of
lectures on Experimental Mechanics which I delivered
in the Royal College of Science at Dublin eighteen
years ago. The audience was a large evening class
consisting chiefly of artisans.

The teacher of Elementary Mechanics, whether
he be in a Board School, a Technical School, a
Public School, a Science College, or a University,
frequently desires to enforce his lessons by exhibiting
working apparatus to his pupils, and by making
careful measurements in their presence.

He wants for this purpose apparatus of substantial
proportions visible from every part of his lecture
room. He wants to have it of such a universal
character that he can produce from it day after
day combinations of an ever-varying type. He
wishes it to be composed of well-designed and well-
made parts that shall be strong and durable, and that

2066709

viii PREFACE.

will not easily get out of order. He wishes those parts
to be such that even persons not specially trained in
manual skill shall presently learn how to combine them
with good effect. Lastly, he desires to economize his
money in the matters of varnish, mahogany, and glass
cases.

I found that I was able to satisfy all these require-
ments by a suitable adaptation of the very ingenious
system of mechanical apparatus devised by the late
Professor Willis of Cambridge. The elements of the
system I have briefly described in an Appendix, and
what adaptations I have made of it are shown in
almost every page and every figure of the book.

In revising the present edition I have been aided
by my friends Mr. G. L. Cathcart, the Rev. M. H.
Close, and Mr. E. P. Culvenvell.

ROBERT S. BALL.

OBSERVATORY, Co. DUBLIN,
yd August, 1888.

TABLE OF CONTENTS.

LECTURE I.
THE COMPOSITION OF FORCES.

PAGE

Introduction. — The Definition of Force. — The Measurement of
Force. — Equilibrium of Two Forces. — Equilibrium of Three
Forces. — A Small Force can sometimes balance Two Larger
Forces i

LECTURE II.
THE RESOLUTION OF FORCES.

Introduction. — One Force resolved into Two Forces. — Experi-
mental Illustrations. — Sailing. — One Force resolved into Three
Forces not in the same Plane. — The Jib and Tie-rod. ... 16

LECTURE III.
PARALLEL FORCES.

Introduction. — Pressure of a Loaded Beam on its Supports. —
Equilibrium of a Bar supported on a Knife-edge. — The Com-
position of Parallel Forces. — Parallel Forces acting in opposite
directions.— The Couple.— The Weighing Scales .... 34

LECTURE IV.
THE FORCE OF GRA VITY.

Introduction. — Specific Gravity. — The Plummet and Spirit
Level.— The Centre of Gravity.— Stable and Unstable Equili-
brium.— Property of the Centre of Gravity in a Revolving
Wheel 50

x TABLE OF CONTENTS.

LECTURE V.
THE FORCE OF FRICTION.

PAGE

The Nature of Friction. — The Mode of Experimenting. — Fric-
tion is proportional to the pressure. — A more accurate form
of the Law. — The Coefficient varies with the weights used. —
The Angle of Friction. — Another Law of Friction. — Con-
cluding Remarks 65

LECTURE VI.
THE PU LLE Y.

Introduction. — Friction between a Rope and an Iron Bar. — The
Use of the Pulley. —Large and Small Pulleys.— The Law of
Friction in the Pulley. — Wheels.— Energy 85

LECTURE VII.

THE PULLEY-BLOCA:

Introduction. — The Single Movable Pulley. — The Three-sheave
Pulley-block.— The Differential Pulley-block. —The Epicy-
cloidal Pulley-block 99

LECTURE VIII.
THE LEVER.

The Lever of the First Order.— The Lever of the Second Order.—

The Shears.— The Lever of the Third Order 119

LECTURE IX.
THE INCLINED PLANE AND THE SCREW.

The Inclined Plane without Friction. — The Inclined Plane with

Friction.— The Screw.— The Screw-jack.— The Bolt and Nut 131

TABLE OF CONTENTS. si

LECTURE X.
THE WHEEL AND AXLE.

PAGE

Introduction. — Experiments upon the Wheel and Axle. — Friction
upon the Axle.— The Wheel and Barrel. — The Wheel and
Pinion. — The Crane. — Conclusion 149

LECTURE XI.
THE MECHANICAL PROPERTIES OF TIMBER.

Introduction. — The General Properties of Timber. — Resistance to
Extension. — Resistance to Compression. — Condition of a
Beam strained by a Transverse Force 169

LECTURE XII.
THE STRENGTH OF A BEAM.

A Beam free at the Ends and loaded in the Middle. — A Beam
uniformly loaded. — A Beam loaded in the Middle, whose
Ends are secured. — A Beam supported at one end and loaded
at the other . . , 188

LECTURE XIII.
THE PRINCIPLES OF FRAMEWORK.

Introduction. — Weight sustained by Tie and Strut. — Bridge with
Two Struts.— Bridge with Four Struts, —Bridge with Two
Ties. — Simple Form of Trussed Bridge 203

LECTURE XIV.
THE MECHANICS OF A BRIDGE.

Introduction.— The Girder.— The Tubular Bridge.— The Sus-
pension Bridge . , 218

xii TABLE OF CONTENTS.

LECTURE XV.
THE MOTION OF A FALLING BODY.

PAGE

Introduction.— The First Law of Motion.— The Experiment of
Galileo from the Tower of Pisa.— The Space is proportional
to the Square of the Time.— A Body falls 16' in the First
Second.— The Action of Gravity is independent of the Motion
of the Body.— How the Force of Gravity is defined.— The
Path of a Projectile is a Parabola 230

LECTURE XVI.
INERTIA.

Inertia.— The Hammer.— The Storing of Energy.— The Fly-
wheel.— The Punching Machine 250

LECTURE XVII.
CIRCULAR MOTION.

The Nature of Circular Motion. — Circular motion in Liquids. —

The Applications of Circular Motion. — The Permanent Axes 267

LECTURE XVIII.
THE SIMPLE PENDULUM.

Introduction. — The Circular Pendulum. — Law connecting the Time
of Vibration with the Length.— The Force of Gravity deter-
mined by the Pendulum. — The Cycloid 284

LECTURE XIX.

THE COMPOUND PENDULUM AND THE COMPOSITION
OF VIBRATIONS.

The Compound Pendulum. — The Centre of Oscillation. — The
Centre of Percussion. — The Conical Pendulum. — The Com-
position of Vibrations 299

TABLE OF CONTENTS. xiii

LECTURE XX.
THE MECHANICAL PRINCIPLES OF A CLOCK.

PAGE

Introduction. — The Compensating Pendulum. — The Escapement.

—The Train of Wheels.— The Hands.— The Striking Tarts .318

APPENDIX I.

The Method of Graphical Construction 339

The Method of Least Squares 342

APPENDIX II.

Details of the Willis Apparatus used in illustrating the foregoing

lectures • 345

INDEX . 355

EXPERIMENTAL MECHANICS

EXPERIMENTAL MECHANICS.

LECTURE I.
THE COMPOSITION OF FORCES.

Introduction. — The Definition of Force. — The Measurement of Force.
— Equilibrium of Two Forces. — Equilibrium of Three Forces. — A
Small Force can sometimes balance Two Larger Forces.

INTRODUCTION.

i. I SHALL endeavour in this course of lectures to illus-
trate the elementary laws of mechanics by means of experi-
ments. In order to understand the subject treated in this
manner, you need not possess any mathematical knowledge
beyond an acquaintance with the rudiments of algebra and
with a few geometrical terms and principles. But even
to those who, having an acquaintance with mathematics,
have by its means acquired a knowledge of mechanics,
experimental illustrations may still be useful. By actually
seeing the truth of results with which you are theoretically
familiar, clearer conceptions may be produced, and perhaps
new lines of thought opened up. Besides, many of the
mechanical principles which lie rather beyond the scope of
elementary works on the subject are very susceptible of

2 EXPERIMENTAL MECHANICS. [LECT.

being treated experimentally; and to the consideration of
these some of the lectures of this course will be devoted.

Many of our illustrations will be designedly drawn from
very commonplace sources : by this means I would try to
impress upon you that mechanics is not a science that exists
in books merely, but that it is a study of those principles
which are constantly in action about us. Our own bodies,
our houses, our vehicles, all the implements and tools which
are in daily use — in fact all objects, natural and artificial,
contain illustrations of mechanical principles. You should
acquire the habit of carefully studying the various mechanical
contrivances which may chance to come before your notice.
Examine the action of a crane raising weights, of a canal
boat descending through a lock. Notice the way a roof is
made, or how it is that a bridge can sustain its load. Even
a well-constructed farm-gate, with its posts and hinges, will
give you admirable illustrations of the mechanical principles
of frame-work. Take some opportunity of examining the
parts of a clock, of a sewing-machine, and of a lock and
key; visit a saw-mill, and ascertain the action of all the
machines you see there ; try to familiarize yourself with the
principles of the tools which are to be found in any work-
shop. A vast deal ef interesting and useful knowledge is
to be acquired in this way.

THE DEFINITION OF FORCE.

2. It is necessary to know the answer to this question,
What is a force ? People who have not studied mechanics
occasionally reply, A push is a force, a steam-engine is a
force, a horse pulling a cart is a force, gravitation is a force,
a movement is a force, &c., &c. The true definition of
force is that which tends to produce or to destroy motion. You

I.] THE DEFINITION OF FORCE. 3

may probably not fully understand this until some further
explanations and illustrations shall have been given ; but,
at all events, put any other notion of force out of your mind.
Whenever I use the word Force, do you think of the words
" something which tends to produce or to destroy motion,"
and I trust before the close of the lecture you will under-
stand how admirably the definition conveys what force
really is.

3. When a string is attached to this small weight, I can,
by pulling the string, move the weight along the table. In
this case, there is something transmitted from my hand along
the string to the weight in consequence of which the weight
moves : that something is a force. I can also move the
weight by pushing it with a stick, because force is transmitted
along the stick, and makes itself known by producing
motion. The archer who has bent his bow and holds the
arrow between his finger and thumb feels the string pulling
until the impatient arrow darts off. Here motion has been
produced by the force of elasticity in the bent bow. Before
he released the arrow there was no motion, yet still the bow
was exerting force and tending to produce motion. Hence
in defining force we must say " that which tends to pro-
duce motion," whether motion shall actually result or not.

4. But forces may also be recognized by their capability
or tendency to prevent or to destroy motion. Before I re-
lease the arrow I am conscious of exerting a force upon it
in order to counteract the pull of the string. Here my force
is merely manifested by destroying the motion that, if it
were absent, the bow would produce. So when I hold a
weight in my hand, the force exerted by my hand destroys
the motion that the weight would acquire were I to let it
fall ; and if a weight greater than I could support were
placed in my hand, my efforts to sustain it would still be

B 2

4 EXPERIMENTAL MECHANICS. [LECT.

properly called force, because they tended to destroy motion,
though unsuccessfully. We see by these simple cases that
a force may be recognized either by producing motion or by
trying to produce it, by destroying motion or by tending to
destroy it; and hence the propriety of the definition of
force must be admitted.

THE MEASUREMENT OF FORCE.

5. As forces differ in magnitude, it becomes necessary to
establish some convenient means of expressing their measure-
ments. The pressure exerted by one pound weight at London
is the standard with which we shall compare other forces.
The piece of iron or other substance which is attracted to
the earth with this force in London, is attracted to the
earth with a greater force at the pole and a less force at the
equator ; hence, in order to define the standard force, we
have to mention the locality in which the pressure of the
weight is exerted.

It is easy to conceive how the magnitude of a pushing or
a pulling force may be described as equivalent to so many
pounds. The force which the muscles of a man's arm can
exert is measured by the weight which he can lift. If a
weight be suspended from an india-rubber spring, it is
evident the spring will stretch so that the weight pulls
the spring and the spring pulls the weight ; hence the
number of pounds in the weight is the measure of the force
the spring is exerting. In every case the magnitude of a
force can be described by the number of pounds expressing
the weight to which it is equivalent. There is another but
much more difficult mode of measuring force occasionally
used in the higher branches of mechanics (Art. 497), but
the simpler method is preferable for our present purpose.

I.] EQUILIBRIUM OF TWO FORCES. 5

6. The straight line in which a force tends to move the
body to which it is applied is called the direction of the
force. Let us suppose, for example, that a force of 3 Ibs.
is applied at the point A, Fig. i, tending to make A move
in the direction AB. A

standard line c of certain A c

length is to be taken. It is
supposed that a line of this
length represents a force of

i Ib. The line AB is to be measured, equal to three times
c in length, and an arrow-head is to be placed upon it to
show the direction, in which the force acts. Hence, by
means of a line of certain length and direction, and hav-
ing an arrow-head attached, we are able completely to
represent a force.

EQUILIBRIUM OF TWO FORCES.

7. In Fig. 2 we have represented two equal weights to which
strings are attached ;
these strings, after pass-
ing over pulleys, are '••'-' — •

fastened by a knot c.

The- knot is pulled

by equal and opposite

forces. I mark off parts

CD, CE, to indicate the

forces ; and since there

is no reason why c

should move to one

side more than the other, it remains at rest. Hence, we

learn that two equal and directly opposed forces counteract

each other, and each may be regarded as destroying the

6 EXPERIMENTAL MECHANICS. [LECT.

motion which the other is striving to produce. If I make
the weights unequal by adding to one of them, the knot is
no longer at rest ; it instantly begins to move in the direction
of the larger force.

8. When two equal and opposite forces act at a point,
they are said to be in equilibrium. More generally this
word is used with reference to any set of forces which
counteract each other. When a force acts upon a body, at
least one more force must be present in order that the body
should remain at rest. If two forces acting on a point be
not opposite, they will not be in equilibrium ; this is easily
shown by pulling the knot c in Fig. 2 .downwards. When
released, it flies back again. This proves that if two forces
be in equilibrium their directions must be opposite, for
otherwise they will produce motion. We have already seen
that the two forces must be equal.

A book lying on the table is at rest. This book is acted
upon by two forces which, being equal and opposite, destroy
each other. One of these forces is the gravitation of the
earth, which tends to draw the book downwards, and which
would, in fact, make the book fall if it were not sustained
by an opposite force. The pressure of the book on the
table is often called the action, while the resistance offered
by the table is the force of reaction. We here see an
illustration of an important principle in nature, which says
that action and reaction are equal and opposite.

EQUILIBRIUM OF THREE FORCES.

9. We now come to the important case where three forces
act on a point: this is to be studied by the apparatus
represented in Fig. 3. It consists essentially of two pulleys

i.] EQUILIBRIUM OF THREE FORCES. 7

H,H, each about 2" diameter,1 which are capable of turning
very freely on their axles ; the distance between these
pulleys is about 5', and they are supported at a height of 6'

FIG. 3.

1 We shall often, in these lectures, represent feet or inches in the
manner usual among practical men — i' is one foot, i" is one inch.
Thus, for example, 3' 4" is to be read " three feet four inches." When
it is necessary to use fractions we shall always employ decimals. For
example, o"'5 is the mode of expressing a length of half an inch;
3' i"'9 is to be read "three feet one inch and nine-tenths of an inch."

EXPERIMENTAL MECHANICS. [LECT.

by a frame, which will easily be understood from the figure.
Over these pulleys passes a fine cord, 9' or 10' long, having
a light hook at each of the ends E,F. To the centre of this
cord D a short piece is attached, which at its free end G is also
furnished with a hook. A number of iron weights, 0-5 lb.}
i lb., 2 Ibs., &c., with rings at the top, are used ; one or more
of these can easily be suspended from the hooks as occasion
may require.

10. We commence by placing one pound on each of the
hooks. The cords are first seen to make a few oscillations
and then to settle into a definite position. If we disturb the
cords and try to move them into some new position they
will not remain there ; when released they will return to the
places they originally occupied. We now concentrate our
attention on the central point D, at which the three forces
act. Let this be represented by o in Fig. 4, and the lines
OP, OQ, and OS will be the directions of the three cords.

On examining these postions, we find
R that the three angles p o s, Q o s,

p o Q, are all equal. This may very
easily be proved by holding behind
the cords a piece of cardboard on
which three lines meeting at a point
and making equal angles have been
drawn ; it will then be seen that the
cords coincide with the three lines on
the cardboard.

11. A little reflection would have led us to anticipate
this result. For the three cords being each stretched by a
tension of a pound, it is obvious that the three forces pulling
at o are all equal. As o is at rest, it seems obvious that the
three forces must make the angles equal, for suppose that
one of the angles, P o Q for instance, was less than either of

I.] EQUILIBRIUM OF THREE FORCES. 9

the others, experiment shows that the forces o P and o Q would
be too strong to be counteracted by o s. The three angles
must therefore be equal, and then the forces are arranged
symmetrically.

12. The forces being each i lb., mark off along the three
lines in Fig. 4 (which represent their directions) three equal
parts o P; o Q, o s, and place the arrowheads to show the
direction in which each force is acting ; the forces are then
completely represented both in position and in magnitude.

Since these forces make equilibrium, each of them may
be considered to be counteracted by the other two. For
example, o s is annulled by o Q and o P. But o s could be
balanced by a force o R equal and opposite to it. Hence
o R is capable of producing by itself the same effect as the
forces o P and OQ taken together. Therefore o R is equiva-
lent to o P and o Q. Here we learn the important truth
that two forces not in the same direction can be replaced by
a single force. The process is called the composition of
forces, and the single force is called the resultant of the two
forces, o R is only one pound, yet it is equivalent to the
forces o P and o Q together, each of which is also one
pound. This is because the forces o P and o Q partly
counteract each other.

13." Draw the lines P R and Q R ; then the angles P o R and
Q o R are equal, because they are the supplements of the
equal angles P o s and Q o s ; and since the angles P o R and
Q o R together make up one-third of four right angles, it
follows that each of them is two-thirds of one right angle,
and therefore equal to the angle of an equilateral triangle.
Also o P being equal to o Q and o R common, the triangles
o P R and o Q R must be equilateral. Therefore the angle
? R o is equal to the angle R o Q ; thus p R is parallel to o Q :
similarly Q R is parallel to o p ; that is, o P R Q is a parallelo-

EXPERIMENTAL MECHANICS.

[LECT.

gram. Here we first perceive the great law that the resultant
of two forces acting at a point is the diagonal of a parallelo-
gram, of which they are the two sides.

14. This remarkable geometrical figure is called the
parallelogram of forces. Stated in its general form, the pro-
perty we have discovered asserts that two forces acting at a
point have a resultant, and that this resultant is represented
both in magnitude and in direction by the diagonal of the
parallelogram, of which two adjacent sides are the lines which
represent the forces.

15. The parallelogram of forces may be illustrated in
various ways by means of the apparatus of Fig. 3. Attach,
for example, to the middle hook 01-5 lb., and place i Ib. on
each of the remaining hooks E, F. Here the three weights
are not equal, and symmetry will not enable us, as it did in
the previous case, to foresee the condition which the cords

will assume ; but they will be observed
to settle in a definite position, to which
they will invariably return if withdrawn
from it.

Let o P, o Q (Fig. 5) be the directions
of the cords ; o P and o Q being each of
the length which corresponds to i lb.,
while o s corresponds to i'5lb. Here,
as before, o P and o Q together may be
considered to counteract o s. But o s
could have been counteracted by an equal
and opposite force o R. Hence OR may be
regarded as the single force equivalent to
o P and o Q, that is, as their resultant ;
and thus it is proved experimentally that
these forces have a resultant. We can
further verify that the resultant is the diagonal of the

FIG. 5.

I.] EQUILIBRIUM OF THREE FORCES. n

parallelogram of which the equal forces are the sides.
Construct a parallelogram on a piece of cardboard having its
four sides equal, and one of the diagonals half as long again
as one of the sides. This may be done very easily by first
drawing one of the two triangles into which the diagonal
divides the parallelogram. The diagonal is to be produced
beyond the parallelogram in the direction o s. When the
cardboard is placed close against the cords, the two cords
will lie in the directions o P, o Q, while the produced diagonal
will be in the vertical o s. Thus the application of the
parallelogram of force is verified.

1 6. The same experiment shows that two unequal forces
may be compounded into one resultant. For in Fig. 5 the
two forces o P and o s may be considered to be counter-
balanced by the force o Q ; in other words, o Q must be
equal and opposite to a force which

is the resultant of o P and o S. ^

17. Let us place on the central
hook G a weight of 5 Ibs., and weights
of 3 Ibs. on the hook E and 4 Ibs.
on F. This is actually the case
shown in Fig. 3. The weights being
unequal, we cannot immediately infer
anything with reference to the position
of the cords, but still we find, as be-
fore, that the cords assume a definite
position, to which they return when
temporarily displaced. Let Fig. 6
represent the positions of the cords.
No two of the angles are in this case
equal. Still each of the forces is
counterbalanced by the other two. Each is therefore equal
and opposite to the resultant of the other two. Construct

s

FIG. 6.

12 EXPERIMENTAL MECHANICS. [LECT.

the parallelogram on cardboard, as can be easily done by form-
ing the triangle o P R, whose sides are 3, 4, and 5, and then
drawing o Q and R Q parallel to R P and o p. Produce the
diagonal o R to s. This parallelogram being placed behind
the cords, you see that the directions of the cords coincide
with its sides and diagonal, thus verifying the parallelogram
of forces in a case where all the forces are of different
magnitudes.

1 8. It is easy, by the application of a set square, to
prove that in this case the cords attached to the 3 Ib. and
4lb. weights are at right angles to each other. We could
have inferred, from the parallelogram of force, that this
must be the case, for the sides of the triangle o P R
are 3, 4, and 5 respectively, and since the square of 5 is 25,
and the squares of 3 and of 4 are 9 and 16 respectively, it
follows that the square of one side of this triangle is equal
to the sum of the squares of the two opposite sides, and
therefore this is a right-angled triangle (Euclid, i. 48).
Hence, since P R is parallel to o Q, the angle P o Q must also
be a right angle.

A SMALL FORCE SOMETIMES BALANCES TWO
LARGER FORCES.

19. Cases might be multiplied indefinitely by placing
various amounts of weight on the hooks, constructing the
parallelogram on cardboard, and comparing it with the cords
as before. We shall, however, confine ourselves to one
more illustration, which is capable of very remarkable appli-
cations. Attach i Ib. to each of the hooks E and F ; the cord
joining them remains straight until drawn down by placing a
weight on the centre hook. A very small weight will suffice to
do this. Let us put on half-a-pound ; the position the cords

i.] A SMALL FORCE BALANCING TWO LARGER. 13

then assume is indicated in Fig. 7. As before, each force is

equal and opposite to the resultant of the other two. Hence

a force of half-a-pound is the K

resultant of two forces each of

i Ib. The apparent paradox

is explained by noticing that the

forces of i Ib. are very nearly

opposite, and therefore to a

large extent counteract each

other. Constructing the cardboard parallelogram we may

easily verify that the principle of the parallelogram of

forces holds in this case also.

20. No matter how small be the weight we suspend from
the middle of a horizontal cord, you see that the cord is
deflected : and no matter how great a tension were applied,
it would be impossible to straighten the cord. The cord
could break, but it could not again become horizontal.
Look at a telegraph wire ; it is never in a straight line
between two consecutive poles, and its curved form is more
evident the greater be the distance between the poles. But
in putting up a telegraph wire great straining force is used,
by means of special machines for the purpose ; yet the wires
cannot be straightened: because the weight of the heavy wire
itself acts as a force pulling it downwards. Just as the cord in
our experiments cannot be straight when any force,' however
small, is pulling it downwards at the centre, so it is impos-
sible by any exertion of force to straighten the long wire.
Some further illustrations of this principle will be given in
our next lecture, and with one application of it the present
will be concluded.

21. One of the most important practical problems in
mechanics is to make a small force overcome a greater.
There are a number of ways in which this may be

14 EXPERIMENTAL MECHANICS. [LECT.

accomplished for different purposes, and to the consideration
of them several lectures of this course will be devoted.
Perhaps, however, there is no arrangement more simple
than that which is furnished by the principles we have
been considering. We shall employ it to raise a 28 Ib.
weight by means of a 2 Ib. weight. I do not say that
this particular application is of much practical use. I show
it to you rather as a remarkable deduction from the
parallelogram of forces than as a useful machine.

A rope is attached at one end of an upright, A (Fig. 8),

and passes over a pulley B at the same vertical height
about 1 6' distant. A weight of 28 Ibs. is fastened to the
free end of the rope, and the supports must be heavily
weighted or otherwise secured from moving. The rope AB is
apparently straight and horizontal, in consequence of its weight
being inappreciable in comparison with the strain (28 Ibs.)
to which it is subjected; this position is indicated in the
figure by the dotted line AB. We now suspend from c at
the middle of the rope a weight of 2 Ibs. Instantly the
rope moves to the position represented in the figure. But
this it cannot do without at the same moment raising
slightly the 28 Ibs., for, since two sides of a triangle, CB,

I.] A SMALL FORCE BALANCING TWO LARGER. 15

CA, are greater than the third side, AB, more of the rope
must lie between the supports when it is bent down by
the 2 Ib. weight than when it was straight. But this can
only have taken place by shortening the rope between
the pulley B and the 28 Ib. weight, for the rope is
firmly secured at the other end. The effect on the heavy
weight is so small that it is hardly visible to you from a
distance. We can, however, easily show by an electrical
arrangement that the big weight has been raised by the
little one.

22. When an electric current passes through this alarum
you hear the bell ring, and the moment I stop the current
the bell stops. I have fastened one piece of brass to the
28 Ib. weight, and another to the support close above it,
but unless the weight be raised a little the two will not
be in contact ; the electricity is intended to pass from one
of these pieces of brass to the other, but it cannot pass
unless they are touching. When the rope is straight the
two pieces of brass are separated, the current does not
pass, and our alarum is dumb ; but the moment I hang
on the 2 Ib. weight to the middle of the rope it raises the
weight a little, brings the pieces of brass in contact, and
now you all hear the alarum. On removing the 2 Ibs. the
current is interrupted and the noise ceases.

23. I am sure you must all have noticed that the 2 Ib.
weight descended through a distance of many inches, easily
visible to all the room ; that is to say, the small weight
moved through a very considerable distance, while in so
doing it only raised the larger one a very small distance.
This is a point of the very greatest importance ; I there-
fore take the first opportunity of calling your attention
to it.

LECTURE II.
THE RESOLUTION OF FORCES.

Introduction. — One Force resolved into Two
Forces. — Experimental Illustrations. —
Sailing. — One Force resolved into Three
Forces not in the same Plane. — The Jib
and Tie-rod.

INTRODUCTION.

24. As the last lecture was princi-
pally concerned with discussing how
one force could replace two forces, so
in the present we shall examine the
converse question, How may two forces
replace one force ? Since the diagonal
of a parallelogram represents a single
force equivalent to those represented
by the sides, it is obvious that one force
may be resolved into two others, pro-
vided it be the diagonal of the paral-
lelogram formed by them.

25. We shall frequently employ in
the present lecture, and in some of
those that follow, the spring balance,
which is represented in Fig. 9 : the
weight is attached to the hook, and
when the balance is suspended by the

L. li.] ONE FORCE RESOLVED INTO TWO FORCES. 17

ring, a pointer indicates the number of pounds on a scale.
This balance is very convenient for showing the strain along
a cord ; for this purpose the balance is held by the ring
while the cord is attached to the hook. It will be noticed
that the balance has two rings and two corresponding hooks.
The hook and ring at the top and bottom will weigh up to
300 Ibs., corresponding to the scale which is seen. TJie
hook and ring at the side correspond to another scale on
the other face of the plate : this second scale weighs up to
about 50 Ibs., consequently for a weight under 50 Ibs. the
side hook and ring are employed, as they give a more
accurate result than would be obtained by the top and bot-
tom hook and ring, which are intended for larger weights.
These ingenious and useful balances are sufficiently accurate,
and can easily be tested by raising known weights. Besides
the instrument thus described, we shall sometimes use one
of a smaller size, and we shall be able with this aid to trace
the existence and magnitude of forces in a most convenient
manner.

ONE FORCE RESOLVED INTO TWO FORCES.

26. We shall first illustrate how a single force may be
resolved into a pair of forces ; for this purpose we shall
use the arrangement shown in Fig. 10 (see next page).

The ends of a cord are fastened to two small spring
balances ; to the centre E of this cord a weight of 4 Ibs.
is attached. At A and B are pegs from which the balances
can be suspended. Let the distances AE, BE be each 12",
and the distance AB 16". When the cord is thus placed,
and the weight allowed to hang freely, each of the cords EA,
EB is strained by an amount of force that is shown to be very
nearly 3 Ibs. by the balances. But the weight of 4 Ibs. is the

i a

EXPERIMENTAL MECHANICS.

[LECT.

only weight acting ; hence it must be equivalent to two
forces of very nearly 3 Ibs. each along the directions AE and
BE. Here the two forces to which 4 Ibs. is equivalent are
each of them less than 4 Ibs., though taken together they
exceed it.

27. But remove the cords from AB and hang them on CD,
the length CD being i' 10", then the forces shown along
FC and FD are each 5 Ibs. ; here, therefore, one force of

4 Ibs. is equivalent to two forces each of 5 Ibs. In the last
lecture (Art. 19) we saw that one force could balance two
greater forces ; here we see the analogous case of one force
being changed into two greater forces. Further, we learn
that the number of pairs of forces into which one force
may be decomposed is unlimited, for with every different
distance between the pegs different forces will be indicated
by the balances.

Whenever the weight is suspended from a point half-
way between the balances, the forces along the cords are

I!.] EXPERIMENTAL ILLUSTRATIONS. 19

equal ; but by placing the weight nearer one balance than
the other, a greater force will be indicated on that balance
to which the weight is nearest.

EXPERIMENTAL ILLUSTRATIONS.
28. The resolution or decomposition of one force into
two forces each greater than itself is capable of being

illustrated in a variety of ways, two of which will be here
explained. In Fig. n an arrangement for this purpose is
shown. A piece of stout twine AB, able to support from
20 Ibs. to 30 Ibs., is fastened at one end A to a fixed support,
and at the other end B to the eye of a wire-strainer. A

0 2

EXPERIMENTAL MECHANICS.

[LECT.

wire-strainer consists of an iron rod, with an eye at one end
and a screw and a nut at the other ; it is used for tightening
wires in wire fencing, and is employed in this case for the
purpose of stretching the cord. This being done, I take a
piece of ordinary sewing-thread, which is of course weaker
than the stout twine. I tie the thread to the middle of the

cord at c, catch the other end in my fingers, and pull ;
something must break — something has broken : but what
has broken ? Not the slight thread, it is still whole ; it is
the cord which has snapped. Now this illustrates the point
on which we have been dwelling. The force which I
transmitted along the thread was insufficient to break it ;

ii.] SAILING. 21

the thread transferred the force to the cord, but under such
circumstances that the force was greatly magnified, and the
consequence was that this magnified force was able to break
the cord before the original force could break the thread.
We can also see why it was necessary to stretch the cord.
In Fig. 10 the strains along the cords are greater when the
cords are attached at c and D than when they are attached
at A and B ; that is to say, the more the cord is stretched
towards a straight line, the greater are the forces into which
the applied force is resolved.

29. We give a second example, in illustration of the same
principle.

In Fig. 12 is shown a chain 8' long, one end of which B
is attached to a wire-strainer, while the other end is fastened
to a small piece of pine A, which is o"'5 square in section,
and 5" long between the two upright irons by which it is
supported. By means of the nut of the wire-strainer I
straighten the chain as I did the string of Fig. n, and for
the same reason. I then put a piece of twine round the
chain and pull it gently. The strain brought to bear on
the wood is so great that it breaks across. Here, the
small force of a few pounds, transmitted to the chain by
pulling the string, is magnified to upwards of a hundred-
weight, for less than this would not break the wood. The
explanation is precisely the same as when the string was
broken by the thread.

SAILING.

30. The action of the wind upon the sails of a vessel
affords a very instructive and useful example of the de-
composition of forces. By the parallelogram of forces we
are able to explain how it is that a vessel is able even to sail

EXPERIMENTAL MECHANICS.

[LECT.

against the wind. A force is that which tends to produce
motion, and motion generally takes place in the line of the
force. In the case of the action of wind on a vessel through
the medium of the sails, we have motion produced which is
not necessarily in the direction of the wind, and which may
be to a certain extent opposed to it. This apparent paradox
requires some elucidation.

FIG. 13.

31. Let us first suppose the wind to be blowing in a
direction shown by the arrows of Fig. 13, perpendicular to
the line AB in which the ship's course lies.

In what direction must the sail be set ? It is clear that
the sail must not be placed along the line AB, for then the
only effect of the wind would be to blow the vessel sideways ;
nor could the sail be placed with its edge to the wind, that

II.] SAILING. 23

is, along the line o w, for then the wind would merely glide
along the sail without producing a propelling force. Let,
then, the sail be placed between the two positions, as in the
direction p Q. The line o w represents the magnitude of the
force of the wind pressing on the saiL

We shall suppose for simplicity that the sail extends on
both sides of o. Through o draw o R perpendicular to p Q,
and from w let fall the perpendicular w x on p Q, and w R on
o R. By the principle of the parallelogram of forces, the
force o w may be decomposed into the two forces o x and
o R, since these are the sides of the parallelogram of which
o w, the force of the wind, is the diagonal. We may then
leave o w out of consideration, and imagine the force of the
wind to be replaced by the pair of forces o x and o R ; but
the force ox cannot produce an effect, it merely represents
a" force which glides along the surface of the sail, not one
which pushes against it ; so far as this component goes, the
sail has its edge towards it, and therefore the force produces
no effect. On the other hand, the sail is perpendicular to
the force o R, and this is therefore the efficient component.

The force of the wind is thus measured by o R, both in
magnitude and direction : this force represents the actual
pressure on the mast produced by the sail, and from the
mast communicated to the ship. Still o R is not in the
direction in which the ship is sailing : we must again de-
compose the force in order to find its useful effect. This
is done by drawing through R the lines R L and R M parallel
to OA and ow, thus forming the parallelogram OMRL.
Hence, by the parallelogram of forces, the force. OR is equi-
valent to the two forces o L and o M.

The effect of o L upon the vessel is to propel it in a
direction perpendicular to that in which it is sailing. We
must, therefore, endeavour to counteract this force as far as

24 EXPERIMENTAL MECHANICS. [LECT.

possible. This is accomplished by the keel, and the form
of the ship is so designed as to present the greatest
possible resistance to being pushed sideways through the
water : the deeper the keel the more completely is the effect
of o L annulled. Still o L would in all cases produce some
leeway were it not for the rudder, which, by turning the
head of the vessel a little towards the wind, makes her sail in
a direction sufficiently to windward to counteract the small
effect of o L in driving her to leeward.

Thus o L is disposed of, and the only force remaining is
o M, which acts directly to push the vessel in the required
direction. Here, then, we see how the wind, aided by the
resistance of the water, is able to make the vessel move in
a direction perpendicular to that in which the wind blows.
We have seen that the sail must be set somewhere between
the direction of the wind and that of the ship's motion. It
can be proved that when the direction of the sail supposed
to be flat and vertical, is such as to bisect the angle w o B,
the magnitude of the force o M is greater than when the sail
has any other position.

32. The same principles show how a vessel is able to
sail against the wind : she cannot, of course, sail straight
against it, but she can sail within half a right angle of it, or
perhaps even less. This can be seen from Fig. 14.

The small arrows represent the wind, as before. Let o w
be the line parallel to them, which measures the force of the
wind, and let the sail be placed along the line P Q ; o w is
decomposed into o x and o Y, o x merely glides along the
sail, and o Y is the effective force. This is decomposed into
o L and o M ; o L is counteracted, as already explained, and
o M is the farce that propels the vessel onwards. Hence we
see that there is a force acting to push the vessel onwards,
even though the movement be partly against the wind.

SAILING.

-5

It will be noticed in this case that the force o L acting to
leewards exceeds OM pushing onwards. Hence it is that
vessels with a very deep keel, and therefore opposing very
great resistance to moving leewards, can sail more closely to
the wind than others not so constructed ; a vessel should be
formed so that she shall move as freely as possible in the
direction of her length, for which reason she is sharpened at
the bow, and otherwise shaped for gliding through the water
easily ; this is in order that o M may have to overcome as

little resistance as possible. If the sail were flat and vertical
it should bisect the angle A o w for the wind to act in the
most efficient manner. Since, then, a vessel can sail to-
wards the wind, it follows that, by taking a zigzag course,
she can proceed from one port to another, even though
the wind be blowing from the place to which she would go
towards the place from which she comes. This well-
known manoeuvre is called "tacking." You will understand
that in a sailing-vessel the rudder has a more important part

EXPERIMENTAL MECHANICS.

[LECT.

to play than in a steamer : in the latter it is only useful for
changing the direction of the vessel's motion, while in the
former it is not only necessary for changing the direction,
but must also be used to keep the vessel to her course by
counteracting the effect of leeway.

ONE FORCE RESOLVED INTO THREE FORCES
NOT IN THE SAME PLANE.

33. Up to the present we have only been considering
forces which lie in the same plane, but in nature we meet

Fia. 15.

with forces acting in all directions, and therefore we must
not be satisfied with confining our inquiries to the simpler
case. We proceed to show, in two different ways, how a
force can be decomposed into three forces not in the same
plane, though passing through the same point. The first
mode of doing so is as follows. To three points A, B, c

ONE FORCE RESOLVED INTO THREE.

(Fig. 15) three spring balances are attached; A, B, c are not
in the same straight line, though they are at the same ver-
tical height : to the spring balances cords are attached, which
unite in a point o, from which a weight w is suspended.
This weight is supported by the three cords, and the strains
along these cords are indicated by the spring balances. The
greatest strain is on the shortest cord and the least strain on
the longest Here the force w Ibs. produces three forces
which, taken together, exceed its own amount. If I add an
equal weight w, I find, as we might have anticipated, that
the strains indicated by the scales are precisely double what
they were before. Thus we see that the proportion of the
force to each of the components into which it is decomposed
does not depend on the actual magnitude of the force, but
on the relative direction of the force and its components.

34. Another mode of show-
ing the decomposition of one
force into three forces not in
the same plane is represented in
Fig. 1 6. The tripod is formed
of three strips of pine, 4' x
o"-5 x o//-5, secured by a piece
of wire running through each
at the top ; one end of this
wire hangs down, and carries a
hook to which is attached a
weight of 28 Ibs. This weight is
supported by the wire, but the
strain on the wire must be borne
by the three wooden rods :
hence there is a force acting
downwards through the wooden
rods. We cannot render this manifest by a contrivance like

28 EXPERIMENTAL MECHANICS. [LECT.

the spring scales> because it is a push instead of a pull.
However, by raising one of the legs I at once become
aware that there is a force acting downwards through it.
The weight is, then, decomposed into three forces, which
act downwards through the legs ; these three forces are riot
in a plane, and the three forces taken together are larger
than the weight.

35. The tripod is often used for supporting weights ; it is
convenient on account of its portability, and it is very steady.
You may judge of its strength by the model represented in
the figure, for though the legs are very slight, yet they sup-
port very securely a considerable weight. The pulleys by
means of which gigantic weights are raised are often sup-
ported by colossal tripods. They possess stability and
steadiness in addition to great strength.

36. An important point may be brought out by contrast-
ing the arrangements of Figs. 15 and 16. In the one case
three cords are used, and in the other three rods. Three
rods would have answered for both, but three cords would
not have done for the tripod. In one the cords are strained,
and the tendency of the strain is to break the cords, but in
the other the nature of the force down the rods is entirely
different ; it does not tend to pull the rod asunder, it is
trying to crush the rod, and had the weight been large
enough the rods would bend and break. I hold one end of
a pencil in each hand and then try to pull the pencil asunder ;
the pencil is in the condition of the cords of Fig. 15;
but if instead of pulling I push my hands together, the pencil
is like the rods in Fig. 16.

37. This distinction is of great importance in mechanics.
A rod or cord is in a state of tension is called a " tie " ; while
a rod in a state of compression is called a " strut." Since a
rod can resist both tension and compression it can serve

ii.] THE JIB AND TIE ROD. 29

either as a tie or as a strut, but a cord or chain can only act
as a tie. A piilar is always a strut, as the superincumbent
load makes it to be in a state of compression. These
distinctions will be very frequently used during this course
of lectures, and it is necessary that they be thoroughly
understood.

THE JIB AND TIE ROD.

38. As an illustration of the nature of the "tie" and
" strut," and also for the purpose of giving a useful example
of the decomposition of forces, I use the apparatus of
Fig. 17 (see next page).

It represents the principle of the framework in the common
lifting crane, and has numerous applications in practical
mechanics. A rod cf wood BC 3' 6" long and i" X i'
section is capable of turning round its support at the bottom
B by means of a joint or hinge : this rod is called the "jib " ;
it is held at its upper end by a tie AC 3' long, which is
attached to the support above the joint. A B is one foot
long. From the point c a wire descends, having a hook at
the end on which a weight can .be hung. The tie is attached
to the spring balance, the index of which shows the strain.
The Spring balance is secured by a wire-strainer, by turn-
ing the nut of which the length of the wire can be shortened
or lengthened as occasion requires. This is necessary,
because when different weights are suspended from the hook
the spring is stretched more or less, and the screw is then
employed to keep the entire length of the tie at 3'. The
remainder of the tie consists of copper wire.

39. Suppose a weight of 20 Ibs. be suspended from the
hook w, it endeavours to pull the top of the jib downwards :
but the tie holds it back, consequently the tie is put into a

EXPERIMENTAL MECHANICS.

[LECT.

state of tension, as indeed its name signifies, and the magni-
tude of that tension is shown to be 60 Ibs. by the spring-
balance. Here we find again what we have already so often
referred to , namely, one force developing another force that
is greater than itself, for the strain along the tie is three times

Flu. 17.

as great as the strain in the vertical wire by which it was
produced.

40. What is the condition of the jib ? It is evidently
being pushed downwards on its joint at B ; it is therefore in
a state of compression ; it is a strut. This will be evident
if we think for a moment how absurd it would be to

11.] THE JIB AND TIE ROD. 31

deavour to replace the jib by a string or chain : the whole
arrangement would collapse. The weight of 20 Ibs. is there-
fore decomposed by this contrivance into two other forces,
one of which is resisted by a tie and the other by a strut.

41. We have no means of showing the magnitude of the
strain along the strut, but we shall prove that it can be
computed by means of the parallelogram of force ; this will
also explain how it is that the tie is strained by a force three
times that of the weight which is used. Through c (Fig. 18)
draw c P parallel to the tie A B, and P Q parallel to the strut

C B then B p is the diagonal of the parallelogram whose sides
are each equal to B c and B Q. If therefore we consider the
force of 20 Ibs. to be represented by B p, the two forces into
which it is decomposed will be shown by B Q and B c ; but
A B is equal to B Q, since each of them is equal to c P ; also
E F is equal to A c. Hence the weight of 20 Ibs. being re-
presented by A c, the strain along the tie will be represented
by the length A B, and that along the strut by the length B c.
Remembering that AB is 3' long, c B 3' 6", and AC i', it
follows that the strain along the tie is 60 Ibs., and along the

32 EXPERIMENTAL MECHANICS. [LECT.

strut 70 Ibs., when the weight of 20 Ibs. is suspended from
the hook.

42. In every other case the strains along the tie and strut
can be determined, when the suspended weight is known,
by their proportionality to the sides of the triangle formed
by the tie, the jib, and the upright post, respectively.

43. In this contrivance you will recognize, no doubt, the
framework of the common lifting crane, but that very essential
portion of the crane which provides for the raising and
lowering is not shown here. To this we shall return again
in a subsequent lecture (Art. 332). You will of course
understand that the tie rod we have been considering is
entirely different from the chain for raising the load.

44. It is easy to see of what importance to the engineer
the information acquired by means of the decomposition of
forces may become. Thus in the simple case with which we
are at present engaged, suppose an engineer were required
to erect a frame which was to sustain a weight of 10 tons,
let us see how he would be enabled to determine the strength
of the tie and jib. It is of importance in designing any
structure not to make any part unnecessarily strong, as
doing so involves a waste of valuable material, but it is of
still more vital importnnce to make every part strong enough
to avoid the risk of accident, not only under ordinary circum-
stances, but also under the exceptionally great shocks and
strains to which every machine is liable.

45. According to the numerical proportions we have
employed for illustration, the strain along the tie rod would
be 30 tons when the load was 10 tons, and therefore the
tie must at least be strong enough to bear a pull of 30 tons ;
but it is customary, in good engineering practice, to make
the machine of about ten times the strength that would just
be sufficient to sustain the ordinary load. Hence the crank

II.] THE JIB AND TIE ROD. 33

must be so strong that the tie would not break with a
tension less than 300 tons, which would be produced when
the crane was lifting 100 tons. So great a margin of safety
is necessary on account of the jerks and other occasional
great strains that arise in the raising and the lowering of
heavy weights. For a crane intended to raise 10 tons, the
engineer must therefore design a tie rod which not less
than 300 tons would tear asunder. It has been proved by
actual trial that a rod of wrought iron of average quality, one
square inch in section, can just withstand a pull of twenty
tons. Hence fifteen such rods, or one rod the section of
which was equal to fifteen square inches, would be just able
to resist 300 tons ; and this is therefore the proper area of
section for the tie rod of the crane we have been
considering.

46. In the same way we ascertain the actual thrust down
the jib; it amounts to 35 tons, and the jib should be ten
times as strong 'as a strut which would collapse under a
strain of 35 tons.

47. It is easy to see from the figure that the tie rod is
pulling the upright, and tending, in fact, to make it snap
off near B. It is therefore necessary that the upright support
A B (Fig. 17) be secured very firmly.

LECTURE III.
PARALLEL FORCES.

Introduction. — Pressure of a Loaded Beam on its Supports. — Equi-
librium of a Bar supported on a Knife-edge. — The Composition or
Parallel Forces. — Parallel Forces acting in opposite directions. —
The Couple.— The Weighing Scales.

INTRODUCTION.

48. THE parallelogram of forces enables us to find the
resultant of two forces which intersect: but since parallel
forces do not intersect, the construction does not avail
to determine the resultant of two parallel forces. We
can, however, find this resultant very simply by other
means.

49. Fig. 1 9 represents a wooden rod 4' long, sustained by
resting on two supports A and B, and having the length A B
divided into 14 equal parts. Let a weight of 14 Ibs. be
hung on the rod at its middle point c ; this weight must be
borne by the supports, and it is evident that they will bear

LECT. HI.] PRESSURE OF A LOADED BEAM. 35

the load in equal shares, for since the weight is at the middle
of the rod there is no reason why one end should be
differently circumstanced from the other. Hence the total
pressure on each of the supports will be 7 Ibs., together
with half the weight of the wooden bar.

50. If the weight of 14 Ibs. be placed, not at the centre
of the bar, but at some other point such as D, it is not
then so easy to see in what proportion the weight is
distributed between the supports. We can easily understand
that the support near the weight must bear more than the
remote one, but how much more ? When we are able
to answer this question, we shall see that it will lead us to
a knowledge of the composition of parallel forces.

PRESSURE OF A LOADED BEAM ON ITS
SUPPORTS.

51. To study this question we shall employ the apparatus
shown in Fig. 20. An iron bar 5' 6" long, weighing 10
Ibs., rests in the hooks of the spring balances A,C, in the
manner shown in the figure. These hooks are exactly five
feet apart, so that the bar projects 3" beyond each end.
The space between the hooks is divided into twenty equal
portions, each of course 3" long. The bar is sufficiently
strong"to bear the weight B of 20 Ibs. suspended from it by
an S hook, without appreciable deflection. Before the
weight of 20 Ibs. is suspended, the spring balances each
show a strain of 5 Ibs. We would expect this, for it is
evident that the whole weight of the bar amounting to
i o Ibs. should be borne equally by the two supports.

52. When I place the weight in the middle, 10 divisions
from each end, I find the balances each indicate 15 Ibs.
But 5 Ibs. is due to the weight of the bar. Hence the
20 Ibs. is divided equally, as we have already stated that it

T) 2

EXPERIMENTAL MECHANICS.

[LECT.

should be. But let the 20 Ibs. be moved to any other
position, suppose 4 divisions from the right, and 16 from
the left; then the right-hand scale reads 21 Ibs., and the
left-hand reads 9 Ibs. To get rid of the weight of the bar
itself, we must subtract 5 Ibs. from each. We learn there-
fore that the 20 Ib. weight pulls the right-hand spring
balance with a strain of 16 Ibs., and the left with a strain of

4 Ibs. Observe this closely; you see I have made the
number of divisions in the bar equal to the number of
pounds weight suspended from it, and here we find that
when the weight is 16 divisions from the left, the strain of
1 6 Ibs. is shown on the right. At the same time the weight
is 4 divisions from the right, and 4 Ibs. is the strain shown
to the left.

in.] PRESSURE OF A LOADED BEAM. 37

53. I will state the law of the distribution of the load
a little more generally, and we shall find that the bar will
prove the law to be true in all cases. Divide the bar into
as many equal parts as there are pounds in the load, then the
pressure in pounds on one end is the number of divisions that
the load is distant from the other.

54. For example, suppose I place the load 2 divisions
from one end : I read by the scale at that end 23 Ibs. ;
subtracting 5 Ibs. for the weight of the bar, the pressure
due to the load is shown to be 18 Ibs., but the weight is
then exactly 18 divisions distant from the other end. We
can easily verify this rule whatever be the position which
the load occupies.

55. If the load be placed between two marks, instead of
being, as we have hitherto supposed, exactly at one, the
partition of the load is also determined by the law. Were
it, for example, 3^5 divisions from one end, the strain
on the other would be 3*5 Ibs. ; and in like manner for
other cases.

56. We have thus proved by actual experiment this useful
and instructive law of nature ; the same result could have
been inferred by reasoning from the parallelogram of force,
but the purely experimental proof is more in accordance
with our scheme. The doctrine of the composition of
parallel forces is one of the most fundamental parts of
mechanics, and we shall have many occasions to employ
it in this as well as in subsequent lectures.

57. Returning now to Fig. 19, with which we com-
menced, the law we have discovered will enable us to
find how the weight is distributed. We divide the length
of the bar between the supports into 14 equal parts because
the weight is 14 Ibs. ; if, then, the weight be at D, 10
divisions from one end A, and 4 from the other B, the

38 EXPERIMENTAL MECHANICS. [LECT.

pressure at the corresponding ends will be 4 and 10. If
the weight were 2-5 divisions from one end, and therefore
1 1 -5 from the other, the shares in which this load would be
supported at the ends are 11*5 Ibs. and 2-5 Ibs. The
actual pressure sustained by each end is, however, about
6 ounces greater if the weight of the wooden bar itself be
taken into account.

58. Let us suspend a second weight from another point
of the bar. We must then calculate the pressures at the
ends which each weight separately would produce, and those
at the same end are to be added together, and to half the
weight of the bar, to find the total pressure. Thus, if one
weight of 20 Ibs. were in the middle, and another of 14 Ibs.
at a distance of n divisions from one end, the middle weight
would produce 10 Ibs. at each end and the 14 Ibs. would pro-
duce 3 Ibs. and 1 1 Ibs., and remembering the weight of the
bar, the total pressures produced would be 13 Ibs. 6 oz.
and 2 1 Ibs. 6 oz. The same principles will evidently apply to
the case of several weights : and the application of the rule
becomes especially easy when all the weights are equal, for
then the same divisions will serve for calculating the effect
of each weight.

59. The principles involved in these calculations are of
so much importance that we shall further examine them by a
different method, which has many useful applications.

EQUILIBRIUM OF A BAR SUPPORTED ON A
XNIFE-EDGE.

60. The weight of the bar has hitherto somewhat com-
plicated our calculations ; the results would appear more
simply if we could avoid this weight ; but since we want a
strong bar, its weight is not so small that we could afford to

in.] EQUILIBRIUM OF A BAR. 39

overlook it altogether. By means of the arrangement of

•M

Fig. 21, we can counterpoise the weight of the bar. To

40 EXPERIMENTAL MECHANICS. [LECT.

the centre of A B a cord is attached, which, passing over
a fixed pulley D, carries a hook at the other end. The bar,
being a pine rod, 4 feet long and i inch square, weighs
about 12 ounces ; consequently, if a weight of twelve ounces
be suspended from the hook, the bar will be counterpoised,
and will remain at whatever height it is placed.

61. A B is divided by lines drawn along it at distances of
i* apart ; there are thus 48 of these divisions. The weights
employed are furnished with rings large enough to enable
them to be slipped on the bar and thus placed in any desired
position.

62. Underneath the bar lies an important portion of
the arrangement ; namely, the knife-edge c. This is a blunt
edge of steel firmly fastened to the support which carries it.
This support can be moved along underneath the bar so
that the knife-edge can be placed under any of the divisions
required. The bar being counterpoised, though still un-
loaded with weights, may be brought down till it just touches
the knife-edge ; it will then remain horizontal, and will
retain this position whether the knife-edge be at either end
of the bar or in any intermediate position. I shall hang
weights at the extremities of the rod, and we shall find that
there is for each pair of weights just one position at which,
if the knife-edge be placed, it will sustain the rod horizon-
tally. We shall then examine the relations between these
distances and the weights that have been attached, and we
shall trace the connection between the results of this
method and those of the arrangement that we last used.

63. Supposing that 6 Ibs. be hung at each end of the
rod, we might easily foresee that the knife-edge should be
placed in the middle, and we find our anticipations verified.
When the edge is exactly at the middle, the rod remains
horizontal ; but if it be moved, even through a very small

in.] .EQUILIBRIUM OF A BAR. 41

distance, to either side, the rod instantly descends on the
other. The knife-edge is 24 inches distant from each end ;
and if I multiply this number by the number of pounds
in the weight, in this case 6, I find 144 for the product,
and this product is the same for both ends of the bar. The
importance of this remark will be seen directly.

64. If I remove one of the 6 Ib. weights and replace it
by 2 Ibs., leaving the other weight and the knife-edge un-
altered, the bar instantly descends on he side of the heavy
weight ; but, by slipping the knife-edge along the bar, I find
that when I have moved it to within a distance of 12 inches
from the 6 Ibs., and therefore 36 inches from the 2 Ibs., the
bar will remain horizontal. The edge must be put carefully
at the right place ; a quarter of an inch to one side or the
other would upset the bar. The whole load borne by
the knife edge is of course 8 Ibs., being the sum of the
weights. If we multiply 2, the number of pounds at one
end, by 36, the distance of that end from the knife-edge,
we obtain the product 72 ; and we find precisely the same
product by multiplying 6, the number of pounds in the
other weight, by 12, its distance from the knife-edge. To
express this result concisely we shall introduce the word
moment, a term of frequent use in mechanics. The 2 Ib.
weight produces a force tending to pull its end of the bar
downwards by making the bar turn round the knife-edge.
The magnitude of th'S force, multiplied into its distance

from the knife-edge, is called the moment of the force. We
can express the result at which we have arrived by saying
that, when the knife-edge has been so placed that the bar
remains horizontal, the moments of the forces about tJie
knife-edge are equal.

65. We may further illustrate this law by suspending
weights of 7 Ibs. and 5 Ibs. respectively from the ends of

42 EXPERIMENTAL MECHANICS. [LECT.

the bar ; it is found that the knife-edge must then be placed
20 inches from the larger weight, and, therefore, 28 inches
from the smaller, but 5 x 28 = 140, and 7 x 20 = 140,
thus again verifying the law of equality of the moments.

From the equality of the moments we can also deduce
the law for the distribution of the load given in Art. 53.
Thus, taking the figures in the last experiment, we have
loads of 7 Ibs. and 5 Ibs. respectively. These produce a
pressure of 7 + 5 = 12 Ibs. on the knife-edge. This edge
presses on the bar with an equal and opposite reaction. To
ascertain the distribution of this pressure on the ends of the
beam, we divide the whole beam into 12 equal parts of
4 inches each, and the 7 Ib. weight is 5 of these parts, i.e.,
20 inches distant from the support. Hence the edge
should be 20 inches from the greater weight, which is the
condition also implied by the equality of the moments.

THE COMPOSITION OF PARALLEL FORCES.

66. Having now examined the subject experimentally, we
proceed to investigate what may be learned from the results
we have proved.

The weight of the bar being allowed for in the way we
have explained, by subtracting one half of it from each
of the strains indicated by the spring balance (FiG. 20), we
may omit it from consideration. As the balances are pulled
downwards by the bar when it is loaded, so they will react
to pull the bar upwards. This will be evident if we
think of a weight — say 14 Ibs. — suspended from one of
these balances : it hangs at rest ; therefore its weight, which
is constantly urging it downwards, must be counteracted by
an equal force pulling it upwards. The balance of course
shows 14 Ibs. ; thus the spring exerts in an upward pull a

in.] PARALLEL FORCES. 43

force which is precisely equal to that by which it is itself
pulled downwards.

67. Hence the springs are exerting forces at the ends of
the bar in pulling them upwards, and the scales indicate
their magnitudes. The bar is thus subject to three forces,
viz. : the suspended weight of 20 Ibs., which acts ver-
tically downwards, and the two other forces which act
vertically upwards, and the united action of the three make
equilibrium.

68. Let lines be drawn, representing s
the forces in the manner already explained.

We have then three parallel forces AP,
BQ, CR acting on a rod in equilibrium
(Fig. 22). The two forces AP andBQ may Q
be considered as balanced by the force |

CR in the position shown in the figure, but B(

the force CR would be balanced by the y

equal and opposite force cs, represented i

by the dotted line. Hence this last

force is equivalent to AP and BQ. In

other words, it must be their resultant.

Here then we learn that a pair of parallel

forces, acting in the same direction, can FIG. 22.

be compounded into a single resultant.

69. We also see that the magnitude of the resultant is
equal to the sum of the magnitudes of the forces, and further
we find the position of the resultant by the following rule.
Add the two forces together ; divide the distance between
them into as many equal parts as are contained in the sum,
measure off from the greater of these two forces as many
parts as there are pounds in the smaller force, and that is
the point required. This rule is very easily inferred from
that which we were taught by the experiments in Art. 51.

44 EXPERIMENTAL MECHANICS. [LECT.

PARALLEL FORCES ACTING IN OPPOSITE
DIRECTIONS.

70. Since the forces AP, BQ, CR (Fig. 22) are in equili-
brium, it follows that we may look on BQ as balancing in
the position which it occupies the two forces of AP and CR
in their positions. This may remind us of the numerous
instances we have already met with, where one force balanced
two greater forces : in the present case AP and CR are acting
in opposite directions, and the force BQ which balances them
is equal to their difference. A force BT equal and opposite
to BQ must then be the resultant of CR and AP, since it is
able to produce the same effect. Notice that in this case
the resultant of the two forces is not between them, but that
it lies on the side of the larger. When the forces act in the
same direction, the resultant is always between them.

71. The actual position which the resultant of two oppo-
site parallel forces occupies is to be found by the following
rule. Divide the distance between the forces into as many
equal parts as there are pounds in their difference, then
measure outwards from the point of application of the larger
force as many of these parts as there are pounds in the
smaller force ; the point thus found determines the position
of the resultant. Thus, if the forces be 14 and 20, the
difference between them is 6, and therefore the distance
between their directions is divided into six parts ; from the
point of application of the force of 20, 14 parts are measured
outwards, and thus the position of the resultant is deter-
mined. Hence we have the means of compounding two
parallel forces in general

THE COUPLE.

72. In one case, however, two parallel forces have no
resultant ; this occurs when the two forces are equal, and in

in.] THE COUPLE. 45

opposite directions. A pair of forces of this kind is called a
couple; there is no single force which could balance a
couple, — it can only be counterbalanced by another couple
acting in an opposite manner. This remarkable case, may
be studied by the arrangement of Fig. 23.

A wooden rod, A B 48 " x o" -5 x o"-5, has strings
attached to it at points A and D, one foot distant. The
string at D passes over a fixed pulley E, and at the end p
a hook is attached for the purpose of receiving weights,
while a similar hook depends from A ; the weight of the rod

itself, which only amounts to three ounces, may be neglected,
as it is very small compared with the weights which will be used.
73. Supposing 2 Ibs. to be placed at P, and i Ib. at Q, we
have two parallel forces acting in opposite directions ; and
since their difference is i Ib., it follows from our rule that
the point F, where D F is equal to A D, is the point where
the resultant is applied. You see this is easily verified, for
by placing my finger over the rod at F it remains horizontal
and in equilibrium; whereas, when I move my finger to one
side or the other, equilibrium is impossible. If I move it
nearer to B, the end A ascends. If I move it towards A, the
end B ascends.

46 EXPERIMENTAL MECHANICS. [LgCT.

74. To study the case when the two forces are equal, a
load of 2 Ibs. may be placed on each of the hooks p and Q.
It will then be found that the finger cannot be placed in
any position along the rod so as to keep it in equilibrium ;
that is to say, no single force can counteract the two forces
which form the couple. Let o be the point midway be-
tween A and D. The forces evidently tend to raise o B and
turn the part o A downwards ; but if I try to restrain o B
by holding my finger above, as at the point x, instantly the
rod begins to turn round x and the part from A to x
descends. I find similarly that any attempt to prevent the
motion by holding my finger underneath is equally un-
successful. But if at the same time I press the rod down-
wards at one point, and upwards at another with suitable
force, I can succeed in producing equilibrium ; in this case
the two pressures form a couple ; and it is this couple which
neutralizes the couple produced by the weights. We learn,
then, the important result that a couple can be balanced by a
couple, and by a couple only.

75. The moment of a couple is the product ot one of the
two equal forces into their perpendicular distance. Two
couples tending to turn the body to which they are applied ,
in the same direction will be equivalent if their moments are
equal. Two couples which tend to turn the body in opposite
directions will be in equilibrium if their moments are equal.
We can also compound two couples in the same or in
opposite directions into a single couple of which the
moment is respectively either the sum or the difference of
the original moments.

THE WEIGHING SCALES.

76. Another apparatus by which the nature of parallel
forces may be investigated is shown in Fig. 24; it con-

IIl.j

THE WEIGHING SCALES.

sists of a slight frame of wood ABC, 4' long. At E, a pair
of steel knife-edges is clamped to the frame. The knife-
edges rest on two pieces of steel, one of which is shown at
o F. When the knife-edges are suitably placed the frame
is balanced, so that a small piece of paper laid at A will
cause that side to descend.

77. We suspend two small hooks from the points A and
B : these are made of fine wire, so that their weight may be

FIG. 24.

left out of consideration. With this apparatus we can in the
first place verify the principle of equality of moments : for
example, if I place the hook A at a distance of 9" from the
centre o and load it with i lb., I find that when B is laden
with 0-5 lb. it must be at a distance of 18" from o in order
to counterbalance A ; the moment in the one case is 9 x i,
in the other 18 x 0-5, and these are obviously equal

78. Let a weight of i lb. be placed on each of the
hooks, the frame will only be in equilibrium when the

48 EXPERIMENTAL MECHANICS. [LECT.

hooks are at precisely the same distance from the centre.
A familiar application of this principle is found in the
ordinary weighing scales ; the frame, which in this case is
called a beam, is sustained by two knife-edges, smaller, how-
ever, than those represented in the figure. The pans p, p
are suspended from the extremities of the beam, and should
be at equal distances from its centre. These scale-pans
must be of equal weight, and then, when equal weights are
placed in them, the beam will remain horizontal. If the
weight in one slightly exceed that in the other, the pan
containing the heavier weight will of course descend.

79. That a pair of scales should weigh accurately, it is
necessary that the weights be correct ; but even with correct
weights, a balance of defective construction will give an in-
accurate result. The error f equently arises from some
inequality in the lengths of the arms of the beam. When
this is the case, the two weights which really balance are
not equal. Supposing, for instance, that with an imperfect
balance I endeavour to weigh a pound of shot. If I put the
weight on the short side, then the quantity of shot balanced
is less than i Ib. ; while if the i Ib. weight be placed at the
long side, it will require mere than i Ib. of shot to produce
equilibrium. The mode of testing a pair of scales is then
evident. Let weights be placed in the pans which balance
each other ; if the weights be interchanged and the balance
still remains horizontal, it is correct.

80. Suppose, for example, that the two arms be i o inches
and 1 1 inches long, then, if i Ib. weight be placed in the pan
of the lo-inch end, its moment is 10 ; and if |£ of i Ib. be
placed in the pan belonging to the n-inch end, its moment
is also 10 : hence i Ib. at the short end balances {° of i Ib.
at the long end ; and therefore, if the shopkeeper placed
his weight in the short arm, his customers would lose T!T

in.] THE WEIGHING SCALES. 49

part of each pound for which they paid ; on the other hand,
if the shopkeeper placed his i Ib. weight on the long arm,
then not less than -]-£ Ib. would be required in the pan
belonging to the short arm. Hence in this case the cus-
tomer would get y1^ Ib. too much. It follows, that if a
shopman placed his weights and his goods alternately in the
one scale and in the other he would be a loser on the
whole ; for, though every second customer gets r:T Ib.
less than he ought, yet the others get -^ Ib. more than they
have paid for.

LECTURE IV.
THE FORCE OF GRA VITY.

Introduction. — Specific Gravity. — The Plummet and Spirit Level. —
The Centre of Gravity. — Stable and Unstable Equilibrium. —
Property of the Centre of Gravity in a Revolving Wheel.

INTRODUCTION.

8 1. IN the last three lectures we considered forces
in the abstract ; we saw how they are to be represented by
straight lines, how compounded together and how decom-
posed into others ; we have explained what is meant by forces
being in equilibrium, and we have shown instances where
the forces lie in the same plane or in different planes, and
where they intersect or are parallel to each other. These
subjects are the elements of mechanics ; they form the
framework which in this and subsequent lectures we shall
try to present in a more attractive garb. We shall commence
by studying the most remarkable force in nature, a force
constantly in action, and one to which all bodies are subject,
a force which distance cannot annihilate, and one the pro-
perties of which have led to the most sublime discoveries of
human intellect. This is the force of gravity.

LECT. iv.] THE FORCE OF GRAVITY. 51

82. If I drop a stone from my hand, it falls to the ground.
That which produces motion is a force : hence the stone
must have been acted upon by a force which drew it
to the ground. On every part of the earth's surface experi-
ence shows that a body tends to fall. This fact proves
that there is an attractive force in the earth tending to draw
all bodies towards it.

83. Let ABCD (Fig. 25) be points from which stones are
let fall, and let the circle represent the section of the earth ;

let P Q R s be the points at the surface of the earth upon which
the stones will drop when allowed to do so. The four
stones will move in the directions of the arrows : from A to P
the stone moves in an opposite direction to the motion from
c to R ; from B to Q it moves from right to left, while from
L) to s it moves from left to right. The movements are in
different directions; but if I produce these directions, as
indicated by the dotted lines, they each pass through the
centre o.

52 EXPERIMENTAL MECHANICS. [LECT.

84. Hence each stone in falling moves towards the centre
of the earth, and this is consequently the direction of the
force. We therefore assert that the earth has an attraction
for the stone, in consequence of which it tries to get as
near the earth's centre as possible, and this attraction is
called the force of gravitation.

85. We are so excessively familiar with the phenomenon
of seeing bodies fall that it does not excite our astonishment
or arouse our curiosity. A clap of thunder, which every
one notices, because much less frequent, is not really more
remarkable. We often look with attention at the attraction
of a piece of iron by a magnet, and justly so, for the
phenomenon is very interesting, and yet the falling of a
stone is produced by a far grander and more important force
than the force of magnetism.

86. It is gravity which causes the weight of bodies. I
hold a piece of lead in my hand : gravity tends to pull it
downwards, thus producing a pressure on my hand which
I call weight. Gravity acts with slightly varying intensity at
various parts of the earth's surface. This is due to two
distinct causes, one of which may be mentioned here, while
the other will be subsequently referred to. The earth is not
perfectly spherical ; it is flattened a little at the poles ; con-
sequently a body at the pole is nearer the general mass of
the earth than a body at the equator ; therefore the body at
the pole is more attracted, and seems heavier. A mass
which weighs 200 Ibs. at the equator would weigh one pound
more at the pole : about one-third of this increase is due to
the cause here pointed out. (See Lecture XVII.)

87. Gravity is a force which attracts every particle of
matter ; it acts not merely on those parts of a body which
lie on the surface, but it equally affects those in the interior.
This is proved by observing that a body has the same

iv.] SPECIFIC GRAVITY. 53

weight, however its shape be altered : for example,
suppose I take a ball of putty which weighs i lb., I shall
find that its weight remains unchanged when the ball is
flattened into a thin plate, though in the latter case the
surface, and therefore the number of superficial particles, is
larger than it was in the former.

SPECIFIC GRAVITY.

88. Gravity produces different effects upon different sub-
stances. This is commonly expressed by saying that some
substances are heavier than others ; for example, I have here
a piece of wood and a piece of lead of equal bulk. The lead
is drawn to the earth with a greater force than the wood.
Substances are usually termed heavy when they sink in water,
and light when they float upon it. But a body sinks in water if
it weigh more than an equal bulk of water, and floats if it
weigh less. Hence it is natural to take water as a standard
with which the weights of other substances may be compared.

89. I take a certain volume, say a cubic inch of cast
iron such as this I hold in my hand, and which has been
accurately shaped for the purpose. This cube is heavier
than one cubic inch of water, but I shall find that a certain
quantity of water is equal to it in weight ; that is to say, a
certain number of cubic inches of water, and it may be
fractional parts of a cubic inch, are precisely of the same
weight. This number is called the specific gravity of cast
iron.

90. It would be impossible to counterpoise water with the
iron without holding the water in a vessel, and the weight of
the vessel must then be allowed for. I adopt the following
plan. I have here a number of inch cubes of wood
(Fig. 26), which would each be lighter than a cubic inch
of water, but I have weighted the wooden cubes by placing

54

EXPERIMENTAL MECHANICS.

[LECT.

grains of shot into holes bored into the wood. The
weight of each cube has thus been accurately adjusted to be
equal to that of a cubic inch of water. This may be tested
by actual weighing. I weigh one of the cubes and find it
to be 252 grains, which is well known to be the weight of a
cubic inch of water.

91. But the cubes maybe shown to be identical in weight

with the same bulk of water by a simpler method. One of
them placed in water should have no tendency to sink, since
it is not heavier than water, nor on the other hand, since it
is not lighter, should it have any tendency to float. It
should then remain in the water in whatever position it may
be placed. It is difficult to prepare one of these cubes
so accurately that this result should be attained, and it is

iv.] SPECIFIC GRAVITY. 55

impossible to ensure its continuance for any time owing to
changes of temperature and the absorption of water by the
wood. We can, however, by a slight modification, prove
that one of these cubes is at all events nearly equal in
weight to the same bulk of water. In Fig. 26 is shown a
tall glass jar filled with a fluid in appearance like plain
water, but it is really composed in the following manner.
I first poured into the jar a very weak solution of salt
and water, which partially filled it ; I then poured gently
upon this a little pure water, and finally filled up the
jar with water containing a little spirits of wine : the salt
and water is a little heavier than pure water, while the spirit
and water is a little lighter. I take one of the cubes and
drop it gently into the glass ; it falls through the spirit and
water, and after making a few oscillations settles itself at
rest in the stratum shown in the figure. This shows that
our prepared cube is a little heavier than spirit and water,
and a little lighter than salt and water, and hence we infer
that it must at all events be very near the weight of pure
water which lies between the two. We have also a number
of half cubes, quarter cubes, and half-quarter cubes, which
have been similarly prepared to be of equal weight with an
equal bulk of water.

92. We shall now be able to measure the specific gravity
of a substance. In one pan of the scales I place the inch
cube of cast iron, and I find that 7^ of the wooden cubes,
which we may call cubes of water, will balance it. We
therefore say that the specific gravity of iron is 7^. The
exact number found by more accurate methods is 7*2. It
is often convenient to remember that 23 cubic inches of
cast iron weigh 6 Ibs., and that therefore one cubic inch
weighs very nearly ^ Ib.

93. I have also cubes of brass, lead, and ivory; by

56 EXPERIMENTAL MECHANICS. [LECT.

counterpoising them with the cubes of water, we can easily
find their specific gravities ; they are shown together with
that of cast iron in the following table : —

Substance. Specific Gravity.

Cast Iron 7'2

Brass 8'i

Lead ii'3

Ivory I '8

94. The mode here adopted of finding specific gravities
is entirely different from the far more accurate methods
which are commonly used, but the explanation of the latter
involve more difficult principles than those we have been
considering. Our method rather offers an explanation of
the nature of specific gravity than a good means of determin-
ing it, though, as we have seen, it gives a result sufficiently
near the truth for many purposes.

THE PLUMMET AND SPIRIT-LEVEL.

95. The tendency of the earth to draw all bodies towards
it is well illustrated by the useful "line and plummet.'
This consists merely of a string to one end of which a
leaden weight is attached. The string when at rest hangs
vertically ; if the weight be drawn to one side, it will, when
released, swing backwards and forwards, until it finally
settles again in the vertical ; the reason is that the weight
always tries to get as near the earth as it can, and this is
accomplished when the string hangs vertically downwards.

96. The surface of water in equilibrium is a horizontal
plane ; that is also a consequence of gravity. All the
particles of water try to get as near the earth as possible,
and therefore if any portion of the water were higher than
the rest, it would immediately spread, as by doing so it
could get lower.

THE CENTRE OF GRAVITY.

57

97. Hence the surface of a fluid at rest enables us to find
a perfectly horizontal plane, while the plummet gives us a
perfectly vertical line : both these consequences of gravity are
of the utmost practical importance.

98. The spirit-level is another common and very useful
instrument which depends on gravity. It consists of a glass
tube slightly curved, with its convex surface upwards, and
attached to a stand with a flat base. This tube is nearly
filled with spirit, but a bubble of air is allowed to remain.
The tube is permanently adjusted so that, when the plate is
laid on a perfectly horizontal surface, the bubble will stand
in the middle : accordingly the position of the bubble gives
a means of ascertaining whether a surface is level.

THE CENTRE OF GRAVITY.

99. We proceed to an experiment which will give an
insight into a curious property of gravity. I have here a
plate of sheet iron ; it has the irregular
shape shown in Fig 27. Five small
holes A B c D E are punched at different
positions on the margin. Attached to
the framework is a small pin from which
I can suspend the iron plate by one
of its holes A : the plate is not sup-
ported in any other way; it hangs
freely from the pin, around which it
can be easily turned. I find that there
is one position, and one only, in which
the plate will rest ; if I withdraw it from
that position it returns there after a
few oscillations. In order to mark
this position, I suspend a line and plummet from the pin,

58 EXPERIMENTAL MECHANICS. [LECT.

having rubbed the line with chalk. I allow the line to come
to rest in front of the plate. I then flip the string against
the plate, and thus produce a chalked mark : this of course
traces out a vertical line A p on the plate.

I now remove the plummet and suspend the plate from
another of its holes B, and repeat the process, thus drawing
a second chalked line B p across the plate, and so on with the
other holes : I thus obtain five lines across the plate, repre-
sented by dotted lines in the figure. It is a very remarkable
circumstance that these five lines all intersect in the same
point P; and if additional holes were bored in the plate,
whether in the margin or not, and the chalk line drawn from
each of them in the manner described, they would one and
all pass through the same point. This remarkable point is
called the centre of gravity of the plate, and the result at
which we have arrived may be expressed by saying that
the vertical line from the point of suspension always
passes through the centre of gravity.

100. At the centre of gravity P a hole has been bored, and
when I place the supporting pin through this hole you see
that the plate will rest indifferently in all positions : this is a
curious property of the centre of gravity. The centre of
gravity may in this respect be contrasted with another hole
Q, which is only an inch distant : when I support the plate
by this hole, it has only one position of rest, viz. when the
centre of gravity p is vertically beneath Q. Thus the centre
of gravity differs remarkably from any other point in the
plate.

1 01. We may conceive the force of gravity on the plate to
act as a force applied at P. It will then be easily seen why
this point remains vertically underneath the point of suspen-
sion when the body is at rest. If I attached a string to the
plate and pulled it, the plate would evidently place itself so

iv.] STABLE AND UNSTABLE EQUILIBRIUM. 59

that the direction of the string would pass through the point
of suspension ; in like manner gravity so places the plate
that the direction of its force passes through the point of
suspension.

102. Whatever be the form of the plate it always contains
one point possessing these remarkable properties, and we may
state in general that in every body, no matter what be its
shape, there is a point called the centre of gravity, such that
if the body be suspended from this point it will remain in
equilibrium indifferently in any position, and that if the body
be suspended from any other point, then it will be in
equilibrium when the centre of gravity is directly underneath

' the point of suspension. In general, it will be impossible to
support a body exactly at its centre of gravity, as this point is
within the mass of the body, and it may also sometimes happen
that the centre of gravity does not lie in the substance of
the body at all, as for example in a ring, in which case the
centre of gravity is at the centre of the ring. We need not,
however, dwell on these exceptional cases, as sufficient
illustrations of the truth of the laws mentioned will present
themselves subsequently.

STABLE AND UNSTABLE EQUILIBRIUM.

103. An iron rod A B, capable of revolving round an axis
passing through its centre P, is shown in Fig. 28.

The centre of gravity lies at the centre B, and consequently,
as is easily seen, the rod will remain at rest in whatever posi-
tion it be placed. But let a weight R be attached to the rod
by means of a binding screw. The centre of gravity of the
whole is no longer at the centre of the rod ; it has moved to
a point s nearer the weight ; we may easily ascertain its
position by removing the rod from its axle and then ascer-
taining the point about which it will balance. This may be

60 EXPERIMENTAL MECHANICS. [LECT.

done by placing the bar on a knife-edge, and moving it to
and fro until the right position be secured ; mark this posi-
tion on the rod, and return it to its axle, the
weight being still attached. We do not now find
that the rod will balance in every position. You
see it will balance if the point s be directly
underneath the axis, but not if it lie to one
side or the other. But if s be directly over the
axis, as in the figure, the rod is in a curious
condition. It will, when carefully placed,
remain at rest ; but if it receive the slightest
displacement, it will tumble over. The rod is
in equilibrium in this position, but it is what is •
called unstable equilibrium. If the centre of
gravity be vertically below the point of suspen-
sion, the rod will return again if moved away :
this position is therefore called one of stable
equilibrium. It is very important to notice
the distinction between these two kinds of
equilibrium.

104. Another way of stating the case is as
follows. A body is in stable equilibrium when
its centre of gravity is at the lowest point :
unstable when it is at the highest. This may
be very simply illustrated by an ellipse, which I
hold in my hand. The centre of gravity of this figure is at its
centre. The ellipse, when resting on its side, is in a posi-
tion of stable equilibrium and its centre of gravity is then
clearly at its lowest point. But I can also balance the
ellipse on its narrow end, though if I do so the smallest
touch suffices to overturn it. The ellipse is then in unstable
equilibrium ; in this case, obviously, the centre of gravity is
at the highest point.

iv.] CENTRE OF GRAVITY. 6r

105. I have here a sphere, the centre of gravity of which
is at its centre ; in whatever way the sphere is placed on a
plane, its centre is at the same height, and therefore cannot
be said to have any highest or lowest point ; in such a case
as this the equilibrium is neutral. If the body be displaced,
it will not return to its old position, as it would have done
had that been a position of stable equilibrium, nor will it
deviate further therefrom as if the equilibrium had been
unstable : it will simply remain in the new position to
which it is brought.

1 06. I try to balance an iron ring upon the
end of a stick H, Fig. 29, but I cannot
easily succeed in doing so. This is because its
centre of gravity s is above the point of support ;
but if I place the stick at F, the ring is in stable
equilibrium, for now the centre of gravity is FIG. 29.
below the point of support.

PROPERTY OF THE CENTRE OF GRAVITY IN A
REVOLVING WHEEL.

107. There are other curious consequences which follow
from the properties of the centre of gravity, and we
shall conclude by illustrating one of the most remarkable,
which is at the same time of the utmost importance in
machinery.

1 08. It is generally necessary that a machine should
work as steadily as possible, and that undue vibration and
shaking of the framework should be avoided : this is par-
ticularly the case when any parts of the machine rotate with
great velocity, as, if these be heavy, inconvenient vibration
will be produced when the proper adjustments are not
made. The connection between this and the centre of

EXPERIMENTAL MECHANICS.

[LECT.

gravity will be understood by reference to the apparatus
represented in the accompanying figure (Fig. 30). We
have here an arrangement consisting of a large cog wheel
C working into a small one B, whereby, when the handle H
is turned, a velocity of rotation can be given to the iron

FIG. 3c.

disk D, which weighs i4lbs, and is 18" in diameter. This
disk being uniform, and being attached to the axis at its
centre, it follows that its centre of gravity is also the centre
of rotation. The wheels are attached to a stand, which,
though massive, is still unconnected with the floor. By
turning the handle I can rotate the disk very rapidly, even

iv.] CENTRE OF GRAVITY. 63

as much as twelve times in a second. Still the stand re-
mains quite steady, and even the shutter bell attached to it
at E is silent.

109. Through one of the holes in the disk D I fasten a
small iron bolt and a few washers, altogether weighing
about i Ib. ; that is, only one-fourteenth of the weight of
the disk. When I turn the handle slowly, the machine
works as smoothly as before ; but as I increase the speed
up to one revolution every two seconds, the bell begins to
ring violently, and when I increase it still more, the stand
quite shakes about on the floor. What is the reason of
this ? By adding the bolt, I slightly altered the position of
the centre of gravity of the disk, but I made no change of
the axis about which the disk rotated, and consequently
the disk was not on this occasion turning round its centre
of gravity : this it was which caused the vibration. It is
absolutely necessary that the centre of gravity of any heavy
piece, rotating rapidly about an axis, should lie in the axis
of rotation. The amount of vibration produced by a high
velocity may be very considerable, even when a very small
mass is the originating cause.

no. In order that the machine may work smoothly again,
it is not necessary to remove the bolt from the hole. If by
any means I bring back the centre of gravity to the axis,
the same end will be attained. This is very simply effected
by placing a second bolt of the same size at the opposite
side of the disk, the two being at equal distances from the
axis; on turning the handle, the machine is seen to work as
smoothly as it did in the first instance.

in. The most common rotating pieces in machines are
wheels of various kinds, and in these the centre of gravity
is evidently identical with the centre of rotation ; but if
from any cause a wheel, which is to turn rapidly, has an

64 EXPERIMENTAL MECHANICS. [LECT. iv.

extra weight attached -to one part, this weight must be
counterpoised by one or more on other portions of the
wheel, in order to keep the centre of gravity of the whole
in its proper place. Thus it is that the driving wheels of
a locomotive are always weighted so as to counteract the
effect of the crank and restore the centre of gravity to the
axis of rotation. The cause of the vibration will be under-
stood after the lecture on centrifugal force (Lect XVII.).

LECTURE V.
THE FORCE OF FRICTION.

The Nature of Friction.— The Mode of Experimenting.— Friction is
proportional to the pressure. — A more accurate form of the Law.
—The Coefficient varies with the weights used.— The Angle of
Friction. — Another Law of Friction. — Concluding Remarks.

THE NATURE OF FRICTION.

1 1 2. A DISCUSSION of the force of friction is a necessary
preliminary to the study of the mechanical powers which we
shall presently commence. Friction renders the inquiry into
the mechanical powers more difficult than it would be if this
force were absent; but its effects are too important to be
overlooked.

113. The nature of friction may be understood by Fig.
31, which represents a section of the top of a table of wood

F.G. 31.

or any other substance levelled so that c D is horizontal ; on
the table rests a block A of wood or any other substance.
To A a cord is attached, which, after passing over a pulley p,

F

66 EXPERIMENTAL MECHANICS. [LECT.

is stretched by a suspended weight B. If the magnitude of B
exceeds a certain limit, then A is pulled along the table and
B descends ; but if B be smaller than this limit, both A and
B remain at rest. When B is not heavy enough to produce
motion it is supported by the tension of the cord, which is
itself neutralized by the friction produced by a certain
coherence between A and the table. Friction is by this
experiment proved to be a force, because it prevents the
motion of B. Indeed friction is generally manifested as a force
by destroying motion, though sometimes indirectly pro-
ducing it.

114. The true source of the force lies in the inevitable
roughness of all known surfaces, no matter how they may
have been wrought. The minute asperities on one surface
are detained in corresponding hollows in the other, and con-
sequently force must be exerted to make one surface slide
upon the other. By care in polishing the surfaces the amount
of friction may be diminished ; but it can only be decreased
to a certain limit, beyond which no amount of polishing
seems to produce much difference.

115. The law of friction under different conditions must
be inquired into, in order that we may make allowance when
its effect is of importance. The discussion of the experi-
ments is sometimes a little difficult, and the truths arrived
at are principally numerical, but we shall find that some
interesting laws of nature will appear.

THE MODE OF EXPERIMENTING.

1 1 6. Friction is present between every pair of surfaces
which are in contact : there is friction between two pieces of
wood, and between a piece of wood and a piece of iron ;
and the amount of the force depends upon the character of
both surfaces. We shall only experiment upon the friction

v.] THE MODE OF EXPERIMENTING. 67

of wood upon wood, as more will be learned by a careful
study of a special case than by a less minute examination
of a number of pairs of different substances.

117. The apparatus used is shown in Fig. 32. A plank
of pine 6' x n'' x 2" is planed on its upper surface, levelled
by a spirit-level, and firmly secured to the framework at a
height of about 4' from the ground. On it is a pine slide
9" x 9", the grain of which is crosswise to that of the plank ;
upon the slide the load A is placed. A rope is attached to
the slide, which passes over a very freely mounted cast iron
pulley c, 14" diameter, and carries at the other end a
hook weighing one pound, from which weights B can be
suspended.

1 1 8. The mode of experimenting consists in placing a cer-
tain load upon A, and then ascertaining what weight applied
to B will draw the loaded slide along the plane. As several
trials are generally necessary to determine the power, a rope
is attached to the back of the slide, and passes over the two
pulleys D ; this makes it easy for the experimenter, when
applying the weights at B, to draw back the slide to the end
of the plane by pulling the ring E : this rope is of course left
quite slack during the process of the experiment, since the
slide must not be retarded. The loads placed upon A during
the series of experiments ranged between one stone and
eight stone. In the loads stated the weight of the slide
itself, which was less than i lb., is always included. A
variety of small weights were provided for the hook B ; they
consisted of o-i, 0-5, i, 2, 7, and i4lbs. There is some
friction to be overcome in the pulley c, but as the pulley is
comparatively large its friction is small, though it was always
allowed for.

119. An example of the experiments made is thus described.
A weight of 56 Ibs. is placed upon the slide, and it is found

68 EXPERIMENTAL MECHANICS. [LECT.

on trial that 2 gibs, on B (including the weight of the hook
itself) is sufficient to start the slide ; this weight is placed

THE MODE OF EXPERIMENTING.

69

upon the hook pound by pound, care being taken to make
each addition gently.

120. Experiments were made in this way with various
weights upon A, and the results are recorded in Table I.

TABLE I.— FRICTION.

Smooth horizontal surface of pine 72" x 1 1" ; slide also of pine 9" x
9" ; grain crosswise ; slide is not started ; force acting on slide
is gradually increased until motion commences.

Number of
Experiment.

Load on sl.de in
Ibs., including
weight of slide.

Force necessary
to move slide,
ist Series.

Force necessary
to move slide.
2nd Series.

Mean
values.

j

14

5

8

6'S

2

28

15

16

IS'S

3

42

20

15

i7'5

4

56

29

24

265

5

70

33

31

320

6

84

43

33

3»x>

7

98

42

38

40 x>

8

112

5°

33

4i-5

In the first column a number is given to each experiment
for convenience of reference. In the second column the
load on the slide is stated in Ibs. In the third column is
found the force necessary to overcome the friction. The
fourth column records a second series of experiments per-
formed in the same manner as the first series ; while the
last column shows the mean of the two frictions.

121. The first remark to be made upon this table is, that
the results do not appear satisfactory or concordant. Thus
from 6 and 7 of the ist series it would appear that the fric-
tion of 84 Ibs. was 43 Ibs., while that of 98 Ibs. was 42 Ibs.,
so that here the greater weight appears to have the less
friction, which is evidently contrary to the whole tenor of
the results, as a glance will show. Moreover the frictions in the
ist and the 2nd series do not agree, being generally greater

70 EXPERIMENTAL MECHANICS. [LECT.

in the former than in the latter, the discordance being espe-
cially noticeable in experiment 8, where the results were
50 Ibs. and 33 Ibs. In the final column of means these
irregularities are lessened, for this column shows that the
friction increases with the weight, but it is sufficient to
observe that as the difference of the ist and the 2nd is 9 Ibs.,
and that of the 2nd and the 3rd is only 2 Ibs., the discovery
of any law from these results is hopeless.

122. But is friction so capricious that it is amenable to
no better law than these experiments appear to indicate ?
We must look a little more closely into the matter. When
two pieces of wood have remained in contact and at rest
for some time, a second force besides friction resists their
separation : the wood is compressible, the surfaces become
closely approximated, and the coherence due to this cause
must be overcome before motion commences. The initial
coherence is uncertain ; it depends probably on a multitude
of minute circumstances which it is impossible to estimate,
and its presence has vitiated the results which we have
found so unsatisfactory.

123. We can remove these irregularites by starting the
slide at the commencement. This may be conveniently
effected by the screw shown at F in Fig. 32 ; a string
attached to its end is fastened to the slide, and by giving
the handle of the screw a few turns the slide begins to
creep. A body once set in motion will continue to move
with the same velocity unless acted upon by a force ;
hence the weight at B just overcomes the friction when
the slide moves uniformly after receiving a start : this
velocity was in one case of average speed measured to be
1 6 inches per minute.

124. Indeed in no case can the slide commence to move
unless the force exceed the friction. The amount of this

V-]

THE MODE OF EXPERIMENTING.

excess is indeterminate. It is certainly greater between
wooden surfaces than between less compressible surfaces
like those of metals or glass. In the latter case, when the
force exceeds the friction by a small amount, the slide starts
off with an excessively slow motion ; with wood the force
must exceed the friction by a larger amount before the slide
commences to move, but the motion is then comparatively
rapid.

125. If the power be too small, the load either does not
continue moving after the start, or it stops irregularly. If
the power be too great, the load is drawn with an accelerated
velocity. The correct amount is easily recognised by the
uniformity of the movement, and even when the slide is
heavily laden, a few tenths of a pound on the power hook
cause an appreciable difference of velocity.

126. The accuracy with which the friction can be measured
may be appreciated by inspecting Table II.

TABLE II. — FRICTION.

Smooth horizontal surface of pine 72" x 1 1" ; slide also of pine 9" x
9" ; grain crosswise ; slide started ; force applied is sufficient to
maintain uniform motion of the slide.

Number of

Load on slide in
Ibs. , including

Force necessary to
maintain motion.

Force necessary to
maintain motion.

Mean

Expenment.

weight of slide.

ist Series.

2nd Series.

I

14

4-9

4'9

4'9

2

28

8-5

8-6

»-5

3

42

12-6

12-4

I2'5

4

56

163

16-2

162

5

70

197

200

I9-8

6

84

23 7

23-0

23 "4

7

98

26-5

26' I

263

8

112

297

29-9

29-8

127. Two series of experiments to determine the power
necessary to maintain the motion have been recorded.

72 EXPERIMENTAL MECHANICS. [LECT.

Thus, in experiment 7, the load on the slide being 98 Ibs.,
it was found that 26-5 Ibs. was sufficient to sustain the
motion, and a second trial being made independently,
the power found was 26'! Ibs. : a mean of the two values,
26-3 Ibs., is adopted as being near the truth. The greatest
difference between the two series, amounting to 07 lb., is
found in experiment 6 ; a third value was therefore ob-
tained for the friction of 84 Ibs. : this amounted to 23-5 Ibs.,
which is intermediate between the two former results, and
23 '4 Ibs., a mean of the three, is adopted as the final result.

128. The close accordance of the experiments in this
table shows that the means of the fifth column are probably
very near the true values of the friction for the correspond-
ing loads upon the slide.

129. The mean frictions must, however, be slightly di-
minished before we can assert that they represent only the
friction of the wood upon the wood. The pulley over which
the rope passes turns round its axle with a small amount of
friction, which must also be overcome by the power. The
mode of estimating this amount, which in these experiments
never exceeds o'5 lb., need not now be discussed. The
corrected values of the friction are shown in the third
column of Table III. Thus, for example, the 4-9 Ibs. of
friction in experiment i consists of 47, the true friction of
the wood, and 0-2, which is the friction of the pulley ; and
26-3 of experiment 7 is similarly composed of 25-8 and 0*5.
It is the corrected frictions which will be employed in our
subsequent calculations.

FRICTION IS PROPORTIONAL TO THE PRESSURE.

130. Having ascertained the values of the force of friction
for eight different weights, we proceed to inquire into the

v.] FRICTION PROPORTIONAL TO PRESSURE. 73

laws which may be founded on our results. It is evident
that the friction increases with the load, of which it is
always greater than a fourth, and less than a third. It
is natural to surmise that the friction (f) is really a
constant fraction of -the load (1?) — in other words, that
F = kR, where k is a constant number.

131. To test this supposition we must try to determine k ;
it may be ascertained by dividing any value of F by the
corresponding value of R. If this be done, we shall find
that each of the experiments yields a different quotient ; the
first gives 0*336, and the last 0*262, while the other experi-
ments give results between these extreme values. These
numbers are tolerably close together, but there is still
sufficient discrepancy to show that it is not strictly true
to assert that the friction is proportional to the load.

132. But the law that the friction varies proportionally
to the pressure is so approximately true as to be sufficient
for most practical purposes, and the question then arises,
which of the different values of k shall we adopt ? By a
method which is described in the Appendix we can deter-
mine a value for k which, while it does not represent any
one cf the experiments precisely, yet represents them collec-
tively better than it is possible for any other value to do.
The number thus found is 0*27. It is intermediate be-
tween the two values already stated to be extreme. The
character of this result is determined by an inspection of
Table III.

The fourth column of this table has been calculated from
the formula F = 0*27 R. Thus, for example, in experiment
5, the friction of a load of 70 Ibs. is 19-4 Ibs., and the
product of 70 and 0-27 is 18-9, which is 0*5 Ib. less than the
true amount. In the last column of this table the discrepancies
between the observed and the calculated values are recorded,

74

EXPERIMENTAL MECHANICS.

[LECT.

for facility of comparison. It will be observed that the
greatest difference is under i Ib.

TABLE III.— FRICTION.

Friction of pine upon pine ; the mean values of the friction given in
Table II. (corrected for the friction of the pulley) compared with the
formula F = o'2"j R.

Number of
Experiment.

R.
Total load on
slide in Ibs.

Corrected
mean value of
friction.

F.

Calculated value
of friction.

Discrepancies
between the
observed and
calculated frictions.

I

H

47

3-8

-0'9

2

28

8'2

7-6

-0-6

3

42

1 2 '2

ii'3

-0-9

4

56

15-8

-07

5

7°

19-4

18-9

-°'5

6

84

23-0

227

-0-3

7

98

25-8

26-5

+ 07

8

112

29 "3

30-2

+ 0-9

133. Hence the law F = 0-27 R represents the experi-
ments with tolerable accuracy; and the numerical ratio 0*27
is called the coefficient of friction. We may apply this law to
ascertain the friction in any case where the load lies between
14 Ibs. and 112 Ibs. ; for example, if the load be 63 Ibs., the
friction is 63 x 0*27 = 17-0.

134. The coefficient of friction would have been slightly
different had the grain of the slide been parallel to that of
the plank ; and it of course varies with the nature of the
surfaces. Experimenters have given tables of the coefficients
of friction of various substances, wood, stone, metals, £c.
The use of these coefficients depends upon the assumption
of the ordinary law of friction, namely, that the friction is
proportional to the pressure : this law is accurate enough for
most purposes, especially when used for loads that lie be-
tween the extreme weights employed in calculating the value
of the coefficient which is employed.

v.] A MORE ACCURATE LAW OF FRICTION. 75

A MORE ACCURATE LAW OF FRICTION.

135. In making one of our measurements with care, it
is unusual to have an error of more than a few tenths of i Ib.
and it is hardly possible that any of the mean frictions we
have found should be in error to so great an extent as o'5 Ib.
But with the value of the coefficient of friction which is used
in Table III., the discrepancies amount sometimes to 0-9
Ibs. With any other numerical coefficient than 0^27, the
discrepancies would have been even still more serious. As
these are too great to be attributed to errors of experiment,
we have to infer that the law of friction which has
been assumed cannot be strictly true. The signs of the
discrepancies indicate that the law gives frictions which for
small loads are too small, and for large loads are too great.

136. We are therefore led to inquire whether some other
relation between ^and-/? may not represent the experiments
with greater fidelity than the common law of friction. If we
diminished the coefficient by a small amount, and then added
a constant quantity to the product of the coefficient and
the load, the effect of this change would be that for small
loads the calculated values would be increased, while for
large loads they would be diminished. This is the kind of
change -which we have indicated to be necessary for re-
conciliation between the observed and calculated values.

137. We therefore infer that a relation of the form
F = x + y R will probably express a more correct law,
provided we can find x and y. One equation between x
and y is obtained by introducing any value of R with the
corresponding value of F, and a second equation can be
found by taking any other similar pair. From these two
equations the values of x and of y may be deduced by
elementary algebra, but the best formula will be obtained

76 EXPERIMENTAL MECHANICS. [LECT.

by combining together all the pairs of corresponding values.
For this reason the method described in the Appendix
must be used, which, as it is founded on all the experiments,
must give a thoroughly representative result. The formula
thus determined, is

f= i '44 4- 0*252 ^?.

This formula is compared with the experiments in
Table IV.

TABLE IV. — FRICTION.

Friction of pine upon pine ; the mean values of the friction given in
Table II. (corrected for the friction of the pulley) compared with the
formula F = I -44 + 0*252 Jf.

Number of
Experiment.

R.

Total load on
slide in Ibs.

Corrected
mean value of
friction.

F.

Calculated value
of friction.

Discrepancies
between the
observed and
calculated frictions

j

H

47

S'°

1-0-3

2

28

8-2

8-5

+ 0-3

3

42

1 2 '2

I2'O

-O'2

4

56

IS '8

I5-6

-0'2

|

e

19-4
23-0

I9-I
22-6

-03
-0-4

7

98

25-8

26-1

+ 0'3

8

112

29-3

297

+ 0'4

The fourth column contains the calculated values : thus,
for example, in experiment 4, where the load is 56 Ibs., the
calculated value is i -44 + 0-252 x 56=15-6; the difference
o'2 between this and the observed value 15 '8 is shown in the
last column.

138. It will be noticed that the greatest discrepancy in
this column is 0-4 Ibs., and that therefore the formula repre-
sents the experiments with considerable accuracy. It is
undoubtedly nearer the truth than the former law (Art. 132) ;
in fact, the differences arc now such as might really belong
to errors unavoidable in making the experiments.

v.] COEFFICIENT VARIES WITH WEIGHTS. 77

139. This formula maybe used for calculating the friction
for any load between 14 Ibs. and 112 Ibs. Thus, if
the load be 63 Ibs., the friction is 1-44 + 0-252 x 63 := 17-3
Ibs., which does not differ much from 17-0 Ibs., the value
found by the more ordinary law. We must, however, be
cautious not to apply this formula to weights which do not
lie between the limits of the greatest and least weight used
in those experiments by which the numerical values in the for-
mula have been determined ; for example, to take an extreme
case, if R = o, the formula would indicate that the friction
was i '44, which is evidently absurd; here the formula errs
in excess, while if the load were very large it is certain
the formula would err in defect.

THE COEFFICIENT VARIES WITH THE WEIGHTS
USED.

140. In a subsequent lecture we shall employ as an
inclined plane the plank we have been examining, and we
shall require to use the knowledge of its friction which we are
now acquiring. The weights which we shall then employ
range from 7 Ibs. to 56 Ibs. Assuming the ordinary law of
friction, we have found that 0-27 is the best value of its
coefficient when the loads range between 14 Ibs. and 112
Ibs. Suppose we only consider loads up to 56 Ibs., we find
that the coefficient 0*288 will best represent the experiments
within this range, though for 112 Ibs. it would give an error
of nearly 3 Ibs. The results calculated by the formula F =
0-288 J? are shown in Table V., where the greatest differ-
ence is 0-7 Ib.

141. But we can replace the common law of friction by
the more accurate law of Art. 137, and the formula computed
so as to best harmonise the experiments up to 56 Ibs.,
disregarding the higher loads, is F— 0-9 + 0-266 £. This

EXPERIMENTAL MECHANICS.

[LECT.

TABLE V.— FRICTION.

Friction of pine upon pine ; the mean values of the friction given in
Table II. (corrected for the friction of the pulley) compared with the
formula F = 0-288 R

Number of
Experiment.

R.
Total load on
slide in Ibs.

Corrected
mean value of
friction.

F.

Calculated value
of friction.

Discrepancies
between the
observed and
calculated frictions

I
2

14
28

47
8'2

4'0

8-1

-07
-O'l

3

4

42
56

I2'2
158

16-1

-O'l

formula is obtained by the method referred to in Art. 137.
We find that it represents the experiments better than
that used in Table V. Between the limits named, this
formula is also more accurate than that of Table IV. It is
compared with the experiments in Table VI., and it will be
noticed that it represents them with great precision, as no
discrepancy exceeds o"i.

TABLE VI. — FRICTION.
Friction of pine upon pine ; the mean values of the friction given in

Table II. (corrected for the friction of the pulley) compared with the

formula F = 0-9 + 0-266 R.

Number of
Experiment.

R.
Total Load on
slide in Ibs.

Corrected
mean value of
friction.

F.

Calculated value
of friction.

Discrepancies
between the
observed and
calculated friction^

I

2

14
28

13

4-6

-o-i

3

4

42
56

1 2 '2
15-8

Is*

-O'l
O'O

THE ANGLE OF FRICTION.

142. There is another mode of examining the action of
friction besides that we have been considering. The appa-
ratus for this purpose is shown in Fig. 33, in which B c
represents the plank of pine we have already used. . It is

THE ANGLE OF FRICTION.

79

8o EXPERIMENTAL MECHANICS. [LECT.

now mounted so as to be capable of turning about one end
B ; the end c is suspended from one hook of the chain from
the " epicycloidal" pulley-block E. This block is very con-
venient for the purpose. By its means the inclination of the
plank can be adjusted with the greatest nicety, as the raising
chain G is held in one hand and the lowering chain F in the
other. Another great convenience of this block is that the
load does not run down when the lifting chain is left free.
The plank is clamped to the hinge about which it turns.
The frames by which both the hinge and the block are
supported are weighted in order to secure steadiness. The
inclination of the plane is easily ascertained by measuring
the difference in height of its two ends above the floor, and
then making a drawing on the proper scale. The starting-
screw D, whose use has been already mentioned, is also
fastened to the frame-work in the position shown in the
figure.

143. Suppose the slide A be weighted and placed upon
the inclined plane B c ; if the end c be only slightly elevated,
the slide remains at rest ; the reason being that the friction
between the slide and the plane neutralizes the force of
gravity. But suppose, by means of the pulley-block, c be
gradually raised ; an elevation is at last reached at which the
slide starts off, and runs with an accelerating velocity to the
bottom of the plane. The angle of elevation of the plane
when this occurs is called the angle of statical friction.

144. The weights with which the slide was laden in these
experiments were 14 Ibs., 56 Ibs., and 112 Ibs., and the
results are given in Table VII.

We see that a load of 56 Ibs. started when the plane
reached an inclination of 20°'! in the first series, and of
i7°'2 in the second, the mean value i8°'6 being given in the
fifth column. These means for the three different weights

v.]

THE ANGLE OF FRICTION.

Si

agree so closely that we assert the remarkable law that
the angle of friction does not depend upon the magnitude of
the load.

TABLE VII. — ANGLE OF STATICAL FRICTION.

A smooth plane of pine 72" x n" carries a loaded slide of pine 9" x 9";
one end of the plane is gradually elevated until the slide starts off.

Number of
Experiment.

Total load en
the sl.de in Ibs.

Angle of
elevation.
is: Series.

Angle of
elevation.
2nd Series.

Mean values
of the angles.

I

H

1 9° '5

i9°-5

2

56

20° 'I

if -2 i8°-6

3

112

20°-3

1 8° -9

19-6

145. We might, however, proceed differently in deter-
mining the angle of friction, by giving the load a start, and
ascertaining if the motion will continue. To do so requires
the aid of an assistant, who will start the load with the
help of the screw, while the elevation of the plane is being
slowly increased. The result of these experiments is given
in Table VIII.

TABLE VIII.— ANGLE OF FRICTION.

A smooth plane of pine 72" x n" carries a loaded slide of pine 9" x 9" ;
one end of the plane is gradually elevated until the slide, having
received a start, moves off uniformly.

Number of
Experiment.

Total load on
the slide in Ibs.

Angle of
inclination.

I
2

3

H
56
112

14° '3
i3°-o
I3"o

tVe see from this table also that the angle of friction is
independent of the load, but the angle is in this case less
by 5° or 6° than was found necessary to impart motion
when a start was not given.

G

8^ EXPERIMENTAL MECHANICS. [LECT.

146. It is commonly stated that the coefficient of friction
equals the tangent of the angle of friction, and this can be
proved to be true when the ordinary law of friction is
assumed. But as we have seen that the law of friction is
only approximately correct, we need not expect to find this
other law completely verified.

147. When the slide is started, the mean value of the
angle of friction is i3°'4. The tangent of this angle is
0-24 : this is about n per cent, less than the coefficient of
friction 0*27, which we have already determined. The
mean value of the angle of friction when the slide is not
started is i9°'2, and its tangent is 0-35. The experiments
of Table I. are, as already pointed out, rather unsatis-
factory, but we refer to them here to show that, so far as
they go, the coefficient of friction is in no sense equal to
the tangent of the angle of friction. If we adopt the mean
values given in the last column of Table I., the best
coefficient of friction which can be deduced is 0-41.
Whether, therefore, the slide be started or not started, the
tangent of the angle of friction is smaller than the corre-
sponding coefficient of friction. When the slide is started,
the tangent is about 1 1 per cent, less than the coefficient ;
and when the slide is not started, it is about 14 per cent,
less. There are doubtless many cases in which these differ-
ences are sufficiently small to be neglected, and in which,
therefore, the law may be received as true.

ANOTHER LAW OF FRICTION.

148. The area of the wooden slide is 9" x 9", but we would
have found that the friction under a given load was practically
the same whatever were the area of the slide, so long as its
material remained unaltered. This follows as a consequence of

v.j CONCLUDING REMARKS. 83

the approximate law that the friction is proportional to the
pressure. Suppose that the weight were 100 Ibs., and the
area of the slide 100 inches, there would then be a pressure
cf i Ib. per square inch over the surface of the slide, and
therefore the friction to be overcome on each square inch
would be 0-27 Ib., or for the whole slide 27 Ibs. If, how-
ever, the slide had only an area of 50 square inches, the load
would produce a pressure of 2 Ibs., per square inch ; the
friction would therefore be 2X0-27= 0-54 Ib. for each square
inch, and the total friction would be 50X0-54 = 27 Ibs., the
same as before : hence the total friction is independent of
the extent of surface. This would remain equally true even
though the weight were not, as we have supposed, uniformly
distributed over the surface of the slide.

CONCLUDING REMARKS.

149. The importance of friction in mechanics arises from
its universal presence. We often recognize it as a destroyer
or impeder of motion, as a waster of our energy, and as a
source of loss or inconvenience. But, on the other hand,
friction is often indirectly the means of producing motion,
and of this we have a splendid example in the locomotive
engine. - The engine being very heavy, the wheels are
pressed closely to the rails ; there is friction enough to
prevent the wheels slipping, consequently when the
engines force the wheels to turn round they must roll
onwards. The coefficient of friction of wrought iron upon
wrought iron is about 0-2. Suppose a locomotive weigh 30
tons, and the share of this weight borne by the driving
wheels be 10 tons, the friction between the driving wheels
and the rails is 2 tons. This is the greatest force the engine
can exert on a level line. A force of 10 Ibs. for every ton

G 2

84 EXPERIMENTAL MECHANICS. [LECT. v.

weight of the train is known to be sufficient to sustain the
motion, consequently the engine we have supposed should
draw along the level a load of 448 tons.

150. But we need not invoke the steam engine to show the
use of friction. We could not exist without it. In the first
place we could not move about, for walking is only possible
on account of the friction between the soles of our boots and
the ground ; nor if we were once in motion could we stop
without coming into collision with some other object, or
grasping something to hold on by. Objects could only be
handled with difficulty, nails would not remain in wood, and
screws would be equally useless. Buildings could hardly be
erected, nay, even hills and mountains would gradually dis-
appear, and finally dry land would be immersed beneath the
level of the sea. Friction is, so far as we are concerned,
quite as essential a law of nature as the law of gravitation.
We must not seek to evade it in our mechanical discussions
because it makes them a little more difficult. Friction
obeys laws ; its action is not vague or uncertain. When
inconvenient it can be diminished, when useful it can be
increased ; and in our lectures on the mechanical powers, to
which we now proceed, we shall have opportunities of
describing machines which have been devised in obedience
to its laws.

LECTURE VI.
THE PULLEY.

Introduction. — Friction between a Rope and an Iron Bar. — The use of
the Pulley. — Large and Small Pulleys. — The Law of Friction in
the Pulley.— Wheels.— Energy.

INTRODUCTION.

151. THE pulley forms a good introduction to the im-
portant subject of the mechanical powers. But before
entering on the discussions of the next few chapters, it will
be necessary for us to explain what is meant in mechanics by
"work," and by "energy, "which is the capacity for performing
work, and we shall therefore include a short outline of this
subject in the present lecture.

152. The pulley is a machine which is employed for the
purpose of changing the direction of a force. We frequently
wish to apply a force in a different direction from that in
which it is convenient to exert it, and the pulley enables us
to do so. We are not now speaking of these arrangements
for increasing power in which pulleys play an important part ;
these will be considered in the next lecture : we at present
refer only to change of direction. In fact, as we shall shortly

86 EXPERIMENTAL MECHANICS. [LECT.

see, some force is even wasted when the single fixed pulley
is used, so that this machine certainly cannot be called a
mechanical power.

153. The occasions upon which a single fixed pulley is
used are numerous and familiar. Let us suppose a sack of
corn has to be elevated from the lower to one of the upper
ttories of a building. It may of course be raised by a man
who carries it, but he has to lift his own weight in addition
to that of the sack, and therefore the quantity of exertion
used is greater than absolutely necessary. But supposing
there be a pulley at the top of the building over which a rope
passes ; then, if a man attach one end of the rope to the
sack and pull the other, he raises the sack without raising
his own weight. The pulley has thus provided the means
by which the downward force has been changed in direction
to an upward force.

154. The weights, ropes, and pulleys which are used in
our windows for counterpoising the weight of the sash afford
a very familiar instance of how a pulley changes the direction
of a force. Here the downward force of the weight is
changed by means of the pulley into an upward force, which
nearly counterbalances the weight of the sash.

FRICTION BETWEEN A ROPE AND AN IRON BAR.

155. Every one is familiar with the ordinary form of the
pulley ; it consists of a wheel capable of turning freely on its
axle, and it has a groove in its circumference in which the
rope lies. But why is it necessary to give the pulley this
form ? Why could not the direction of the rope be changed
by simply passing it over a bar, as well as by the more com-
plicated pulley? We shall best answer this question by
actually trying the experiment, which we can do by means of
the apparatus of Fig. 34 (see page 90).- In this are shown

vi.] A ROPE AND AN IRON BAR. 87

two iron studs, G, H, o"-6 diameter, and about 8" apart ; over
these passes a rope, which has a hook at each end. If I
suspend a weight of 14 Ibs. from one hook A, and pull the
hook B, I can by exerting sufficient force raise the weight on
A, but with this arrangement I am conscious of having to
exert a very much larger force than would have been
necessary to raise 14 Ibs. by merely lifting it.

156. In order to study the question exactly, we shall
ascertain what weight suspended from the hook B will suffice
to raise A. I find that in order to raise 14 Ibs. on A no less
than 47 Ibs. is necessary on B, consequently there is an enor-
mous loss of force : more than two-thirds of the force which
is exerted is expended uselessly. If instead of the 14 Ibs.
weight I substitute any other weight, I find the same result,
viz. that more than three times its amount is necessary to raise
it by means of the rope passing over the studs. If a labourer,
in raising a sack, were to pass a rope over two bars such as
these, then for every stone the sack weighed he would have
to exert a force of more than three stones, and there would
be a very extravagant loss of power.

157. Whence arises this loss? The rope in moving slides
over the -surface of the iron studs. Although these are quite
smooth and polished, yet when there is a strain on the rope
it presse~s closely upon them, and there is a certain amount
of force necessary to make the rope slide along the iron. In
other words, when I am trying to raise up 14 Ibs. with this
contrivance, I not only have its weight opposed to me, but
also another force due to the sliding of the rope on the iron :
this force is due to friction. Were it not for friction, a force
of 14 Ibs. on one hook would exactly balance 14 Ibs. on the
other, and the slightest addition to either weight would make
it descend and raise the other. If, then, we are obliged to
change the direction of a force, we must devise some means

88 EXPERIMENTAL MECHANICS. [LECT.

of doing so which does not require so great a sacrifice as the
arrangement we have just used.

THE USE OF THE PULLEY.

158. We shall next inquire how it is that we are enabled
to obviate friction by means of a pulley. It is evident we
must provide an arrangement in which the rope shall not be
required to slide upon an iron surface. This end is attained
by the pulley, of which we may take i, Fig. 34, as an
example. This represents a cast iron wheel 14" in diameter,
with a V~snaPed groove in its circumference to receive the
rope : this wheel turns on a f -inch wrought iron axle, which
is well oiled. The rope used is about o"'2$ in diameter.

159. From the hooks E, F at each end of the rope a i4lb.
weight is suspended. These equal weights balance each
other. According to our former experiment with the
studs, it would be necessary for me to treble the weight on
one of these hooks in order to raise the other, but now I
find that an additional 0-5 Ib. placed on either hook causes
it to descend and make the other ascend. This is a great
improvement ; 0-5 Ib. now accomplishes what 33 Ibs. was
before required for. We have avoided a great deal of
friction, but we have not got rid of it altogether, for 0-25 Ib.
is incompetent, when added to either weight, to make that
weight descend.

160. To what is the improvement due ? When the
weight descends the rope does not slide upon the wheel,
but it causes the wheel to revolve with it, consequently
there is little or no friction at the circumference of the
pulley ; the friction is transferred to the axle. We still
have some resistance to overcome, but for smooth oiled iron
axles the friction is very small, hence the advantage of the
pulley.

vi.] LARGE AND SMALL PULLEYS. 89

There is in every pulley a small loss of power from the
force expended in bending the rope ; this need not concern
us at present, for with the pliable plaited rope that we
have employed the effect is inappreciable, but with large
strong ropes the loss becomes of importance. The
amount of loss by using different kinds of ropes has been
determined by careful experiments.

LARGE AND SMALL PULLEYS.

161. There is often a considerable advantage obtained
by using large rather than small pulleys. The amount of
force necessary to overcome friction varies inversely as the
size of the pulley. We shall demonstrate this by actual
experiment with the apparatus of Fig. 34. A small pulley
K is attached to the large pulley i ; they are iri fact one
piece, and turn together on the same axle. Hence if we
first determine the friction with the rope over the large
pulley, and then with the rope over the small pulley, any
difference can only be due to the difference in size, as all
the other circumstances are the same.

162. In making the experiments we must attend to the
following point. The pulleys and the socket on which they
are mounted weigh several pounds, and consequently there
is friction on the axle arising from the weight of the pulleys,
quite independently of any weights that may be placed on
the hooks. We must then, if possible, evade the friction of
the pulley itself, so that the amount of friction which is
observed will be entirely due to the weights raised. This
can be easily done. The rope and hooks being on the
large pulley i, I find that 0*16 Ib. attached to one of the
hooks, E, is sufficient to overcome the friction of the
pulley, and to make that hook descend and raise F. If
therefore we leave 0-16 Ib. on E, we may consider the

EXPERIMENTAL MECHANICS.

[LECT.

friction due to the weight of the pulley, rope, and hooks as
neutralized.

163. I now place a stone weight on each of the hooks E
and F. The amount necessary to make the hook E and
its load descend is 0-28 Ib. This does not of course

include the weight of 0-16 Ib. already referred to. We see
therefore that with the large pulley the amount of friction to
be overcome in raising one stone is 0-28 Ib.

164. Let us now perform precisely the same experiment
with the small pulley. I transfer the same rope and hooks

VI.] LARGE AND SMALL PULLEYS. 91

to K, and I find that 0-16 Ib. is not now sufficient to over-
come the friction of the pulley, but I add on weights until
c will just descend, which occurs when the load reaches
0-95 Ib. This weight is to be left on c as a counterpoise,
for the reasons already pointed out. I place a stone weight
on c and another on D, and you see that c will descend when
it receives an additional load of 1*35 Ibs. ; this is therefore
the amount of friction to be overcome when a stone weight
is raised over the pulley K.

165. Let us compare these results with the dimensions
of the pulleys. The proper way to measure the effective
circumference of a pulley when carrying a certain rope is
to measure the length of that rope which will just embrace
it. The length measured in this way will of course depend
to a certain extent upon the size of the rope. I find that
the circumferences of the two pulleys are 43"'o and g"$.
The ratio of these is 4*5 ; the corresponding resistances
from friction we have seen to be o'28 Ib. and 1*35 Ibs. The
larger of these quantities is 4 '8 times the smaller. This
number is very close to 4*5 ; we must not, as already
explained, expect perfect accuracy in experiments in friction.
In the present case the agreement is within the i-i6th of
the whole, and we may regard it as a proof of the law
that the friction of a pulley is inversely proportional to its
circumference.

1 66. It is easy to see the reason why friction should
diminish when the size of the pulley is increased. The
friction acts at the circumference of the axle about which
the wheel turns ; it is there present as a force tending to
retard motion. Now the larger the wheel the greater will
be the distance from the axis at which the force acts which
overcomes the friction, and therefore the less need be the
magnitude of the force. You will perhaps understand

92 EXPERIMENTAL MECHANICS. [LECT.

this better after the principle of the lever has been
discussed.

167. We may deduce from these considerations the prac-
tical maxim that large pulleys are economical of power.
This rule is well known to engineers ; large pulleys should
be used, not only for diminishing friction, but also to avoid
loss of power by excessive bending of the rope. A rope is
bent gradually around the circumference of a large pulley
with far less force than is necessary to accommodate it to a
smaller pulley : the rope also is apt to become injured by
excessive bending. In coal pits the trucks laden with coal
are hoisted to the surface by means of wire ropes which
pass from the pit over a pulley into the engine-house : this
pulley is of very large dimensions, for the reasons we have
pointed out.

THE LAW OF- FRICTION IN THE PULLEY.

168. I have here a wooden pulley 3^-5 in diameter ; the
hole is lined with brass, and the pulley turns very freely on
an iron spindle. I place the rope and hooks upon the
groove. Brass rubbing on iron has but little friction, and
when 7 Ibs. is placed on each hook, 0*5 Ib. added to either
will make it descend and raise up the other. Let 14 Ibs. be
placed on each hook, 0-5 Ib. is no longer sufficient ; i Ib. is
required : hence when the weight is doubled the friction is
also doubled. Repeating the experiment with 21 Ibs. and
28 Ibs. on each side, the corresponding weights necessary
to overcome friction are 1*5 Ibs. and 2 Ibs. In the four
experiments the weights used are in the proportion i, 2, 3, 4 ;
and the forces necessary to overcome friction, 0^5 Ib., i Ib.,
1-5 Ibs., and 2 Ibs., are in the same proportion. Hence the
friction is proportional to the load.
WHEELS.
169. The wheel is one of the most simple and effective

vi.] WHEELS. 93

devices for overcoming friction. A sleigh is an admirable
vehicle on a smooth surface such as ice, but it is totally
unadapted for use on common roads ; the reason being that
the amount of friction between the sleigh and the road is so
great that to move the sleigh the horse would have to exert
a force which would be very great compared with the load
he was drawing. But a vehicle properly mounted on wheels
moves with the greatest ease along the road, for the circum-
ference of the wheel does not slide, and consequently there
is no friction between the wheel and the road ; the wheel
however turns on its axle, therefore there is sliding, and
consequently friction, at the axle, but the axle and the wheel
are properly fitted to each other, and the surfaces are
lubricated with oil, so that the friction is extremely
small.

170. With large wheels the amount of friction on the
axle is less than with small wheels ; other advantages of
large wheels are that they do not sink much into depres-
sions in the roads, and that they have also an increased
facility in surmounting the innumerable small obstacles from
which even the best road is not free.

171. When it is desired to make a pulley turn with
extremely small friction, its axle, instead of revolving in
fixed bfarings, is mounted upon what are called friction
wheels. A set of friction wheels is shown in the apparatus
of Fig. 66 : as the axle revolves, the friction between the
axles and the wheels causes the latter to turn round with a
comparatively slow motion; thus all the friction is trans-
ferred to the axles of the four friction wheels ; these revolve
in their bearings with extreme slowness, and consequently
the pulley is but little affected by friction. The amount of
friction in a pulley so mounted may be understood from the
following experiment. A silk cord is placed on the pulley,

94 EXPERIMENTAL MECHANICS. [LECT.

and i Ib. weight is attached to each of its ends : these of
course balance. A number of fine wire hooks, each weigh-
ing o-ooi Ib., are prepared, and it is found that
when a weight of 0-004 Ib. is attached to either side
it is sufficient to overcome friction and set the weights in
motion.

ENERGY.

172. In connection with the subject of friction, and also
as introductory to the mechanical powers, the notion of
"work," or as it is more properly called "energy," is of
great importance. The meaning of this word as employed
in mechanics will require a little consideration.

173. In ordinary language, whatever a man does that
can cause fatigue, whether of body or mind, is called work.
In mechanics, we mean by energy that particular kind
of work which is directly or indirectly equivalent to raising
weights.

174. Suppose a weight is lying on the floor and a stool
is standing beside it : if a man raise the weight and place it
upon the stool, the exertion that he expends is energy in the
sense in which the word is used in mechanics. The amount
of exertion necessary to place the weight upon the stool de-
pends upon two things, the magnitude of the weight and
the height of the stool. It is clear that both these things
must be taken into account, for although we know the
weight which is raised, we cannot tell the amount of exer-
tion that will be required until we know the height through
which it is to be raised ; and if we know the height, we can-
not appreciate the quantity of exertion until we know the
weight.

175. The following plan has been adopted for expressing
quantities of energy. The small amount of exertion
necessary to raise i Ib. avoirdupois through one British

vi.] ENERGY. 95

foot is taken as a standard, compared with which all other
quantities of energy are estimated. This quantity of exer-
tion is called in mechanics the unit of energy, and some-
times also the "foot-pound."

176. If a weight of i Ib. has to be raised through a height of
2 feet, or a weight of 2 Ibs. through a height of i foot, it will
be necessary to expend twice as much energy as would have
raised a weight of i Ib. through i foot, that is, 2 foot-
pounds.

If a weight of 5 Ibs. had to be raised from the floor up to
a stool 3 feet high, how many units of energy would be re-
quired? To raise 5 Ibs. through i foot requires 5 foot-
pounds, and the process must be again repeated twice before
the weight arrive at the top of the stool. For the
whole operation 15 foot-pounds will therefore be neces-
sary.

If 100 Ibs. be raised through 20 feet, too foot-pounds of
energy is required for the first foot, the same for the second,
third, &c., up to the twentieth, making a total of 2,000
foot-pounds.

Here is a practical question for the sake of illustration.
Which would it be preferable to hoist, by a rope passing
over a single fixed pulley, a trunk weighing 40 Ibs. to a
height of 20 feet, or a trunk weighing 50 Ibs. to a height of
1 5 feet ? We shall find how much energy would be necessary
in each case : 40 times 20 is 800 ; therefore in the first case
the energy would be 800 foot-pounds. But 50 times 15
is 750 ; therefore the amount of work, in the second
case, is only 750 Ibs. Hence it is less exertion to carry
50 Ibs. up 15 feet than 40 Ibs. up 20 feet.

177. The rate of working of every source of energy,
whether it lie in the muscles of men or other animals, in
water-wheels, steam-engines, or other prime movers, is to

96 EXPERIMENTAL MECHANICS. [LECT.

be measured by the number of foot-pounds produced in
the unit of time.

The power of a steam-engine is denned by its equivalent
in horse-power. For example, it is meant that a steam-
engine of 3 horse-power, could, when working for an hour,
do as much work as 3 horses could do when working for
the same time. The power of a horse is, however, an un-
certain quantity, differing in different animals and not
quite uniform in the same individual; accordingly the
selection of this measure for the efficiency of the steam-
engine is inconvenient. We replace it by a convenient
standard horse-power, which is, however, a good deal larger
than that continuously obtainable from any ordinary horse.
A one horse-power steam-engine is capable of accomplishing
33,000 foot-pounds per minute.

178. We shall illustrate the numerical calculation of
horse-power by an example : if a mine be 1,000 feet deep, how
much water per minute would a 50 horse-power engine be
capable of raising to the surface? The engine would
yield 50 x 33,000 units of work per minute, but the weight
has to be raised 1,000 feet, consequently the number of
pounds of water raised per minute is

5° X 33.0QO = Ij650.

1,000

179. We shall apply the principle of work to the con-
sideration of the pulley already described (p. 90). In
order to raise the weight of 14 Ibs., it is necessary that
the rope to which the power is applied should be pulled
downwards by a force of 15 Ibs., the extra pound being on
account of the friction. To fix our ideas, we shall suppose
the 1 4 Ibs. to be raised i foot ; to lift this load directly, with-
out the intervention of the pulley, 14 foot-pounds would be
necessary, but when it is raised by means of the pulley, 15,

vi.] ENERGY. 97

foot-pounds are necessary. Hence there is an absolute loss
of ^-th of the energy when the pulley is used. If a steam-
engine of i horse-power were employed in raising weights
by a rope passing over a pulley similar to that on which we
have experimented, only -rfths of the work would be use-
fully employed ; but we find

33,000 x — = 30,800.

The engine would therefore perform 30,800 foot-pounds of
useful work per minute.

1 80. The effect of friction on a pulley, or on any other
machine, is always to waste energy. To perform a piece of
work directly requires a certain number of foot-pounds,
while to do it by a machine requires more, on account of the
loss by friction. This may at first sight appear somewhat
paradoxical, as it is well known that, by levers, pulleys, &c.,
an enormous mechanical advantage may be gained. This
subject will be fully explained in the next and following
lectures, which relate to the mechanical powers.

181. We shall conclude with a few observations on a
point of the greatest importance. We have seen a case
where 15 foot-pounds of energy only accomplished 14 foot-
pounds of- work, and thus i foot-pound appeared to be lost.
We say that this was expended upon the friction ; but what
is the friction ? The axle is gradually worn away by rub-
bing in its bearings, and, if it be not properly oiled, it
becomes heated. The amount of energy that seems to dis-
appear is partly expended in grinding down the axle, and
is partly transformed into heat ; it is thus not really lost, but
unfortunately assumes a form which we do not require and
in which it is rather injurious than otherwise. Indeed we
know that energy cannot be destroyed, however it may be

H

98 EXPERIMENTAL MECHANICS. [LECT. vi.

transformed; if it disappear in one shape, it is only to
reappear in another. A so-called loss of energy by friction
only means a diversion of a part of the work to some pur-
pose other than that which we wish to accomplish. It has
long been known that matter is indestructible : it is now
equally certain that the same may be asserted of energy.

LECTURE VII.
THE PULLEY-BLOCK.

Introduction. — The Single Moveablc Pulley. — The Three-sheave
Pulley-block.— The Differential Pulley-block. —The Epicycloidal
Pulley-block.

INTRODUCTION.

182. IN the first lecture I showed how a large weight
could be raised by a smaller weight, and I stated that this
subject would again occupy our attention. I now fulfil this
promise. The questions to be discussed involve the most
advantageous methods of employing a small force to
overcome" a greater. Here is a subject of practical impor-
tance. A man of average strength cannot raise more than
a hundredweight without great exertion, yet the weights
which it is necessary to lift and move about often weigh many
hundredweights, or even many tons. It is not always
practicable to employ numerous hands for the purpose, nor is
a steam-engine or other great source of power at all times
available. But what are called the mechanical powers
enable the forces at our disposal to be greatly increased.
One man, by their aid, can exert as much force as several

H 2

loo EXPERIMENTAL MECHANICS. [LECT.

could without such assistance; and when they are employed
to augment the power of several men or of a steam-engine,
gigantic weights, amounting sometimes to hundreds of tons,
can be managed with facility.

183. In the various arts we find innumerable cases where
great resistances have to be overcome ; we also find a cor-
responding number and variety of devices contrived by
human skill to conquer them. The girders of an iron bridge
have to be lifted up to their piers ; the boilers and
engines of an ocean steamer have to be placed in position ;
a great casting has to be raised from its mould ; a railway
locomotive has to be placed on the deck of a vessel for
transit ; a weighty anchor has to be lifted from the bottom
of the sea ; an iron plate has to be rolled or cut or punched :
for all of these cases suitable arrangements must be devised
in order that the requisite power may be obtained.

184. We know but little of the means which the ancients
employed in raising the vast stones of those buildings whick
travellers in the East have described to us. It is sometimes
thought that a large number of men could have transported
these stones without the aid of appliances which we would
now use for a similar purpose. But it is more likely that
some of the mechanical powers were used, as, with a multi-
tude of men, it is difficult fo ensure the proper application
of their united strength. In Easter Island, hundreds of
miles distant from civilised land, and now inhabited by
savages, vast idols of stone have been found in the hills
which must have been raised by human labour. It is useless
to speculate on the extinct race by whom this work was
achieved, or on the means they employed.

185. The mechanical powers are usually enumerated as
follows : — The pulley, the lever, the wheel and axle, the wedge,
the inclined plane, the screw. These different powers are so

VIL] THE SINGLE MOVEABLE PULLEY. 101

frequently used in combination that the distinctions cannot be
always maintained. The classification will, however, suffice
to give a general notion of the subject.

186. Many of the most valuable mechanical powers are
machines in which ropes or chains play an important part.
Pulleys are usually employed wherever it is necessary to
change the direction of a rope or chain which is transmitting
power. In the present lecture we shall examine the most
important mechanical powers that are produced by the
combination of pulleys.

THE SINGLE MOVEABLE PULLEY.

187. We commence wit a the most simple case, that of
the single moveable pulley (Fig. 35). The rope is firmly
secured at one end A; it then passes down under the move-
able pulley B, and upwards over a fixed pulley. To the free
end, which depends from the fixed pulley, the power is
applied while the load to be raised is suspended from the
moveable pulley. We shall first study the relation between
the power and the load in a simple way, and then we shall
describe a few exact experiments.

1 88. When the load is raised the moveable pulley itself
must of course be also raised, and a part of the power is
expended* for this purpose. But we can eliminate the weight
of the moveable pulley, so far as our calculations are con-
cerned, by first attaching to the power end of the rope a
sufficient weight to lift up the moveable pulley when not
carrying a load. The weight necessary for doing this is found
by trial to be a little over 1-5 Ibs. So that when a load
is being raised we must reduce the apparent power by 1-5
Ibs. to obtain the power really effective.

189. Let us suspend 14 Ibs. from the load hook at B, and
ascertain what power will raise the load. We leave the weight

EXPERIMENTAL MECHANICS.

[LECT.

of the moveable pulley and i '5 Ibs. of the power at c out of
consideration. I then find by experiment that 7 Ibs. of effec-
tive power is not sufficient to raise the load, but if one pound
more be added, the power descends, and the load is raised.

FlQ. 35.

Here, then, is a remarkable result ; a weight of 8 Ibs. has
overcome 14 Ibs. In this we have the first application of
the mechanical powers to increase our available forces.

190. Let us examine the reason of this mechanical
advantage. If the load be raised one foot, it is plain that
the power must descend two feet : for in order to raise the

vii.] THE SINGLE MOVEABLE PULLEY. 103

load the two parts of the rope descending to the moveable
pulley must each be shortened one foot, and this can only
be done by the power descending two feet. Hence when
the load of 14 Ibs. is lifted by the machine, for every foot it
is raised the power must descend two feet : this simple
point leads to a conception of the greatest importance, on
which depends the efficiency of the pulley. In the study of
the mechanical powers it is essential to examine the number of
feet through which the power must act in order to raise the load
one foot : this number we shall always call the velocity ratio.

191. To raise 14 Ibs. one foot requires 14 foot-pounds of
energy. Hence, were there no such, thing as friction, 7 Ibs.
on the power hook would be sufficient to raise the load ;
because 7 Ibs. descending through two feet yields 14 foot-
pounds. But there is a loss of energy on account of friction,
and a power of 7 Ibs. is not sufficient : 8 Ibs. are necessary.
Eight Ibs. in descending two feet performs 1 6 foot-pounds ; of
these only 14 are utilised on the load, the remainder being
the quantity of energy that has been diverted by friction.
We learn, then, that in the moveable pulley the quantity of
energy employed is really greater than that which would lift
the weight directly, but that the actual force which has to be
exerted is less.

192. Suppose that 28 Ibs. be placed on the load hook, a
few trials assure us that a power of 16 Ibs. (but not less) will
be sufficient for motion ; that is to say, when the load is
doubled, we find, as we might have expected, that the power
must be doubled also. It is easily seen that the loss of energy
by friction then amounts to 4 foot-pounds. We thus verify,
in the case of the moveable pulley, the approximate law
that the friction is proportional to the load.

193. By means of a moveable pulley a man is able to
raise a weight nearly double as great as he could lift

104

EXPERIMENTAL MECHANICS.

[LECT.

directly. From a series of careful experiments it has been
found that when a man is employed in the particular exer-
tion necessary for raising weights over a pulley, he is able
to work most efficiently when the pull he is required to
make is about 40 Ibs. A man could, of course, exert
greater force than this, but in an ordinary day's work he is
able to perform more foot-pounds when the pull is 40 Ibs.
than when it is larger or smaller. If therefore the weights to
be lifted amount to about So Ibs., energy may really be econo-
mized by the use of the single moveable pulley, although by
so doing a greater quantity of energy would be actually
expended than would have been necessary to raise the
weights directly.

194. Some experiments on larger loads have been tried
with the moveable pulley we have just described ; the
results are recorded in Table IX.

TABLE IX. — SINGLE MOVEABLE PULLEY.

Moveable pulley of cast iron 3" "25 diameter, groove o" "6 wide, wrought
iron axle o"'6 diameter ; fixed pulley of cast iron 5" diameter, groove
o"'4 wide, wrought iron axle o""6 diameter, axles oiled ; flexible
plaited rope o"'25 diameter ; velocity ratio 2, mechanical efficiency
I '8, useful effect 90 per cent. ; formula P = 2'2i + 0-5453 A".

Discrepancies be-

Number of
Experiment.

K.
Load in Ibs.

Observed
power in Ibs.

Calculated
power i.n Ibs.

tween observed
and calculated
powers.

1

28

i7'5 175

O'O

2

57

33'5 33 '3

- 0-2

3

^5

485 48-6

+ O'l

4

"3

64 o

638

- O'2

5

142

80-0

796

- 0'4

6

170

94-5

94-9

+ o'4

I

ic,8
226

110-5
125-5

IIO'2
I25-5

- 03

O'O

The dimensions of the pulleys are precisely stated
because, for pulleys of different construction, the numerical

VIL] THE SINGLE MOVEABLE PULLEY. 105

coefficients would not necessarily be the same. An attentive
study of this table will, however, show the general character
of the relation between the power and the load in all
arrangements of this class.

The table consists of five columns. The first contains
merely the numbers of the experiments for convenience of
reference. In the second column, headed ^?, the loads,
expressed in pounds, which are raised in each experiment,
are given ; that is, the weight attached to the hook, not
including the weight of the lower pulley. The weight of this
pulley is not included in the stated loads. In the third
column the powers are recorded, which were found to be
sufficient to raise the corresponding loads in the second
column. Thus, in experiment 7, it is found that a power
of 110-5 Ibs. will be sufficient to raise a load of 198 Ibs.
The third column has thus been determined by gradually
increasing the power until motion begins.

195. From an examination of the columns showing the
power and the load, we see that the power always amounts to
more than half the load. The excess is partly due to a
small portion of the power (about 1-5 Ibs.) being employed
in raising the lower block, and partly to friction. For
example, in experiment 7, if there had been no friction and
if the knver block were without weight, a power of 99 Ibs.
would have been sufficient ; but, owing to the presence of
these disturbing causes, 110-5 Ibs. are necessary: of this
amount 1-5 Ibs. is due to the weight of the pulley, 10 Ibs. is
the force of friction, and the remaining 99 Ibs. raises the load.

196. By a calculation based on this table we have
ascertained a certain relation between the power and the
load ; they are connected by the formula which may be
enunciated as follows •

The power is found by multiplying the weight of the load

106 EXPERIMENTAL MECHANICS. [LECT.

into 0-5453, and adding 2*21 to the product. Calling P the
power and R the load, we may express the relation thus :
P = 2-2i -f 0-5453 R. For example, in experiment 5,
the product of 142 and 0-5453 is 77-43, to which, when
2'2i is added, we find for P 79-64, very nearly the same ar,
80 Ibs., the observed value of the power.

In the fourth column the values of P calculated by
means of this formula are given, and in the last we exhibit
the discrepancies between the observed and the calculated
values for the sake of comparison. It will be seen that the
discrepancy in no case amounts to 0-5 lb., consequently
the formula expresses the experiments very well. The mode
of deducing it is given in the Appendix.

197. The quantity 2 '21 is partly that portion of the
power expended in overcoming the weight of the moveable
pulley, and partly arises from friction.

198. We can readily calculate from the formula how
much power will be required to raise a given weight ; for
example, suppose 200 Ibs. be attached to the moveable
pulley, we find that in Ibs. must be applied as the power.
But in order to raise 200 Ibs. one foot, the power exerted
must act over two feet ; hence the number of foot-pounds
required is 2 x in = 222. The quantity of energy that
is lost is 22 foot-pounds. Out of every 222 foot-pounds
applied, 200 are usefully employed ; that is to say, about
90 per cent, of the applied energy is utilized, while the
remaining 10 per cent, is lost by friction.

THE THREE-SHEAVE PULLEY-BLOCK.

199. The next arrangement we shall employ is a pair of
pulley-blocks s T, Fig. 35, each containing three sheaves, as
the small pulleys are termed. A rope is fastened to the
upper block, s ; it then passes down to the "lower block T

vii.] THE THREE-SHEAVE PULLEY-BLOCK. 107

under one sheave, up again to the upper block and over a
sheave, and so on, as shown in the figure. To the end of
the rope from the last of the upper sheaves the power H is
applied, and the load G is suspended from the hook attached
to the lower block. When the rope is pulled, it gradually
raises the lower block • and to raise the load one foot, each
of the six parts of the rope from the upper block to
the lower block must be shortened one foot, and therefore
the power must have pulled out six feet of rope. Hence,
for every foot that the load is raised the power must have
acted through six feet ; that is to say, the Telocity ratio is 6.

200. If there were no friction, the power would only be
one-sixth of the load. This follows at once from the prin-
ciples already explained. Suppose the load be 60 Ibs., then
to raise it one foot would require 60 foot-pounds ; and the
power must therefore exert 60 foot-pounds ; but the power
moves over six feet, therefore a power of 10 Ibs. would be
sufficient. Owing, however, to friction, some energy is lost,
and we must have recourse to experiment in order to test
the real efficiency of the machine. The single moveable
pulley nearly doubled our power ; we shall prove that the
three-sheave pulley-block will quadruple it. In this case we
deal with larger weights, with reference to which we may
leave the weight cf the lower block out of consideration.

20 1. Let us first attach i cwt. to the load hook ; we find
that 29 Ibs. on the power hook is the smallest weight that
can produce motion : this is only i Ib. more than one-quarter
of the load raised. If 2 cwt. be the load, we find that 56
Ibs. will just raise it : this time the power is exactly one-
puarter of the load. The experiment has been tried of
placing 4 cwt. on the hook ; it is then found that 109 Ibs.
will raise it, which is only 3 Ibs. short of i cwt. These
experiments demonstrate that for a three-sheave pulley-

io8

EXPERIMENTAL MECHANICS.

[LECT.

block of this construction we may safely apply th rule, that
the power is one-quarter of the load.

202. We are thus enabled to see how much of our ex-
ertion in raising weights must be expended in merely over-
coming friction, and how much may be utilized. Suppose
for example that we have to raise a weight of 100 Ibs. one
foot by means of the pulley-block ; the power we must
apply is 25 Ibs., and six feet of rope must be drawn out
from between the pulleys: therefore the power exerts 150
foot-pounds of energy. Of these only 100 foot-pounds are
usefully employed, and thus 50 foot-pounds, one-third of
the whole, have been expended on friction. Here we see that
notwithstanding a small force overcomes a large one, there is
an actual loss of energy in the machine. The real advantage
of course is that by the pulley-block I can raise a greater
weight than I could move without assistance, but I do not
create energy ; I merely modify it, and lose by the process.

203. The result of another series of experiments made
with this pair of pulley-blocks is given in Talle X.

TABLE X. — THREE-SHEAVE PULLEY-BLOCKS.

Sheaves cast iron 2" '5 diameter; plaited rope o"'25 diameter; velocity
ratio 6 ; mechanical advantage 4 ; useful effect 67 per cent. ; formula
/'= 2-36 + 0-238 A'.

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

p.

Calculated
pjwer in Ibs.

Discrepancies
between observed
and
calculated power-.

I

57

I5'5

I5'9

+ 0-4

2

114

29'5

29'5

O 0

3

171

4T5

43'i

- 04

4

228

56-0

566

+ O6

281

70 -o

69-2

- 0-8

6

338

83-0

82-8

- 0-2

7

395

97 -o

964

-0-6

8

452

109 -o

109-9

+ 0-9

vii.] THE THREE SHEAVE PULLEY-BLOCK. 109

204. This table contains five columns ; the weights raised
(shown in the second column) range up to somewhat over
4 cwt. The observed values of the power are given in the
third column ; each of these is generally about one-quarter
of the corresponding value of the load. There is, however,
a more accurate rule for finding the power ; it is as follows.

205. To find the power necessary to raise a given load,
multiply the loads in Ibs. by 0*238, and add 2-36 Ibs. to the
product. We may express the rule by the formula P= 2-36
+ 0-238 R.

206. To find the power which would raise 228 Ibs. : the
product of 228 and 0-238 is 54-26; adding 2-36, we find
56-6 Ibs. for the power required ; the actual observed power
is 56 Ibs., so that the rule is accurate to within about half a
pound. In the fourth column will be found the values of P
calculated by means of this rule. In the fifth column, the
discrepancies between the observed and the calculated values
of the powers are given, and it will be seen that the diffe-
rence in no case reaches i Ib. Of course it will be understood
that this formula is only reliable for loads which lie between
those employed in the first and last of the experiments.
We can calculate the power for any load between 57 Ibs.
and 452 Ibs., but for loads much larger than 452 or less than
57 it would probably be better to use the simple fourth of
the load rather than the power computed by the formula.

207. I will next perform an experiment with the three-
sheave pulley-block, which will give an insight into the
exact amount of friction without calculation by the help of
the velocity ratio. We first counterpoise the weight of
the lower block by attaching weights to the power. It is
found that about r6 Ibs. is sufficient for this purpose. I
attach a 56 Ib. weight as a load, and find that 13-1 Ibs. ii>
sufficient power for motion. This amount is partly com

i io EXPERIMENTAL MECHANICS. [LECT.

posed of the force necessary to raise the load if there were
no friction, and the rest is due to the friction. I next
gradually remove the power weights : when I have taken off
a pound, you see the power and the load balance each
other; but when I have reduced the power so low as 5^5
Ibs. (not including the counterpoise for the lower block), the
load is just able to overhaul the power, and run down. We
have therefore proved that a power of 13-1 Ibs. or greater raises
56 Ibs. , that any power between 13-1 Ibs. and 5-5 Ibs. balances
56 Ibs., and that any power less than 5*5 Ibs. is raised by
56 Ibs.

When the power is raised, the force of friction, together
with the power, must be overcome by the load. Let us call
X the real power that would be necessary to balance 56 Ibs.
in a perfectly frictionless machine, and Y the force of
friction. We shall be able to determine X and F by the
experiments just performed. When the load is raised a
power equal to X + Y must be applied, and therefore X +
Y= 13'!. On the other hand, when the power is raised,
the force X is just sufficient to overcome both the friction Y
and the weight 5-5 ; therefore X= Y + 5-5.

Solving this pair of equations, we find that X = 9-3 and
K= 3-8. Hence we infer that the power in the frictionless
machine would be 9*3 ; but this is exactly what would have
been deduced from the velocity ratio, for 56 + 6 — 9-3 Ibs.
In this result we find a perfect accordance between theory
and experiment.

THE DIFFERENTIAL PULLEY-BLOCK.

208. By increasing the number of sheaves in a pair of
pulley-blocks the power may be increased ; but the length
of rope (or chain) requisite for several sheaves becomes a
practical inconvenience. There are also other reasons

vii.] THE DIFFERENTIAL PULLEY-BLOCK. in

which make the differential pulley-block, which we shall
now consider, more convenient for many purposes than the
common pulley blocks when a considerable augmentation of
power is required.

209. The principle of the differential pulley is very
ancient, and in modern times it has been embodied in a
machine of practical utility. The object is to secure, that
while the power moves over a considerable distance, the
load shall only be raised a short distance. When this has
been attained, we then know by the principle of energy
that we have gained a mechanical advantage.

210. Let us consider the means by which this is effected
in that ingenious contrivance, Weston's differential pulley-
block. The principle of this machine will be understood
from Fig. 36 and Fig. 37.

It consists of three parts,-
moveable pulley, and an
endless chain. We shall
briefly describe them. The
upper block p is furnished
with a hook for attachment
to a support. The sheave
it contains resembles two
sheaves,~one a little smaller
than the other, fastened
together : they are in fact
one piece. The grooves
are provided with ridges,
adapted to prevent the
chain from slipping. The
lower pulley Q consists
of one sheave, which is
also furnished with a groove

-an upper pulley-block, a

hook,

112 EXPERIMENTAL MECHANICS. [LECT.

which the load is attached. The endless chain performs
a part that will be understood from the sketch of
the principle in Fig. 36. The chain passes from the
hand at A up to L over the larger groove in the upper
pulley, then downwards at B, under the lower pulley, up again
at c, over the smaller groove in the upper pulley at A, and
then back again by D to the hand at A. When the hand
pulls the chain downwards, the two grooves of the upper
pulley begin to turn together in the direction shown by the
arrows on the chain. The large groove is therefore winding
up the chain, while the smaller groove is lowering.

211. In the pulley which has been employed in the
experiments to be described, the effective circumference
of the large groove is found to be 1 i"'84, while that of the
small groove is io"-^6. When the upper pulley has made
one revolution, the large groove must have drawn up u"'84
of chain, since the chain cannot slip on account of the
ridges ; but in the same time the small groove has lowered
io"'36 of chain : hence when the upper pulley has revolved
once, the chain between the two must have been shortened
by the difference between n"'84 and io"-^6, that is by
i"'48 ; but this can only have taken place by raising the
moveable pulley through half i*'48, that is, through a space
o"'74. The power has then acted through n'/<84, and has
raised the resistance o//74. The power has therefore moved
through a space 16 times greater than that through which
the load moves. In fact, it is easy to verify by actual
trial that the power must be moved through 16 feet in order
that the load may be raised i foot. We express this by
saying that the velocity ratio is 16.

212. By applying power to the chain at D proceeding from
the smaller groove, the chain is lowered by the large groove
faster than it is raised by the small one, and the lower

vii.] THE DIFFERENTIAL PULLEY-BLOCK. 113

pulley descends. The load is thus raised or lowered by
simply pulling one chain A or the other D.

213. We shall next consider the me-
chanical efficiency of the differential
pulley-block. The block (Fig. 37) which
we shall use is intended to be worked
by one man, and will raise any weight
not exceeding a quarter of a ton.

We have already learned that with this
block the power must act through six-
teen feet for the load to be raised one
foot. Hence, were it not for friction, the
power need only be the sixteenth part of
the load. A few trials will show us that
the real efficiency is not so large, and that
in fact more than half the work exerted
is merely expended upon overcoming
friction. This will lead afterwards to a
result of considerable practical import-
ance.

214. Placing upon the load-hook a
weight of 200 Ibs., I find that 38 Ibs.
attached to a hook fastened on the power-
chain -is sufficient to raise the load ; that is
to say, the power is about one-sixth of the
load. If I make the load 400 Ibs. I find
the requisite power to be 64 Ibs., which is

only about 3 Ibs. less than one-sixth of 400 Ibs. We may
safely adopt the practical rule, that with this differential
pulley-block a man would be able to raise a. weight six
times as great as he could raise without such assistance.

215. A series of experiments carefully tried with different
loads have given the results shown in Table XI.

FIG. 37.

EXPERIMENTAL MECHANICS.

[LECT.

TABLE XL— THE DIFFERENTIAL PULLEY-BLOCK.

Circumference of large groove u"'84, of small groove io"'36 ; velocity
ratio 1 6 ; mechanical efficiency 6 '07 ; useful effect 38 per cent ;
formula P = 3*87 + 0-1508 R.

Number cf
Experiment.

/?.

Load in Ibs.

Observed
power in Ibs.

p.

Calculated
power in Ibs.

Difference of the
observed and
calculated values.

j

56

IO

12-3

+ 2-T.

2

112

20

20-8

+ 0'8

3

168

31

29-2

- 1-8

4

224

38

377

- 0-3

280

48

46-1

- I-Q

I

336
392
448

g

72

54-6
63-1
71'S

+ 0-6
- 0-9
-0-5

9

504

80

80-0

O'O

,0

86

88-4

+ 2-4

The first column contains the numbers of the experiments,
the second the weights raised, the third the observed values
of the corresponding powers. From these the following
rule for finding the power has been obtained : —

216. To find the power, multiply the load by 0-1508,
and add 3-87 Ibs. to the product; this rule may be
expressed by the formula P = 3-87 + 0-1508 ^?. (See
Appendix.)

217. The calculated values of the powers are given in
the fourth column, and the differences between the observed
and calculated values in the last column. The differences
do not in any case amount to 2-5 Ibs., and considering that
the loads raised are up to a quarter of a ton, the formula
represents the experiments with satisfactory precision.

218. Suppose for example 280 Ibs. is to be raised; the
product of 280 and 0-1508 is 42-22, to which, when 3-87 is

vii.] THE DIFFERENTIAL PULLEY-BLOCK. 115

added, we find 46-09 to be the requisite power. The
mechanical efficiency found by dividing 46*09 into 280
is 6*07.

219. To raise 280 Ibs. one foot 280 foot-pounds of
energy would be necessary, but in the differential pulley-
block 46-09 Ibs. must be exerted for a distance of 16 feet
in order to accomplish this object. The product of 46-09
and 1 6 is 737-4. Hence the differential pulley-block
requires 737-4 foot-pounds of. energy to be applied in
order to yield 280 useful foot-pounds ; but 280 is only
38 per cent, of 737-4, and therefore with a load of
280 Ibs. only 38 per cent, of the energy applied to a differen-
tial pulley-block is utilized. In general, we may state
that not more than about 40 per cent, is profitably
used, and that the remainder is expended in overcoming
friction.

220. It is a remarkable and useful property of the
differential pulley, that a weight which has been hoisted
will remain suspended when the hand is removed, even
though the chain be not secured in any manner. The
pulleys we have previously considered do not possess this
convenient property. The weight raised by the three-sheave
pulley-block, for example, will run down unless the free
end of" the rope be properly secured. The difference in
this respect between these two mechanical powers is not
a consequence of any special mechanism; it is simply
caused by the excessive friction in the differential pulley-
block.

221. The reason why the load does not run down in the
differential pulley may be thus explained. Let us suppose
that a weight of 400 Ibs. is to be raised one foot by the
differential pulley-block ; 400 units of work are necessary,
and therefore 1,000 units of work must be applied to the

I 2

u6 EXPERIMENTAL MECHANICS. [LECT.

power chain to produce the 400 units (since only 40 per
cent, is utilized). The friction will thus have consumed
600 units of work when the load has been raised one foot.
If the power-weight be removed, the pressure supported by
the upper pulley-block is diminished. In fact, since the
power- weight is about £th of the load, the pressure on the
axle when the power-weight has been removed is only 4ths
of its previous value. The friction is nearly proportional to
that pressure : hence when the power has been removed the
friction on the upper axle is -Jyths of its previous value, while
the friction on the lower pulley remains unaltered.

We may therefore assume that the total friction is
at least |ths of what it was before the power-weight
was removed. Will friction allow the load to descend ?
600 foot-pounds of work were required to overcome the
friction in the ascent: at least f x 600 = 514 foot-pounds
would be necessary to overcome friction in the descent.
But where is this energy to come from ? The load
in its descent could only yield 400 units, and thus
descent by the mere weight of the load is impossible.
To enable the load to descend we have actually to aid the
movement by pulling the chain D (Figs. 36 and 37),
which proceeds from the small groove in the upper pulley.

222. The principle which we have here established
extends to other mechanical powers, and may be stated
generally. Whenever more than half the applied energy is
consumed by friction, the load will remain without running
down when the machine is left free.

THE EPICYCLOIDAL PULLEY-BLOCK.

223. We shall conclude this lecture with some experi-
ments upon a useful mechanical power introduced by Mr.
Eade under the name of the epicycloidal pulley-block. It

vii.] THE EPICYCLOIDAL PULLEY-BLOCK. 117

is shown in Fig. 33, and also in Fig. 49. In this machine
there are two chains : one a slight endless chain to which
the power is applied ; the other a stout chain which has a
hook at each end, from either of which the load may be
suspended. Each of these chains passes over a sheave in
the block : these sheaves are connected by an ingenious
piece of mechanism which we need not here describe.
This mechanism is so contrived that, when the power
causes the sheave to revolve over which the slight chain
passes, the sheave which carries the large chain is also made
to revolve, but very slowly.

224. By actual trial it is ascertained that the power must
be exerted through twelve feet and a half in order to raise
the load one foot; the velocity ratio of the machine is
therefore 12-5.

225. If the machine were frictionless, its mechanical
efficiency would be of course equal to its velocity ratio;
owing to the presence of friction the mechanical efficiency
is less than the velocity ratio, and it will be necessary to
make experiments to determine the exact value. I attach
to the load-hook a weight of 280 Ibs., and insert a few
small hooks into the links of the power chain in order
to receive weights : 56 Ibs. is sufficient to produce
motion-, hence the mechanical efficiency is 5. Had there
been no friction a power of 56 Ibs. would have been
capable of overcoming a load of 12-5x56 = 700 Ibs.
Thus 700 units of energy must be applied to the
machine in order to perform 280 units of work. In
other words, only 40 per cent, of the applied energy is
utilized.

226. An extended series of experiments upon the
epicycloidal pulley-block is recorded in Table XII.

Ii8 EXPERIMENTAL MECHANICS. [LECT. vii.

TABLE XII.— THE EPICYCLOIDAL PULLEY-BLOCK.

Size adapted for lifting weights up to 5 cwt. ; velocity ratio 12-5 ;
mechanical efficiency 5 ; useful effect 40 per cent. ; calculated formula
P = 5-8 + 0-185 •#•

Number of
Experiment.

R.

Loads in Ibs.

Observed
power in Ibs.

P.
Calculated
power in Ibs.

Difference of the
observed and
calculated values.

I

56

'5

16-2

+ 1-2

2

112

27

26-5

~ °'5

3

168

40

- 3'i

4

224

47

47 -2

+ O'2

280

56

57-6

+ 1-6

6

336

66

680

+ 2'O

7

392

78

78-3

+ O*3

8

448

88

88-6

+ o'6

9

504

100

99-0

- I'D

10

560

IIO

109-4

-0-6

The fourth column shows the calculated values of the
powers derived from the formula. It will be seen by the
last column that the formula represents the experiments
with but little error.

227. Since 60 per cent, of energy is consumed by friction,
this machine, like the differential pulley-block, sustains its
load when the chains are free. The differential pulley-
block gives a mechanical efficiency of 6, while the epicy-
cloidal pulley-block has only a mechanical efficiency of 5,
and so far the former machine has the advantage ; on the
other hand, that the epicycloidal pulley contains but one
block, and that its lifting chain has two hooks, are practical
conveniences strongly in its favour.

LECTURE VIII.

THE LEVER.

The Lever of the First Order.— The Lever of the Second Order.—
The Shears.— The Lever of the Third Order.

THE LEVER OF THE FIRST ORDER.

228. THERE are many cases in which a machine for over-
coming great resistance is necessary where pulleys would be
quite inapplicable. To meet these various demands a
correspondingly various number of contrivances has been
devised. Amongst these the lever in several different forms
holds "an important place.

229. The lever of the first order will be understood by
reference to Fig. 38. It consists of a straight rod, to
one end of which the power is applied by means of the
weight c. At another point B the load is raised, while at A
the rod is supported by what is called the fulcrum. In the
case represented in the figure the rod is of iron, i" x i" in
section and 6' long; it weighs 19 Ibs. The power is pro-
duced by a 56 Ib. weight : the fulcrum consists of a
moderately sharp steel edge firmly secured to the framework.

EXPERIMENTAL MECHANICS.

[LECT.

The load in this case is replaced by a spring balance H,
and the hook of the balance is attached to the frame. The
spring is strained by the action of the lever, and the index

FIG. 38.

records the magnitude of the force produced at the short
end. This is the lever with which we shall commence our
experiments.

VIIL] THE LEVER OF THE FIRST ORDER. 121

230. In examining the relation between the power and
the load, the question is a little complicated by the weight
of the lever itself (19 Ibs.), but we shall be able to evade
the difficulty by means similar to those employed on a
former occasion (Art. 60) ; we can counterpoise the weight
of the iron bar. This is easily done by applying a hook to
the middle of the bar at D, thence carrying a rope over a
pulley F, and suspending a weight G of 19 Ibs. from its free
extremity. Thus the bar is balanced, and we may leave its
weight out of consideration.

231. We might also adopt another plan analogous to
that of Art. 51, which is however not so convenient. The
weight of the bar produces a certain strain upon the spring
balance. I may first read off the strain produced by the
bar alone, and then apply the weight c and read again.
The observed strain is due both to the weight c and to the
weight of the bar. If I subtract the known effect of the
bar, the remainder is the effect of c. It is, however, less
complicated to counterpoise the bar, and then the strains
indicated by the balance are entirely due to the power.

232. The lever is 6' long ; the point B is 6" from the end,
and B c is 5' long. B c is divided into 5 equal portions of i';
A is at one of these divisions, i' distant from B, and c is 5' dis-
tant, frem B in the figure ; but c is capable of being placed
at any position, by simply sliding its ring along the bar.

233. The mode of experimenting is as follows : — The
weight is placed on the bar at the position c : a strain is
immediately produced upon H ; the spring stretches a little,
and the bar becomes inclined. It may be noticed that the
hook of the spring balance passes through the eye of a wire-
strainer, so that by a few turns of the nut upon the strainer
the lever can be restored to the horizontal position.

234. The power of 56 Ibs. being 4' from the fulcrum,

122 EXPERIMENTAL MECHANICS. [LECT.

while the load is i' from the fulcrum, it is found
that the strain indicated by the balance is 224 Ibs. ; that is,
four times the amount of the power. If the weight be
moved, so as to be 3' from the fulcrum, the strain is observed
to be 168 Ibs.; and whatever be the distance of the power from
the fulcrum, we find that the strain produced is obtained by
multiplying the magnitude of the power in pounds by the
distance expressed in feet, and fractional parts of a foot.
This law may be expressed more generally by stating that
tfie pouter is to the load as the distance of the load from the
fulcrum is to the distance of the power from the fulcrum.

235. We can verify this law under varied circumstances.
I move the steel edge which forms the fulcrum of the lever
until the edge is 2' from B, and secure it in that position. I
place the weight c at a distance of 3' from the fulcrum. I
now find that the strain on the balance is 84 Ibs. ; but 84
is to 56 as 3 is to 2, and therefore the law is also verified in
this instance.

236. There is another aspect in which we may express
the relation between the power and the load. The law in
this form is thus stated : The power multiplied by its distance

from the fulcrum is equal to the load multiplied by its
distance from the fulcrum. Thus, in the case we have just
considered, the product of 56 and 3 is 168, and this is equal
to the product of 84 and 2. The distances from the
fulcrum are commonly called the arms of the lever, and the
rule is expressed by stating that The power multiplied into
its arm is equal to the load multiplied into its arm : hence
the load may be found by dividing the product of the power
and the power arm by the load arm. This simple law gives
a very convenient method of calculating the load, when we
know the power and the distances of the power and the load
from the fulcrum.

VIIL] THE LEVER OF THE FIRST ORDER. 123

237. When the power arm is longer than the load arm,
the load is greater than the power ; but when the power
arm is shorter than the load arm, the power is greater than
the load.

We may regard the strain on the balance as a power
which supports the weight, just as we regard the weight to
be a power producing the strain on the balance. We see,
then, that for the lever of the first order to be efficient as a
mechanical power it is necessary that the power arm be
longer than the load arm.

238. The lever is an extremely simple mechanical power ;
it has only one moving part. Friction produces but little
effect upon it, so .that the laws which we have given may
be actually applied in practice, without making any allow-
ance for friction. In this we notice a marked difference
between the lever and the pulley-blocks already de-
scribed.

239. In the lever of the first order we find an excellent
machine for augmenting power. A power of 14 Ibs. can by
its means overcome a resistance of a hundredweight, if the
power be eight times as far from the fulcrum as the load is
from the fulcrum. This principle it is which gives utility to
the crowbar. The end of the bar is placed under a heavy
stone, -which it is required to raise ; a support near that end
serves as a fulcrum, and then a comparatively small force
exerted at the power end will suffice to elevate the stone.

240. The applications of the lever are innumerable. It
is used not only for increasing power, but for modifying and
transforming it in various ways. The lever is also used in
weighing-machines, the principles of which will be readily
understood, for they are consequences of the law we have
explained. Into these various appliances it is not our
intention to enter at present ; the great majority of them

124 EXPERIMENTAL MECHANICS. [LECT.

may, when met with, be easily understood by the principle
we have laid down.

THE LEVER OF THE SECOND ORDER.

241. In the lever of the second order the power is at
one end, the fulcrum at the other end, and the load lies
between the two : this lever therefore differs from the lever
of the first order, in which the fulcrum lies between the two
forces. The relation between the power and the load in
the lever of the second order may be studied by the
arrangement in Fig. 39.

242. The bar A c is the same rod of iron 72" x i" x i"
which was used in the former experiment. The fulcrum A
is a steel edge on which the bar rests ; the power consists of
a spring balance H, in the hook of which the end c of the
bar rests ; the spring balance is sustained by a wire-strainer,
by turning the nut of which the bar may be adjusted hori-
zontally. The part of the bar between the fulcrum A and
the power c is divided into five portions, each i' long, and
the points A and c are each 6" distant from the extremities
of the bar. The load employed is 56 Ibs. ; through the
ring of this weight the bar passes, and thus the bar sup-
ports the load. The bar is counterpoised by the weight of
19 Ibs. at G, in the manner already explained (Art. 231).

243. The mode of experimenting is as follows : — Let the
weight B be placed i' from the fulcrum; the strain shown
by the spring balance is about 1 1 Ibs. If we calculate the
value of the power by the rule already given, we should
have found the same result. The product of the load by its
distance from the fulcrum is 56, the distance of the power
from the fulcrum is 5 ; hence the value of the power
should be 56 -f- 5 = 1 1 -2.

244. If the weight be placed 2' from the fulcrum the

vin.] THE LEVER OF THE SECOND ORDER. 125

strain is about 22-5 Ibs. and it is easy to ascertain that this
is the same amount as would have been found by the appli-
cation of the rule. A similar result would have been

FIG. 39.

obtained if the 56 Ib. weight had been placed upon any
other part of the bar ; and hence we may regard the rule
proved for the lever of the second order as well as for the

126

EXPERIMENTAL MECHANICS.

[LECT.

lever of the first order : that, the power multiplied by
its distance from the fulcrum is equal to the load mul-
tiplied by its distance from the fulcrum. In the present
case the load is uniformly 56 Ibs., while the power by
which it is sustained is always less than 56 Ibs.

FIG. 40.

245. The lever of the second order is frequently applied
to practical purposes ; one of the most instructive of these
applications is illustrated in the shears shown in Fig. 40.

The shears consist of two levers of the second order,
which by their united action enable a man to exert a greatly
increased force, sufficient, for example, to cut with ease a
rod of iron 0^-25 squarg. The mode of action is simple.
The first lever A F has a handle at one end F, which is 22"
distant from the other end A, where the fulcrum is placed.

vili.] THE LEVER OF THE SECOND ORDER. 127

At a point B on this lever, i"-8 distant from the fulcrum A,
a short link B c is attached ; the end of the link c is jointed
to a second lever c D ; this second lever is 8" long ; it forms
one edge of the cutting shears, the other edge being fixed
to the framework.

246. I place a rod of iron o"-25 square between the jaws
of the shears in the position E, the distance D E being 3//g5,
and proceed to cut the iron by applying pressure to the
handle. Let us calculate the amount by which the levers
increase the power exerted upon F. Suppose for example
that I press downwards on the handle with a force of
10 Ibs., what is the magnitude of the pressure upon the
piece of iron ? The effect of each lever is to be calculated
separately. We may ascertain the power exerted at B by
the rule of moments already explained ; the product of the
power and its arm is 22 x 10=220: this divided by the
number of inches, 1*8 in the line A B, gives a quotient 122,
and this quotient is the number of pounds pressure which
is exerted by means of the link upon the second lever. We
proceed in the same manner to find the magnitude of the
pressure upon the iron at E. The product of 122 and 8
is 976. This is divided by 3-5, and the quotient found is
279. Hence the exertion of a pressure of 10 Ibs. at F
produces a pressure of 279 Ibs. at E. In round numbers,
we may say that the pressure is magnified 28-fold by means
of this combination of levers of the second order.

247. A pressure of 10 Ibs. is not sufficient to shear across
the bar of iron, even though it be magnified to 279 Ibs. I
therefore suspend weights from F, and gradually increase
the load until the bar is cut. I find at the first trial that
112 Ibs. is sufficient, and a second trial with the same bar
gives 114 Ibs.; 113 Ibs., the mean between these results,
may be considered an adequate force. This is the load on

128 EXPERIMENTAL MECHANICS. [LECT.

F; the real pressure on the bar is 113 x 27'9=3i53 Ibs. :
thus the actual pressure which was necessary to cut the bar
amounted to more than a ton.

248. We can calculate from this experiment the amount
of force necessary to shear across a bar one square inch in
section. We may reasonably suppose that the necessary
power is proportional to the section, and therefore the
power will bear to 3153 Ibs. the proportion which a square
of one inch bears to the square of a quarter inch ; but this
ratio is 16: hence the force is 16x3153 Ibs., equal to
about 22-5 tons.

249. It is noticeable that 22-5 tons is nearly the force
which would suffice to tear the bar in sunder by actual
tension. We shall subsequently return to the subject
of shearing iron in the lecture upon Inertia (Lecture
XVI.).

THE LEVER OF THE THIRD ORDER.

250. The lever of the third order may be easily under-
stood from Fig. 39, of which we have already made use.
In the lever of the third order the fulcrum is at one end,
the load is at the other end, while the power lies between
the two. In this case, then, the power is represented by
the 56 Ib. weight, while the load is indicated by the spring
balance. The power always exceeds the load, and con-
sequently this lever is to be used where speed is to be
gained instead of power. Thus, for example; when the
power, 56 Ibs., is 2' distant from the fulcrum, the load
indicated by the spring balance is about 23 Ibs.

251. The treadle of a grindstone is often a lever of the
third order. The fulcrum is at one end, the load is at the
other end, and the foot has only to move through a small
distance.

vin.] THE LEVER OF THE THIRD ORDER. 129

252. The principles which have been discussed in Lecture
III. with respect to parallel forces explain the laws now laid
down for levers of different orders, and will also enable us
to express these laws more concisely.

253. A comparison between Figs. 20 and 39 shows that
the only difference between the contrivances is that in
Fig. 20 we have a spring balance c in the same place as the
steel edge A in Fig 39. We may in Fig. 20 regard one
spring balance as the power, the other as the fulcrum, and
the weight as the load. Nor is there any essential difference
between the apparatus of Fig. 38 and that of Fig. 20. In
Fig. 38 the bar is pulled down by a force at each end, one
a weight, the other a spring balance, while it is supported
by the upward pressure of the steel edge. In Fig. 20 the
bar is being pulled upwards by a force at each end, and
downwards by the weight. The two cases are substantially
the same. In each of them we find a bar acted upon by a
pair of parallel forces applied at its extremities, and retained
in equilibrium by a third force.

254. We may therefore apply to the lever the principles of
parallel forces already explained. We showed that two
parallel forces acting upon a bar could be compounded into
a resultant, applied at a certain point of the bar. We have
defined the moment of a force (Art. 64), and proved that
the moments of two parallel forces about the point of
application of their resultant are equal.

255. In the lever of the first order there are two parallel
forces, one at each end ; these are compounded into a
resultant, and it is necessary that this resultant be applied
exactly over the steel edge or fulcrum in order that the bar
may be maintained at rest. In the levers of the second and
third orders, the power and the load are two parallel forces
acting in opposite directions ; their resultant, therefore, does

130 EXPERIMENTAL MECHANICS. [LECT. vin.

not lie between the forces, but is applied on the side of the
greater, and at the point where the steel edge supports
the bar. In all cases the moment of one of the forces
about the fulcrum must be equal to that of the other. From
the equality of moments it follows that the product of the
power and the distance of the power from the fulcrum equals
the product of the load, and the distance of the load from
the fulcrum : this principle suffices to demonstrate the rules
already given.

256. The laws governing the lever may be deduced
from the principle of work ; the load, if nearer than the
power to the fulcrum, is moved through a smaller distance
than the power. Thus, for example, in the lever of the
first order : if the load be 1 2 times as far as the power from
the fulcrum, then for every inch the load moves it can be
demonstrated that the power must move 12 inches. The
number of units of work applied at one end of a machine
is equal to the number yielded at the other, always excepting
the loss due to friction, which is, however, so small in the
lever that we may neglect it. If then a power of i Ib. be
applied to move the power end through 12 inches, one unit
of work will have been put into the machine. Hence one
unit of work must be done on the load, but the load only
moves through T^ of a foot, and therefore a load of 12 Ibs.
could be overcome : this is the same result as would be
given by the rule (Art. 236).

257. To conclude : we have first determined by actual
experiment the relation between the power and the load
in the lever; we have seen that the law thus obtained
harmonizes with the principle of the composition of
parallel forces ; and, finally, we have shown how the same
result can be deduced from the fertile and important principle
of work.

LECTURE IX.
THE INCLINED PLANE AND THE SCREW.

The Inclined Plane without Friction.— The Inclined Plane with
Friction.— The Screw.— The Screw-jack.— The Bolt and Nut.

THE INCLINED PLANE WITHOUT FRICTION.

258. THE mechanical powers now to be considered are
often used for other purposes beside those of raising great
weights. For example : the parts of a structure have to
be forcibly drawn together, a powerful compression has to
be exerted, a mass of timber or other material has to be
riven asunder by splitting. For purposes of this kind the
inclined plane in its various forms, and the screw, are of
the greatest use. The screw also, in the form of the screw-
jack, is sometimes used in raising weights. It is principally
convenient when the weight is enormously great, and the
distance through which it has to be raised comparatively
small.

259. We shall commence with the study of the inclined
plane. The apparatus used is shown in Fig. 41. A B is a
plate of glass 4' long, mounted on a frame and turning

K 2

132 EXPERIMENTAL MECHANICS. [LECT.

round a hinge at A ; B D is a circular arc, with its centre at
A, by which the glass may be supported ; D c is a vertical
rod, to which the pulley c is clamped. This pulley can be
moved up and down, to be accommodated to the position
of A B ; the pulley is made of brass, and turns very freely.
A little truck R is adapted to run on the plane of glass.
The truck is laden to weigh i lb.,
and this weight is unaltered
throughout the experiments ;
the wheels are very free, so that
the truck runs with but little
friction.

260. But the friction, though
small, is appreciable, and it
will be necessary to measure

the amount and then endeavour to counteract its effect
upon the motion. The silk cord attached to the truck is
very fine, and its weight is neglected. A series of weights
is provided ; they are made from pieces of brass wire, and
weigh o'i lb. and o'oi lb. : these can easily be hooked into
the loop on the cord at P. We first make the plane A B
horizontal, and bring down the pulley c so that the cord
shall be parallel to the plane ; we find that a force must be
applied by the cord in order to draw the truck along the
plane : this force is of course the friction, and1 by a suitable
weight at P the friction may be said to be counterbalanced.
But we cannot expect that the friction will be the same when
the plane is horizontal as when the plane is inclined. We
must therefore examine this question by a method analogous
to that used in Art. 207.

261. Let the plane be elevated until B E, the elevation
of B above A D, is 20" ; let c be properly adjusted : it is
found that when P is 0-45 lb. R is just pulled up ; and on the

IX.] INCLINED PLANE WITHOUT FRICTION. 133

other hand, when P is only 0-40 Ib. the truck descends and
raises P ; and when P has any value intermediate between
these two, the truck remains in equilibrium. Let us
denote the force of gravity acting down the plane by R,
and it follows that R must be 0*425 Ib., and the friction
0^025 Ib. For when P raises R, it must overcome fric-
tion as well as R; therefore the power must be 0*025 +
0-425 = 0-45. On the other hand, when R raises P, it
must also overcome the friction 0-025, therefore P can
only be 0-425 — 0-025 = °'4° > an<^ R ig trms found to be a
mean between the greatest and least values of P consistent
with equilibrium. If the plane be raised so that the height
B E is 33", the greatest and least values of P are o'66 and
0-71 ; therefore R is 0-685 and the friction 0*025, the same
as before. Finally, making the height B E only 2'', the friction
is found to be 0-020, which is not much less than the
previous determinations. These experiments show that we
may consider this very small friction to be practically con-
stant at these inclinations. (Were the friction large, other
methods are necessary, see Art. 265.) As in the experi-
ments R is always raised we shall give P the permanent
load of 0-025 lb-» thus sufficiently counteracting friction,
which we may therefore dismiss from consideration. It is
hardly 'necessary to remark that, in afterwards recording the
weights placed at P, this counterpoise is not to be included.
262. We have now the means of studying the relation
between the power and the load in the frictionless inclined
plane. The incline being set at different elevations, we
shall observe the force necessary to draw up the constant load
of i Ib. Our course will be guided by first making use of
the principle of energy. Suppose B E to be 2' ; when the
truck has been moved from the bottom of the plane to the
top, it will have been raised vertically through a height of 2',

134

EXPERIMENTAL MECHANICS.

[LECT.

and two units of energy must have been consumed. But
the plane being 4' long, the force which draws up the
truck need only be 0*5 lb., for 0*5 Ib. acting over 4' pro-
duces two units of work. In general, if / be the length of
the plane and h its height, R the load, and P the power,
the number of units of energy necessary to raise the load
is R h, and the number of units expended in pulling it
up the plane is PI : hence R h = PI, and consequently
P : h : : R : I ; that is, the power is to the height of the
plane as the load is to its length. In the present case
R = i lb., / = 48" ; therefore P = 0-0208 h, where h is the
height of the plane in inches, and P the power in pounds.

263. We compare the powers calculated by this formula
with the actual observed values : the result is given in
Table XIII.

TABLE XIII. — INCLINED PLANE.

Glass Plane 48" long, truck I lb. in weight, friction counterpoised ;
formula />=O'O2o8 x h".

Number of
Experiment.

Height of
plane.

Observed
power in Ibs.

P.
Calculated
power in Ibs.

Difference of the
observed and cal-
culated powers.

I

2"

0-04

0-04

O'OO

2

4"

0-08

0'08

O'OO

3

6"

0-13

O'I2

-O'OI

4

8"

O'i6

0-17

+ O'OI

5

10"

0-21

0'2I

O'OO

6

15"

0-31

0-31

O'OO

7

20"

0-42

0-42

O'OO

8

33"

071

O'69

-0'02

Thus for example, in experiment 6, where the height B E
is 15", it is observed that the power necessary to draw the
truck is o'3i lb. The truck is placed in the middle of the
plane, and the power is adjusted so as to be sufficient to

ix.] INCLINED PLANE WITH FRICTION. 135

draw the truck to the top with certainty ; the necessary
power calculated by the formula is also 0-31 Ibs., so that
the theory is verified.

264. The fifth column of the table shows the difference
between the observed and the calculated powers. The very
slight differences, in no case exceeding the fiftieth part of
a pound, may be referred to the inevitable errors of
experiment.

THE INCLINED PLANE WITH FRICTION.

265. The friction of the truck upon the glass plate is
always very small, and is shown to have but little varia-
tion at those inclinations of the plane which we used.
But when the friction is large, we shall not be justified in
neglecting its changes at different elevations, and we must
adopt more rigorous methods. For this inquiry we shall
use the pine plank and slide already described in Art. 1*7.
We do not in this case attempt to diminish friction by the
aid of wheels, and consequently it will be of considerable
amount.

266. In another respect the experiments of Table XIII.
are also in contrast with those now to be described. In the
former the load was constant, while the elevation was
changed. In the latter the elevation remains constant while
a succession of different loads are tried. We shall find in
this inquiry also that when the proper allowance has been
made for friction, the theoretical law connecting the power
and the load is fully verified.

267. The apparatus used is shown in Fig. 33 ; the plane,
is, however, secured at one inclination, and the pulley c shown
in Fig. 32 is adjusted to the apparatus, so that the rope
from the pulley to the slide is parallel to the incline. The
elevation of the plane in the position adopted is i7°'2, so

136 EXPERIMENTAL MECHANICS. [LECT.

that its length, base, and height are in the proportions of
the numbers j, 0*955, and o'2()6. Weights ranging from
7 Ibs. to 56 Ibs. are placed upon the slide, and the power is
found which, when the slide is started by the screw, will
draw it steadily up the plane. The requisite power consists
of two parts, that which is necessary to overcome gravity
acting down the plane, and that which is necessary to over
come friction.

268. The forces are shown in Fig. 42. R G, the force of

gravity, is resolved into R L and
R M ; R L is evidently the com-
ponent acting down the plane,
and R M the pressure against the
plane; the triangle GLR is similar
to A B c, hence if R be the load,
the force R L acting down the
plane must be 0-296 R, and the
pressure upon the plane 0*955 R.

269. We shall first make a calculation with the ordinary
law that the friction is proportional to the pressure. The
pressure upon the plane A B, to which the friction is pro-
portional, is not the weight of the load. The pressure is
that component (R M) of the load which is perpendicular
to the plane A B. When the weights do not extend
beyond 56 Ibs., the best value for the coefficient of friction
is 0*288 (Art. 141) : hence the amount of friction upon
the plane is

0-288 x 0-955 & = °'275 &•

This force must be overcome in addition to 0-296 R (the
component of gravity acting down the plane) : hence the
expression for the power is

0-275 R + 0-296 R = 0-571 R.

INCLINED PLANE WITH FRICTION.

137

270. The values of the observed powers compared with
the powers calculated from the expression 0-571 R are
shown in Table XIV.

TABLE XIV.— INCLINED PLANE.

Smooth plane of pine 72" x n" ; angle of inclination I7°'2 ; slide of
pine, grain crosswise ; slide started ; formula P=o~$Jj R.

Number of
Experiment.

R.

Total load on
•slide in Ibs.

Power in Ibs. 1 P.
which just Calculated value
draws up slide. of the power.

Difference of the
observed and cal-
culated powers.

I

7

4-6

40

-0-6

2

H

8-3 8-0

-0'3

3

21

I2'3 12-0

-0-3

4

28

i6'5

i6'o

-0'5

5

35

2O '0

20 x>

O'O

5

42

24'2

24-0

-O'2

7

49

28-0

28-0

O'O

8

56

SI'S

32-0

+ 0-2

271. Thus for example, in experiment 6, a load of
42 Ibs. was raised by a force of 24*2 Ibs., while the cal-
culated value is 24-0 Ibs. ; the difference, 0*2 Ibs., is shown
in the last column.

272. The calculated values are found to agree tolerably
well with the observed values, but the presence of the large
differences in No. i and No. 4 leads us to inquire whether
by employing the more accurate law of friction (Art. 141) a
better result may not be obtained.

In Table VI. we have shown that the friction for weights
not exceeding 56 Ibs. is expressed by the formula F= 0-9 +
0-266 X pressure, but the pressure is in this case =0*955 ^>
and hence the friction is

0-9 + 0-2547?.
To this must be added 0-296 J?s the component of the force

138

EXPERIMENTAL MECHANICS.

[LECT.

of gravity which must be overcome, and hence the total
force necessary is

0-9 + 0-55^.

The powers calculated from this expression are compared
with those actually observed in Table XV.

TABLE XV. — INCLINED PLANE.

Smooth plane of pine 72" x n" ; angle of inclination I7°'2 ; slide of
pine, grain crosswise ; slide started ; formula P— 0-9 + 0-55 R.

Number of
Experiment.

R.
Total load on
slide in Ibs.

Power in Ibs.
which just
draws up slide.

p.

Calculated value
of the power.

Difference of the
observed and cal-
culated powers.

j

7

4'6

47

+ OT

2

H

8'3

8-6

+ 0-3

3

21

12-3

12-5

-t-O'2

4

28

i6'5

16-3

-0-2

5

35

20'0

2O'I

+ 0'I

6

42

24-2

24-0

-O'2

7

49

28-0

27-8

-O*2

8

56

3i-8

317

-O'l

For example : in experiment 5, a load of 35 Ibs. is found
to be raised by a power of 20-0 Ibs., while the calculated
power is 0*9 + 0^55 x 35 = 20*1 Ibs.

273. The calculated values of the powers are shown by
this table to agree extremely well with the observed values,
the greatest difference being only 0.3 Ib. Hence there can
be no doubt that the principles on which the formula has
been calculated are correct. This table may therefore be
regarded as verifying both the law of friction, and the rule
laid down for the relation between the power and the load
in the inclined plane.

274. The inclined plane is properly styled a mechanical
power. For let the weight be 30 Ibs., we calculate by the
formula that 17*4 Ibs. would be sufficient to raise it, so that,

ix.] THE SCREW. 139

notwithstanding the loss by friction, we have here a smaller
force overcoming a larger one, which is the essential feature
of a mechanical power. The mechanical efficiency is
304-17 -4=172.

275. The velocity ratio in the inclined plane is the ratio
of the distance through which the power moves to the
height through which the weight is raised, that is i -H 0*296
= 3-38. To raise 30 Ibs. one foot, a force of 17-4 Ibs. must
therefore be exerted through 3-38 feet. The number of units
of work expended is thus 17 -4x3 '38 = 58*8. Of this 30 units,
equivalent to 51 per cent., are utilized. The remaining
28-8 units, or 49 per cent, are absorbed by friction.

276. We have pointed out in Art. 222 that a machine
in which less than half the energy is lost by friction will
permit the load to run down when free : this is the case in
the present instance ; hence the weight will run down the
plane unless specially restrained. That it should do so agrees
with Art. 147, for it was there shown that at about i3°'4,
and still more at any greater inclination, the slide would
descend when started.

THE SCREW.

277. The inclined plane as a mechanical power is often
used in^the form of a wedge or in the still more disguised
form of a screw. A wedge is an inclined plane which is
forced under the load ; it is usually moved by means of a
hammer, so that the efficiency of the wedge is augmented by
the dynamical effect of a blow.

278. The screw is one of the most useful mechanical
powers which we possess. Its form may be traced by
wrapping a wedge-shaped piece of paper around a cylinder,
and then cutting a groove in the cylinder along the spiral
line indicated by the margin of the paper. Such a groove

140 EXPERIMENTAL MECHANICS. [LECT.

is a screw. In order that the screw may be used it must
revolve in a nut which is made from a hollow cylinder,
the internal diameter of which is equal to that of the
cylinder from which the screw is cut. The nut contains
a spiral ridge, which fits into the corresponding thread in
the screw ; when the nut is turned round, it moves back-
wards or forwards according to the direction of the rotation.
Large screws of the better class, such as those upon which
we shall first make experiments, are always turned in a lathe,
and are thus formed with extreme accuracy. Small screws
are made in a simpler manner by means of dies and other
contrivances.

279. A characteristic feature of a screw is the inclination
of the thread to the axis. This is most conveniently de-
scribed by the number of complete turns which the thread
makes in a specified length of the screw, usually an inch.
For example : a screw is said to have ten threads to the
inch when it requires 10 revolutions of the nut in order to
move it one inch. The shape of the thread itself varies
with the purposes for which the screw is intended; the
section may be square or triangular, or, as is generally the
case in small screws, of a rounded form.

280. There is so much friction in the screw that ex-
periments are necessary for the determination of the law
connecting the power and the load.

281. We shall commence with an examination of the
screw by the apparatus shown in Fig. 43.

The nut A is secured upon a stout frame ; to the end of
the screw hooks are attached, in order to receive the load,
which in this apparatus does not exceed 224 Ibs. ; at the
top of the screw is an arm E by which the screw is turned ;
to the end of the arm a rope is attached, which passing over
a pulley D, carries a hook for receiving the power c.

IX.]

THE SCREW.

141

282. We first apply the principle of work to this screw,
and calculate the relation between the power and the load
as it would be found if friction were absent. The diameter
of the circle described by the end of the arm is 20" -5 ; its
circumference is therefore 64" -4. The screw contains three
threads in the inch, hence in order to raise the load i" the

FIG. 43.

power moves 3 x 64"-4= 193" very nearly; therefore the
velocity ratio is 193, and were the screw capable of working
without friction, 193 would represent the mechanical effi-
ciency. In actually performing the experiments the arm E
is placed at right angles to the rope leading to the pulley,
and the power hook is weighted until, with a slight start,

142

EXPERIMENTAL MECHANICS.

[LECT.

the arm is steadily drawn ; but the power will only move the
arm a few inches, for when the cord ceases to be perpen-
dicular to the arm the power acts with diminished efficiency;
consequently the load is only raised in each experiment
through a small fraction of an inch, but quite sufficient for
our purpose.

TABLE XVI.— THE SCREW.

Wrought iron screw, square thread, diameter i"'25, with 3 threads to
the inch, length of arm io"*25 ; nut of cast iron, bearing surfaces
oiled, velocity ratio 193, useful effect 36 per cent., mechanical
efficiency 70 ; formula P=

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

P.
Calculated
power of Ibs.

Difference of the
observed and cal-
culated powers.

I

28

0'4

0'4

O'O

2

56

0-8

0-8

O'O

3

84

12

I'2

O'O

4

112

1-6

1-6

O'O

1

140

[68

2'O
2'4

2'O
2"4

O'O

o-o

7

196

27

2-8

+ 0-1

8

224

3'3

3'2

-O'l

283. The results of the experiments are shown in Table
XVI. If the motion had not been aided by a start the
powers would have been greater. Thus in experiment 6,
2-4 Ibs. is the power with a start, when without a start 3-2
Ibs. was found to be necessary. The experiments have all
been aided by a start, and the results recorded have been
corrected for the friction of the pulley over which the rope
passes : this correction is very small, in no case exceeding
0-2 Ib. The fourth column contains the values of the
powers computed by the formula P= 0-0143 & This
formula has been deduced from the observations in the

IX.] THE SCREW. 143

manner described in the Appendix. The fifth column
proves that the experiments are truly represented by the
formula : in each of the experiments 7 and 8, the difference
between the calculated and observed values amounts to o-i
lb., but this is quite inconsiderable in comparison with the
weights we are employing.

284. In order to lift 100 Ibs. the expression 0*0143 -^ shows
that i '43 Ibs. would be necessary: hence the mechanical
efficiency of the screw is IOO-T 1-43 =70. Thus this screw
is vastly more powerful than any of the pulley systems
which we have discussed. A machine so capable, so com-
pact, and so strong as the screw, is invaluable for innumerable
purposes in the Arts, as well as in multitudes of appliances
in daily use.

285. It is evident, however, that the distance through
which the screw can raise a weight must be limited by the
length cf the screw itself, and that in the length of lift the
screw cannot compete with many of the other contrivances
used in raising weights.

286. We have seen that the velocity ratio is 193 ; there-
fore, to raise 100 Ibs. i foot, we find that 1-43 x 193 = 276
units of energy must be expended : of this only 100 units,
or 36 per cent., is usefully employed ; the rest being
consumed in overcoming the friction of the screw. Thus
nearly two-thirds of the energy applied to such a screw is
wasted. Hence we find that friction does not permit the
load to run down, since less than fifty per cent, of the applied
energy is usefully employed (Art. 222). This is one of
the valuable properties which the screw possesses.

287. We may contrast the screw with the pulley block
(Art. 199). They are both powerful machines : the latter
is bulky and economical of power, the former is compact
and wasteful of power; the latter is adapted for raising

ix.] THE SCREW-JACK. 145

weights through considerable distances, and the former for
exerting pressures through short distances.

THE SCREW-JACK.

288. The importance of the screw as a mechanical power
justifies us in examining another of its useful forms, the
screw-jack. This machine is used for exerting great pres-
sures, such for example as starting a ship which is reluctant
to be launched, or replacing a locomotive upon the line
from which its wheels have slipped. These machines vary
in form, as well as in the weights for which they are
adapted ; one of them is shown at D in Fig. 44, and a
description of its details is given in Table XVII. We shall
determine the powers to be applied to this machine for
overcoming resistances not exceeding half a ton.

289. To employ weights so large as half a ton would be
inconvenient if not actually impossible in the lecture-room,
but the required pressures can be produced by means of a
lever. In Fig. 44 is shown a stout wooden bar 16' long.
It is prevented from bending by means of a chain ; at E the
lever is attached to a hinge, about which it turns freely ; at
A a tray is placed for the purpose of receiving weights. The
screw-jack is 2' distant from E, consequently the bar is a
lever of-the second order, and any weight placed in the
tray exerts a pressure eightfold greater upon the top of the
screw-jack. Thus each stone in the tray produces a pres-
sure of i cwt. at the point D. The weight of the lever and
the tray is counterpoised by the weight c, so that until the
tray receives a load there is no pressure upon the top of the
screw-jack, and thus we may omit the lever itself from con-
sideration. The screw-jack is furnished with an arm D G ;
at the extremity G of this arm a rope is attached, which
passes over a pulley and supports the power B.

L

EXPERIMENTAL MECHANICS.

[LECT.

290. The velocity ratio for this screw-jack with an arm of
33'', is found to be 414, by the method already described
(Art. 283).

291. To determine its mechanical efficiency we must
resort to experiment. The result is given in Table XVII.

TABLE XVII.— THE SCREW-JACK.

Wrought iron screw, square thread, diameter 2", pitch 2 threads to the
inch, arm 33" ; nut brass, bearing surfaces oiled ; velocity ratio 414 ;
useful effect, 28 per cent.; mechanical efficiency 116; formula
P= o "66 -t- o '0075 'ft.

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

P.

Calculated
power in Ibs.

Difference of the
observed and cal-
culated powers.

I

112

I '4

i '5

+ 0 I

2

224

2 '2

2-3

+ 0 I

3

336

3 "3

3-2

-0 I

4

448

4'i

4-0

-0 I

560

5'°

4'9

-0 I

5

672

57

57

OX)

7

784

6-5

6-5

O'O

8

896

7 '4

7 '4

O'O

9

I008

8'i

8-2

+ 0'I

10

1 1 20

9-0

9-1

+ O'I

292. It may be seen from the column of differences how
closely the experiments are represented by the formula.
The power which is required to raise a given weight,
say 600 Ibs., may be calculated by this formula ; it is
o 66 + 0-0075 x 600 = 5-16. Hence the mechanical efficiency
of the screw-jack is 6004-5-16 = 116. Thus the screw is
very powerful, increasing the force applied to it more than a
hundredfold. In order to raise 600 Ibs. one foot, a quantity
of work represented by 5-16X414=2136 units must be ex-

IX.] THE SCREW-JACK. 147

pended; of this only 600, or 28 per cent., is utilized, so
that nearly three-quarters of the energy applied is expended
upon friction.

293. This screw does not let the load run down, since
less than 50 per cent, of energy is utilised ; to lower the
weight the lever has actually to be pressed backwards.

294. The details of an experiment on this subject will be
instructive, and afford a confirmation of the principles laid
down. In experiment 10 we find that 9*0 Ibs. suffice to
raise 1,120 Ibs.; now by moving the pulley to the other side
of the lever, and placing the rope perpendicularly to the
lever, I find that to produce motion the other way — that is,
of course to lower the screw — a force of 3-4 Ibs. must be
applied. Hence, even with the assistance of the load, a
force of 3 '4 Ibs. is necessary to overcome friction. This
will enable us to determine the amount of friction in the
same manner as we determined the friction in the pulley-
block (Art. 207). Let x be the force usefully employed in
raising, and Y the force of friction, which acts equally in
either direction against the production of motion ; then to
raise the load the power applied must be sufficient to over-
come both x and Y, and therefore we have x+Y=9-o.
When the weight is to be lowered the force x of course aids
in the- lowering, but x alone is not sufficient to overcome
the friction; it requires the addition of 3-4 Ibs.,
and we have therefore x-r-3'4=:Yr and hence x = 2'8,

Y=6'2.

That is, 2-8 is the amount of force which with a friction-
less screw would have been sufficient to raise half a ton.
But in the frictionless screw the power is found by dividing
the load by the velocity ratio. In this case 1120-^414=27,
which is within o'i Ib. of the value of x. The agreement of
these results is satisfactory.

L 2

148

EXPERIMENTAL MECHANICS. [LECT. ix.

THE SCREW BOLT AND NUT.

295. One of the most useful applications of the screw is
met with in the common bolt and nut, shown in Fig. 45.
It consists of a wrought-iron rod with a head at one end
and a screw on the other, upon which the nut works. Bolts
in many different sizes and forms represent the stitches by
which machines and frames are most
readily united. There are several
reasons why the bolt is so convenient.
It draws the parts into close contact
with tremendous force ; it is itself so
strong that the parts united practically
form one piece. It can be adjusted
quickly, and removed as readily. The
same bolt by the use of washers can
be applied to pieces of very different
sizes. No skilled hand is required to
use the simple tool that turns the nut.
Adding to this that bolts are cheap and
durable, we shall easily understand why
they are so extensively used.

296. We must remark in conclusion
that the bolt owes its utility to friction ;
screws of this kind do not overhaul, hence when the nut
is screwed home it does not recoil. If it were not that
more than half the power applied to a screw is consumed
in friction, the bolt and the nut would either be rendered
useless, or at least would require to be furnished with some
complicated apparatus for preventing the motion of the nut.

FIG. 45.

LECTURE X.
THE WHEEL AND AXLE,

Introduction. — Experiments upon the Wheel and Axle. — Friction
upon the Axle.— The Wheel and Barrel.— The Wheel and
Pinion. — The Crane. — Conclusion.

INTRODUCTION.

297. THE mechanical powers discussed in these lec-
tures may be grouped into two classes, — the first where
ropes or chains are used, and the second where ropes or
chains are absent. Belonging to that class in which ropes
are not employed, we have the screw discussed in the last
lecture-, and the lever discussed in Lecture VIII. ; while
among those machines in which ropes or chains form an
essential part of the apparatus, the pulley and the wheel and
axle hold a prominent place. We have already examined
several forms of the pulley, and we now proceed to the not
less important subject of the wheel and axle.

298. Where great resistances have to be overcome, but
where the distance through which the resistance must be
urged is short, the lever or the screw is generally found to
be the most appropriate means of increasing power. When,

150

EXPERIMENTAL MECHANICS.

[LECT.

however, the resistance has to be moved a considerable
distance, the aid of the pulley, or the wheel and axle, or
sometimes of both combined, is called in. The wheel and
axle is the form of mechanical power which is generally used

when the distance is considerable through which a weight
must be raised, or through which some resistance must be
overcome.

299. The wheel and axle assumes very many forms cor-
responding to the various purposes to which it is applied.

x.] THE WHEEL AND AXLE. 151

The general form of the arrangement will be understood
from Fig. 46. It consists of an iron axle B, mounted in
bearings, so as to be capable of turning freely ; to this axle a
rope is fastened, and at the extremity of the rope is a weight
D, which is gradually raised as the axle revolves. Attached
to the axle, and turning with it, is a wheel A with hooks in
its circumference, upon which lies a rope ; one end of this
rope is attached to the circumference of the wheel, and the
other supports a weight E. This latter weight may be called
the power, while the weight D suspended from the axle is the
load. When the power is sufficiently large, E descends,
making the wheel to revolve ; the wheel causes the axle to
revolve, and thus the rope is wound up and the load D is
raised.

300. When compared with the differential pulley as a
means of raising a weight, this arrangement appears rather
bulky and otherwise inconvenient, but, as we shall presently
learn, it is a far more economical means of applying
energy. In its practical application, moreover, the arrange-
ment is simplified in various ways, two of which may be
mentioned.

301. The capstan is essentially a wheel and axle ; the
power is not in this case applied by means of a rope, but by
direct pressure on the part of the men working it ; nor is
there actually a wheel employed, for the pressure is applied
to what would be the extremities of the spokes of the wheel
if a wheel existed.

302. In the ordinary winch, the power of the labourer
is directly applied to the handle which moves round in the
circumference of a circle.

303. There are innumerable other applications of the
principle which are constantly met with, and which can be
easily understood with a little attention. These we shall

152 EXPERIMENTAL MECHANICS. [LECT.

not stop to describe, but we pass on at once to the
important question of the relation between the power and
the load.

EXPERIMENTS UPON THE WHEEL AND AXLE.

304. We shall commence a series of experiments upon
the wheel A and axle B of Fig. 46. We shall first determine
the velocity ratio, and then ascertain the mechanical effi-
ciency by actual experiment The wheel is of wood ; it is
about 30" in diameter. The string to which the power is
attached is coiled round a series of hooks, placed near the
margin of the wheel ; the effective circumference is thus a
little less than the real circumference. I measure a single
coil of the string and find the length to be 88"'5. This
length, therefore, we shall adopt for the effective circum-
ference of the wheel. The axle is o"75 in diameter, but its
effective circumference is larger than the circle of which
this length is the diameter.

305. The proper mode of finding the effective circum-
ference of the axle in a case where the rope bears a
considerable proportion to the axle is as follows. Attach
a weight to the extremity of the rope sufficient to stretch
it thoroughly. Make the wheel and axle revolve suppose
20 times, and measure the height through which the
weight is lifted ; then the one-twentieth part of that
height is the effective circumference of the axle. By
this means I find the circumference of the axle we are
using to be a'^Sy.

306. We can now ascertain the velocity ratio in this
machine. When the wheel and axle have made one com-
plete revolution the power has been lowered through a
distance of 88"'5, and the load has been raised through
2"~&-j. This is evident because the wheel and axle are

x.] THE WHEEL AND AXLE. 153

attached together, and therefore each completes one revolu-
tion in the same time ; hence the ratio of the distance
which the power moves over to that through which the load
is raised is 88"'5 -r- 2"-87 =31 very nearly. We shall there-
fore suppose the velocity ratio to be 31. Thus this wheel
and axle has a far higher velocity ratio than any of the
systems of pulleys which we have been considering.

307. Were friction absent the velocity ratio of 31 would
necessarily express the mechanical efficiency of this wheel
and axle ; owing to the presence of friction the real efficiency
is less than this — how much less, we must ascertain by
experiment. I attach a load of 56 Ibs. to the hook which is
borne by the rope descending from the axle : this load is
shown at D in Fig. 46. I find that a power of 2*6 Ibs.
applied at E is just sufficient to raise D. We infer from this
result that the mechanical efficiency of this machine is
56 -f 2-6 = 21-5. I add a second 56 Ib. weight to the load,
and I find that a power of 5-0 Ibs. raises the load of 112 Ibs.
The mechanical efficiency in this case is 112 ^5 = 22*5.
We adopt the mean value 22. Hence the mechanical
efficiency is reduced by friction from 31 to 22.

308. We may compute from this result the number of
units of energy which are utilized out of every 100 units
appliedr Let us suppose a load of 100 Ibs. is to be raised
one foot ; a force of 100 -r 22 = 4-6 Ibs. will suffice to raise
this load. This force must be exerted through a space of
31', and consequently 31 x 4-6 = 143 units of energy must
be expended ; of thij amount 100 units are usefully em-
ployed, and therefore the percentage of energy utilized is
100 -r- 143 x 100 = 70. It follows that 30 per cent, of
the applied energy is consumed in overcoming friction.

309. We can see the reason why the wheel and axle
overhauls — that is, runs down of its own accord — when

154 EXPERIMENTAL MECHANICS. [LECT.

allowed to do so ; it is because less than half the applied
energy is expended upon friction.

310. A series of experiments which have been carefully
made with this wheel and axle are recorded in Table
XVIII.

TABLE XVIII.— WHEEL AND AXLE.

Wheel of wood ; axle of iron, in oiled brass bearings ; weight of wheel
and axle together, 16-5 Ibs. ; effective circumference of wheel, 88"'5 ;
effective circumference of axle, 2" "87 ; velocity ratio, 31 ; mechanical
efficiency, 22 ; useful effect, 70 per cent. ; formula, P — 0-204 +
0-0426 R.

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

P.

Calculated

Difference of
the observed
and calculated

values.

I

28

I'4

I "4

O'O

2

42

2'O

2-0

O'O

3

56

2'6

2'6

O'O

4
5

11

3'2

37

%

O'O
+ O'l

6

9»

4'4

4'4

O'O

7

112

S'o

S'o

O'O

By <the method of the Appendix a relation connecting
the power and the load has been determined ; it is
expressed in the form —

P= o'204-f- 0-0426 R.

311. Thus for example in experiment 5 a load of 84 Ibs.
was found to be raised by a power of 37 Ibs. The value
calculated by the formula is 0-204 + 0-0426 x 84 = 3*8.
The calculated value only differs from the observed value
by o'i lb., which is shown in the fifth column. It will be
seen from this column that the values calculated from the
formula represents the experiments with fidelity.

312. We have deduced the relation between the power

x.] FRICTION UPON THE AXLE. 155

and the load from the principle of energy, but we might
have obtained it from the principle of the lever. The
wheel and axle both revolve about the centre of the axle ;
we may therefore regard the centre as the fulcrum of a
lever, and the points where the cords meet the wheel and
nxle as the points of application of the power and the load
respectively.

313. By the principle of the lever of the first order
(Art. 237), the power is to the load in the inverse pro-
portion of the arms ; in this case, therefore, the power is
to the load in the inverse proportion of the radii of the
wheel and the axle. But the circumferences of circles are
in proportion to their radii, and therefore the power must
be to the load as the circumference of the axle is to the
circumference of the wheel.

314. This mode of arriving at the result is a little arti-
ficial ; it is more natural to deduce the law directly from
the principle of energy. In a mechanical power of any
complexity it would be difficult to trace exactly the trans-
mission of power from one part to the next, but the
principle of energy evades this difficulty ; no matter what
be the mechanical arrangement, simple or complex, of few
parts or of many, we have only to ascertain by trial how
many fe~et the power must traverse in order to raise the load
one foot; the number thus obtained is the theoretical
efficiency of the machine.

FRICTION UPON THE AXLE.

315. In the wheel and axle upon which we have been
experimenting, we have found that about 30 per cent, of
the power is consumed by friction. We shall be able to
ascertain to what this loss is due, and then in some degree
to remove its cause. From the experiments of Art. 165

IS6

EXPERIMENTAL MECHANICS.

[LECT.

we learned that the friction of a small pulley was very
much greater than that of a large pulley— in fact, the
friction is inversely proportional to the diameter of the
pulley. We infer from this that by winding the rope upon
a barrel instead of upon the axle, the friction may be
diminished.

FIG. 47.

316. We can examine experimentally the effect of friction
on the axle by the apparatus of Fig. 47. B is a shaft o"'JS
diameter, about which a rope is coiled several times ; the
ends of this rope hang down freely, and to each of them
hooks E, F are attached. This shaft revolves in brass

X.] FRICTION UPON THE AXLE. 157

bearings, which are oiled. In order to investigate the
amount of power lost by winding the rope upon an axle
of this size, I shall place a certain weight — suppose 56 Ibs. —
upon one hook F, and then I shall ascertain what amount
of power hung upon the other hook E will be sufficient to
raise F. There is here no mechanical advantage, so that
the excess of load which E must receive in order to raise F
is the true measure of the friction.

317. I add on weights at E until the power reaches
85 Ibs., when E descends. We thus see that to raise 56 Ibs.
an excess of 29 Ibs. was necessary to overcome the friction.
We may roughly enunciate the result by stating that to
raise a load in this way, half as much again is required
for the power. This law is verified by suspending 28 Ibs.
at F, when it is found that a power of 43 Ibs. at E is re-
quired to lift it. Had the power been 42 Ibs., it would have
been exactly half as much again as the load.

318. Hence in raising F upon this axle, about one-third
of the power which must be applied at the circumference
of the axle is wasted. This experiment teaches us where
the loss lies in the wheel and axle of Art. 304, and explains
how it is that about a third of its efficiency is lost. 85 Ibs.
was only able to raise two-thirds of its own weight, owing
to the friction ; and hence we should expect to find, as we
actually have found, that the power applied at the cir-
cumference of the wheel has an effect which is only two-
thirds of its theoretical efficiency.

319. From this experiment we should infer that the
proper mode of avoiding the loss by friction is to wind the
rope upon a barrel of considerable diameter rather than
upon the axle itself. I place upon a similar axle to that
on which we have been already experimenting a barrel of
about 15" circumference. I coil the rope two or three

158 EXPERIMENTAL MECHANICS. [LECT.

times about the barrel, and let the ends hang down as
before. I then attach to each end 56 Ibs. weight, and I
find that 10 Ibs. added to either of the weights is sufficient
to overcome friction, to make it descend, and raise the
other weight. The apparatus is shown in Fig. 47. A is the
barrel, c and D are the weights. In this arrangement
10 Ibs. is sufficient to overcome the friction which required
29 Ibs. when the rope was simply coiled around the axle.
In other words, by the barrel the loss by friction is reduced
to one-third of its amount.

THE WHEEL AND BARREL.

320. We next place the barrel upon the axis already
experimented upon and shown in Fig. 46 at B. The cir-
cumference of the wheel is 88^5 ; the circumference of
the barrel is i4"'9. The proper mode of finding the cir-
cumference of the barrel is to suspend a weight from the
rope, then raise this weight by making one revolution of
the wheel, and the distance through which the weight is
raised is the effective circumference of the barrel. The
velocity ratio of the wheel and barrel is then found, by
dividing 14-9 into 88-5, to be 5-94.

321. The mechanical efficiency of this machine is deter-
mined by experiment. I suspend a weight of 56 Ibs. from
the hook, and apply power to the wheel. I find that io'i_lbs.
is just sufficient to raise the load.

322. The mechanical efficiency is to be found by dividing
io'i into 56 ; the quotient thus obtained is 5-54. The
mechanical efficiency does not differ much from 5-94, the
velocity ratio ; and consequently in this machine but little
power is expended upon friction.

323. We can ascertain the loss by computing the per-
centage of applied energy which is utilized. Let us sup-

X.]

THE WHEEL AND BARREL.

159

pose a weight of 100 Ibs. has to be raised one foot : for this
purpose a force of 100-7-5-54 = 18-1 Ibs. must be applied.
This is evident from the definition of the mechanical
efficiency ; but since the load has to be raised one foot, it
is clear from the meaning of the velocity ratio that the
power must move over 5^94 : hence the number of units
of work to be applied is to be measured by the product of
5-94 and i8'i, that is, by 107-5 ; in order therefore to
accomplish 100 units of work 107*5 units of work must be
applied. The percentage of energy usefully employed is
100^-107-5 x I0° = 93- This is far more than 70, which
is the percentage utilized when the axle was used without
the barrel (Art. 309).

324. A series of experiments made with care upon the
wheel and barrel are recorded in Table XIX.

TABLE XIX.— THE WHEEL AND BARREL.

Wheel of wood, 88" -5 in circumference, on the same axle as a cast-iron
barrel of I4""9 circumference ; axle is of wrought iron, o"'75 in
diameter, mounted in oiled brass bearings ; power is applied to the
circumference of the wheel, load raised by rope round barrel ; velocity
ratio, 5 "94 ; mechanical efficiency, 5 -54 ; useful effect, 93 per cent. ;
formula, P = 0*5 + 0*169 R-

Difference of

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

Calculated
power in Ibs.

the observed
and calculated
values.

I

14

27

2'9

-f 0'2

2

28

5'3

5 '2

-O'l

3

42

77

7-6

-O'l

4

56

IO'I

IO'O

-O'l

5

70

12-4

12-4

O'O

6

84

H7

147

O'O

7

98

17-1

17-1

cro

8

112

19-4

19-5

+ O'I

160 EXPERIMENTAL MECHANICS. [LECT.

The formula which represents the experiments with the
greatest amount of accuracy is P= 0-5 + 0-169 •#• This
formula is compared with the experiments, and the column
of differences shows that the calculated and the observed
values agree very closely. The constant part 0-5 is partly
due to the constant friction of the heavy barrel and wheel,
and partly, it may be, to small irregularities which have
prevented the centre of gravity of the whole mass from
being strictly in the axle.

325. Though this machine is more economical of power
than the wheel and axle of Art. 305, yet it is less powerful ;
in fact, the mechanical efficiency, 5*54, is only about one-
fourth of that of the wheel and axle. It is therefore
necessary to inquire whether we cannot devise some method
by which to secure the advantages of but little friction, and
at the same time have a large mechanical efficiency : this
we shall proceed to investigate.

THE WHEEL AND PINION.

326. By means of what are called cog-wheels or toothed-
wheels, we are enabled to combine two or more wheels and
axles together, and thus greatly to increase the power which
can be produced by a single wheel and axle. Toothed-
wheels are used for a great variety of purposes in mechanics ;
we have already had some illustration of their use during
these lectures (Fig. 30). The wheels which we shall employ
are those often used in lathes and other small machines;
they are what are called lo-pitch wheels, — that is to say, a
wheel of this class contains ten times as many teeth in its
circumference as there are inches in its diameter. I have
here a wheel 20" diameter, and consequently it has 200
teeth; here is another which is 2 '"-5 diameter, and which
consequently contains 25 teeth. We shall mount these

x.] THE WHEEL AND PINION. 161

wheels upon two parallel shafts, so that they gear one into
the ether in the manner shown in Fig 46 : F is the large
wheel containing 200 teeth, and G the pinion of 25 teeth.
The axles are v""]$ diameter; around each of them a rope
is wound, by which a hook is suspended.

327. A small weight at K is sufficient to raise a much larger
weight on the other shaft ; but before experimenting on
the mechanical efficiency of this arrangement, we shall as
usual calculate the velocity ratio. The wheel contains
eight times as many teeth as the pinion ; it is therefore
evident that when the wheel has made one revolution, the
pinion will have made eight revolutions, and conversely the
pinion must turn round eight times to turn the wheel round
once: hence the power which is turning the pinion round
must be lowered through eight times the circumference of
the axle, while the load is raised through a length equal to
one circumference of the axle. We thus find the velocity
ratio of the machine to be 8.

328. We determine the mechanical efficiency by trial.
Attaching a load of 56 Ibs. to the axle of the large wheel, it
is observed that a power of 137 Ibs. at K will raise it; the
mechanical efficiency of the machine is therefore about 4*1,
which is almost exactly half the velocity ratio. We note that
the load-wii! only just run down when the power is removed ;
from this we might have inferred, by Art. 222, that nearly
half the power is expended on friction, and that therefore
the mechanical efficiency is about half the velocity ratio.
The actual percentage of energy that is utilised with this
particular load is 51. If we suspend 112 Ibs. from the
load hook, 26 Ibs. is just enough to raise it ; the mecha-
nical efficiency that would be deduced from this result is
1 12 H- 26=4*3, which is slightly in excess of the amount
obtained by the former experiment. It is often found to be

1 62

EXPERIMENTAL MECHANICS.

[LECT.

a property of the mechanical powers, that as the load in-
creases the mechanical efficiency slightly improves.

329. In Table XX. will be found a record of experiments up-
on the relation between the power and the load with the wheel
and pinion ; the table will sufficiently explain itself, after the
description of similar tables already given (Arts. 310, 324).

TABLE XX. — THE WHEEL AND PINION.

Wheel (lo-pitch), 200 teeth; pinion, 25 teeth ; axles equal, effective
circumference of each being 2"'87 ; oiled brass bearings ; velocity
ratio, 8; mechanical efficiency, 4/1 ; useful effect, 51 per cent.;
formula, P = 2-46 + O'2I Jf.

Number of
Experiment,

x.

Load in Ibs.

Observed
power in Ibs.

P.

Calculated
power in Ibs.

Difference of
the observed
and calculated
powers.

,

14

5 '4

5'4

O'O

2

28

87

8'3

-0'4

3

42

II'O

1 1 -3

+ 0'3

4

56

137

14-2

+ 0'5

5

70

I7'5

17-2

-°'3

6

84

200

2O'I

+ 0-1

7

98

23-0

23-0

O'O

8

112

26 x>

26-0

O'O

330. The large amount of friction present in this con-
trivance is the consequence of winding the rope directly
upon the axle instead of upon a barrel, as already pointed
out in Art. 3 1 9. We might place barrels upon these axles
and demonstrate the truth of this statement ; but we need
not delay to do so, as we use the barrel in the machines
which we shall next describe.

THE CRANE.

331. We have already explained (Arf. 38) the construction
of the lifting crane, so far as its framework is concerned. We
now examine the mechanism by which the load is raised.
We shall employ for this purpose the model which is repre-

X.] THE CRANE. 163

sented in Fig. 48. The jib is supported by a wooden bar as
a tie, and the crane is steadied by means of the weights
placed at H : some such counterpoise is necessary, for
otherwise the machine would tumble over when a load is
suspended from the hook.

332. The load is supported by a rope or chain which
passes over the pulley E and thence to the barrel D, upon
which it is to be wound. This barrel receives its motion
from a large wheel A, which contains 200 teeth.

The wheel A is turned by the pinion B which contains
25 teeth. In the actual use of the crane, the axle which
carries this pinion would be turned round by means of a
handle ; but for the purpose of experiments upon the relation
of the power to the load, the handle would be inconvenient,
and therefore we have placed upon the axle of the pinion a
wheel c containing a groove in its circumference. Around
this groove a string is wrapped, so that when a weight G is
suspended from the string it will cause the wheel to revolve.
This weight G will constitute the power by which the load
may be raised.

333. Let us compute the velocity ratio of this machine
before commencing experiments upon its mechanical
efficiency. The effective circumference of the barrel D is
found by trial to be i4*'9. Since there are 200 teeth on A
and 25 on B, it follows that the pinion B must revolve eight
times to produce one revolution of the barrel. Hence the
wheel c at the circumference of which the power is applied
must also revolve eight times for one revolution of the barrel
The effective circumference of c is 43'' ; the power must
therefore have been applied through 8 x 43"=344", in order
to raise the load i5'"9. The velocity ratio is 344-:- 14-9 =
23 very nearly. We can easily verify this value of the velocity
ratio by actually raising the load i', when it appears that the

M 2

LECT. X.]

THE CRANE.

165

number of revolutions of the wheel B is such that the power
must have moved 23'.

334. The mechanical efficiency is to be found as usual by
trial. 561bs. placed at F is raised by 3*1 Ibs. at G; hence
the mechanical efficiency deduced from this experiment is
56-r3'i = i8. The percentage of useful effect is easily
shown to be 78 by the method of Art. 323. Here, then,
we have a machine possessing very considerable efficiency,
and being at the same time economical of energy.

TABLE XXI.— THE CRANE.

Circumference of wheel to which the power is applied, 43" ; train of
wheels, 25 -f- 200; circumference of drum on which rope is wound,
I4"'9; velocity ratio, 23 ; mechanical efficiency, 18 ; useful effect, 78
per cent. ; formula, P = 0*0556 A'.

•

Difference of

Number of
Kxperiment.

R.
Load in Ibs,

Observed
power in Ibs.

Calculated
power in Ibs.

the observed
and calculated
values.

I

H

0'9

0-8

-OT

2

28

1-6

1-6

OX)

3

42

2'4

2-3

-OT

4

56

3'i

3'i

O'O

5

70

3'8

3 '9

+ 0'I

6

84

4'5

47

+ 0-2

I _

98
112

H

11

+ 0-2

+ 0'0

335. A series of experiments made with this crane is
recorded in Table XXL, and a comparison of the calculated
and observed values will show that the formula/' =0-05 56 R
represents the experiments with considerable accuracy.

336. It may be noticed that in this formula the term
independent of Jv>, which we frequently meet with in the ex-
pression of the relation between the power and the load,
is absent. The probable explanation is to be found in
the fact that some minute irregularity in the form of the

166

EXPERIMENTAL MECHANICS.

[LECT.

barrel or of the wheel has been constantly acting like a
small weight in favour of the power. In each experiment
the motion is always started from the same position of the
wheels, and hence any irregularity will be constantly acting
in favour of the power or against it ; here the former appears
to have happened. In other cases doubtless the latter has
occurred; the difference is, however, of extremely small amount.
The friction of the machine itself when without a load is
another source for the production of the constant term ; it has
happened in the present case that this friction has been almost
exactly balanced by the accidental influence referred to.

337. In cranes it is usual to provide means of adding a
second train of wheels, when the load is very heavy. In
another model we applied the power to an axle with a pinion
of 25 teeth, gearing into a wheel of 200 teeth ; on the
axle of the wheel with 200 teeth is a pinion of 30 teeth,
which gears into a wheel of 180 teeth ; the barrel is on the
axle of the last wheel. A series of experiments with this
crane is shown in Table XXII.

TABLE XXII.— THE CRANE FOR HEAVY LOADS.
Circumference of wheel to which power is applied, 43"; train of wheels,
25 -r 200 x 30 -j- 1 80 ; circumference of drum on which rope is
wound, 14" '9 ; velocity ratio, 137 ; mechanical efficiency, 87 ; useful
effect, 63 per cent. ; formula, P = 0-185 + 0-00782 R.

Number of
Experiment.

R.
Load in Ibs.

Observed
power in Ibs.

P.
Calculated
power in Ibs.

Difference of
the observed
and calculated
values.

I

'4

0-30

0-29

-O'OI

2

28

0-40

O'4O

O'OO

3

42

0-50

0-51

+ O'OI

4

56

0'60

O-62

+ O'O2

I

i

075
0-85

073
084

-0'02
-O'OI

I

98

112

0'95
1-05

25

O'OO

+ 0'OI

x.] THE CRANE. 167

The velocity ratio is now 137, and the mechanical
efficiency is 87 ; one man could therefore raise a ton
with ease by applying a power of 26 Ibs. to a crane of
this kind.

CONCLUSION.

338. It will be useful to contrast the wheel and axle on
which we have experimented (Art. 304) with the differential
pulley (Art. 209). The velocity ratio of the former machine
is nearly double that of the latter, and its mechanical
efficiency is nearly four times as great. Less than half the
applied power is wasted in the wheel and axle, while more
than half is wasted in the differential pulley. This makes
the wheel and axle both a more powerful machine, and a
more economical machine than the differential pulley. On
the other hand, the greater compactness of the latter,
its facility of application, and the practical conveniences
arising from the property of not allowing the load to run
down, 'do often more than compensate for the superior
mechanical advantage of the wheel and axle.

339. We may also contrast the wheel and axle with the
screw (Art. 277). The screw is remarkable among the
mechanical powers for its very high velocity ratio, and its
excessive friction. Thus we have seen in Art. 291 how the
velocity ratio of a screw-jack with an arm attached amounted
to 414, while its mechanical efficiency was little more than
one-fourth as great. No single wheel and axle could
conveniently be made to give a mechanical efficiency of
116 ; but from Art. 337 we could easily design a combina-
tion of wheels and axles to yield an efficiency of quite this
amount. The friction in the wheel and axle is very much
less than in the screw, and consequently energy is saved
by the use of the former machine.

168 EXPERIMENTAL MECHANICS. [LECT. x.

340. In practice, however, it generally happens that
economy of energy does not weigh much in the selection
of a mechanical power for any purpose, as there are always
other considerations of greater consequence.

341. For example, let us take the case of a lifting crane
employed in loading or unloading a vessel, and inquire
why it is that a train of wheels is generally used for the
purpose of producing the requisite power. The answer
is simple, the train of wheels is convenient, for by their aid
any length of chain can be wound upon the barrel ; whereas
if a screw were used, we should require a screw as long as
the greatest height of lift. This screw would be incon-
venient, and indeed impracticable, and the additional cir-
cumstance that a train of wheels is more economical of
energy than a screw has no influence in the matter.

342. On the other hand, suppose that a very heavy load
has to be overcome for a short distance, as for example in
starting a ship launch, a screw-jack is evidently the proper
machine to employ; it is easily applied, and has a high
mechanical efficiency. The want of economy of energy is
of no consequence in such an operation.

LECTURE XI.
THE MECHANICAL PROPERTIES OF TIMBER.

Introduction. — The General Properties of Timber. — Resistance to
Extension. — Resistance to Compression. — Condition of a Beam
strained by a Transverse Force.

INTRODUCTION

343. IN the lectures on the mechanical powers which
have been just completed, we have seen how great weights
may be raised or other large resistances overcome. We are
now to consider the important subject of the applica-
tion of mechanical principles to structures. These are
fixtures, while machines are adapted for motion ; a roof or
a bridge is a structure, but a crane or a screw-jack is a
machine. Structures are employed for supporting weights,
and the mechanical powers give the means of raising them.

344. A structure has to support both its own weight and
also any load that is to be placed upon it. Thus a railway
bridge must at all times sustain what is called the permanent
load, and frequently, of course, the weight of one or more
trains. The problem which the engineer solves is to design
a bridge which shall be sufficiently strong, and, at the same

I7o EXPERIMENTAL MECHANICS. [LECT.

time, economical ; his skill is shown by the manner in which
he can attain these two ends in the same structure.

345. In the four lectures of the course which will be
devoted to this subject it will only be possible to give a slight
sketch, and therefore but few details can be introduced. An
extended account of the properties of different materials used
in structures would be beyond our scope, but there are some
general principles relating to the strength of materials which
may be discussed. Timber, as a building material, has, in
modern times, been replaced to a great extent by iron in
large structures, but timber is more capable than iron of
being experimented upon in the lecture room. The ele-
mentary laws which we shall demonstrate with reference to
the strength of timber, are also, substantially the same as the
corresponding laws for the strength of iron or any other
material. Hence we shall commence the study of structures
by two lectures on timber. The laws which we shall prove
experimentally will afterwards be applied to a few simple cases
of bridges and other actual structures.

THE GENERAL PROPERTIES OF TIMBER.

346. The uses of timber in the arts are as various as its
qualities. Some woods are useful for their beauty, and others
for their strength or durability under different circumstances.
We shall only employ " pine " in our experimental inquiries.
This wood is selected because it is so well known and so
much used. A knowledge of the properties of pine would
probably be more useful than a knowledge of the properties
of any other wood, and at the same time it must be remem-
bered that the laws which we shall establish by means of
slips of pine may be generally applied.

347. A transverse section of a tree shows a number of
rings, each of which represents the growth of wood in one

Xl.] GENERAL PROPERTIES OF TIMBER. 171

year. The age of the tree may sometimes be approximately
found by counting the number of distinguishable rings. The
outer rings are the newer portions of the wood.

348. When a tree is felled it contains a large quantity of
sap, which must be allowed to evaporate before the wood is
fit for use. With this object the timber is stored in suitable
yards for two or more years according to the purposes for
which it is intended ; sometimes the process of seasoning, as
it is called, is hastened by other means. Wood, when
seasoning, contracts ; hence blocks of timber are often found
split from the circumference to the centre, for the outer rings,
being newer and containing more sap, contract more than
the inner rings. For the same reason a plank is found to
warp when the wood is not thoroughly seasoned. The side
of the plank which was farthest from the centre of the tree
contracts more than the other side, and becomes concave.
This can be easily verified by looking at the edge of
the plank, for we there see the rings of which it is com-
posed.

349. Timber may be softened by steaming. I have here
a rod of pine, 24" x o"-5 Xo"'5, and here a second rod
cut from the same piece and of the same size, which has
been exposed to steam of boiling water for more than an
hour : securing these at one end to a firm stand, I bend them
down together, and you see that after the dry rod has
broken the steamed rod can be bent much farther before it
gives way. This property of wood is utilized in shaping the
timbers of wooden ships. We shall be.;able to understand
the action of steam if we reflect that wood is composed
of a number of fibres ranged side by side and united
together. A rope is composed of a number of fibres laid to
gether and twisted, but the fibres are not coherent as they are
in wood. Hence we find that a rod of wood is stiff, while

172 EXPERIMENTAL MECHANICS. [LECT.

a rope is flexible. The steam finds its way into the interstices
between the fibres of the wood ; it softens their connections,
and increases the pliability of the fibres themselves, and
thus, the operation of steaming tends to soften a piece of
timber and render it tractable.

350. The structure of wood is exhibited by the following
simple experiment : — Here are two pieces of pine, each 9"
x i"x i". One of them I easily snap across with a blow,

while my blows are unable to break the other. The differ-
ence is merely that one of these pieces is cut against the grain,
while the other is with it. In the first case I have only to
separate the connection between the fibres, which is quite
easy. In the other case I would have to tear asunder the fibres
themselves, which is vastly more difficult. To a certain ex-
tent the grained structure is also found in wrought iron, but
the contrast between the strength of iron with the grain and
against the grain is not so marked as it is in wood.

RESISTANCE TO EXTENSION.

351. It will be necessary to explain a little more definitely
what is meant by the strength of timber. We may conceive
a rod to be broken in three different ways. In the first place
the rod may be taken by a force at each end and torn
asunder by pulling, as a thread may be broken. To do this
requires very great power, and the strength of the material
with reference to such a mode of destroying it is called its
resistance to extension. In the second place, it may be
broken by longitudinal pressure at each end, as a pillar may
be crushed by the superincumbent weight being too large ;
the strength that relates to this form of force is called resist-
ance to compression : finally, the rod may be broken by
a force applied transversely. The strength of pine with
reference to these different applications of force will be

XL] RESISTANCE TO EXTENSION. 173

considered successively. The rods that are to be used have
been cut from the same piece of timber, which has been
selected on account of its straightness of grain and freedom
from knots. They are of different rectangular sections,

FIG. 49.

i"Xo'"5 and o"'5 xo"'5 being generally used, but sometimes
i x i ' is employed.

352. With reference to the strength of timber in its
capacity to resist extension, we can do but little in the lecture
room. I have here a pine rod A B, of dimensions 48' x o'"5
x o"-5, Fig. 49. Each end of this rod is firmly secured

174 EXPERIMENTAL MECHANICS. [LECT.

between two cheeks of iron, which are bolted together : the
rod is suspended by its upper extremity from the hook of
the epicyloidal pulley-block (Art. 213), which is itself sup-
ported by a tripod ; hooks are attached to the lower end of
the rod for carrying the weights. By placing 3 cwt.
on these hooks and pulling the hand chain of the pulley-
block, I find that I can raise the weight safely, and therefore
the rod will resist at all events a tension of 3 cwt. From
experiments which have been made on the subject, it is
ascertained that about a ton would be necessary to tear such
a rod asunder ; hence we see that pine is enormously strong
in resisting a force of extension. The tensile strength of the
rod does not depend upon its length, but upon the area of
the cross section. That of the rod we have used is one-
fourth of a square inch, and the breaking weight of a rod
one square inch in section is about four tons.

353. A rod of any material generally elongates to some
extent under the action of a suspended weight ; and we shall
ascertain whether this occurs perceptibly in wood. Before
the rod was strained I had marked two points upon it exactly
2 feet apart. When the rod supports 3 cwt. I find that the
distance between the two points has not appreciably altered,
though by more delicate measurement I have no doubt we
should find that the distance had elongated to an insignifi-
cant extent.

354. Let us contrast the resistance of a rod of timber to
extension with the effect upon a rope under the same cir-
cumstances. I have here a rope about o".25 diameter; it
is suspended from a point, and bears a 14 Ib. weight in
order to be completely stretched. I mark points upon
the rope 2 apart. I now change the stone weight for a
weight of i cwt, and on measurement I find that the two
points which before were 2' apart, are now 2' 2'; thus the

XL] RESISTANCE TO COMPRESSION. 175

rope has stretched at the rate of an inch per foot for
a strain of i cwt., while the timber did not stretch
perceptibly for a strain of 3 cwt.

355. We have already explained in Art, 37 the meaning
of the word " lie." The material suitable for a tie should
be capable of offering great resistance, not only to actual
rupture by tension, but even to appreciable elongation.
These qualities we have found to be possessed by wood.
They are, however, possessed in a much higher degree
by wrought iron, which possesses other advantages in
durability and facility of attachment.

RESISTANCE TO COMPRESSION.

356. We proceed to examine into the capability of timber
to resist forces of longitudinal compression, either as a
pillar or in any other form of " strut," such for instance,
as the jib of the crane represented in Fig. 17. The
use of timber as a strut depends in a great degree upon
the coherence of the fibres to each other, as well as upon
their actual rigidity. The action of timber in resisting forces
of compression is thus very different from its action when
resisting forces of extension ; we can examine, by actual ex-
periment, the strength of timber under the former conditions,
as the weights which it will be necessary to employ are
within the capabilities of our lecture-room apparatus.

357. The apparatus is shown in Fig. 50. It consists of
a lever of the second order, 10' long, the mechanical advan-
tage of which is threefold ; the resistance of the pillar D E to
crushing is the load to be overcome, and the power consists
of weights, to receive which the tray B is used ; every pound
placed in the tray produces a compressive force of 3 Ibs.
on the pillar at D. The fulcrum is at A and guides at G.
The lever and the tray would somewhat complicate our

176 EXPERIMENTAL MECHANICS. [LECT.

calculations unless their weights were counterpoised. A
cord attached to the extremity of the lever passes over a
pulley F ; at the other end of this cord, sufficient weights c
are attached to neutralize the weight of the apparatus. In
fact, the lever and tray now swing as if they had no weight,
and we may therefore leave them out of consideration.
The pillar to be experimented upon is fitted at its lower

FIG. so.

end E into a hole in a cast-iron bracket : this bracket can be
adjusted so as to take in pieces of different lengths ; the upper
end of the pillar passes through a hole in a second piece
of cast-iron, which is bolted to the lever : thus our little
experimental column is secured at each end, and the risk of
slipping is avoided. The stands are heavily weighted to
secure the stability of the arrangement.

358. The first experiment we shall make with this

xi.] RESISTANCE TO COMPRESSION. 177

apparatus is upon a pine rod 40" long and o"'5 square ;
the lower bracket is so placed that the lever is horizontal
when just resting upon the top of the rod. Weights placed
in the tray produce a pressure three times as great down
the rod, the effect of which will first be to bend the rod,
and, when the deflection has reached a certain amount, to
break it across. I place 28 Ibs. in the tray: this produces
a pressure of 84 Ibs. upon the rod, but the rod still remains
perfectly straight, so that it bears this pressure easily.
When the pressure is increased to 96 Ibs. a very slight
amount of deflection may be seen. When the strain
reaches 114 Ibs. the rod begins to bend into a curved
form, though the deflection of the middle of the rod from
its original position is still less than o'-z^. Gradually
augmenting the pressure, I find that when it reaches
132 Ibs. the deviation has reached o"'5 ; and finally, when
48 Ibs. is placed in the tray, that is, when the rod is
subjected to 144 Ibs., it breaks across the middle. Hence
we see that this rod sustained a load of 96 Ibs. without
sensibly bending, but that fracture ensued when the load was
increased about half as much again. Another experiment
\vith a similar rod gave a slightly less value (132 Ibs.)
for the breaking load. If I add these results together,
and divide the sum by 2, I find 138 Ibs. as the mean
value of the breaking load, and this is a sufficiently exact
determination.

359. Let us next try the resistance of a shorter rod of
the same section. I place a piece of pine 20" long and
o"*5 square in the apparatus, firmly securing each end as
in the former case. The lower bracket is adjusted so as
to make the lever horizontal ; the counterpoise, of course,
remains the same, and weights are placed in the tray as
before. No deflection is noticed when the rod supports

N

i;8 EXPERIMENTAL MECHANICS. [LECT.

126 Ibs. ; a very slight amount of bending is noticeable
with 1 86 Ibs. ; with 228 Ibs., the amount by which the
centre of the rod has deviated laterally from its original
position is about o"'2 ; and finally, when the load reaches
294 Ibs., the rod breaks. Fracture first occurs in the
middle, but is immediately followed by other fractures
near where the ends of the rod are secured.

360. Hence the breaking load of a rod 20" long is more
than double the breaking load of a rod of 40" long the same
section ; from this we learn that the sections being equal,
short pillars are stronger than long pillars. It has been
ascertained by experiment that the strength of a square
pillar to resist compression is proportional to the square
of its sectional area. Hence a rod of pine, 40" long and
i" square, having four times the section of the rod of the
same length we have experimented on, would be sixteen
times as strong, and consequently its breaking weight would
amount to nearly a ton. The strength of a rod used as
a tie depends only on its section, while the strength of a
rod used as a strut depends on its length as well as on its
section.

CONDITION OF A BEAM STRAINED BY A
TRANSVERSE FORCE.

361. We next come to the important practical subject of
the strength of timber when supporting a transverse strain ;
that is, when used as a beam. The nature of a transverse
strain may be understood from Fig. 51, which represents a
small beam, strained by a load at its centre. Fig. 52 shows
two supports 40" apart, across which a rod of pine
48" x i" x i" is laid ; at the middle of this rod a hook is
placed, from which a tray for the reception of weights is
suspended. A rod thus supported, and bearing weights, is

XL] TRANSVERSE STRAIN. 179

said to be strained transversely. A rafter of a roof, the
flooring of a room, a gangway from the wharf to a
ship, many forms of bridge, and innumerable other
examples, might be given of beams strained in this
manner. To this important subject we shall devote the
remainder of this lecture and the whole of the next.

362. The first point to be noticed is the deflection of
the beam from which a weight is suspended. The beam
is at first horizontal ; but as the weight in the tray is

augmented, the beam gradually curves downwards until,
when the weight reaches a certain amount, the beam breaks
across in the middle and the tray falls.

For convenience in recording the experiments the tray
chain and hooks have been adjusted to weigh exactly 14 Ibs.
(Fig. 52). A B is a cord which is kept stretched by the little
weights D : this cord gives a rough measure of the deflection
of the beam from its horizontal position when strained by
a load in the tray. In order to observe the deflection

N 2

i8o

EXPERIMENTAL MECHANICS.

[LECT.

accurately an instrument is used called the cathetometer (G).
It consists of a small telescope, always directed horizontally,
though capable of being moved up and down a vertical
triangular pillar ; on one of the sides of the pillar a scale
is engraved, so that the height of the telescope in any

FIG. 52.

position can be accurately determined. The cathetometer
is levelled by means of the screws H H, so that the tri-
angular pillar on which the telescope slides is accurately
vertical: the dotted line shows the direction of the visual
ray when the centre c of the beam is seen by the observer
through the telescope.

XL] TRANSVERSE STRAIN. 181

Inside the telescope and at its focus a line of spider's
web is fixed horizontally ; on the bar to be observed, and
near its middle point c, a cross of two fine lines is marked.
The tray being removed, the beam becomes horizontal ; the
telescope of the cathetometer is then directed towards the
beam, so that the lines marked upon it can be seen dis-
tinctly. By means of a screw the telescope may be raised
or lowered until the spider's web inside the telescope is
observed to pass through the image of the intersection of
the lines. The scale then indicates precisely how high the
telescope is on the pillar.

363. While I look through the telescope my assistant
suspends the tray from the beam. Instantly I see the cross
descend in the field of view. I lower the telescope until
the spider's web again passes through the image of the
intersection of the lines, and then by looking at the scale I
see that the telescope has been moved down o"*i9, that
is, about one-fifth of an inch : this is, therefore, the distance
by which the cross lines on the beam, and therefore the
centre of the beam itself, must have descended. Indeed,
even a simpler apparatus would be competent to measure
the amount of deflection with some degree of precision. By
placing successively one stone after another upon the tray,
the beam is seen to deflect more and more, until even
without the telescope you see the beam has deviated from
the horizontal.

364. By carefully observing with the telescope, and
measuring in the way already described, the deflections
shown in Table XXIII. were determined. The scale along
the vertical pillar was read after the spider's web had been
adjusted for each increase in the weight. The movement
from the original position is recorded as the deflection for
each load.

1 82

EXPERIMENTAL MECHANICS. [LECT.

TABLE XXIII.— DEFLECTION OF A BEAM.

A rod of pine 48" x i" x i" ; resting freely on supports 40" apart ; and
laden in the middle.

Number of
Experiment.

Magnitude
of load.

Deflection.'

I

14

o"'I9

2

28

o"'37

3

42

o""55

4

56

°",'74

0

s

?''?3

7

98

i'-|S

8

112

I *6i

9

126

i"'95

10

I40

2"'37

365. The first column records the number of the experi-
ment. The second represents the load, and the third
contains the corresponding deflections. It will be seen that
up to 98 Ibs. the deflection is about o"*2 for every stone
weight, but afterwards the deflection increases more rapidly.
When the weight reaches 140 Ibs. the deflection at first
indicated is 2"'37 ; but gradually the cross lines are seen to
descend in the field of the telescope, showing that the beam
is yielding and finally it breaks across. This experiment
teaches us that a beam is at first deflected by an amount
proportional to the weight it supports ; but that when two-
thirds of the breaking weight is reached, the beam is
deflected more rapidly.

366. It is a question of the utmost importance to ascertain
the greatest load a beam can sustain without injury to its
strength. This subject is to be studied by examining the
effect of different deflections upon the fibres of a beam.
A beam is always deflected whatever be the load it supports ;

XI.] TRANSVERSE STRAIN. 183

thus by looking through the telescope of the cathetometer
I can detect an increase of deflection when a single pound
is placed in the tray : hence whenever a beam is loaded we
must have some deflection. An experiment will show what
amount of deflection may be experienced without producing
any permanently injurious effect.

367. A pine rod 40" x i x i" is freely supported at
each end, the distances between the supports being 38", and
the tray is suspended from its middle point. A fine pair of
cross lines is marked upon the beam, and the telescope
of the cathetometer is adjusted so that the spider's line
exactly passes through the image of the intersection. 14 Ibs.
being placed in the tray, the cross is seen to descend ; the
weight being removed, the cross returns precisely to its
original position with reference to the spider's line : hence,
after this amount of deflection, the beam has clearly
returned to its initial condition, and is evidently just as
good as it was before. The tray next received 56 Ibs. ; the
beam was, of course, considerably deflected, but when the
weight was removed the cross again returned, — at all events,
to within o"-oi of where the spider's line was left to indicate
its former position. We may consider that the beam is in
this case also restored to its original condition, even though
it has b~brne a strain which, including the tray, amounted to
70 Ibs. But when the beam has been made to carry 84 Ibs.
for a few seconds, the cross does not completely return on
the removal of the load from the tray, but it shows that the
beam has now received a permanent deflection of o"'O3.
This is still more apparent after the beam has carried 98 Ibs.,
for when this load is removed the centre of the beam is
permanently deflected by o"'i3. Here, then, we may
infer that the fibres of the beam are beginning to be
strained beyond their powers of resistance, and this is

1 84 EXPERIMENTAL MECHANICS. [LECT.

verified when we find that with 28 additional pounds in the
tray a collapse ensues.

368. Reasoning from this experiment, we might infer that
the elasticity of a beam is not affected by a weight which is
less than half that which would break it, and that, therefore,
it may bear without injury a weight not exceeding this
amount. As, however, in our experiments the weight was
only applied once, and then but for a short time, we cannot be
sure that a longer-continued or more frequent application of
the same load might not prove injurious ; hence, to be on
the safe side, we assume that one-third of the breaking
weight of a beam is the greatest load it should be made to
bear in any structure. In many cases it is found desirable
to make the beam much stronger than this ratio would
indicate.

369. We next consider the condition of the fibres of a
beam when strained by a transverse force. It is evident
that since the fracture commences at the lower surface of
the beam, the fibres there must be in a state of tension,
while those at the concave upper surface of the beam are
compressed together. This condition of the fibres may be
proved by the following experiment.

370. I take two pine rods, each 48" x i" x i", perfectly
similar in all respects, cut from the same piece of timber,
and therefore probably of very nearly identical strength.
With a fine tenon saw I cut each of the rods half through
at its middle point. I now place one of these beams on the
supports 40" apart, with the cut side of the beam upwards.
I suspend from it the tray, which I gradually load with
weights until the beam breaks, which it does when the total
weight is 8 1 Ibs.

If I were to place the second beam on the same supports
with the cut upwards, then there can be no doubt that it

XL] TRANSVERSE STRAIN. 185

would require as nearly as possible the same weight to break
it. I place it, however, with the cut downwards, I suspend
the tray, and find that the beam breaks with a load of 3 1 Ibs.
This is less than half the weight that would have been
required if the cut had been upwards.

371. What is the cause of this difference? The fibres
being compressed together on the upper surface, a cut has
no tendency to open there ; and if the cut could be made
with an extremely fine saw, so as to remove but little material,
the beam would be substantially the same as if it had not
been tampered with. On the other hand, the fibres at the
lower surface are in a state of tension ; therefore when the
cut is below it yawns open, and the beam is greatly weakened.
It is, in fact, no stronger than a beam of 48"xo"'5Xi",
placed with its shortest dimension vertical. If we remember
that an entire beam of the same size required about 140 Ibs.
to break it (Art. 366), we see that the strength of a beam is
reduced to one-fourth by being cut half-way through and
having the cut underneath.

372. We may learn from this the practical consequence
that the sounder side of a beam should always be placed
downwards. Any flaw on the lower surface will seriously
weaken the beam : so that the most knotty face of the wood
should certainly be placed uppermost. If a portion of the
actual substance of a beam be removed — for example, if a
notch be cut out of it — this will be almost equally injurious
on either side of the beam.

373. We may illustrate the condition of the upper surface
of the beam by a further experiment. I make two cuts o"*5
deep in the middle of a pine rod 48" x i"X i". These cuts
are o"-5 apart, and slightly inclined ; the piece between them
being removed, a wedge is shaped to fit tightly into the
space ; the wedge is long enough to project a little on one

1 86 EXPERIMENTAL MECHANICS. [LECT.

side. If the wedge be uppermost when the beam is placed
on the supports, the beam will be in the same condition as
if it had two fine cuts on the upper surface. I now load the
beam with the tray in the usual manner, and I find it to
bear 70 Ibs. securely. On examining the beam, which has
curved down considerably, I find that the wedge is held in
very tightly by the pressure of the fibres upon it, but, by a
sharp tap at the end, I knock out the wedge, and instantly
the load of 70 Ibs. breaks the beam ; the reason is simple —
the piece being removed, there is no longer any resistance
to the compressive strain of the upper fibres, and con-
sequently the beam gives way.

374. The collapse of a beam by a transverse strain
commences by fracture of the fibres on the lower surface,
followed by a rupture of all fibres up to a considerable
depth. Here we see that by a transverse force the fibres
in a beam of 48" x i" X i" have been broken by a strain
of 140 Ibs. (Art. 366) ; but we have already stated (Art. 353)
that to tear such a rod across by a direct pull at each end
a force of about four tons is necessary. The breaking
strain of the fibres must be a certain definite quantity,
yet we find that to overcome it in one way four tons is
necessary, while by another mode of applying the strain
1 40 Ibs. is sufficient.

375. To explain this discrepancy we may refer to the
experiment of Art. 28, wherein a piece of string was broken
by the transverse pull of a piece of thread in illustration
of the fact that one force may be resolved into two others,
each of them very much greater than itself. A similar
resolution of force occurs in the transverse deflection of
the beam, and the force of 140 Ibs. is changed into two other
forces, each of them enormously greater and sufficiently
strong to rupture the fibres. We need not suppose that

XL] TRANSVERSE STRAIN. 187

the force thus developed is so great as four tons, because
that is the amount required to tear across a square inch of
fibres simultaneously, whereas in the transverse fracture the
fibres appear to be broken row after row; the fracture is
thus only gradual, nor does it extend through the entire
depth of the beam.

376. We shall conclude this lecture with one more
remark, on the condition of a beam when strained by a
transverse force. We have seen that the fibres on the
upper surface are compressed, while those on the lower
surface are extended ; but what is the condition of the
fibres in the interior? There can be no doubt that the
following is the state of the case : — The fibres immediately
beneath the upper surface are in compression ; at a greater
depth the amount of compression diminishes until at the
middle of the beam the fibres are in their natural condition ;
on approaching the lower surface the fibres commence to
be strained in extension, and the amount of the extension
gradually increases until it reaches a maximum at the lower
surface.

LECTURE XII.

THE STRENGTH OF A BEAM.

A Beam free at the Ends and loaded in the Middle. — A Beam uniformly
loaded. — A Beam loaded in the Middle, whose Ends are secured. —
A Beam supported at one end and loaded at the other.

A BEAM FREE AT THE ENDS AND LOADED IN
THE MIDDLE.

377. IN the preceding lecture we have examined some
general circumstances in connection with the condition
of a beam acted on by a transverse force ; we proceed
in the present to inquire more particularly into the strength
under these conditions. We shall, as before, use for our
experiments rods of pine only, as we wish rather to illustrate
the general laws than to determine the strength of different
materials. The strength of a beam depends upon its length,
breadth, and thickness ; we must endeavour to distinguish
the effects of each of these elements on the capacity of the
beam to sustain its load.

We shall only employ beams of rectangular section ;
this being generally the form in which beams of wood are
used. Beams of iron, when large, are usually not rect-
angular, as the material can be more effectively disposed

LECT. xil.] STRENGTH OF A BEAM. 189

in sections of a different form. It is important to distinguish
between the stiffness of a beam in its capacity to resist
flexure, and the strength of a beam in its capacity to resist
fracture. Thus the stiffest beam which can be made from
the cylindrical trunk of a tree i' in diameter is 6" broad and
io"'5 deep, while the strongest beam is 7" broad and
9"- 7 5 deep. We are now discussing the strength (not the
stiffness) of beams.

378. We shall commence the inquiry by making a
number of experiments : these we shall record in a table,
and then we shall endeavour to see what we can learn
from an examination of this table. I have here ten pieces
of pine, of lengths varying from i' to 4', and of three
different sections, viz. i" x i", i" X o"'5, and o"'5 X o"'5.
I have arranged four different stands, on which we can
break these pieces : on the first stand the distance between
the points of support is 40", and on the other stands the
distances are 30," 20", and 10" respectively ; the pieces
being 4', 3', 2', and i' long, will just be conveniently held
on the supports.

379. The mode of breaking is as follows: — The beam
being laid upon the supports, an S hook is placed at its
middle point, and from this S hook the tray is suspended.
Weights a"re then carefully added to the tray until the beam
breaks ; the load in the tray, together with the weight of the
tray, is recorded in the table as the breaking load.

380. In order to guard as much as possible against
error, I have here another set of ten pieces of pine,
duplicates of the former. I shall also break these ; and
whenever I find any difference between the breaking
loads of two similar beams, I shall record in the table
the mean between the two loads. The results are shown
in Table XXIV.

190

EXPERIMENTAL MECHANICS.

[LECT.

TABLE XXIV. — STRENGTH OF A BEAM.

Slips of pine (cut from the same piece) supported freely at each end ;
the length recorded is the distance between the points of support ; the
load is suspended from the centre of the beam, and gradually increased
until the beam breaks ;

Formula, P = 6080

area of section x depth
span

No.of

Dimensions.

Mean of the

p.

Difference of

Ex-

observations

Calculated

the observed

peri-
ment.

Span.

Breadth.

Depth.

ofthebreak-
ng load in Ibs.

breaking load
in Ibs.

and calculated
values.

I

4o"-o

l"'O

l"'O

152

152

O'O

2

40" -o

o"-s

l"'O

77

76

— I'O

3

4o"-o

i"-o

o"-5

38

38

O'O

4

40" -o

o"'5

o"'5

19

19

O'O

•»

30" -o

r 'O

°"'S

59

SI

-8-0

6

7

30" -o

23"'O

°"'5
i"-o

°,75
o"'5

25

74

9

O'O
+ 2'0

8

20" -0

o"'5

o'"5

36

38

+ 2'0

9
10

IO"'O
IO"'O

IX)

°"'5

o"-s
o"-5

•a

152

76

-2'0

+ 8-0

381. In the first column is a series of figures for con-
venience of reference. The next three columns are occupied
with the dimensions of the beams. By span is meant
the distance between the points of support ; the real length
is of course greater; the depth is that dimension of the
beam which is vertical. The fifth column gives the mean
of t\vo observations of the breaking load. Thus for example,
in experiment No. 5 the two beams used were each
36" x i"xo"'5, they were placed on points of support 30"
distant, so the span recorded is 30" : one of the beams

xii.] STRENGTH OF A BEAM. 191

was broken by a load of 58 Ibs., and the second by a load
of 60 Ibs. ; the mean between the two, 59 Ibs., is recorded
as the mean breaking load. In this manner the column of
breaking loads has been found. The meaning of the two
last columns of the table will be explained presently.

382. We shall endeavour to elicit from these observations
the laws which connect the breaking load with the span,
breadth, and depth of the beam.

383. Let us first examine the effect of the span ; for
this purpose we bring together the observations upon beams
of the same section, but of different spans. Sections of
o"'5 x o"-5 will be convenient for this purpose ; Nos. 4, 6,
8, and 10 are experiments upon beams of this section. Let
us first compare 4 and 8. Here we have two beams of the
same section, and the span of one (40") is double that of
the other (20"). When we examine the breaking weights
we find that they are 19 Ibs. and 36 Ibs. ; the former of
these numbers is rather more than half of the latter. In
fact, had the breaking load of 40" been fib. less, 18-25 Ibs.,
and had that of 20" been | Ib. more, 36-5 Ibs., one of the
breaking loads would have been exactly half the other.

384. We must not look for perfect numerical accuracy
in these experiments ; we must only expect to meet with
approximation, because the laws for which we are in
search are in reality only approximate laws. Wood itself is
variable in quality, even when cut from the same piece :
parts near the circumference are different in strength from
those nearer the centre ; in a young tree they are generally
weaker, and in an old tree generally stronger. Minute
differences in the grain, greater or less perfectness in
the seasoning, these are also among the circumstances
which prevent one piece of timber from being identical
with another. We shall, however, generally find that the

192 EXPERIMENTAL MECHANICS. [LECT.

effect of these differences is small, but occasionally this is
not the case, and in trying many experiments upon the
breaking of timber, discrepancies occasionally appear for
which it is difficult to account.

385. But you will find, I think, that, making reasonable
allowances for such difficulties as do occur, the laws on the
whole represent the experiments very closely.

386. We shall, then, assume that the breaking weight of
a bar of 40" is half that of a bar of 20" of the same
section, and we ask, Is this generally true? is it true that
the breaking weight is inversely proportional to the span?
In order to test this hypothesis, we can calculate the
breaking weight of a bar of 30" (No. 6), and then
compare the result with the observed value ; if the
supposition be true, the breaking weight should be given
by the proportion—

30" : 40" :: 19 : Answer.

The answer is 25-3 Ibs. ; on reference to the table we find
25 Ibs. to be the observed value, hence our hypothesis is
verified for this bar.

387. Let us test the law also for the 10" bar, No. 10 —

10" : 40" :: 19 : Answer.

The answer in this case is 76, whereas the observed value
is 68, or 8 Ibs. less ; this does not agree very well with the
theory, but still the difference, though 8 Ibs., is only about
ii or 12 per cent, of the whole, and we shall still retain the
law, for certainly there is no other that can express the
result as well.

388. But the table will supply another verification. In
experiment No. 3 a 40" bar, i" broad, and o"'5 deep,
broke with 38 Ibs. ; and in experiment No. 7 a 20" bar of

XIL] STRENGTH OF A BEAM. 193

the same section broke with 74 Ibs. ; but this is so nearly
double the breaking weight of the 40" bar, as to be an
additional illustration of the law, that for a given section the
breaking load varies inversely as the span.

' 389. We next inquire as to the effect of the breadth of
the beam upon its strength ? For this purpose we compare
experiments Nos. 3 and 4 : we there find that a bar
4o"xi"Xo"'5 is broken by a load of 38 Ibs., while a bar
just half the breadth is broken by 19 Ibs. We might have
anticipated this result, for it is evident that the bar of No. 3
must have the same strength as two bars similar to that of
No. 4 placed side by side.

390. This view is confirmed by a comparison of Nos. 7
and 8, where we find that a 20" bar takes twice the load to
break it that is required for a bar of half its breadth. The
law is not quite so well verified by Nos. 5 and 6,
for half the breaking weight of No. 5, namely 29-5 Ibs.,
is more than 25, the observed breaking weight of No. 6 : a
similar remark may be made about Nos. 9 and TO.

391. Supposing we had a beam of 40" span, 2" broad, and
o'"5 deep, we can easily see that it is equivalent to two bars
like that of No. 3 placed side by side ; and we infer generally
that the strength of a bar is proportional to its breadth ; or
to speak- more definitely, if hvo beams have the same span
and depth, the ratio of their breaking loads is the same as the
ratio of their breadths.

392. We next examine the effect of the depth of a beam
upon its strength. In experimenting upon a beam placed
edgewise, a precaution must be observed, which would not
be necessary if the same beam were to be broken flatwise.
When the load is suspended, the beam, if merely laid
edgewise on the supports, would almost certainly turn over ;
it is therefore necessary to place its extremities in recesses in

O

194 EXPERIMENTAL MECHANICS. [LECT.

the supports, which will obviate the possibility of this
occurrence; at the same time the ends must not be prevented
from bending upwards, for we are at present discussing a beam
free at each end, and the case where the ends are not free
will be subsequently considered.

393. Let us first compare together experiments Nos. 2 and
3 ; here we have two bars of the same dimensions, the section
in each being i"-o x o'"5, but the first bar is broken
edgewise, and the second flatwise. The first breaks with
77 Ibs., and the second with 38 Ibs. ; hence the same bar is
twice as strong placed edgewise as flatwise when one
dimension of the section is twice as great as the other. We
may generalize this law, and assert that the strength of a
rectangular beam broken edgewise is to the strength of a beam
of like span and section broken flatwise, as the greater dimen-
sion of the section is to the lesser dimension.

394. The strength of a beam 4o"Xo'"5x"i is four
times as great as the strength of 4o"Xo'"5Xo"-5, though
the quantity of wood is only twice as great in one as in the
other. In general we may state that if a beam were bisected
by a longitudinal cut, the strength of the beam would be
halved when the cut was horizontal, and unaltered when the
cut was vertical; thus, for example, two beams of experiment
No. 4, placed one on the top of the other, would break
with about 40 Ibs., whereas if the same rods were in one
piece, the breaking load would be nearly 80 Ibs.

395. This may be illustrated in a different manner. I
have here two beams of 4o"Xi"Xo'"5 superposed;
they form one beam, equivalent to that of No. i in
bulk, but I find that they break with 80 Ibs., thus showing
that the two are only twice as strong as one.

396. I take two similar bars, and, instead of laying them
loosely one on the other, I unite them tightly with iron

xii.] STRENGTH OF A BEAM. 195

clamps like those represented in Fig. 56. I now find that
the bars thus fastened together require iO4lbs. for fracture.
We can readily understand this increase of strength. As
soon as the bars begin to bend under the action of the
weight, the surfaces which are in contact move slightly one
upon the other in order to accommodate themselves to the
change of form. By clamping I greatly impede this motion
hence the beams deflect less, and require a greater load
before they collapse ; the case is therefore to some extent
approximated to the state of things when the two rods form
one solid piece, in which case a load of 152 Ibs. would be
required to produce fracture.

397. We shall be able by a little consideration to under-
stand the reason why a bar is stronger edgewise than flatwise.
Suppose I try to break a bar across my knee by pulling the
ends held one in each hand, what is it that resists the
breaking? It is chiefly the tenacity of the fibres on the
convex surface of the bar. If the bar be edgewise, these
fibres are further away from my knee and therefore resist
with a greater moment than when the bar is flatwise : nor
is the case different when the bar is supported at each end,
and the load placed in the centre ; for then the reactions of
the supports correspond to the forces with which I pulled
the ends" of the bar.

398. We can now calculate the strength of any rectangular
beam of pine:

Let us suppose it to be 12' long, 5" broad, and 7" deep.
This is five times as strong as a beam i" broad and 7" deep
for we may conceive the original beam to consist of
5 of these beams placed side by side (Art 391); the beam
i" broad and 7" deep, is 7 times as strong as a beam 7"
broad, i" deep (Art. 393). Hence the original beam must
be 35 times as strong as a beam 7" broad, i" deep ; but the

o 2

196 EXPERIMENTAL MECHANICS. [LECT.

beam 7" broad and i" deep is seven times stronger than a
beam the section of which is i" X i", hence the original
beam is 245 times as strong as a beam 12' long and i" x i"
in section ; of which we can calculate the strength, by Art.
388, from the proportion —

144" : 40" : : 152 : Answer.

The answer is 42*2 Ibs., and thus the breaking load of the
original beam is about 10,300 Ibs.

399. It will be useful to deduce the general expression
for the breaking load of a beam /' span, b" broad, and d"
deep, supported freely at the ends and laden in the
centre.

Let us suppose a bar /" long, and i" x i" in section.
The breaking load is found by the proportion —

/ : 40 : : 152 : Answer;
and the result obtained is ° A beam which is

d" broad, /" span, and i" deep, would be just as strong as
d of the beams /" X i" X i placed side by side ; of which
the collective strength would be —

If such a beam, instead of resting flatwise, were placed edge-
wise, its strength would be increased in the ratio of its
depth to its breadth— that is, it would be increased </-fold —
and would therefore amount to

We thus leam the strength of a beam i" broad, d" deep,
and /" span. The strength of b of these beams placed
side by side, would be the same as the strength of one

xii.] STRENGTH OF A BEAM. 197

beam b" broad, d" deep, and /" span, and thus we finally
obtain

Since b d is the area of the section, we can express this
result conveniently by saying that the breaking load in Ibs.
of a rectangular pine beam is equal to

, 0 vx area of section X depth ;
oooo X - ---
span

the depth and span being expressed in inches linear
measure, and the section in square inches.

400. In order to test this formula, we have calculated from
it the breaking loads of all the ten beams given in Table
XXIV. and the results are given in the sixth column. The
difference between the amount calculated and the observed
mean breaking weight is shown in the last column.

401. Thus, for example, in experiment No. 7 the span
is 20", breadth, i", depth o" -5 ; the formula gives, since the
area is o'"5,

20

This agrees sufficiently with 74 Ibs., the mean of two
observed values.

402. Except in experiments Nos. 5 and 10, the differences
are very small, and even in these two cases the differences are
not sufficient to make us doubt that we have discovered the
correct expression for the load generally sufficient to produce
fracture.

403. We have already pointed out that a beam begins to
sustain permanent injury when it is subjected to a load
greater than half that which would break it (Art. 368), and
we may infer that it is not in general prudent to load a beam

198 EXPERIMENTAL MECHANICS. [LECT.

which is part of a permanent structure with more than about
a third of a fourth of the breaking weight. Hence if we
wanted to calculate a fair working load in Ibs. for a beam of
pine, we might obtain it from the formula.

. area of section x depth

1500 X — '

span

Probably a smaller coefficient than 1500 would often be used
by the cautious builder, especially when the beam was liable
to sudden blows or shocks, The coefficient obtained from
small selected rods such as we have used would also be
greater than that found from large beams in which imper-
fections are inevitable.

404. Had we adopted any other kind of wood we should
have found a similar formula for the breaking weight, but
with a different numerical coefficient. For example,
had the beams been made of oak the number 6080 must
be replaced by a larger figure.

A BEAM UNIFORMLY LOADED.

405. We have up to the present only considered the case
where the load is suspended from the centre of the beam.
But in the actual employment of beams the load is not
generally applied in this manner. See in the rafters which
support a roof how every inch in the entire length has its
burden of slates to bear. The beams which support a ware-
house floor have to carry their load in whatever manner the
goods are disposed : sometimes, as for example in a grain-store,
the pressure will be tolerably uniform along the beams, while
if the weights be irregularly scattered on the floor, there will
be corresponding inequalities in the mode in which the loads
are distributed over the beams. It will therefore be useful for
us to examine the strength of a beam when its load is applied
otherwise than at the centre.

BEAM UNIFORMLY LOADED.

199

406. We shall employ, in the first place, a beam 40" span,
o'"5 broad, and i" deep ; and we shall break it by apply-
ing a load simultaneously at two points, as maybe most con-
veniently done by the contrivance shown in the diagram, Fig.
53. A B is the beam resting en two supports ; c and D
are the points of trisection of the span ; from whence loops
descend, which carry an iron bar P Q ; at the centre R of
which a weight w is suspended. The load is thus di-
vided equally between the two points c and D, and we may
regard A B as a beam loaded at its two points of trisection.

FIG. 53.

The tray and weights are employed which we have used in
the apparatus represented in Fig. 58.

407. We proceed to break this beam. Adding weights to
the tray, we see that it yields with 117 Ibs., and cracks across
between c and D. On reference to Table XXIV. we find
from experiment No. 2 that a similar bar was broken by
7 7 Ibs. at the centre; now 4x77 = 115 '5; hence we may state
with sufficient approximation that the bar is half as strong
again when the load is suspended from the two points of
trisection as it is when suspended from the centre. It is
remarkable that in breaking the beam in this manner the frac-
ture is equally likely to occur at any point between C and D.

200 EXPERIMENTAL MECHANICS. [LECT.

408. A beam uniformly loaded requires twice as much
load to break it as would be sufficient if the load were
merely suspended from the centre.""] The mode of applying a
load uniformly is shown in Fig. 54.

FIG. S4.

In an experiment actually tried, a beam 4o"Xo'"5Xi//
placed edgewise was found to support ten i4lb. weights
ranged as in the figure ; one or two stone more would,
however, doubtless produce fracture.

409. We infer from these considerations that beams loaded
in the manner in which they are usually employed are con-
siderably stronger than would be indicated by the results
in Table XXIV.

EFFECT OF SECURING THE ENDS CF A BEAM
UPON ITS STRENGTH.

410. It has been noticed during the experiments that
when the weights are suspended from a beam and the
beam begins to deflect, the ends curve upwards from the
supports. This bending of the ends is for example shown
in Fig. 54. If we restrain the ends of the beam from bending
up in this manner, we shall add very considerably to its
strength. This we can do by clamping them down to the
supports.

411. Let us experiment upon a beam 40" X i"X i". We
clamp each of the ends and then break the beam by a weight

xii.] BEAM SECURED AT ONE END. 201

suspended from the centre. It requires 238 Ibs. to accom-
plish fracture. This is a little more than half as much again
as 152 Ibs., which we find from Table XXIV. was the
weight required to break this bar when its ends were free.
Calculation shows that the strength of a beam may be even
doubled when the ends are kept horizontal by more perfect
methods than we have used.

412. When the beam gives way under these circum-
stances, there is not only a fracture in the centre, but each
of the halves are also found to be broken across near
the points of support ; the necessity for three fractures
instead of one explains the increase of strength obtained
by restraining the ends to the horizontal direction.

413. In structures the beams are generally more or less
secured at each end, and are therefore more capable of
bearing resistance than would be indicated by Table XXIV.
From the consideration of Arts. 408 and 411, we can infer
that a beam secured at each end and uniformly loaded would
require three or four times as much load to break it as
would be sufficient if the ends were free and if the load were
applied at the centre.

BEAMS SECURED AT ONE END AND LOADED AT
THE OTHER.

414. A beam, one end of which is firmly imbedded in
masonry or otherwise secured, is occasionally called upon to
support a weight suspended from its extremity. Such a beam
is shown in Fig. 55.

In the case we shall examine, A B is a pine beam of
dimensions 20" X o"'5 x o'"5, and we find that, when w reaches
10 Ibs., the beam breaks. In experiment No. 8, Table
XXIV., a similar beam required 36 Ibs.; hence we see that

202 EXPERIMENTAL MECHANICS. [LECT. xn.

the beam is broken in the manner of Fig. 55, by about one-
fourth of the load which would have been required if the
beam had been supported at each end and laden in the
centre.

FIG. 55.

We shall presently have occasion to apply some of the
results obtained by the experiments made in the lecture
now terminated.

LECTURE XIII.
THE PRINCIPLES OF FRAMEWORK.

Introduction. — Weight sustained by Tie and Strut. — Bridge with Two
Struts. — Bridge with Four Struts. — Bridge with Two Ties. — Simple
Form of Trussed Bridge.

INTRODUCTION.

415. IN this lecture and the next we shall experiment
upon some of the arts of construction. We shall employ slips
of pine o"'5Xo".5 in section for the purpose of making
models of simple framework: these

slips can be attached to each other by
means of-the small clamps about 3" long,
shown in Fig. 56, and the general
appearance of the models thus pro-
duced may be seen from Figs. 58 and 62.

416. The following experiment shows the tenacity with
which these clamps hold. Two slips of pine, each 1 2" x
o"'5 Xo"'5, are clamped together, so that they overlap about
2", thus forming a length of 22": this composite rod is raised
by a pulley-block as in Fig. 49, while a load of 2 cwt. is
suspended from it. Thus the clamped rods bear a direct

204 EXPERIMENTAL MECHANICS. [LECT.

tension of 2 cwt. The efficiency of the clamps depends
principally upon friction, aided doubtless by a slight crushing
of the wood, which brings the surfaces into perfect contact.

417. These slips of pine united by the clamps are
possessed of strength quite sufficient for the experiments
now to be described. Models thus constructed have
the great advantage of being erected, varied or pulled down,
with the utmost facility.

We have learned that the compressive strength, and,
still more, the tensile strength of timber, is much greater
than its transverse strength. This principle is largely used
in the arts of construction. We endeavour by means of
suitable combinations to turn transverse forces into forces
of tension or compression, and thus strengthen our con-
structions. We shall illustrate the mode of doing so by
simple forms of framework.

WEIGHT SUSTAINED BY TIE AND STRUT.

418. We begin with the study of a very simple contrivance,
represented in Fig. 57.

A B is a rod of pine 2o// long. In the diagram it is
represented, for simplicity, imbedded at the end A in the
support. In reality, however, it is clamped to the support,
and the same remark may be made about some other dia-
grams used in this lecture. Were A B unsupported except
at its end A, it would of course break when a weight of
10 Ibs. was suspended at B, as we have already found in
Art. 414.

419. We must ascertain whether the transverse force
on A B cannot be changed into forces of tension and
compression. The tie B c is attached by means of clamps ;
A B is sustained by this tie; it cannot bend downwards
under the action of the weight w, because we should then

Xill.] WEIGHT SUSTAINED BY TIE AND STRUT. 205

require to have on the same base and on the same side of
it two triangles having their conterminous sides equal,
but this we know from Euclid (I. 7) is impossible. Hence
B is supported, and we find that 112 Ibs. may be safely
suspended, so that the strength is enormously increased.
In fact the transverse force is changed into a compressive
force or thrust down A B, and a tensile force on B c.

420. The actual magnitudes of these can be com-
puted. Draw the parallelogram c D E B ; if B D represent

\

FIG. 57.

the weight w, it may be resolved into two forces, — one,
B c, a force of extension on the tie; the other, B E, a
compressive force on A B, which is therefore a strut. Hence
the forces are proportional to the sides of the triangle,
ABC. In the present case

A B = 2o//, A c = 18", B c == 27";

therefore, when w is 112 Ibs., we calculate that the
force on A B is 124 Ibs., and on c B 168 Ibs. A B would
require about 300 Ibs. to crush it, and c B about 2,000
Ibs. to tear it asunder, consequently the tie and strut can

206 EXPERIMENTAL MECHANICS. [LECT.

support i cwt. with ease. If, however, w were increased
to about 270 Ibs., the force on A B would become too
great, and fracture would arise from the collapse ot this
strut.

421. When a structure is loaded up to the breaking
point of one part, it is proper for economy that all the
other parts should be so designed that they shall be as near
as possible to their breaking points. In fact, since nothing
is stronger than its weakest part, any additional strength
which the remaining parts may possess adds no strength
to the whole, and is only so much material wasted.
Hence our structure would be just as strong, and would
be more properly designed if the section of B c were
reduced to one-fifth, for the tie would then break when the
tension upon it amounted to 400 Ibs. When w is 270 Ibs.
the compression on A B is 300 Ibs., and the tension on B c is
405 Ibs., so that both tie and strut attain their breaking
loads together. The principle of duly apportioning the
strength of each piece to the load it has to carry, involves
the essence of sound engineering. In that greatest of
mechanical feats, the construction of a mighty railway bridge
across a wide span, attention to this principle is of vital
importance. Such a bridge has to bear the occasional load
of a passing train, but it has always to support the far greater
load of the bridge materials. There is thus every induce-
ment to make the weight of each part of the bridge as light
as may be consistent with safety.

A BRIDGE WITH TWO STRUTS.

422. We shall next examine the structure of a type of
bridge, shown in Fig. 58.

It consists of two beams, A B, 4' long, placed parallel to

XIIL] A BRIDGE WITH TWO STRUTS. 207

each other at a distance of 3'"5, and supported at each
end ; they are firmly clamped to the supports, and a road-
way of short pieces is laid upon them. At the points of

FIG. 58.

trisection of the beams c, D, struts c F and D E are clamped,
their lower ends being supported by the framework : these
struts are 2' long, and there are two of them supporting
each of the beams. The tray G is attached by a chain to

2oS EXPERIMENTAL MECHANICS. [LECT.

a stout piece of wood, which rests upon the roadway at
the centre of the bridge.

423. We shall first determine the strength of this bridge
by actual experiment, and then we shall endeavour to ex-
plain the results in accordance with mechanical principles. We
can observe the deflection of the bridge by the cathetometer
in the manner already described (Art. 362). By this means
we shall ascertain whether the load has permanently injured
the elasticity of the structure (Art. 367). We begin by testing
the deflection when a load is distributed uniformly, as
the weights are disposed in the case of Fig. 62. A cross is
marked upon one of the beams, and is viewed in the
cathetometer. We arrange 1 1 stone weights along the bridge,
and the cathetometer shows that the deflection is only
o" -09 : the elasticity of the bridge remains unaltered, for
when the weights are removed the cross on the beam returns
to its original position ; hence the bridge is well able to
bear this load.

424. We remove the row of weights from the bridge and
suspend the tray from the roadway. I take my place at
the cathetometer to note the deflection, while my assistant
places weights H H on the tray, i cwt. being the load, I
see that the deflection amounts to o"-2 ; with 2 cwt. the
deflection reaches 0*43"; and the bridge breaks with
238 Ibs.

425. Let us endeavour to calculate the additional
strength which the struts have imparted to the bridge.
By Table XXIV. we see that a rod 40" x o'"5 x o"'5
is broken by a load of 19 Ibs.: hence the beams of
the bridge would have been broken by a load of 38 Ibs.
if their ends had been free. As, however, the ends
of the beams had been clamped down, we learn from
Art. 411 that a double load would be necessary.

xin.] A BRIDGE WITH TWO STRUTS. 209

We may, however, be confident that about So Ibs. would
have broken the unsupported bridge. The strength is,
therefore, increased threefold by the struts, for a load
of 238 Ibs. was required to produce fracture.

426. We might have anticipated this result, because
the points c and D being supported by the struts may be
considered as almost fixed points ; in fact, we see that c
cannot descend, because the triangle A c F is unalterable,
and for a similar reason D remains fixed : the beam
breaks between c and D, and the force required must
therefore be sufficient to break a beam supported at the
points c and D, whose ends are secured. But c D is one-
third of A B, and we have already seen that the strength
of a beam is inversely as its length (Art. 388) ; hence the
force required to break the beam when supported by the
struts is three times as large as would have been necessary
to break the unsupported beam. Thus the strength of the
bridge is explained.

427. As a load of 238 Ibs. applied near the centre is
necessary to break this bridge, it follows from the prin-
c;ple of Art. 408 that a load of about double this amount
must be placed uniformly on the roadway before it
succumbs ; we can, therefore, understand how a load of
ii stone was easily borne (Art. 423) without permanent
injury to the elasticity of the structure. If we take the
factor of safety as 3, we see that a bridge of the form we
have been considering may carry, as its ordinary working
load, a far greater weight than would have crushed it if
unsupported by the struts and with free ends.

428. The strength of the bridge in Fig. 58 is greater
in some parts than in others. At the points c and D a
maximum load could be borne ; the weakest places on the
bridge are in the middle points of the segments A c, D c,

p

210 EXPERIMENTAL MECHANICS. [LECT.

and D B. The load applied by the tray was principally
borne at the middle of D c, but owing to the piece of wood
which sustained the chain being about 18" long, the load
was to some extent distributed.

The thrust upon the struts is not so easy to calculate
accurately. That down c F for example must be less
than if the part c D were removed, and half the load were
suspended from c. The force in this case can be de-
termined by principles already explained (Art. 420).

A BRIDGE WITH FOUR STRUTS.

429. The same principles that we have employed in the
construction of the bridge of Fig. 58 may be extended
further, as shown in the diagram of Fig. 59.

FIG. 59.

We have here two horizontal rods, 48" x o"-5 x o"'5, each
end being secured to the supports ; one of these rods is
shown in the figure. It is divided into five equal parts in
the points B, c, c', B'. We support the rod in these four
points by struts, the other extremities of which are fastened
to the framework. The points B, c, C', B' are fixed, as they
are sustained by the struts : hence a weight suspended from
p, which is to break the bridge, must be sufficiently strong

xiii.] A BRIDGE WITH TWO TIES. 211

to break a piece c c', which is secured at the ends ; the rod
A A' would have been broken with 38 Ibs., hence 190 Ibs.
would be necessary to break c c'. There is a similar beam
on the other side of the bridge, and therefore to break the
bridge 380 Ibs. would be necessary, but this force must be
applied exactly at the centre of c c' ; and if the weights be
spread over any considerable length, a heavier load will
be necessary. In fact, if I were to distribute the weight
uniformly over the distance c c', it appears from Art. 408
that double the load would be necessary to produce fracture.

430. We shall now break this model. I place 18 stone
upon it ranged uniformly, and the cathetometer tells me
that the bridge only deflects o"'i, and that its elasticity is
not injured. Placing the tray in position, and loading the
bridge by this means, I find with a weight of 2 cwt. that
there is a deflection of o" 15 ; with 4 cwt. the deflection
amounts to o'72. We therefore infer that the bridge is
beginning to yield, and the clamps give way when the load
is increased to 500 Ibs.

A BRIDGE WITH TWO TIES.

431. It might happen that circumstances would not make
it convenient to obtain points of support below the bridge
on which to erect the struts. In such a case, if suitable
positions for ties can be obtained, a bridge of the form
represented in Fig. 60 may be used.

A D is a horizontal rod of pine 4o"Xo'"5 Xo"'5 ; it is
trisected in the points B and K, from which points the ties
B E and c F are secured to the upper parts of the frame-
work. A D is then supported in the points B and c, which
may therefore be regarded as fixed points. Hence, for the
reasons we have already explained, the strength of the
bridge should be increased nearly threefold. Remembering

p 2

EXPERIMENTAL MECHANICS.

[LECT.

that the bridge has two beams we know it would require
about yolbs. or 80 Ibs. to produce fracture without the ties,
and therefore we might expect that over 200 Ibs. would be
necessary when the beams were supported by the ties. I
perform the experiment, and you see the bridge yields when
the load reaches 194 Ibs.: this is somewhat less than the
amount we had calculated ; the reason being, I think, that
one of the clamps slipped before fracture.

A SIMPLE FORM OF TRUSS.

432. It is often not convenient, or even possible, to
sustain a bridge by the methods we have been considering.
It is desirable therefore to inquire whether we cannot arrange
some plan of strengthening a beam, by giving to it what
shall be equivalent to an increase of depth.

433. We shall only be able to describe here some very
simple methods for doing this. Superb examples are to be
found in railway bridges all over the country, but the full
investigation of these complex structures is a problem of
no little difficulty, and one into which it would be quite

XIII.]

A SIMPLE FORM OF TRUSS.

213

beyond our province to enter. We shall, however, show
how by a judicious combination of several parts a structure
can offer sufficient resistance. The most complex lattice
girder is little more than a network of ties and struts.

434. Let A B (Fig. 61) be a rod of pine 40" x o"-5"Xo"'5,
secured at each end. We shall suppose that the load is
applied at the two points G and H, in the manner shown in
the figure. The load which a bridge must bear when a
train passes over it is distributed over a distance equal to the
length of the train, and the weight of the bridge itself is of
course arranged along the entire span ; hence the load which

a bridge bears is at all times more or less distributed and
never entirely concentrated at the centre in the manner we
have been considering. In the present experiment we shall
apply the breaking load at the two points G and H, as this
will be a variation from the mode we have latterly used.
E F is an iron bar supported in the loops E G and F H. Let
us first try what weight will break the beam. Suspending
the tray from E F, I find that a load of 48 Ibs. is suffi-
cient ; much less would have done had not the ends
been clamped. We have already applied a load in this
manner in Art. 406.

214

EXPERIMENTAL MECHANICS.

[LECT.

435. You observed that the beam, as usual, deflected
before it broke ; if we could prevent deflection we might
reasonably expect to increase the strength. Thus if we support
the centre of the beam c, deflection would be prevented.
This can be done very simply. We clamp the pieces D A,
D B, D c, on a similar beam, and it is evident that c cannot
descend so long as the joints at A, B, D, c remain firmly
secured. We now find that even with a weight of 1 1 2 Ibs.

in the tray, the bar is unbroken. An arrangement of this kind
is frequently employed in engineering, for it seems to be able
to bear more than double the load which is sufficient to
break the unsupported beam.

436. Two frames of this kind, with a roadway laid between
them, would form a bridge, or if the frames were turned up-
side down they would answer equally well, though of course
in this case D A and D B would become ties, and D c a strut,
but a better arrangement for a bridge will be next described.

XIII.] THE TRUSSED BRIDGE. 215

THE WYE BRIDGE.

437. An instructive bridge was erected by the late Sir
I. Brunei over the Wye, for the purpose of carrying a rail-
way. The essential parts of the bridge are represented in
the model shown in Fig. 62, which as before is made of
slips of pine clamped together.

438. Our model is composed of two similar frames,
one of which we shall describe. A B is a rod of pine
48" x o"-5 x o"'5, supported at each extremity. This rod
is sustained at its points of trisection D, c by the uprights
D E and c F, while E and F are supported by the rods B E,
F E, and A F ; the rectangle D E F c is stiffened by the piece
c E, and it would be proper in an actual structure to have
a piece connecting D and F, but it has not been introduced
into the model.

439. We shall understand the use of the diagonal c E
by an inspection of Fig. 63. Suppose the quadrilateral
A B c D be formed of four pieces of

wood hinged at the corners. It is
evident that this quadrilateral can
be deformed by pressing A and c
together, or by pulling them asunder.
Evenjf there were actual joints at
the corners, it would be almost im-
possible to make the quadrilateral
stiff by the strength of the joints. '
You see this by the frame which I hold in my hand ; the
pieces are clamped together at the corners, but no matter
how tightly I compress the clamps, I am able with the
slightest exertion to deform the figure.

440. We must therefore look for some method of stiffening
the frame. I have here a triangle of three pieces, which

216 EXPERIMENTAL MECHANICS. [LECT.

have been simply clamped together at the corners ; this
triangle is unalterable in form ; in fact, since it is impossible
to make two different triangles with the same three sides, it
is evident the triangle cannot be deformed. This points to a
guiding principle in all bridgework. The quadrilateral is not
stiff because innumerable different quadrilaterals can be made
with the same four sides. But if we draw the diagonal A c of
the quadrilateral it isdividedinto two triangles, and hence when
we attach to the quadrilateral, which has been clamped at the
four corners, an additional piece in the direction of one of
the diagonals, it becomes unalterable in shape.

441. In Fig. 63 we have drawn the two diagonals A c and
B D : one would be theoretically sufficient, but it is desirable
to have both, and for the following reason. If I pull A and
c apart, I stretch the diagonal A c and compress B D. If I
compress A and c together, I compress the line A c and ex-
tend B D ; hence in one of these cases A c is a tie, and in the
other it is a strut. It therefore follows that in all cases
one of the diagonals is a tie, and the other a strut. If then
we have only one diagonal, it is called upon to perform
alternately the functions of a tie and of a strut. This is not
desirable, because it is evident that a piece which may act
perfectly as a tie may be very unsuitable for a strut, and vice
versa. But if we insert both diagonals we may make both
of them ties, or both of them struts, and the frame must be
rigid. Thus for example, I might make A c and B D slender
bars of wrought iron, which form admirable ties, though quite
incapable of acting as struts.

442. What we have said with reference to the necessity
for dividing a quadrilateral figure into triangles applies still
more to a polygon with a large number of sides, and we may
lay down the general principle that every such piece of frame-
work should be composed of triangles.

xill.] THE TRUSSED BRIDGE. 217

443. Returning to Fig. 62, we see the reason why the
rectangle E D c F should have one or both of its diagonals
introduced. A load placed, for example, at D would tend
to depress the piece D E, and thus deform the rectangle,
but when the diagonals are introduced this deformation is
impossible.

444. Hence one of these frames is almost as strong as
a beam supported at the points c and D, and therefore, from
the principles of Art. 388, its strength is three times as great
as that of an unsupported beam.

445. The two frames placed side by side and carrying a
roadway form an admirable bridge, quite independent of any
external support, except that given by the piers upon which
the extremities of the frames rest. It would be proper to
connect the frames together by means of braces, which are
not, however, shown in the figure. The model is repre-
sented as carrying a uniform load in contradistinction to
Fig. 58, where the weight is applied at a single point.

446. With eight stone ranged along it, the bridge of
Fig. 62 did not indicate an appreciable deflection.

LECTURE XIV.
THE MECHANICS OF A BRIDGE.

Introduction. — The Girder. — The Tubular Bridge. — The Suspension
Bridge.

INTRODUCTION.

447. PERHAPS it may be thought that the structures we
have been lately considering. are not those which are most
universally used, and that the bridges which are generally
referred to as monuments of engineering skill are of quite
a different construction. Every one is familiar with the
arch, and most of us have seen suspension bridges and the
celebrated Menai tube. We must therefore allude further to
some of these structures, and this we propose to do in the
present lecture. It will only be possible to take a very
slight survey of an extensive subject to which elaborate
treatises have been devoted.

We shall first give a brief account of the use of iron in
the arts of construction. We shall then explain simply the
principle of the tubular bridge, and also of the suspension
bridge. The more complex forms are beyond our scope.

LECT. xiv.] THE GIRDER. 219

THE GIRDER.

448. A horizontal beam supported at each end, and
perhaps at intermediate points, and designed to support a
heavy load is called a girder. Those rods upon which we
have performed experiments, the results of which have
been given in Table XXIV., are small girders; but the
term is generally understood to relate to structures of iron :
the greatest girders for railway bridges are made of bars or
plates of iron riveted together.

449. We shall first consider the application of cast iron
to girders, and show what form they should assume.

450. A beam of cast iron, supposing its section to be
rectangular, has its strength determined by the same laws
as the beams of pine. Thus, supposing the section of two
beams to be the same, their strengths are inversely pro-
portional to their lengths, and the strength of a beam placed
edgewise is to its strength placed flatwise in the proportion
of the greater dimension of its section to the less dimension.
These laws determine the strength of every rectangular beam
of cast iron when that of one beam is known, and we must
perform an experiment in order to find the breaking load in
a particular case.

45i_._I take here a piece of cast iron, which is 2' long,
and o"'5 x o"'5 in section. I support this beam at each
end upon a frame ; the distance between the supports is
20". I attach the tray to the centre of the beam and load
it with weights. The ends of the beam rest freely upon the
supports, but I have taken the precaution of tying each end
by a piece of wire, so that they may not fly about when the
fracture occurs. Loading the trajr, I find that with 280 Ibs.
the crash comes.

452. Let us compare this result with No. 8 of Table XXIV.

220 EXPERIMENTAL MECHANICS. [LECT.

(p. 190). There we find that a piece of pine, the same size
as the cast iron, was broken with 36 Ibs. : the ratio of 280
to 36 is nearly 8, so that the beam of cast iron is about
8 times as strong as the piece of pine of the same size.
This result is a little larger than we would have inferred
from an examination of tables of the strength of large bars
of cast iron ; the reason may be that a very small casting,
such as this bar, is stronger in proportion than a larger
one, owing to the iron not being so uniform throughout
the larger mass.

453. I hold here a bar of cast iron 12" long and i" x i"
in section. I have not sufficient weights at hand to break
it, but we can compute how much would be necessary by
our former experiment.

454. In the first place a bar 12" long, and o"'5 x o"'5 of
section, would require 20 x 280 -f 12 = 467 Ibs. by the
law that the strength is inversely as the length. We also
know that one beam 12" x i" x i" is just as strong as two
beams 12" x i" x o"'5, each placed edgewise; each of
these latter beams is twice as strong as 12" x i" x o"'5
placed flatwise, because the strength when placed edgewise
is to the strength when placed flatwise, as the depth to the
breadth, that is as 2 to i : hence the original beam is four
times as strong as one beam 12" x i" x o"'5 placed flat-
wise : but this last beam is twice as strong as a beam
12" x o"'5 x o'"5, and hence we see that a beam
12" x i" x i" must be 8 times as strong as a beam of
12" x o'"5 X o"'5, but this last beam would require a
load of 467 Ibs. to break it, and hence the beam of
12" x i" x i" would require 467 x 8 = 3736 Ibs. to
produce fracture. This amounts to more than a ton
and a half.

455. It is a rule sometimes useful to practical men that

XIV.] THE GIRDER. 221

a cast iron bar one foot long by one inch square would break
with about a ton weight. If the iron be of the same quality
as that which we have used, this result is too small, but the
error is on the safe side ; the real strength will then be
generally a little greater than the strength calculated from
this rule. What we have said (Art. 403) with reference to
the precaution for safety in bars of wood applies also to cast
iron. The load which the beam has to bear in ordinary
practice should only be a small fraction of that which would
break it.

456. In making any description of girder it is desirable
on very special grounds that as little material as possible be
uselessly employed. It will of course be remembered that
a girder has to support its own weight, besides whatever
may be placed upon it : and if the girder be massive, its
own weight is a serious item. Of two girders, each capable
of bearing the same total load, the lighter, besides employ-
ing less material, will be able to bear a greater weight placed
upon it. It is therefore for a double reason desirable to
diminish the weight. This remark applies especially to such
a material as cast iron, which can be at once given the
form in which it shall be capable of offering the greatest
resistance.

457.~The principles which will guide us in ascertaining
the proper form to give a cast iron girder, are easily de-
duced from what we have laid down in Lectures XI. and
XII. We have seen that depth is very desirable for a strong
beam. If therefore we strive to attain great depth in a light
beam, the beam must be very thin. Now an extremely
thin beam will not be safe. In the first place it would
be flexible and liable to displacement sideways; and, in
the second place, there is a still more fatal difficulty. We
have shown that when a beam of wood is supporting a

EXPERIMENTAL MECHANICS.

[LECT.

weight, the fibres at the bottom of the beam are extended,
the tendency being to tear them (Art. 3 7 6). The fibres on the
top of the beam are compressed, while the centre of the
beam is in its natural state. The condition of strain in
a cast-iron beam is precisely similar ; the bottom portions
are in a state of extension, while the top is compressed.
If therefore a beam be very thin, the material at the
lower part may not be sufficient to withstand the forces
of extension, and fracture is produced. To obviate this,
we strengthen the bottom of the beam by placing extra
material there. Thus we are led to the idea of a thin
beam with an excess of iron at the bottom.

458. E F (Fig. 64) is the thin
iron beam along the bottom of
which is the stout flange shown
at CD; rupture cannot commence
at the bottom unless this flange
be torn asunder ; for until this
happens it is clear that fracture
cannot begin to attack the upper
and slender part of the beam
E F.

459. But the beam is in a

state of compression along its upper side, just as in the
wooden beams which we have already considered. If
therefore the upper parts were not powerful enough to
resist this compression, they would be crushed, and the
beam would give way. The remedy for this source of
weakness is obvious ; a second flange runs along the top
of the beam, as shown at A B. If this be strong enough
to resist the compression, the stability of the beam is
ensured.

460. The upper flange is made very much smaller than

FIG. 64.

xiv.] THE TUBULAR BRIDGE. 223

the lower one, in consequence of a property of cast iron.
This metal is more capable of resisting forces of compres-
sion than forces of extension, and it is only necessary to
use one-sixth of the iron on the upper flange that is
required for the lower. When the section has been thus
proportioned, the beam is equally strong at both top and
bottom ; adding material to either flange without strengthen-
ing the other, will not benefit the girder, but will rather
prove a source of weakness, by increasing the weight which
has to be supported.

461. I have here a small girder made of what we are familiar
with under the name of " tin," but which is of course sheet
iron thinly covered over with tin. It has the shape shown in
Fig. 64, and it is 12" long. I support it at each end, and you
see it bears two hundred weight without apparent deflection.

THE TUBULAR BRIDGE.

462. I shall commence the description of the principle of
this bridge by performing some experiments upon a tube,
which I hold in my hand. The tube is square, i" x i" in
section, and 38" long. It is made of " tin," and weighs
rather less than a pound.

463. Here is a solid rod of iron of the same length as
the tube, but containing considerably more metal. This is
easily verified by weighing the tube and the rod one against
the other. I shall regard them as two girders, and experi-
ment upon their strength, and we shall find that, though
the tube contains less substance than the rod, it is much
the stronger.

464. I place the rod on a pair of supports about 3' apart ;
I then attach the tray to the middle of the rod : 14 Ibs.
produce a deflection of o'"5i, and 42 Ibs. bends down the
rod through 3'"! 8. This is a large deflection ; and when

224 EXPERIMENTAL MECHANICS. [LECT.

I remove the load, the rod only returns through i"78, thus
showing that a permanent deflection of i'"4o is produced.
This proves that the rod is greatly injured, and demonstrates
its unsuitability for a girder.

465. Next we place the tube upon the same supports,
and treat it in the same manner. A load of 56 Ibs. only
produces a deflection of ©"-op, and, when this load is
removed, the tube returns to its original position : this is
shown by the cathetometer, for a cross is marked on the tube,
and I bring the image of it on the horizontal wire of the
telescope before the load of 56 Ibs. is placed in the tray.
When the load is removed, I see that the cross returns
exactly to where it was before, thus proving that the
elasticity of the tube is unimpaired. We double the load,
thus placing i cwt. in the tray, the deflection only reaches
o"-26, and, when the load is removed, the tube is found to
be permanently deflected by a quantity, at all events not
greater than o"*oo4 ; hence we learn that the tube bears
easily and without injury a load more than twice as great as
that which practically destroyed a rod of wrought iron, con-
taining more iron than the tube. We load the tube still further
by placing additional weights in the tray, and with 140 Ibs. the
tube breaks ; the fracture has occurred at a joint which was
soldered, and the real breaking strength of the tube, had it
been continuous, is doubtless far greater. Enough, how-
ever, has been borne to show the increase of strength
obtained by the tubular form.

466. We can explain the reason of this remarkable
result by means of Fig. 64. Were the thin portion of
the girder E F made of two parts placed side by side,
the strength would not be altered. If we then imagine the
flange A B widened to the width of c D, and the two
parts which form E F opened out so as to form a tube,

xiv.] THE SUSPENSION BRIDGE. 225

the strength of the girder is still retained in its modified
form.

467. A tube of rectangular section has the advantage of
greater depth than a solid rod of the same weight ; and if
the bottom of the tube be strong enough to resist the ex-
tension, and the top strong enough to resist the compression,
the girder will be stiff and strong.

468. In the Menai Tubular Bridge, where a gigantic tube
supported at each end bridges over a span of four hundred
and sixty feet, special arrangements have been made for
strengthening the top. It is formed of cells, as wrought
iron disposed in this way is especially adapted for resisting
compression.

469. We have only spoken of rectangular tubes, but it is
equally true for tubes of circular or other sections that when
suitably constructed they are stronger than the same quantity
of material, if made into a solid rod.

470. We find this principle in nature ; bones and quills
are often found to be hollow in order to combine lightness
with strength, and the stalks of wheat and other plants are
tubular for the same reason.

THE SUSPENSION BRIDGE.

47i._ Where a great span is required, the suspension
bridge possesses many advantages. It is lighter than a
girder bridge of the same span, and consequently cheaper,
while its singular elegance contrasts very favourably with
the appearance of more solid structures. On the other
hand, a suspension bridge is not so well suited for railway
traffic as the lattice girder.

472. The mechanical character of the suspension bridge
is simple. If a rope or a chain be suspended from two
points to which its ends are attached, the chain hangs in a

Q

226 EXPERIMENTAL MECHANICS. [LECT. xiv.

certain curve known to mathematicians as the catenary. The
form of the catenary .varies with the length of the rope, but
it would not be possible to make the chain lie in a straight
line between the two points of support, for reasons pointed
out in Art. 20. No matter how great be the force applied, it
will still be concave. When the chain is stretched until the
depression in the middle is small compared with the dis-
tance between the points of support, the curve though
always a catenary, has a very close resemblance to the
parabola.

473. In Fig. 65 a model of a suspension bridge is shown.
The two chains are fixed one on each side at the points E and
F ; they then pass over the piers A, D, and bridge a span of
nine feet. The vertical line at the centre B c shows the
greatest amount by which the chain has deflected from the
horizontal A D. When the deflection of the middle of the
chain is about one-tenth part of A D, the curve A c D becomes
for all practical purposes a parabola. The roadway is
suspended by slender iron rods from the chains, the
lengths of the suspension rods being so regulated as to
make it nearly horizontal.

474. The roadway in the model is laden with 8 stone
weights. We have distributed them in this manner in
order to represent the permanent load which a great suspen-
sion bridge has to carry. The series of weights thus arranged
produces substantially the same effect as if it were actu-
ally distributed uniformly along the length. In a real
suspension bridge the weight of the chain itself adds
greatly to the tension.

475. We assume that the chain hangs in the form of a
parabola, and that the load is uniformly ranged along the
bridge. The tension upon the chains is greatest at their
highest points, and least at their lowest points, though

228 EXPERIMENTAL MECHANICS. [LECT.

the difference is small. The amount of the tension can
be calculated when the load, span, and deflection are
known. We cannot give the steps of the calculation, but
we shall enunciate the result.

476. The magnitude of the tension at the lowest
point c of each chain is found by multiplying the total
weight (including chains, suspension rods, and roadway)
by the span, and dividing the product by sixteen times the
deflection.

The tension of the chain at the highest point A exceecs
that at the lowest point c, by a weight found by multiplying
the total load by the deflection, and dividing the product
by twice the span.

477. The total weight of roadway, chains, and load in
the model is 120 Ibs. ; the deflection is 10", the span 108";
the product of the weight and span is 12,960; sixteen times
the deflection is 160 ; and, therefore, the tension at the point
c is found, by dividing 12,960 by 160, to be 81 Ibs.

To find the tension at the point A, we multiply 120 by 10,
and divide the product by 216; the quotient is nearly 6.
This added to 81 Ibs. gives 87 Ibs. for the tension on the
chain at A.

478. One chain of the model is attached to a spring-
balance at A ; by reference to the scale we see the tension
indicated to be 90 Ibs. : a sufficiently close approximation
to the calculated tension of 87 Ibs.

479. A large suspension bridge has its chains strained
by an enormous force. It is therefore necessary that the
ends of these chains be very firmly secured. A good
attachment is obtained by anchoring the chain to a large
iron anchor imbedded in solid rock.

480. In Art. 45 we have pointed out how the dimensions
of the tie rod could be determined when the tension was

xiv.] THE SUSPENSION BRIDGE. 229

known. Similar considerations will enable us to calculate
the size of the chain necessary for a suspension bridge
when we have ascertained the tension to which it will be
subjected.

481. We can easily determine by trial what effect is pro-
duced on the tension of the chain, by placing a weight upon
the bridge in addition to the permanent load. Thus an
additional stone weight in the centre raises the tension of the
spring-balance to 100 Ibs. ; of course the tension in the other
chain is the same : and thus we find a weight of 14 Ibs. has
produced additional tensions of 10 Ibs. each in the two
chains. With a weight of 28 Ibs. at the centre we find a
strain of no Ibs. on the chain.

482. These additional weights may be regarded as an-
alogous to the weights of the vehicles which the suspension
bridge is required to carry. In a large suspension bridge
the tension produced by the passing loads is only a small
fraction of the permanent load.

LECTURE XV.
THE MOTION OF A FALLING BOD Y.

Introduction. — The First Law of Motion. — The Experiment of Galileo
from the Tower of Pisa. — The Space is proportional to the Square
of the Time.— A Body falls 16' in the First Second.— The Action
of Gravity is independent of the Motion of the Body. — How the
Force of Gravity is defined.— The Path of a Projectile is a
Parabola.

INTRODUCTION.

483. Kinetics is that branch of mechanics which treats
of the action of forces in the production of motion. We
shall find it rather more difficult than the subjects with which
we have been hitherto occupied ; the difficulties in kinetics
arise from the introduction of the element of time, into our
calculations. The principles of kinetics were unknown to
the ancients. Galileo discovered some of its truths in the
seventeenth century ; and, since his time, the science has
grown rapidly. The motion of a falling body was first
correctly apprehended by Galileo ; and with this subject we
can appropriately commence.

THE FIRST LAW OF MOTION.

484. Velocity, in ordinary language, is supposed to con-
vey a notion of rapid motion. Such is not precisely the

LECT. xv.] THE FIRST LAW OF MOTION. 231

meaning of the word in mechanics. By velocity is merely
meant the rate at which a body moves, whether the rate be
fast or be slow. This rate is most conveniently measured
by the number of feet moved over in one second. Hence
when it is said the velocity of a body is 25, it is meant that
if the body continued to move for one second with its
velocity unaltered, it would in that time have moved over
25 feet.

485. The first law of motion may be stated thus. If no
force act upon a body, it will, if at rest, remain for ever at
rest ; or if in motion, it will continue for ever to move with
a uniform velocity. We know this law to be true, and yet
no one has ever seen it to be true for the simple reason that
we cannot realise the condition which it requires. We can-
not place a body in the condition of being unacted upon by
any forces. But we may convince ourselves of the truth of
the law by some such reasoning as the following. If a stone
be thrown along the road, it soon comes to rest. The
body leaves the hand with a certain initial velocity and is
not further acted upon by it. Hence, if no other force acted
on the stone, we should expect, if the first law be true, that
it would continue to run on for ever with the original
velocity at the moment of leaving the hand. But other
forces do act upon the stone ; the attraction of the earth
pulls it down ; and, when it begins to bound and roll upon
the ground, friction comes into operation, deprives the stone
of its velocity, and brings it to rest. But let the stone be
thrown upon a surface of smooth ice; when it begins to
slide, the force of gravity is counteracted by the reaction of
the ice : there is no other force acting upon the stone
except friction, which is small. Hence we find that the
stone will run on for a considerable distance. It requires
but little effort of the imagination to suppose a lake whose

[LECT.

surface is an infinite plane,
perfectly smooth, and that
the stone is perfectly
smooth also. In such a
case as this the first law of
motion amounts to the
assertion that the stone
would never stop.

486. We may, in the lec-
ture room, see the truth of
this law verified to a certain
extent by Atwood's machine
(Fig. 66). This machine
has been devised for the
purpose of investigating the
laws of motion by actual
experiment. It consists
principally of a pulley c,
mounted so that its axle
rests upon two pairs of
wheels, as shown in the
figure ; it being the object
of this contrivance to en-
able the wheel to revolve
with the utmost freedom. A
pair of equal weights A, B, are
attached by a silken thread,
which passes over the pul-
ley; each of the weights
is counterbalanced by the
other : so that when the
two are in motion, we may
consider either as a body

XV.] THE EXPERIMENT OF GALILEO. 233

not acted upon by any forces, and it . will be found
that it moves uniformly, as far as the size of the apparatus
will permit.

487. If we try to conceive a body free in space, and not
acted upon by any force, it is more natural to suppose that
such a body, when once started, should go on moving
uniformly for ever, than that its velocity should be altered.
The true proof of the first law of motion is, that all con-
sequences properly deduced from it, in combination with
other principles, are found to be verified. Astronomy
presents us with the best examples. The calculation of
the time of an eclipse is based upon laws which in them-
selves assume the first law of motion ; hence, when we
invariably find that an eclipse occurs precisely at the
moment for which it has been predicted, we have a splendid
proof of the sublime truth which the first law of motion
expresses.

THE EXPERIMENT OF GALILEO FROM THE
TOWER OF PISA.

488. The contrast between heavy bodies and light bodies is
so marked that without trial we hardly believe that a heavy
body and a light body will fall from the same height in
the same time. That they do so Galileo proved by drop-
ping a heavy ball and a light ball together from the top
of the Leaning Tower at Pisa. They were found to reach
the ground simultaneously. We shall repeat this experi-
ment on a scale sufficiently reduced to correspond with the
dimensions of the lecture room.

489. The apparatus used is shown in Fig. 67. It con-
sists of a stout framework supporting a pulley H at a
height of about 20 feet above the ground. This pulley
carries a rope; one end of the rope is attached to a

FIG. 67-

LECT. XV.] THE EXPERIMENT OF GALILEO. 235

triangular piece of wood, to which two electro-magnets G
are fastened. The electro-magnet is a piece of iron in
the form of a horse-shoe, around which is coiled a long
wire. The horse-shoe becomes a magnet immediately an
electric current passes through the wire; it remains a
magnet as long as the current passes, and returns to its
original condition the moment the current ceases. Hence,
if I have the means of controlling the current, I have
complete control of the magnet ; you see this ball of iron
remains attached to the magnet as long as the current
passes, but drops the instant I break the current. The
same electric circuit includes both the magnets ; each of
them will hold up an iron ball F when the current passes,
but the moment the current is broken both balls will be
released. Electricity travels along a wire with prodigious
velocity. It would pass over many thousands of miles in a
second ; hence the time that it takes to pass through the
wires we are employing is quite inappreciable. A piece of
thin paper interposed between the magnets and the balls
will ensure that they are dropped simultaneously; when
this precaution is not taken one or both balls may hesitate
a little before commencing to descend. A long pair of
wires E, B, must be attached to the magnets, the other ends
of the wires communicating with the battery D ; the triangle
and its load is hoisted up by means of the rope and pulley
and the magnets thus carry the balls to a height of
20 feet : the balls we are using weigh about 0-25 Ib.
and i Ib. respectively.

490. We are now ready to perform the experiment. I
break the circuit ; the two balls are disengaged simul-
taneously ; they fall side by side the whole way, and reach
the ground together, where it is well to place a cushion
to receive them. Thus you see the heavy ball and the

236 EXPERIMENTAL MECHANICS. [LECT.

light one each require the same amount of time to fall from
the same height.

491. But these balls are both of iron; let us compare
together balls made of different substances, iron and wood
for example. A flat-headed nail is driven into a wooden
ball of about 2"'5 in diameter, and by means of the iron
in the nail I can suspend this ball from one of the magnets ;
while either of the iron balls we have already used hangs
from the other. I repeat the experiment in the same manner,
and you see they also fall together. Finally, when an iron ball
and a cork ball are dropped, the latter is within two or
three inches of its weighty companion when the cushion is
reached : this small differenc e is due to the greater effect
of the resistance of the air on the lighter of the two bodies.
There can be no doubt that in a vacuum all bodies of
whatever size or material would fall in precisely the same
time.

492. How is the fact that all bodies fall in the same
time to be explained ? Let us first consider t\yo iron balls.
Take two equal particles of iron : it is evident that these
fall in the same time ; they would do so if they were very
close together, even if they were touching, but then they
might as well be in one piece : and thus we should find that
a body consisting of two or more iron particles takes the same
time to fall as one (omitting of course the resistance of
the air). Thus it appears most reasonable that two balls
of iron, even though unequal in size, should fall in the
same time.

493. The case of the wooden ball and the iron ball will
require more consideration before we realise thoroughly how
much Galileo's experiment proves. We must first explain
the meaning of the word mass in mechanics.

494. It is hot correct to define mass by the introduction

xv.] THE EXPERIMENT OF GALILEO. 237

of the idea of weight, because the mass of a body is some-
thing independent of the existence of the earth, whereas
weight is produced by the attraction of the earth. It is true
that weight is a convenient means of measuring mass, but
this is only a consequence of the property of gravity which
the experiment proves, namely, that the attraction of gravity
for a body is proportional to its mass.

495. Let us select as the unit of mass the mass of a piece
of platinum which weighs i Ib. at London ; it is then evident
that the mass of any other piece of platinum should be ex-
pressed by the number of pounds it contains : but how are
we to determine the mass of some other substance, such as
iron ? A piece of iron is defined to have the same mass as a
piece of platinum, if the same force acting on either of the
bodies for the same time produces the same velocity. This
is the proper test of the equality of masses. The mass of
any other piece of iron will be represented by the number
of times it contains a piece equal to that which we have just
compared with the platinum ; similarly of course for other
substances.

496. The magnitude of a force acting for the time unit is
measured by the product of the mass set in motion and the
velocity which it has acquired. This is a truth established,
like thejirst law of motion, by indirect evidence.

497. Let us apply these principles to explain the ex-
periment which demonstrated that a ball of wood and a ball
of iron fall in the same time. Forces act upon the two
bodies for the same time, but the magnitudes of the forces
are proportional to the mass of each body multiplied into its
velocity, and, since the bodies fall simultaneously, their
velocities are equal. The forces acting upon the bodies are
therefore proportional to their masses ; but the force acting
on each body is the attraction of the earth, therefore, the

238 EXPERIMENTAL MECHANICS. [LECT.

gravitation to the earth of different bodies is proportional
to their masses.

498. We may here note the contrast between the attraction
of gravitation and that of a magnet. A magnet attracts
iron powerfully and wood not at all ; but the earth draws
all bodies with forces depending on their masses and their
distances, and not on their chemical composition.

THE SPACE DESCRIBED BY A FALLING BODY IS
PROPORTIONAL TO THE SQUARE OF THE TIME.

499. We have next to discover the law by which we ascer-
tain the distance a body falling from rest will move in a given
time ; it is not possible to experiment directly upon this
subject, as in two seconds a body will drop 64 feet and acquire
an inconveniently large velocity ; we can, however, resort to
Atwood's machine (Fig. 66) as a means of diminishing the
motion. For this purpose we require a clock with a seconds
pendulum.

500. On one of the equal cylinders A I place a slight brass
rod, whose weight gives a preponderance to A, which will
consequently descend. I hold the loaded weight in my
hand, and release it simultaneously with the tick of the pen-
dulum. I observe that it descends 5" before the next tick.
Returning the weight to the place from whence it started, I
release it again, and I find that at the second tick of the
pendulum it has travelled 20". Similarly we find that in
three seconds it descends 45". It greatly facilitates these
experiments to use a little stage which is capable of being
slipped up and down the scale, and which can be clamped
to the scale in any position. By actually placing the stage
at the distance of 5", 20", or 45" below the point from which
the weight starts, the coincidence of the tick of the pendulum

xv.] FALL OF A BODY IN THE FIRST SECOND. 239

with the tap of the weight on its arrival at the stage is very
marked.

501. These three distances are in the proportion of
1,4, 9 ; that is, as the squares of the numbers of seconds
i, 2, 3. Hence we may infer that the distance traversed
by a body falling from rest is proportional to the square of the
time.

502. The motion of the bodies in Atwood's machine is
much slower than the motion of a body falling freely, but
the law just stated is equally true in both cases so that in a
free fall the distance traversed is proportional to the
square of the time. Atwood's machine cannot directly
tell us the distance through which a body falls in one
second. If we can find this by other means, we shall
easily be able to calculate the distance through which a
body will fall in any number of seconds.

A BODY FALLS 16' IN THE FIRST SECOND.

503. The apparatus by which this important truth maybe
demonstrated is shown in Fig. 67. A part of it has been
already employed in performing the experiment of Galileo,
but two other parts must now be used which will be briefly
explained.

504. At A a pendulum is shown which vibrates once every
second ; it need not be connected with any clockwork to
sustain the motion, for when once set vibrating it will con-
tinue to swing some hundreds of times. When this pendulum
is at the middle of its swing, the bob just touches a slender
spring, and presses it slightly downwards. The electric
current which circulates about the magnets G (Art. 489)
passes through this spring when in its natural position ; -but
when the spring is pressed down by the pendulum, the
current is interrupted. The consequence is that, as the

240 EXPERIMENTAL MECHANICS. [LECT.

pendulum swings backwards and forwards, the current is
broken once every second. There is also in the circuit an
electric alarm bell c, which is so arranged that, when the
current passes, the hammer is drawn from the bell ; but,
when the current ceases, a spring forces the hammer
to strike the bell. When the circuit is closed, the
hammer is again drawn back. The pendulum and the
bell are in the same circuit, and thus every vibration of the
pendulum produces a stroke of the bell. We may regard
the strokes from the bell as the ticks of the pendulum
rendered audible to the whole room.

505. You will now understand the mode of experimenting.
I draw the pendulum aside so that the current passes un-
interruptedly. An iron ball is attached to one of the
electro-magnets, and it is then gently hoisted up until the
height of the ball from the ground is about 16'. A cushion
is placed on the floor in order to receive the falling body.
You are to look steadily at the cushion while you listen
for the bell. All being ready, the pendulum, which has
been held at a slight inclination, is released. The moment
the pendulum reaches the middle of its swing it touches the
spring, rings the bell, breaks the current which circulated
around the magnet, and as there is now nothing to sustain
the ball, it drops down to the cushion ; but just as it arrives
there, the pendulum has a second time broken the electric
circuit, and you observe the falling of the ball upon the
cushion to be identical with the second stroke of the bell.
As these strokes are repeated at intervals of a second, it
follows that the ball has fallen 16' in one second. If the
magnet be raised a few feet higher, the ball may be seen to
reach the cushion after the bell is heard. If the magnet be
lowered a few feet, the ball reaches the cushion before the
bell is heard.

xv.] THE ACTION OF GRAVITY. 241

506. We have previously shown that the space is propor-
tional to the square of the time. We now see that when the
time is one second, the space is 16 feet. Hence if the time
were two seconds, the space would be 4 X 1 6 = 64 feet ;
and in general the space in feet is equal to 16 multiplied by
the square of the time in seconds.

507. By the help of this rule we are sometimes enabled
to ascertain the height of a perpendicular cliff, or the depth
of a well. For this purpose it is convenient to use a stop-
watch, which will enable us to measure a short interval of
time accurately. But an ordinary watch will do nearly as
well, for with a little practice it is easy to count the beats,
which are usually at the rate of five a second. By observing
the number of beats from the moment the stone is released
till we see or hear its arrival at the bottom, we determine
the time occupied in the act of falling. The square of
the number of seconds (taking account of fractional parts)
multiplied by 16 gives the depth of the well or the height
of the cliff in feet, provided it be not high.

THE ACTION CF GRAVITY IS INDEPENDENT OF
THE MOTION OF THE BODY.

5o8._We have already learned that the effect of gravity
does not depend upon the actual chemical composition of the
body. We have now to learn that its effect is uninfluenced
by any motion which the body may possess. Gravity pulls
a body down 16' per second, if the body starts from rest
But suppose a stone be thrown upwards with a velocity of
20 feet, where will it be at the end of a second? Did
gravity not act upon the stone, it would be at a height of
20 feet. The principle we have stated tells us that gravity
will draw this stone towards the earth through a distance of

R

242 EXPERIMENTAL MECHANICS. [LECT.

16', just as it would have done if the stone had started from
rest. Since the stone ascends 20' in consequence of its own
velocity, and is pulled back 1 6' by gravity, it will, at the end
of a second, be found at the height of 4'. If, instead of
being shot up vertically, the body had been projected in any
other direction, the result would have been the same ;
gravity would have brought the body at the end of one
second 16' nearer the earth than it would have been had
gravity not acted. For example, if a body had been shot
vertically downwards with a velocity of 20', it would in one
second have moved through a space of 36'.

509. We shall illustrate this remarkable property by an
experiment. The principle of doing so is as follows : —
Suppose we take two bodies, A and B. If these be held at
the same height, and released together, of course they reach
the ground at the same instant ; but if A, instead of being
merely dropped, be projected with a horizontal velocity at
the same moment that B is released, it is still found that A
and B strike the floor simultaneously.

510. You may very simply try this without special ap-
paratus. In your left hand hold a marble, and drop it at
the same instant that your right hand throws another marble
horizontally. It will be seen that the two marbles reach the
ground together.

511. A more accurate mode of making the experiment
is shown by the contrivance of Fig. 68.

In this we have an arrangement by which we ensure that
one ball shall be released just as the other is projected.
At A B is shown a piece of wood about 2" thick ; the circular
portion (2' radius) on which the ball rests is grooved, so
that the ball only touches the two edges and not the bottom
of the groove. Each edge of the groove is covered
with tinfoil c, but the pieces of tinfoil on the two sides

XV.]

THE ACTION OF GRAVITY.

must not communicate. One edge is connected with one
pole of the battery K, and the other edge with the other
pole, but the current is unable to pass until a communication
by a conductor is opened between the two edges. The
ball G supplies the bridge ; it is covered with tinfoil, and
therefore, as long as it rests upon the edges, the circuit is

complete ; the groove is so placed that the tangent to it at
the lowest point B is horizontal, and therefore, when the ball
rolls down the curve, it is projected from the bottom in a
horizontal direction. An india-rubber spring is used to
propel the ball ; and by drawing it back when embraced by
the spring, I can communicate to the missile a velocity which

R 2

244 EXPERIMENTAL MECHANICS. [LECT.

can be varied at pleasure. At H we have an electro-
magnet, the wire around which forms part of the circuit we
have been considering. This magnet is so placed that a
ball suspended from it is precisely at the same height above
the floor as the tinned ball is at the moment when it leaves
the groove.

512. We now understand the mode of experimenting. So
long as the tinned ball G remains on the curve the bridge is
complete, the current passes, and the electro-magnet will sus-
tain H, but the moment G leaves the curve, H is allowed to fall.
We invariably find that whatever be the velocity with which
G is projected, it reaches the ground at the same instant as
H arrives there. Various dotted lines in the figure show
the different paths which G may traverse ; but whether it
fall at D, at E, or at F, the time of descent is the same as that
taken by H. Of course, if G were not projected horizontally,
we should not have arrived at this result : all we assert is
that whatever be the motion of a body, it will (when possible)
be at the end of a second, sixteen feet nearer the earth than
it would have been if gravity had not acted. If the body
be projected horizontally, its descent is due to gravity alone,
and is neither accelerated nor retarded by the horizontal
velocity. What this experiment proves is, that the mere fact
of a body having velocity does not affect the action of
gravity thereon.

513. Though we have only shown that a horizontal
velocity does not affect the action of gravity, yet neither
does a velocity in any direction. This is verified, like the
first law of motion, by the accordance between the con-
sequences deduced from it and the facts of observation.

514. We may summarize these results by saying that no
matter what be the material of which a particle is composed,
whether it be heavy or light, moving or at rest, if no force

XV.] THE ACTION OF GRAVITY. 245

but gravity act upon the particle for / seconds, it will then
be i6/2 feet nearer the earth than it would have been had
gravity not acted.

515. A proposition which is of some importance may be
introduced here. Let us suppose a certain velocity and a
certain force. Let the velocity be such that a point starting
from A, Fig. 69, would in one second move uniformly to B.

Let the force be such that if it acted on a particle originally
at rest at A, it would in one second draw the particle to D ;
if then the force act on a particle having this velocity
where will it be at the end of the second ? Complete the
parallelogram A B c D, and the particle will be found at c.
By what we have stated the force will equally discharge
its duty whatever be the initial velocity. The force will
therefore make the particle move to a distance equal and
parallel to A D from whatever position the particle would
have assumed, had the force not acted ; but had the force
not acted, the particle would have been found at B : hence,
when the force does act, the particle must be found at c,
since B c is equal and parallel to A D.

246 EXPERIMENTAL MECHANICS. [LECT.

HOW THE FORCE OF GRAVITY IS DEFINED.

516. From the formula

Distance = 16/2,

we learn that a body falls through 64' in 2 seconds ; and as
we know that it falls 16' in the first second, it must fall 48'
in the next second. Let us examine this. After falling for
one second, the body acquires a certain velocity, and with
that velocity it commences the next second. Now, accord-
ing to what we have just seen, gravity will act during the next
second quite independently of whatever velocity the body
may have previously had. Hence in the second second
gravity pulls the body down 16', but the body moves alto-
gether through 48'; therefore it must move through 32' in
consequence of the velocity which has been impressed upon
it by gravity during the first second. We learn by this that
when gravity acts for a second, it produces a velocity such
that, if the body be conceived to move uniformly with the
velocity acquired, the body would in one second move
over 32'.

517. In three seconds the body falls 144', therefore in the
third second it must have fallen

144'— 64' = 80';

but of this 80' only 16' could be due to the action of gravity
impressed during that second ; the rest,

80'— 1 6' = 64',

is due to the velocity with which the body commenced the
third second.

518. We see therefore that after the lapse of two seconds
gravity has communicated to the body a velocity of 64' per
second ; we should similarly find, that at the end of the third
second, the body has a velocity of 96', and in general at the

XV.] THE PATH OF A PROJECTILE. 247

end of / seconds a velocity of 32^. Thus we illustrate the
remarkable law that the velocity developed by gravity is
proportional to the time,

519. This law points out that the most suitable way of
measuring gravity is by the velocity acquired by a falling
body at the end of one second. Hence we are accustomed
to say that g (as gravity is generally designated) is 32. We
shall afterwards show in the lecture on the pendulum
(XVIII.) how the value of g can be obtained accurately.
From the two equations, v = $2t and J=i6/2it is easy to
infer another very well known formula, namely, i? — 645.

THE PATH OF A PROJECTILE IS A PARABOLA.

520. We have already seen, in the experiments of Fig. 68,
that a body projected horizontally describes a curved path
on its way to the ground, and we have to determine the
geometrical nature of the curve. As the movement is rapid,
it is impossible to follow the projectile with the eye so as to
ascertain the shape of its path with accuracy ; we must there-
fore adopt a special contrivance, such as that represented
in Fig. 70.

B c is a quadrant of wood 2" thick ; it contains a groove,
along-which the ball B will run when released. A series of
cardboard hoops are properly placed on a black board,
and the ball, when it leaves the quadrant, will pass through
all these hoops without touching any, and finally fall into
a basket placed to receive it. The quadrant must be
secured firmly, and the ball must always start from precisely
the same place. The hoops are easily adjusted by trial.
Letting the ball run down the quadrant two or three times,
we can see how to place the first hoop in its right position,
and secure it by drawing pins ; then by a few more trials

248

EXPERIMENTAL MECHANICS.

[LECT.

the next hoop is to be adjusted, and so on for the whole
eight.

521. The curved line from the bottom of the quadrant,
which passes through the centres of the hoops, is the path in

FIG. 70

which the ball moves ; this curve is a parabola, of which F
is the focus and the line A A the directrix.

It is a property of the parabola that the distance of any
point on the curve from the focus is equal to its perpen-
dicular distance from the directrix. This is shown in the
figure, For example, the dotted line F D, drawn from F to

xv.] THE PATH OF A PROJECTILE. 249

the centre of the lowest hoop D, is equal in length to the
perpendicular D p let fall from D on the directrix A A.

522. The direction in which the ball is projected is in
this case horizontal, but, whatever be the direction of pro-
jection, the path is a parabola. This can be proved mathe-
matically as a deduction from the theorem of Art. 515.

LECTURE XVI.
INERTIA.

Inertia. — The Hammer. — The Storing of Energy. — The Fly-wheel. —
The Punching Machine.

INERTIA.

523. A BODY unacted upon by force will continue for ever
at rest, or for ever moving uniformly in a straight line. This
is asserted by the first law of motion (Art. 485). It is
usual to say that Inertia is a property of all matter, by which
it is meant that matter cannot of itself change its state of
rest or of motion. Force is accordingly required for this
purpose. In the present chapter we shall discuss some
important mechanical considerations connected with the
application of force in changing the state of a body from
rest or in altering its velocity when in motion. In the next
chapter we shall study the application of force in compelling
a body to swerve from its motion in a straight line.

524. We have in earlier lectures been concerned with the
application of force either to raise a weight or to overcome
friction. We have now to consider the application of force

LECT. xvi.] INERTIA, 251

to a body, not for the purpose of raising it, nor for pushing
it along against a frictional resistance, but merely for the
purpose of generating a velocity. Unfortunately there is a
practical difficulty in the way of making the experiments
precisely in the manner we should wish. We want to get a
mass isolated both from gravitation and from friction, but
this is just what we cannot do — that is, we cannot do it
perfectly. We have, however, a simple appliance which
will be sufficiently isolated for our present purpose. Here
is a heavy weight of iron, about 25 Ibs., suspended by an
iron wire from the ceiling about 32 feet above the floor (see
Fig. 82). This weight may be moved to and fro by the
hand. It is quite free from friction, for we need not at
present remember the small resistance which the air offers.
We may also regard the gravity of the weight as neutralized
by the sustaining force of the wire, and accordingly as the
body now hangs at rest it may for our purposes be regarded
as a body unacted upon by any force.

525. To give this ball a horizontal velocity I feel that I
must apply force to it. This will be manifest to you all
when I apply the force through the medium of an india-
rubber spring. If I pull the spring sharply you notice how
much it stretches ; you see therefore that the body will not
move quickly unless a considerable force is applied to it. It
thus follows that merely to generate motion in this mass
force has been required.

526. So, too, when the body is in motion as it now is I
cannot stop it without the exertion of force. See how the
spring is stretched and how strong a pull has thus been
exerted to deprive the body of motion. Notice also that
while a small force applied sufficiently long will always
restore the body to rest, yet that to produce rest quickly a
large force will be required.

252 EXPERIMENTAL MECHANICS. [LECT.

527. It is an universal law of nature that action and
reaction are equal and opposite. Hence when any agent
acts to set a body in motion, or to modify its motion in
any way, the body reacts on the agent, and this force has
been called the Kinetic reaction.

528. For example. When a railway train starts, the loco-
motive applies force to the carriages, and the speed generated
during one second is added to that produced during the
next, and the pace improves. The kinetic reaction of the
train retards the engine from attaining the speed it would
acquire if free from the train.

THE HAMMER.

529. The hammer and other tools which give a blow
depend for their action upon inertia. A gigantic hammer
might force in a nail by the mere weight of the head resting
on the nail, but with the help of inertia we drive the nail by
blows from a small hammer. We have here inertia aiding
in the production of a mechanical power to overcome the con-
siderable resistance which the wood opposes to the entrance
of the nail. To drive in the nail usually requires a direct
force of some hundreds of pounds, and this we are able
conveniently to produce by suddenly checking the velocity
of a small moving body.

530. The theory of the hammer is illustrated by the
apparatus in Fig. 71. It is a tripod, at the top of which,
about 9' from the ground, is a stout pulley c ; the rope is
about 15' long, and to each end of it A and B are weights
attached. These weights are at first each 1 4 Ibs. I raise A
up to the pulley, leaving B upon the ground ; I then let go
the rope, and down falls A : it first pulls the slack rope
through, and then, when A is about 3' from the ground, the
rope becomes tight, B gets a violent chuck and is lifted into

xvi.] THE HAMMER. 253

the air. What has raised B ? It cannot be the mere weight
of A, because that being equal to B, could only just balance
B, and is insufficient to raise it. It must have been a force
which raised B ; that force must have been something more

FIG. 71.

than the weight of A, which was produced when the motion
was checked. A was not stopped completely; it only lost
some of its velocity, but it could not lose any velocity with-
out being acted upon by a force. This force must have been

=54 EXPERIMENTAL MECHANICS. [LECT.

applied by the rope by which A was held back, and the
tension thus arising was sufficient to pull up B.

531. Let us remove the 14 Ib. weight from B, and attach
there a weight of 28 Ibs., A remaining the same as before
(14 Ibs.). I raise A to the pulley; I allow it to fall. You
observe that B, though double the weight of A, is again
chucked up after the rope has become tight. We can only
explain this by the supposition that the tension in the rope
exerted in checking the motion of A is at least 28 Ibs.

532. Finally, let us remove the 28 Ibs. from B, put on
56 Ibs., and perform the experiment again ; you see that even
the 56 Ibs. is raised up several inches. Here a tension in the
rope has been generated sufficient to overcome a weight four
times as heavy as A. We have then, by the help of inertia,
been able to produce a mechanical power, for a small force
has overcome a greater.

533. After B is raised by the chuck to a certain height it
descends again, if heavier than A, and raises A. The height
to which B is raised is of course the same as the height
through which A descends. You noticed that the height
through which 28 Ibs. was raised was considerably greater than
that through which the 56 Ibs. was raised. Hence we may
draw the inference, that when A was deprived of its velocity
while passing through a short space, it required to be opposed
by a greater force than when it was gradually deprived of its
velocity through a longer space. This is a most important
point. Supposing I were to put a hundredweight at B, I
have little doubt, if the rope were strong enough to bear the
strain, that though A only weighs 14 Ibs., B would yet be
raised a little : here A would be deprived of its motion in a
very short space, but the force required to arrest it would
be very great.

534. It is clear that matters would not be much altered

xvi.] THE HAMMER. 255

if A were to be stopped by some force, exerted from below
rather than above; in fact, we may conceive the rope
omitted, and suppose A to be a hammer-head falling upon a
nail in a piece of wood. The blow would force the nail to
penetrate a small distance, and the entire velocity of A
would have to be destroyed while moving through that small
distance : consequently the force between the head of the
nail and the hammer would be a very large one. This
explains the effect of a blow.

535. In the case that we have supposed, the weight merely
drops upon the nail : this is actually the principle of the
hammer used in pile-driving machines. A pile is a large
piece of timber, pointed and shod with iron at one end : this
end is driven down into the ground. Piles are required for
various purposes in engineering operations. They are often
intended to support the foundations of buildings ; they are
therefore driven until the resistance with which the ground
opposes their further entrance affords a guarantee that they
shall be able to bear what is required.

536. The machine for driving piles consists essentially of
a heavy mass of iron, which is raised to a height, and
allowed to fall upon the pile. The resistance to be over-
come depends upon the depth and nature of the soil : a
pile may be driven two or three inches with each blow, but
the less the distance the pile enters each time, the greater is
the actual pressure with which the blow forces it downwards.
In the ordinary hammer, the power of the arm imparts
velocity to the hammer-head, in addition to that which is
due to the fall ; the effect produced is merely the same as if
the hammer had fallen from a greater height.

537. Another point may be mentioned here. A nail will
only enter a piece of wood when the nail and the wood are
pressed together with sufficient force. The nail is urged by

256 EXPERIMENTAL MECHANICS. [LECT.

the hammer. If the wood be lying on the ground, the re-
action of the ground prevents the wood from getting away
and the nail will enter. In other cases the element of
time is all-important. If the wood be massive less fcrce
will make the nail penetrate than would suffice to move
the wood quickly enough. If the wood be thin and un-
supported, less force may be required to make it yield than
to make the nail penetrate. The usual remedy is obvious.
Hold a heavy mass close at the back of the wood. The
nail will then enter because the augmented mass cannot
now escape as rapidly as before.

THE STORING OF ENERGY.

538. Our study of the subject will be facilitated by some
considerations founded on the principles of energy. In the
experiment of Fig. 71 let Abe 14 Ibs., and B, on the ground,
be 56 Ibs. Since the rope is 15' long, A is 3' from the
ground, and therefore 6' from the pulley. I raise A to the
pulley, and, in doing so, expend 6 x 14 = 84 units of energy.
Energy is never lost, and therefore I shall expect to recover
this amount. I allow A to fall ; when it has fallen 6', it is
then precisely in the same condition as it was before being
raised, except that it has a considerable velocity of descent.
In fact, the 84 units of energy have been expended in giving
velocity to A. B is then lifted to a maximum height x,
in which 56 x x units of energy have been consumed.
At the instant when B is at the summit x, A must be at a
distance of 6 + x feet from the pulley ; hence the quantity
of work performed by A is 14 x (6 + x). But the work
done by A must be equal to that done upon B, and therefore

i4(6 + x) = 56x,

whence x = 2. If there were no loss by friction, B would there-
fore be raised 2'; but owing to friction,and doubtless also.to the

THE STORING OF ENERGY. 257

imperfect flexibility of the rope, the efifect is not so great.
We may regard the work done in raising A as so much
energy stored up, and when A is allowed to fall, the energy
is reproduced in a modified form.

539. Let us apply the principle of energy to the pile-
driving engine to which we have referred (Art. 536) ; we shall
then be able to find the magnitude of the force developed
in producing the blow. Suppose the " monkey," that is the
heavy hammer head, weighs 560 Ibs. (a quarter of a ton).
A couple of men raise this by means of a small winch to a
height of 15'. It takes them a few minutes to do so; their
energy is then saved up, and they have accumulated a store
of 560 x 15=8,400 units. When the monkey falls upon the
top of the pile it transfers thereto nearly the whole of the
8,400 units of energy, and this is expended in forcing the pile
into the ground. Suppose the pile to enter one inch, the re-
action of the pile upon the monkey must be so great that
the number of units of energy consumed in one inch is 8,400.
Hence this reaction must be 8,400 x 12 = 100,800 Ibs. If
the reaction did not reach this amount, the monkey could
not be brought to rest in so short a distance. The re-
action of the pile upon the monkey, and therefore the
action of the monkey upon the pile, is about 45 tons. This
is the actual pressure exerted.

546. If the soil which the pile is penetrating be more
resisting than that which we have supposed, — for example,
if the pile require a direct pressure of 100 tons to force it
in, — the same monkey with the same fall would still be
sufficient, but the pile would not be driven so far with
each blow. The pressure required is 2 24,000 Ibs. : this
exerted over a space of o'"45 would be 8,400 units of energy ;
hence the pile would be driven o'"45. The more the re-
sistance, the less the penetration produced by each blow.

258 EXPERIMENTAL MECHANICS. [LECT.

A pile intended to bear a very heavy load permanently
must be driven until it enters but little with each blow.

541. We may compare the pile-driver with the mechanical
powers in one respect, and contrast it in another. In each,
we have machines which receive energy and restore it
modified into a greater power exerted through a smaller
distance ; but while the mechanical powers restore the
energy at one end of the machine, simultaneously with
their reception of it at the other, the pile-driver is a
reservoir for keeping energy which will restore it in the
form wanted.

542. We have, then, a class of mechanical powers, of
which a hammer may be taken as the type, which depend
upon the storage of energy ; the power of the arm is accumu-
lated in the hammer throughout its descent, to be instantly
transferred to the nail in the blow. Inertia is the property
of matter which qualifies it for this purpose. Energy is
developed by the explosion of gunpowder in a cannon.
This energy is transferred to the ball, from which it is again
in large part passed on to do work against the object which
is struck. Here we see energy stored in a rapidly moving
body, a case to which we shall presently return.

543. But energy can be stored in many ways; we might
almost say that gunpowder is itself energy in a compact and
storable form. The efforts which we make in forcing air into
an air-cane are preserved as energy there stored to be re-
produced in the discharge of a number of bullets. During
the few seconds occupied in winding a watch, a small
charge of energy is given to the spring which it expends
economically over the next twenty-four hours. In using a
bow my energy is stored up from the moment I begin to pull
the string until I release the arrow.

544. Many machines in extensive use depend upon these

XVI.] THE STORING OF ENERGY. 259

principles. In the clock or watch the demand for energy to
sustain the motion is constant, while the supply is only occa-
sional; in other cases the supply is constant, while the demand
is only intermittent. We may mention an illustration of the
latter. Suppose it be required occasionally to hoist heavy
weights rapidly up to a height. If an engine sufficiently power-
ful to raise the weights be employed, the engine will be idle
except when the weights are being raised ; and if the machinery
were to have much idle time, the waste of fuel in keeping
up the fire during the intervals would often make the
arrangement uneconomical. It would be a far better plan to
have a smaller engine ; and even though this were not
able to raise the weight directly with sufficient speed, yet
by keeping the engine continually working and storing up
its energy, we might produce enough in the twenty-four
hours to raise all the weights which it would be necessary
to lift in the same time.

545. Let us suppose we want to raise slates from the
bottom of a quarry to the surface. A large pulley is
mounted at the top of the quarry, and over this a rope
is passed : to each end of the rope a bucket is attached,
so that when one of these is at the bottom the other is at
the top, and their sizes and .that of the pulley are so ar-
ranged that they pass each other with safety. A reservoir
is established at the top of the quarry on a level with the
pulley, and an engine is set to work constantly pumping up
water from the bottom of the quarry into the reservoir.
Each of the buckets is partly composed of a large tank,
which can be quickly filled or emptied. The lower bucket
is loaded with slates, and when ready for work, the man at
the top fills the tank of the upper bucket with water : this
accordingly becomes so heavy that it descends and raises
the slates. When the heavier one reaches the bottom, the

S 2

260 EXPERIMENTAL MECHANICS. [LECT.

water from its tank is let out into the lower reservoir, from
which the engine pumps, and the slates are removed from
the bucket which has been raised. All is then ready
for a repetition of the same operation. If the slates be
raised at intervals of ten minutes, the energy of the engine
will be sufficient when in ten minutes' work it can pump up
enough water to fill one tank ; by the aid of this contrivance
we are therefore able to accumulate for one effort the whole
power of the engine for ten minutes.

THE FLY-WHEEL.

546. One of the best means of storing energy is by setting
a heavy body in rapid motion. This has already been re-
ferred to in the case of the cannon-ball. In order to render
this method practically available for the purposes of ma-
chinery, the heavy body we use is a fly-wheel, and the
rapid motion imparted to it is that of rotation about its axis.
A very large amount of energy can by this means be stored
in a manageable form.

547. We shall illustrate the principle by the apparatus of
Fig. 72. This represents an iron fly-wheel B : its diameter
is 1 8", and its weight is 26 Ibs. ; the fly is carried upon a shaft
(A) of wrought iron f" in diameter. We shall store up a
quantity of energy in this wheel, by setting it in rapid motion,
and then we shall see how we can recover from it the energy
we have imparted.

548. A rope is coiled around the shaft ; by pulling this
rope the wheel is made to turn round : thus the rope is the
medium by which my energy shall be imparted to the wheel.
To measure the operation accurately, I attach the rope to
the hook of the spring balance (Fig. 9) ; and by taking the
ring of the balance in my hand, I learn from the index the
amount of the force I am exerting. I find that when I walk

XVI.]

THE FLY-WHEEL.

261

backwards as quickly as is convenient, pulling the rope all
the time, the scale shows a tension of about 50 Ibs. To set
the wheel rapidly in motion, I pull about 20' of rope from
the axle, so that I have imparted to the wheel somewhere
about 50 x 20 = 1,000 units of energy. The rope is
fastened to the shaft, so that, after it has been all unwound,
the wheel now rapidly rotating winds it in. By measuring
the time in which the wheel made a certain number of coils
of the rope around the shaft, I find that it makes about 600
revolutions per minute.

549. Let us see how the stored-up energy can be exhibited.
A piece of pine 24" x i" X i" of which both ends are supported,
requires a force of 300 Ibs. applied to its centre to produce
fracture (See p. 190). I arrange such a piece of pine near
the wheel. As the shaft is winding in the rope, a tremendous
chuck would be given to anything which tried to stop the

262 EXPERIMENTAL MECHANICS. [LECT.

motion. If I tied the end of the rope to the piece of pine, the
chuck would break the rope ; therefore I have fastened one
end of a 10' length of chain to the rope, and the other has
been tied round the middle of the wooden bar. The wheel
first winds in the rope, then the chain takes a few turns
before it tightens, and crack goes the rod of pine. The
wheel had no choice ; it must either stop or break the
rod : but nature forbids it to be stopped, unless by a great
force, which the rod was not strong enough to apply.
Here I never exerted a force greater than 50 Ibs. in setting
the wheel in motion. The wheel stored up and modified
my energy so as to produce a force of 300 Ibs., which
had, however, only to be exerted over a very small distance.

550. But we may show the experiment in another way,
which is that represented in the figure (72). We see the
chain is there attached to two 56 Ib. weights. The mode of
proceeding is that already described. The rope is first
wound round the shaft, then by pulling the rope the wheel
is made to revolve ; the wheel then begins to wind in the
rope again, and when the chain tightens the two 56
Ibs. are raised up to a height of 3 or 4 feet. Here,
again, the energy has been stored and recovered. But though
the fly-wheel will thus preserve energy, it does so at some
cost : the store is continually being frittered away by friction
and the resistance of the air ; in fact, the energy would
altogether disappear in a little time, and the wheel would
come to rest ; it is therefore economical to make the wheel
yield up what it has received as soon as possible.

551. These principles are illustrated by the function of the
fly-wheel in a steam-engine. The pressure of the steam upon
the piston varies according to the different parts of the stroke :
and the fly-wheel obviates the inconvenience which would arise
from such irregularity. Its great inertia makes its velocity but

xvi.] THE PUNCHING MACHINE. 263

little augmented by the exuberant action of the piston when
the pressure is greatest, while it also sustains the motion
when the piston is giving no assistance. The fly-wheel is a
vast reservoir into which the engine pours its energy, sudden
floods alternating with droughts; but these succeed each
other so rapidly, and the area of the reservoir is so vast, that
its level remains sensibly uniform, and the supplies sent out
to the consumers are regular and unvaried. The consumers
of the energy stored in the fly-wheel of an engine are the
machines in the mill ; they are supplied by shafts which
traverse the building, conveying, by their rotation, the energy
originally condensed within the coal from which combustion
has set it free.

THE PUNCHING MACHINE.

552. When energy has been stored in a fly-wheel, it can
be withdrawn either as a small force acting over a great
distance, or as a large force over a small distance. In the
latter case the fly-wheel acts as a mechanical power, and in
this form it is used in the very important machine to be next
described. A model of the punching machine is shown in

Fig- 73-

The punching machine is usually worked by a steam-
engine, a handle will move our small model. The handle
turns a shaft on which the fly-wheel F is mounted. On the
shaft is a small pinion D of 40 teeth : this works into a large
wheel E of 200 teeth, so that, when the fly and the pinion
have turned round 5 times, E will have turned round once,
c is a circular piece of wood called a cam, which has a hole
bored through it, between 'the centre and circumference : by
means of this hole, the cam is mounted on the same axle as
E, to which it is rigidly fastened, so that the two must revolve
together. A is a lever of the first order, whose fulcrum is at

EXPERIMENTAL MECHANICS.

[LECT.

A : the remote end of this lever rests upon the cam c ; the
other end B contains the punch. As the wheel E revolves it
carries with it the cam : this raises the lever and forces the
punch down a hole in a die, into which it fits exactly. The
metal to be operated on is placed under the punch
before it is depressed by the cam, and the pressure drives
the punch through, cutting out a cylindrical piece of metal
from the plate : this model will, as you see, punch through
ordinary tin.

553. Let us examine the mode of action. The machine
being made to rotate rapidly, the punch is depressed once
for every 5 revolutions of the fly ; the resistance which the
metal opposes to being punched is no doubt very great, but
the lever acts at a twelve-fold advantage. When the
punch comes down on the surface of the metal, one of three
things must happen : either the motion must stop suddenly,
or the machine must be strained and injured, or the metal
must be punched. But the motion cannot be stopped
suddenly, becai'se, before this could happen, an infinite force

xvi.] THE PUNCHING MACHINE. 265

would be developed, which must make something yield. If
therefore we make the structure sufficiently massive to
prevent yielding, the metal must be punched. Such
machines are necessarily built strong enough to make the
punching of the metal easier than breaking the machine.

554. We shall be able to calculate, from what we have
already seen in Art. 248, the magnitude of the force re-
quired for punching. We there learned that about 22-5 tons
of pressure was necessary to shear a bar of iron one square
inch in section. Punching is so far analogous to shearing
that in each case a certain area of surface has to be cut ;
the area in punching is measured by the cylinder of iron
which is removed.

555. Suppose it be required to punch a hole o"'5 in
diameter through a plate o"-8 thick, the area of iron that
has to be cut across is -2T3- x £ X | • = 1*26 square inches:
and as 22^5 tons per square inch are required for shearing,
this hole will require 22*5 x i'26 = 28-4 tons. A force of this
amount must therefore be exerted upon the punch : which
will require from the cam a force of more than 2 tons
upon its end of the lever. Though the iron must be pierced
to a depth of o"'8, yet it is obvious that almost immediately
after the punch has penetrated the surface of the iron,
the cylinder must be entirely cut and begin to emerge from
the other side of the plate. We shall certainly be correct
in supposing that the punching is completed before the punch
has penetrated to a depth of o"'2, and that for not more than
this distance has the great pressure of 28 tons been exerted ;
for a small pressure is afterwards sufficient to overcome the
friction which opposes the motion of the cylinder of iron.
Hence, though so great a pressure has been required, yet
the number of units of energy consumed is not very large ;
it is -$\; x 2240 x 28-4= 1062.

266 EXPERIMENTAL MECHANICS. [LECT. xvi.

The energy actually required to punch a hole of half
an inch diameter through a plate eight-tenths of an inch
thick is therefore less than that which would be expended
in raising i cwt. to a height of ten feet.

556. The fly-wheel may be likened to the reservoir in
Art 545. The time that is actually occupied in the
punching is extremely small, and the sudden expenditure
of 1062 units is gradually reimbursed by the engine. If
the rotating fly-wheel contain 50,000 units of energy, the
abstraction of 1362 units will not perceptibly affect its velo-
city. There is therefore an advantage in having a heavy fly
sustained at a high speed for the working of a punching
machine.

LECTURE XVII.
CIRCULAR MOTION.

The Nature of Circular Motion. — Circular -.notion in Liquids. — The
Applications of Circular Motion. — The Permanent Axes.

THE NATURE OF CIRCULAR MOTION.

557. To compel a body to swerve from motion in a straight
line force must be exercised. In this chapter we shall
study the comparatively simple case of a body revolving in a
circle.

558. When a body moves round uniformly in a circle
force must be continuously applied, and the first question
for us to examine is, as to the direction of that force. We
have to demonstrate the important fact, that it constantly
tends towards the centre.

559. The direction of the force can be exhibited by
actual experiment, and its magnitude will be at the same
time clearly indicated by the extent to which a spring
is stretched. The apparatus we use is shown in Fig. 74.

The essential parts of the machine consist of two balls

268

EXPERIMENTAL MECHANICS.

A, B, each 2" in diameter : these are thin hollow spheres of
silvered brass. The balls are supported on arms p A, Q B,
which are attached to a piece of wood, p Q, capable of
turning on a socket at c. The arm A P is rigidly fixed
to P Q ; the other arm, B Q, is capable of turning round
a pin at Q. An india-rubber door-spring is shown at F ;

FIG. 74.

one end of this is secured to P Q, the other end to the
movable arm, Q B. If the arm Q B be turned so as to
move B away from c, the spring F must be stretched.

A small toothed wheel is mounted on the same socket
as c ; this is behind P o, and is therefore not seen in the

xvil.] NATURE OF CIRCULAR MOTION. 269

figure : the whole is made to revolve rapidly by the large
wheel E, which is turned by the handle D.

560. The room being darkened, a beam from the lime-
light is allowed to fall on the apparatus : the reflections of
the light are seen in the two silvered balls as two bright
points. When D is turned, the balls move round rapidly,
and you see the points of light reflected from them describe
circles. The ball B when at rest is 4" from c, while A is
8" from c ; hence the circle described by B is smaller than
that described by A. The appearance presented is that of
two concentric luminous circles. As the speed increases,
the inner circle enlarges till the two circles blend into one.
By increasing the speed still more, you see the circle whose
diameter is enlarging actually exceeding the fixed circle, and
its size continues to increase until the highest velocity which
it is safe to employ has been communicated to the machine.

561. What is the explanation of this? The arm A is
fixed and the distance A c cannot alter, hence A describes
the fixed circle. B, on the other hand, is not fixed ; it
can recede from c, and we find that the quicker the speed
the further it recedes. The larger the circle described by
B the more is the spring stretched, and the greater is the
force with which B is attracted towards the centre. This
experiment proves that the force necessary to retain a body
in a circular path must be increased when the speed is
increased.

562. Thus we see that uniform motion of a body in a
circle can only be produced by an uniform force directed to
the centre.

If the motion, even though circular, have variable speed
the law of the force is not so simple.

563. We can measure the magnitude of this force by the
same apparatus. The ball B weighs o'i Ib. I find that I

270

EXPERIMENTAL MECHANICS

LECT.

must pull it with a force of 3 Ibs. in order to draw it
to a distance of 8" from c ; that is, to the same distance
as A is from c. Hence, when the diameters of the circles
in which the balls move are equal, the central force must
be 3 Ibs. ; that is, it must be nearly thirty times as great
as gravity.

c

FIG. 75.

564. The necessity for the central force is thus shown :
Let us conceive a weight attached to a string to be swung
round in a circle, a portion of which is shown in Fig. 75.

Suppose the weight be at s and moving towards p, and
let a tangent to the circle be drawn at P. Take two points

XVII.] ACTION OF CIRCULAR MOTION. 271

on the circle, A and B, very near p ; the small arc A B does
not differ perceptibly from the part A B on the tangent line ;
hence, when the particle arrives at A, it is a matter of
indifference whether it travels in the arc A B, or along the
line A B. Let us suppose it to move along the line. By
the first law of motion, a particle moving in the line A B
would continue to do so ; hence, if the particle be allowed,
it will move on to Q : but the particle is not allowed to move
to Q ; it is found at R. Hence it must have been withdrawn
by some force.

565. This force is supplied by the string to which the
weight is attached. The incessant change from the recti-
linear motion of the weight requires a constantly applied
force, and this is always directed to the centre. Should the
string be released, the body flies off in the direction of the
tangent to the circle at the point which the body occupied at
the instant of release.

566. The central force increases in proportion to the
square of the velocity. If I double the speed with which
the weight is whirled round in the circle, I quadruple the
force which the string must exert on the body. If the speed
be trebled, the force is increased ninefold, and so on.
When the speeds with which two equal masses are re-
volving in two circles are equal, the central force in the
smafler circle is greater than that of the larger circle, in the
proportion of the radius of the larger circle to that of the
smaller.

THE ACTION OF CIRCULAR MOTION UPON
LIQUIDS.

567. I have here a small bucket nearly filled with water :
to the handle a piece of string is attached. If I whirl the

272

EXPERIMENTAL MECHANICS.

bucket round in a vertical plane sufficiently fast, you see no
water escapes, although the bucket is turned upside down
once in every revolution. This is because the water has
not time to fall out during such a brief interval. A body
would not fall half an inch from rest in the twentieth of a
second.

568. The action of circular motion upon liquids is
illustrated by the experiment which is represented in
Fig. 76.

A glass beaker about half full of water is mounted so that
it can be spun round rapidly. The motion is given by
means of a large wheel turned by a handle, as shown in
the figure. When the rotation commences, the water is
seen to rise up against the glass sides and form a hollow in
the centre,

569. In order to demonstrate this clearly, I turn upon
the vessel a beam from the lime-light. I have previously

ACTION OF CIRCULAR MOTION. 273

dissolved a little quinine in the water. The light from the
lamp is transmitted through a piece of dense blue glass.
When the light thus coloured falls on the water, the
presence of the quinine makes the entire liquid glow with a
bluish hue. This remarkable property of quinine, which is
known as fluorescence, enables you to see distinctly the
hollow form caused by the rotation.

570. You observe that as' the speed becomes greater the
depth of the hollow increases, and that if I turn the wheel
sufficiently fast the water is actually driven out of the glass.
The shape of the curve which the water assumes is that
which would be produced by the revolution of a parabola
about its axis.

571. The explanation is simple. As soon as the glass
begins to revolve, the friction of its sides speedily imparts
a revolving motion to the water ; but in this case there is
nothing to keep the particles near the centre like the string
in the revolving weight, so the liquid rises at the sides of
the glass.

572. But you may ask why all the particles of the water
should not go to the circumference, and thus line the in-
side of the glass with a hollow cylinder of water instead
of the parabola. Such an arrangement could not exist
in a liquid acted on by gravity. The lower parts of the
cylinder must bear the pressure of the water above, and
therefore have more tendency to flatten out than the
upper portions. This tendency could not be overcome by
any consequences of the movement, for such must be alike
on all parts at the same distance from the axis.

573. A very beautiful experiment was devised by
Plateau for the purpose of studying the revolution of a
liquid removed from the action of gravity.

The apparatus employed is represented in Fig. 77. A

T

274

EXPERIMENTAL MECHANICS.

[LECT.

glass vessel 9" cube is filled with a mixture of alcohol and
water. The relative quantities ought to be so proportioned
that the fluid has the same specific gravity as olive oil,
which is heavier than alcohol and lighter than water. In
practice, however, it is found so difficult to adjust the com-
position exactly that the best plan is to make two alcoholic
mixtures so that olive oil will just float on one of them, and just

sink in the other. The lower half of the glass is to be
filled with the denser mixture, and the upper half with the
lighter. If, then, the oil be carefully introduced with a
funnel it will form a beautiful sphere in the middle of the
vessel, as shown in the figure. We thus see that a liquid
mass freed from the action of terrestrial gravity, forms a
sphere by the mutual attraction of its particles.

xvn.] ACTION OF CIRCULAR MOTION. 275

Through the liquid a vertical spindle passes. On this
there is a small disk at the middle of its length, about which
the sphere of oil arranges itself symmetrically. To the end
of the spindle a handle is attached. When the handle is
turned round slowly, the friction of the disk and spindle
communicates a motion of rotation to the sphere of oil.
We have thus a liquid spheroidal mass endowed with a
movement of rotation ; and we can study the effect of
the motion upon its form. We first see the sphere
flatten down at its poles, and bulge at its equator. In
order to show the phenomenon to those who may not be
near to the table, the sphere can be projected on the screen
by the help of the lime-light lamp and a lens. It first
appears as a yellow circle, and then, as the rotation begins,
the circle gradually transforms into an ellipse. But a very
remarkable modification takes place when the handle is
turned somewhat rapidly. The ellipsoid gradually flattens
down until, when a certain velocity has been attained, the
surface actually becomes indented at the poles, and flies from
the axis altogether. Ultimately the liquid assumes the form
of a beautiful ring, and the appearance
on the screen is shown in Fig. 78.

574. The explanation of the de-
velopment of the ring involves some
additional principles : as the sphere
of oil spins round in the liquid, its
surface is retarded by friction ; so that

when the velocity attains a certain FlG 8

amount, the internal portions of the
sphere, which are in the neighbourhood of the spindle,
are driven from the centre into the outer portions, but
the full account of the phenomenon cannot be given here.

575. The earth was, we believe, originally in a fluid

T 2

276 EXPERIMENTAL MECHANICS. [LECT.

condition. It had then, as it has now, a diurnal rotation,
and one of the consequences of this rotation has been to
cause the form to be slightly protuberant at the equator,
just as we have seen the sphere of oil to bulge out under
similar conditions.

576. Bodies lying on the earth are whirled around in a
great circle every day. Hence, if there were not some force
drawing them to the centre, they would fly off at a tangent.
A part of the earth's attraction goes for this purpose, and the
remainder, which is the apparent weight, is thus diminished'
by a quantity increasing from the pole to the equator (Art.
86).

THE APPLICATIONS OF CIRCULAR MOTION.

577. These principles have many applications in the
mechanical arts; we shall mention two of them. The
first is to the governor-balls of a steam-engine ; the second
is to the process of sugar-refining.

An engine which turns a number of machines in a factor}'
should work uniformly. Irregularities of motion may be
productive of loss and various inconveniences. An engine
would work irregularly either from variation in the produc-
tion of steam, or from the demands upon the power being
lessened or increased. Even if the first of these sources of
irregularity could be avoided by care, it is clear that the
second could not. Some machines in the mill are occa-
sionally stopped, others occasionally set in motion, and the
engine generally tends to go faster the less it has to do. It
is therefore necessary to provide means by which the speed
shall be restrained within narrow limits, and it is obviously
desirable that the contrivance used for this purpose should
be self-acting. We must, therefore, have some arrange-
ment which shall admit more steam to the cylinder when

xvn.] APPLICATIONS OF CIRCULAR MOTION. 277

the engine is moving too slowly, and less steam when it is
moving too quickly. The valve which is to regulate this
must, be worked by some agent which depends upon
the velocity of the engine ; this at once points to circular
motion because the force acting on the revolving body de-
pends upon its velocity. Such was the train of reasoning
which led to the happy invention of the govern or -balls :
these are shown in Fig. 79.

A B is a vertical spindle which is turned by the engine.
PP is a piece firmly attached to the spindle and turning
with it. P w, p w are arms terminating in weights w w ;
these are balls of iron, generally very massive : the arms
are free to turn round pins at
PP. At Q Q links are placed,
attached to another piece R R,
which is free to slide up and
down the spindle. When A B
rotates, w and w are carried
round, and therefore fly out-
wards from the spindle; to
do this they must evidently
pull the piece R R up the
shaft. We can easily imagine
an arrangement by which R R
shall be made to shut or open
the steam-valve according as
it ascends or descends. The
problem is then solved, for
if the engine begin to go
too rapidly, the balls fly out
further just as they did in
Fig. 74 : this movement raises

FlG- 79-

the piece RR, which diminishes the supply of steam,

2;8 EXPERIMENTAL MECHANICS. [LECT.

and consequently checks the speed. On the other hand,
when the engine works too slowly, the balls fall in towards
the spindle, the piece R R descends, the valve is opened,
and a greater supply of steam is admitted. The objection
to this governor is that though it moderates, it does not
completely check irregularity. There are other governors
occasionally employed which depend also on circular
motion ; some of these are more sensitive than the governor-
balls ; but they are elaborate machines, only to be employed
under exceptional circumstances.

578. The application of circular motion to sugar-refining
is a very beautiful invention. To explain it I must briefly
describe the process of refining.

The raw sugar is dissolved in water, and the solution is
purified by straining and by filtration through animal char-
coal. The syrup is then boiled. In order to preserve the
colour of the sugar, and to prevent loss, this boiling is
conducted in vacua, as by this means the temperature
required is much less than would be necessary with the
ordinary atmospheric pressure.

The evaporation having been completed, crystals of
sugar form throughout the mass of syrup. To separate
these crystals from the liquor which surrounds them, the aid
of circular motion force is called in. A mass of the mixture
is placed in a large iron tub, the sides of which are per-
forated with small holes. The tub is then made to rotate
with prodigious velocity ; its contents instantly fly off to the
circumference, the liquid portions find an exit through the
perforations in the sides, but the crystals are left behind.
A little clear syrup is then sprinkled over the sugar while
still rotating : this washes from the crystals the last traces
of the coloured liquid, and passes out through the holes ;
when the motion ceases, the inside of the tub contains a

XVIL] THE PERMANENT AXES. 279

layer of perfectly pure white sugar, some inches in thickness,
ready for the market.

579. Circular motion is peculiarly fitted for this pur-
pose ; each particle of liquid strives to get as far away
from the axis as possible. The action on the sugar
is very different from what it would have been had the
mass been subjected to pressure by a screw-press or similar
contrivance ; the particles immediately acted on would then
have to transmit the pressure to those within ; and the
consequence would be that while the crystals of sugar on
the outside would be crushed and destroyed, the water
would only be very imperfectly driven from the interior :
for it could lurk in the interstices of the sugar, which
remain notwithstanding the pressure.

580. But with circular motion the water must go,
not because it is pushed by the crystals, but because of
its own inertia ; and it can be perfectly expelled by a
velocity of rotation less than that which would be
necessary to produce sufficient pressure to make the
crystals injure each other.

THE PERMANENT AXES.

581. There are some curious properties of circular
motion which remain to be considered. These we shall
investigate by means of the apparatus of Fig. 80. This
consists of a pair of wheels B c, by which a considerable
velocity can be given to a horizontal shaft. This shaft
is connected by a pair of bevelled wheels D with a vertical
spindle F. The machine is worked by a handle A, and
the object to be experimented upon is suspended from
the spindle.

582. I first take a disk of wood 18" in diameter; a hole
is bored in the margin of this disk ; through this hole a

280

EXPERIMENTAL MECHANICS.

[LECT,

rope is fastened, by means of which the disk is suspended
from the spindle. The disk hangs of course in a vertical
plane.

583. I now begin to turn the handle round gently,

and you see the disk begins to rotate about the vertical
diameter ; but, as the speed increases, the motion becomes
a little unsteady ; and finally, when I turn the handle very
rapidly, the disk springs up into a horizontal plane, and

XVII.]

THE PERMANENT AXES.

281

you see it like the surface of a small table : the rope
swings round and round in a cone, so rapidly that it is
hardly seen.

584. We may repeat the experiment in a different manner.
I take a piece of iron chain about 2 long, G ; I pass the
rope through the two last links of its extremities, and
suspend the rope from the spindle. When I commence
to turn the handle, you see the chain gradually opens out

into a loop H ; and as the speed increases, the loop
becomes a complete ring. Still increasing the
speed, I find the ring becomes unsteady, till finally it
rises into a horizontal plane. The ring of chain in the
horizontal plane is shown at I. When the motion is
further increased, the ring swings about violently, and so
I cease turning the handle.

585. The principles already enunciated will explain these

282 EXPERIMENTAL MECHANICS. [LECT.

remarkable results ; we shall only describe that of the chain,
as the same explanation will include that of the disk of
wood. We shall begin with the chain hanging vertically
from the spindle : the moment rotation commences, the
chain begins to spin about a vertical axis ; the parts of the
chain fly outwards from this axis just as the ball flies out-
wards in Fig. 74; this is the cause of the looped form H
which the chain assumes. As the speed is increased the
loop gradually opens more and more, just as the diameter
of the circle Fig. 74 increases with the velocity. But
we have also to inquire into the cause of the remarkable
change of position which the ring undergoes ; instead of
continuing to rotate about a vertical diameter, it comes into
a horizontal plane. This will be easily understood with the
help of Fig. 81. Let OP represent the rope attached to
the ring, and o c be the vertical axis. Suppose the ring
to be spinning about the axis o c, when o c was a diameter ;
if then, from any cause, the ring be slightly displaced, we
can show that the circular motion will tend to drive the ring
further from the vertical plane, and force it into the horizontal
plane. Let the ring be in the position represented in the
figure ; then, since it revolves about the vertical line o c, the
tendency of p and Q is to move outwards in the directions
of the arrows, thus evidently tending to bring the ring into
the horizontal plane.

586. In Art. 103 we have explained what is meant by
stable and unstable equilibrium ; we have here found a
precisely analogous phenomenon in motion. The rotation
of the ring about its diameter is unstable, for the minutest
deviation of the ring from this position is fatal ; circumstances
immediately combine to augment the deviation more and
more, until finally the ring is raised into the horizonal
plane. Once in the horizontal plane, the motion there is

xvir.] THE PERMANENT AXES. 283

stable, for if the ring be displaced the tendency is to restore
it to the horizontal.

587. The ring, when in a horizontal plane, rotates per-
manently about the vertical axis through its centre ; this
axis is called permanent, to distinguish it from all other
directions, as being the only axis about which the motion is
stable.

588. We may show another experiment with the chain : if
instead of passing the rope through the links at its ends, I
pass the rope through the centre of the chain, and allow the
ends of the chain to hang downwards. I now turn the
handle ; instantly the parts of the chain fly outwards in a
curved form ; and by increasing the velocity, the parts of the
chain at length come to lie almost in a straight line.

LECTURE XVIII.
THE SIMPLE PENDULUM.

Introduction. — The Circular Pendulum. — Law connecting the Time
of Vibration with the Length. — The Force of Gravity determined
by the Pendulum.— The Cycloid.

INTRODUCTION.

589. IF a weight be attached to a piece of string, the
other end of which hangs from a fixed point, we have what
is called a simple pendulum. The pendulum is of the
utmost importance in science, as well as for its practical
applications as a time-keeper. In this lecture and the next
we shall treat of its general properties ; and the last will be
devoted to the practical applications. We shall commence
with the simple pendulum, as already defined, and prove, by
experiment, the remarkable property which was discovered
by Galileo. The simple pendulum is often called the
circular pendulum.

THE CIRCULAR PENDULUM.

590. We first experiment with a pendulum on a large
scale. Our lecture theatre is 32 feet high, and there is a

LECT. xvill.] THE CIRCULAR PENDULUM.

2S5

wire suspended from the ceiling 27' long; to the end of this
a ball of cast iron weighing 25 Ibs. is attached. This wire
when at rest hangs vertically in the direction o c (Fig. 82).

I draw the ball from its position of rest to A ; when
released, it slowly returns to c, its original position ; it then
moves on the other side to B, and back again to my hand at
A. The ball — or to speak more
precisely, the centre of the ball —
moves in a circle, the centre being
the point o in the ceiling from
which the wire is suspended.

591. What causes the motion
of the pendulum when the weight
is released ? It is the force of
gravity ; for by moving the ball to
A I raise it a little, and therefore^
when I release it gravity compels
it to return to c it being the only
manner in which the mode of
suspension will allow it to fall.
But when it has reached its
original position at c, why does
it continue its motion ? — for
gravity must be acting against the
ball during the journey from c
to B. The first law of motion

explains this. (Art 485). In travelling from A to c the ball
has acquired a certain velocity, hence it has a tendency
to go on, and only by the time it has arrived at B will
gravity have arrested the velocity, and begin to make it
descend.

592. You see, the ball continues moving to and fro —
oscillating, as it is called — for a long time. The fact is that

FIG. 82.

286 EXPERIMENTAL MECHANICS. [LECT.

it would oscillate for ever, were it not for the resistance of
the air, and for some loss of energy at the point of
suspension.

593. By the time of an oscillation is meant the time of
going from A to B, but not back again. The time of our
long pendulum is nearly three seconds.

594. With reference to the time of oscillation Galileo made
a great discovery. He found that whether the pendulum
were swinging through the arc A B, or whether it had been
brought to the point A', and was thus describing the arc
A' B', the time of oscillation remained nearly the same. The
arc through which the pendulum oscillates is called its
amplitude, so that we may enunciate this truth by
saying that the time of oscillation is nearly independent
of the amplitude. The means by which Galileo proved this
would hardly be adopted in modern days. He allowed a
pendulum to perform a certain number of vibrations, say 100,
through the arc A B, and he counted his pulse during the
time ; he then counted the number of pulsations while the
pendulum vibrated 100 times in the arc A' B', and he found
the number of pulsations in the two cases to be equal.
Assuming, what is probably true, that Galileo's pulse
remained uniform throughout the experiment, this result
showed that the pendulum took the same time to perform
100 vibrations, whether it swung through the arc A B, or
through the arc A'B'. This discovery it was which first
suggested the employment of the pendulum as a means of
keeping time.

595. We shall adopt a different method to show that
the time does not depend upon the amplitude. I have
here an arrangement which is represented in Fig. 83. It
consists of two pendulums AD and B c, each 12' long, and
suspended from two points A B, about i' apart, in the same

xvill.] THE CIRCULAR PENDULUM. 287

FIG. 83.

horizontal line. Each of these pendulums carries a weight of
the same size : they are in fact identical.

288 EXPERIMENTAL MECHANICS. [LECT.

596. I take one of the balls in each hand. If I withdraw
each of them from its position of rest through equal
distances and then release them, both balls return to my.
hands at the same instant. This might have been expected
from the identity of the circumstances.

597. I next withdraw the weight c in my right hand to a
distance of i', and the weight D in my left hand to a distance
of 2', and release them simultaneously. What happens ?
I keep my hands steadily in the same position, and I find
that the two weights return to them at the same instant.
Hence, though one of the weights moved through an
amplitude of 2' (c E) while the other moved through an
amplitude of 4' (D F), the times occupied by each in making
two oscillations are identical. If I draw the right-hand ball
away 3', while I draw the left hand only i' from their
respective positions of rest, I still observe the same result.

598. In two oscillations we can see no effect on the time
produced by the amplitude, and we are correct in saying
that, when the amplitude is only a small fraction of the
length of the pendulum, its effect is inappreciable. But if the
amplitude of one pendulum were very large, we should find
that its time of oscillation is slightly greater than that of the
other, though to detect the difference would require a
delicate test. One consequence of what is here remarked
will be noticed at a later page. (Art 655.)

599. We next inquire whether the weight which is
attached to the pendulum has any influence upon the time
of vibration. Using the 12' pendulums of Fig. 83, I place
a weight of 1 2 Ibs. on one hook and one of 6 Ibs. on the
other. I withdraw one in each hand ; I release them ; they
return to my hand at the same moment. Whether I with-
draw the weights through long arcs or short arcs, equal or
unequal, they invariably return together, and both therefore

XVIIL] TIME OF VIBRATION. 289

have the same time of vibration. With other iron weights
the same law is confirmed, and hence we learn that,
besides being independent of the amplitude, the time of
vibration is also independent of the weight.

600. Finally, let us see if the material of the pendulum
can influence its time of vibration. I place a ball of wood
on one wire and a ball of iron on the other ; I swing them
as before : the vibrations are still performed in equal times.
A ball of lead is found to swing in the same time as a ball of
brass, and both in the same time as a ball of iron or of
wood.

601. In this we may be reminded of the experiments
on gravity (Art. 491), where we showed that all bodies fall
to the ground in equal times, whatever be their sizes or their
materials. From both cases the inference is drawn that the
force of gravity upon different bodies is proportional to
their masses, though the bodies be made of various substan-
ces. It was indeed by means of experiments with the
pendulum that Newton proved that gravity had this property,
which is one of the most remarkable truths in nature.

LAW CONNECTING THE TIME OF VIBRATION
WITH THE LENGTH.

602. We have seen that the time of vibration of a
pendulum depends neither upon its amplitude, material,
nor weight ; we have now to learn on what the time does
depend. It depends upon the length of the pendulum.
The shorter a pendulum the less is its time of vibration.
"We shall find by experiment the relation between the
time and the length of the cord by which the weight is
suspended.

603. I have here (Fig. 84) two pendulums AD, B c, one

u

290

EXPERIMENTAL MECHANICS.

[LECT.

of which is 12' long and the other 3'; they are mounted
side by side, and the weights are at the same distance

from the floor. I take one of the weights in each hand,
and withdraw them to the same distance from the position

XVIIL] TIME OF VIBRATION. 291

of rest. I release the balls simultaneously ; c moves off
rapidly, arrives at the end C' while D has only reached D',
and returns to my hand just as D has completed one oscil-
lation. I do not seize c : it goes off again, only to return
at the same moment when D reaches my hand. Thus c
has performed four oscillations while D has made no more
than two. This proves that when one of two pendulums
is a quarter the length of the other, the time of vibration
of the shorter one is half that of the other.

604. We shall repeat the experiment with another pen-
dulum 27' long, suspended from the ceiling, and compare
its vibrations with those of a pendulum 3' long. I draw
the weights to one side and release them as before ; and you
see that the small pendulum returns twice to my hand
while the long pendulum is still absent ; but that, keeping my
hands steadily in the same place throughout the experiment,
the long pendulum at last returns simultaneously with the
third arrival of the short one. Hence we learn that a
pendulum 27' long takes three times as much time for a
single vibration as a 3' pendulum.

605. The lengths of the three pendulums on which we
have experimented (27', 12', 3'), are in the proportions of the
numbers 9, 4, i ; and the times of the oscillations are pro-
portional to 3, 2, i : hence we learn that the period of oscilla-
tion of a pendulum is proportional to the square root of its
length.

606. But the time of vibration must also depend upon
gravity ; for it is only owing to gravity that the pendulum
vibrates at all. It is evident that, if gravity were increased,
all bodies would fall to the earth more than 16' in the first
second. The effect on the pendulum would be to draw
the ball more quickly from D to D' (Fig. 84), and thus the
time of vibration would be diminished.

u a

2Q2 EXPERIMENTAL MECHANICS. [LECT.

It is found by calculation, and the result is confirmed by
experiment, that the time of vibration is represented by the
expression,

' Length
3 I4I V ;porce of gravity.

607. The accurate value of the force of gravity in London
is 3 2 'i 908, so that the time of vibration of a pendulum
there is 0-5537 ^/ length: the length of the seconds pen-
dulum is 3'-26i6.

THE FORCE OF GRAVITY DETERMINED BY THE
PENDULUM.

608. The pendulum affords the proper means of meas-
uring the force of gravity at any place on the earth. We
have seen that the time of vibration can be expressed in
terms of the length and the force of gravity ; so conversely,
when the length and the time of vibration are known, the
force of gravity can be determined and the expression for
it is —

Length X

609. It is impossible to observe the time of a single
vibration with the necessary degree of accuracy ; but sup-
posing we consider a large number of vibrations, say TOO, and
find the time taken to perform them, we shall then learn the
time of one oscillation by dividing the entire period by 100.
The amplitudes of the oscillations may diminish, but they are
still performed in the same time ; and hence, if we are sure
that we have not made a mistake of more than one second

xvill.] MODE OF FINDING GRAVITY. 293

in the whole time, there cannot be an error of more than
o-oi second, in the time of one oscillation. By taking a still
larger number the time may be determined with the utmost
precision, so that this part of the inquiry presents little
difficulty.

6 10. But the length of the pendulum has also to be ascer-
tained, and this is a rather baffling problem. The ideal
pendulum whose length is required, is supposed to be
composed of a very fine, perfectly flexible cord, at the end
of which a particle without appreciable size is attached ; but
this is very different from the pendulum which we must
employ. We are not sure of the exact position of the point
of suspension, and, even if we had a perfect sphere for the
weight, the distance between its centre and the point of
suspension is not the same thing as the length of the
simple pendulum that would vibrate in the same time.
Owing to these circumstances, the measurement of the pen-
dulum is embarrassed by considerable difficulties, which have
however been overcome by ingenious contrivances to be
described in the next chapter.

6 1 1. But we shall perform, in a very simple way, an ex-
periment for determining the force of gravity. I have
here a silken thread which is fastened by being clamped
between two pieces of wood. A cast-iron ball 2"-54 in
diameter is suspended by this piece of silk. The distance
from the point of suspension of the silk to the ball is 24"'07,
as well as it can be measured.

The length of the ideal pendulum which would vibrate
isochronously with this pendulum is 25'"37, being about
°'"°3 greater than the distance from the point of suspension
to the centre of the sphere.

12. The length having been ascertained, the next element
to be determined is the time of vibration. For this purpose

294 EXPERIMENTAL MECHANICS. [LECT.

I use a stop-watch, which can be started or stopped instan-
taneously by touching a little stud : this watch will indicate
time accurately to one-fifth of a second. It is necessary that
the pendulum should swing in a small arc, as otherwise the
oscillations are not strictly isochronous. Quite sufficient
amplitude is obtained by allowing the ball to rotate to and
fro through a few tenths of an inch.

613. In order to observe the movement easily, I have
mounted a little telescope, through which I can view the
top of the ball. In the eye-piece of the telescope a vertical
wire is fastened, and I count each vibration just as the silk-
en thread passes the vertical wire. Taking my seat with
the stop-watch in my hand, I write down the position of the
hands of the stop-watch, and then look through the telescope.
I see the pendulum slowly moving to and fro, crossing the
vertical wire at every vibration ; on one occasion, just as it
passes the wire, I touch the stud and start the watch. I allow
the pendulum to make 300 vibrations, and as the silk arrives
at the vertical wire for the 3ooth time, I promptly stop the
watch ; on reference I find that 241-6 seconds have elapsed
since the time the watch was started. To avoid error, I
repeat this experiment, with precisely the same result : 241 '6
seconds are again required for the completion of 300
vibrations.

614. It is desirable to reckon the vibrations from the
instant when the pendulum is at the middle of its stroke,
rather than when it arrives at the end of the swing. In the
former case the pendulum is moving with the greatest
rapidity, and therefore the time of coincidence between the
thread and the vertical wire can be observed with especial
definiteness.

615. The time of a single vibration is found, by
dividing 241 '6 by 300, to be 0*805 second. This is certainly

xvill.] THE CYCLOID. 295

correct to within a thousandth part of a second. We
conclude that a pendulum whose length is 25"-37 = 2 '114,
vibrates in 0*805 second ; and from this we find that gravity at

/3-i4i6\ 2
Dublin is 2-114 X ( O.g0-' J = 32fig6. This result agrees

with one which has been determined by measurement
made with every precaution.

Another method of measuring gravity by the pen-
dulum will be described in the next lecture (Art. 637).

THE CYCLOID.

6 1 6. If the amplitude of the vibration of a circular
pendulum bear a large proportion to the radius, the time
of oscillation is slightly greater than if the amplitude be
very small. The isochronism of the pendulum is only true
for small arcs.

617. But there is a curve in which a weight may be
made to move where the time of vibration is precisely the
same, whatever be the amplitude. This curve is called
a cycloid. It is the path described by a nail in the cir-
cumference of a wheel, as the wheel rolls along the ground.
Thus, if a circle be rolled underneath the line A B
(Fig. 85), a point on its circumference describes the
cycloid A D c P B. The lower part of this curve does
not differ very much from a circle whose centre is a certain
point o above the curve.

6 1 8. Suppose we had a piece of wire carefully shaped
to the cycloidal curve A D c P B, and that a ring could slide
along it without friction, it would be found that, whether
the ring be allowed to drop from c, P or B, it would fall
to D precisely in the same time, and would then run
up the wire to a distance from D on the other side

296 EXPERIMENTAL MECHANICS. [LECT.

equal to that from which it had originally started. In the
oscillations on the cycloid, the amplitude is absolutely
without effect upon the time.

619. As a frictionless wire is impossible, we cannot
adopt this method, but we can nevertheless construct
a cycloidal pendulum in another way, by utilizing a property
of the curve, o A (Fig. 85) as a half cycloid; in fact,
o A is just the same curve as B D, but placed in a different
position, so also is OB. If a string of length o D be
suspended from the point o, and have a weight attached
to it, the weight will describe the cycloid, provided that

the string wrap itself along the arcs o A and o B ; thus when
the weight has moved from D to p, the string is wrapped
along the curve through the space o x, the part T P only
being free. This arrangement will always force the
point P to move in the cycloidal arc.

620. We are now in a condition to ascertain experi-
mentally, whether the time of oscillation in the cycloid
is independent of the amplitude. We use for this purpose
the apparatus shown in Fig. 86. D c E is the arc of the
cycloid ; two strings are attached at o, and equal weights
A, B are suspended from them ; c is the lowest point of

XVIII.]

THE CYCLOID.

297

the curve. The time A will take to fall through the arc A c is
of course half the time of its oscillation. If, ' therefore,
I can show that A and B both take the same time to fall

FIG. 86.

down to c, I shall have proved that the vibrations are
isochronous.

621. Holding, as shown in the figure, A in one hand and B
in the other, I release them simultaneously, and you see
the result, — they both meet at c : even if I bring A up to E,

298 EXPERIMENTAL MECHANICS. [LECT. xvm.

and bring B down close to c, the result is the same. The
motion of A is so rapid that it arrives at c just at the same
instant as B. When I bring the two balls on the same side
of c, and release them simultaneously, A overtakes B just
at the moment when it is passing c. Hence, under all
circumstances, the times of descent are equal.

622. It will be noticed that the string attached to the ball
B, in the position shown in the figure, is almost as free as if
it were merely suspended from o, for it is only when the ball
is some distance from the lowest point that the side arcs
produce any appreciable effect in curving the string. The
ball swings from B to c nearly in a circle of which the centre
is at o. Hence, in the circular pendulum, the vibrations when
small are isochronous, for in that case the cycloid and the
circle become indistinguishable.

LECTURE XIX.

THE COMPOUND PENDULUM AND THE COMPOSITION
OF VIBRATIONS.

The Compound Pendulum.— The Centre of Oscillation.— The Centre
of Percussion. — The Conical Pendulum. — The Composition of
Vibrations.

THE COMPOUND PENDULUM.

623. PENDULOUS motion must now be studied in other
forms besides that of the simple pendulum, which consists
of a weight and a cord. Any body rotating about an axis
may be made to oscillate by gravity. A body thus vibrating
is called a compound pendulum. The ideal form, which
consists of an indefinitely small weight attached to a perfectly
flexible and imponderable string, is an abstraction which
can only be approximately imitated in nature. It follows
that every pendulum used in our experiments is strictly
speaking compound.

624. The first pendulum of this class which we shall
notice is that used in the common clock (Fig. 87). This
consists of a wooden or steel rod A E, to which a brass or
leaden bob B is attached. This pendulum is suspended by
means of a steel spring c A, which being very flexible, allows

300 EXPERIMENTAL MECHANICS. [LECT.

the vibration to be performed with considerable freedom. The
use of the screw at the end will be explained in Art. 664. A
pendulum like this vibrates isochronously, when the ampli-
tude is small, but it is not easy to see precisely what is the
length of the simple pendulum which would
oscillate in the same time. In the first place, we
are uncertain as to the virtual position of the point
of suspension, for the spring, though flexible,
will not yield at the point c to the same extent
as a string ; thus the effective point of suspen-
sion must be somewhat lower than c. The other
extremity is still more uncertain, for the weight,
so far from being a single point, is not exclu-
sively in the neighbourhood of the bob, inas-
much as the rod of the pendulum has a mass
that is appreciable. This form of pendulum
cannot therefore be used where it is necessary
to determine the length with accuracy.

625. When the length of a pendulum is to
be measured, we must adopt other means of
supporting it than that of suspension by a spring,

fas otherwise we cannot have a definite point from
a which to measure. To illustrate the mode that
is to be adopted, I take here an iron bar 6' long
and i" square, which weighs 19 Ibs. I wish to
support this at one end so that it can vibrate
freely, and at the same time have a definite point
of suspension. I have here two small prisms
of steel E (Fig. 88) fastened to a brass frame ; the faces of
the prisms meet at about an angle of 60° and form the edges
about which the oscillation takes place : this frame and the
edges can be placed on the end of the bar, and can be fixed
there by tightening two nuts. The object of having the

XIX.]

THE COMPOUND PENDULUM.

301

edges on a sliding frame is that they may be applicable to
different parts of the bar with facility. In some instruments
used in experiments requiring extreme delicacy, the edges
which are attached to the pendulum are supported upon
plates of agate; they are to be adjusted on the same
horizontal line, and the pendulum really vibrates about this
line, as about an axis. For our purpose it will be sufficient
to support the edges upon
small pieces of steel. A B,
Fig. 88, represents one side of
the top of the iron bar ; E is
the edge projecting from it,
with its edge perpendicular to
the bar. c D is a steel plate
bearing a knife edge on its up-
per surface ; this piece of steel
is firmly secured to the frame- FIG. 88.

work. There is of course a

similar piece on the other side, supporting the other edge. The
bar, thus delicately poised, will, when once started, vibrate
backwards and forwards for an hour, as there is very little
friction between the edges and the pieces which support them.

626. The general appearance of the apparatus, when
mouQted, is shown in Fig. 89. A B is the bar : at A the
two edges are shown, and also the pieces of steel which
support them. The whole is carried by a horizontal beam
bolted to two uprights ; and a glance at the figure will
explain the arrangements made to secure the steadiness
of the apparatus ; the second pair of edges shown at B will
be referred to presently (Art. 635).

627. This bar, as you see, vibrates to and fro; and we
shall determine the length of a simple pendulum which would
vibrate in the same period of time. The length might be

302 EXPERIMENTAL MECHANICS. [LECT.

deduced by finding the time of vibration, and then calcu-
lating from Art. 606. This would be the most accurate
mode of proceeding, but I have preferred to adopt a direct
method which does not require calculation. A simple

FIG. 89.

pendulum, consisting of a fine cord and a small iron sphere
c, is mounted behind the edge, Fig. 89. The point from
which the cord is suspended lies exactly in the line of the
two edges, and there is an adjustment for lengthening or
shortening the cord at pleasure.

XIX.] THE COMPOUND PENDULUM. 303

628. We first try with 6' of cord, so that the simple pen-
dulum shall have the same length as the bar. Taking the
ball in one hand and the bar in the other, I draw them aside,
and you see, when they are released, that the bar performs
two vibrations and returns to my hand before the ball.
Hence the length of the isochronous simple pendulum is
certainly less than the length of the bar ; for a pendulum
of that length is too slow.

629. I now shorten the cord until it is only half the
length of the bar ; and, repeating the experiment, we find
that the ball returns before the bar, and therefore
the simple pendulum is too short. Hence we learn that
the isochronous pendulum is greater than half the length of
the bar, and less than the whole length.

630. Let us finally try a simple pendulum two-thirds of the
length of the bar. I make the experiment, and find that
the ball and the bar return to my hand precisely at the same
instant. Therefore two- thirds of the length of the bar is
the length of the isochronous simple pendulum.

We may state generally that the time of vibration of a
uniform bar about one end equals that of a simple pendulum
whose length is two-thirds of the bar ; no doubt the bar
we have used is not strictly uniform, because of the
edges ; but in the positions they occupy, their influence on
the time of vibrations is imperceptible.

632. For this rule to be verified, it is essentially neces-
sary that the edges be properly situated on the bar; to
illustrate this we may examine the oscillations of the small
rod, shown at D (Fig. 89). This rod is also of iron
24" x o"'5 xo"-5, and it is suspended from a point near the
centre by a pair of edges ; if the edges could be placed so
that the centre of gravity of the whole lay in the line of the
edges, it is evident that the bar would rest indifferently

304 EXPERIMENTAL MECHANICS. [LECT.

however it were placed, and would not oscillate. If then
the edges be very near the centre of gravity, we can easily
understand that the oscillations may be very slow, and this
is actually the case in the bar D. By the aid of the stop-
watch, I find that one hundred vibrations are performed in
248 seconds, and that therefore each vibration occupies
2-48 seconds. The length of the simple pendulum which
has 2-48 seconds for its period of oscillation, is about 20'.
Had the edges been at one end, the length of the simple
pendulum would have been

24"x§ = 1 6".

A bar 72" long will vibrate in a shorter time when the edge
is i5"'2 from one end than when it has any other position.
The length of the corresponding simple pendulum is 4i"-6.

THE CENTRE OF OSCILLATION.

633. It appears that corresponding to each compound
pendulum we have a specific length equal to that of the
isochronous simple pendulum. To take as example the
6' bar already described (Art. 625), this length is 4'. If
I measure off from the edges a distance of 4', and mark this
point upon fhe bar, the point is called the centre of
oscillation. More generally the centre of oscillation is found
by drawing a line equal to the isochronous simple pendulum
from the centre of oscillation through the centre of gravity.

634. In the bar D the centre of oscillation would be at
a distance of 20' below the edges ; and in general the
position will vary with the position of the edges.

635. In the 6' bar B is the centre of oscillation. I take
another pair of edges and place them on the bar, so that
the line of the edges passes through B. I now lift the bar

Xix.] THE CENTRE OF OSCILLATION. 305

carefully and turn it upside down, so that the edges B rest
upon the steel plates. In this position one-third of the bar
is above the axis of suspension, and the remaining two-
thirds below. A is of course now at the bottom of the
bar, and is on a level with the ball, c : the pendulum is
made to oscillate about the edges B, and the time of its
vibration may be approximately determined by direct com-
parison with c, as already explained. I find that, when I
allow c and the bar to swing together, they both vibrate
precisely in the same time. You will remember, that when
the ball was suspended by a string 4' long, its vibrations were
isochronous with those of the bar when suspended from the
edges A. Without having altered c, but having made the
bar to vibrate about B, I find that the time of oscilla-
tion of the bar is still equal to that of c. Therefore,
the period of oscillation about A is equal to that about B.
Hence, when the bar is vibrating about B, its centre of
oscillation must be 4' from B, that is, it must be at A : so
that when the bar is suspended from A, the centre of os-
cillation is B ; while, when the bar is suspended from B, the
centre of oscillation is A. This is an interesting dynamical
theorem. It may be more concisely expressed by saying
that the centre of oscillation and the centre of suspension are
reciprocal.

636. Though the proof that we have given of this
curious law applies only to a uniform bar, yet the law
is itself true in general, whatever be the nature of the
compound pendulum.

637. We alluded in the last lecture (Art 610) to the
difficulty of measuring with accuracy the length of a
simple pendulum ; but the reciprocity of the centres of
oscillation and suspension, suggested to the ingenious
Captain Kater a method by which this difficulty could be

306 EXPERIMENTAL MECHANICS. [LECT.

evaded. We shall explain the principle. Let one pair of
edges be at A. Let the other pair of edges, B, be moved as
near as possible to the centre of oscillation. We can test
whether B has been placed correctly : for the time taken by
the pendulum to perform 100 vibrations about A should be
equal to the time taken to perform 100 vibrations about B.
If the times are not quite equal, B must be moved slightly
until the times are properly brought to equality. The length
of the isochronous simple pendulum is then equal to the dis-
tance between the edges A and B; and this distance, from one
edge to the other edge, presents none of the difficulties in
its exact measurement which we had before to contend
with : it can be found with precision. Hence, knowing the
length of the pendulum and its time of oscillation, gravity
can be found in the manner already explained (Art. 608).

638. I have adjusted the two edges of the 6' bar as
nearly as I could at the centres of oscillation and suspen-
sion, and we shall proceed to test the correctness of the
positions. Mounting the bar first by the edges at A, I set
it vibrating. I take the stop-watch already referred to
(Art. 612), and record the position of its hands. I then
place my finger on the stud, and, just at the moment when
the bar is at the middle of one of its vibrations, I start
the watch. I count a hundred vibrations ; and when the
pendulum is again at the middle of its stroke, I stop the
watch, and find it records an interval of 110-4 seconds.
Thus the time of one vibration is 1*104 seconds. Revers-
ing the bar, so that it vibrates about its centre of oscillation
B, I now find that no'o is the time occupied by one
hundred vibrations counted in the same manner as before ;
hence 1*100 seconds is the time of one vibration about B :
thus, the periods of the vibrations are very nearly equal, as
they differ only by yi^th part of a second.

xix.] THE CENTRE OF PERCUSSION. 307

639. It would be difficult to render the times of oscil-
lation exactly equal by merely altering the position of B.
In Kater's pendulum the two knife-edges are first placed so
that the periods are as nearly equal as possible. The final
adjustments are given by moving a small sliding-piece
on the bar until it is found that the times of vibration about
the two edges are identical. We shall not, however, use
this refinement in a lecture experiment ; I shall adopt the
mean value of i '102 seconds. The distance of the knife-
edges is about 3''992 ; hence gravity may be found from
the expression (Art. 608)

/3-
-

Vl'

The value thus deduced is 32''4, which is within a small
fraction of the true value.

640 With suitable precautions Kater's pendulum can be
made to give a very accurate result. It is to be adjusted so
that there shall be no perceptible difference in the number
of vibrations in twenty-four hours, whichever edge be the
axis of suspension : the distance between the edges is
then to be measured with the last degree of precision by
comparison with a proper standard.

THE CENTRE OF PERCUSSION.

641. The centre of oscillation in a body free to rotate
about a fixed axis is identical with another remarkable
point, called the centre of percussion. We proceed to
examine some of the properties of a body thus suspended
with reference to the effects of a blow. For the purposes of
these experiments the method of suspension by edges is
however quite unsuited.

642. We shall first use a rod suspended from a pin about

3o8 EXPERIMENTAL MECHANICS. [LECT.

which the rod can rotate. A B, Fig. 90, is a pine rod
48" x i" x i", free to turn round B. Suppose this rod be hanging
vertically at rest. I take a stick in my hand, and, giving
the rod a blow, an impulsive shock will instantly be com-
municated to the pin at B ; but the actual effect
upon B will be very different according to the
position at which the blow is given. If I strike
the upper part of the rod at D, the action of A B
upon the pin is a pressure to the left. If I strike
the lower part at A, the pressure is to the right.
But if I strike the point c, which is distant from B
by two-thirds of the length of the rod, there is no
pressure upon the pin. Concisely, for a blow below
c, the pressure is to the right ; for one above c, it is
to the left ; for one at c it is nothing.

643. We can easily verify this by holding one
extremity of a rod between the finger and thumb of
the left hand, and striking it in different places with
a stick held in the right hand ; the pressure of the
rod, when struck, will be so felt that the circum-
stances already stated can be verified.

644. A more visible way of investigating the
subject is shown in Fig. 91. FB is a rod of wood,
suspended from a beam by the string F c. A
piece of paper is fastened to the rod at F by
means of a small slip of wood clamped firmly to
the rod ; the other ends of this piece of paper are

A similarly clamped at P and Q.
FIG. 90- 645 when the rod receives a blow on the
right-hand side at A, we find that the piece of paper is broken
across at E, because the end F has been driven by the
blow towards Q, and consequently caused the fracture of the
paper at a place, E, where it had been specially narrowed

XIX.]

THE CENTRE OF PERCUSSION.

309

I remove the pieces of paper, and replace them by a new
piece precisely similar. I now strike the rod at B, — a
smart tap is all that is necessary, — and the piece of paper
breaks at D. Finally replacing the pieces of paper by a
third piece, I find that when I
give the rod a tap (not a violent
blow) at c, neither D nor E are
broken.

646. This point c, where the
rod can receive a blow without
producing a shock upon the axis
of suspension is the centre of
percussion. We see, from its
being two-thirds of the length of
the rod distant from F, that it
is identical with the centre of
oscillation of the rod, if vibrating
about knife-edges at F. It is true
in genera], whatever be the shape
of the body, that the centre of
oscillation is identical with the
centre of percussion.

647. The principle embodied
in. ihe property of the centre
of percussion has many practical
applications. Every cricketer
well knows that there is a
part of his bat from which the
ball flies without giving his hands

any unpleasant feeling. The explanation is simple. The bat
is a body suspended from the hands of the batman ;
and if the ball be struck with the centre of percussion
of the bat, there is no shock experienced. The centre of

FIG. QI.

3io EXPERIMENTAL MECHANICS. [LECT.

percussion in a hammer lies in its head, consequently a
nail can receive a violent blow with perfect comfort to
the hand which holds the handle.

THE CONICAL PENDULUM.

648. I have here a tripod (Fig. 92) which supports a
heavy ball of cast iron by a string 6' long. If I withdraw
the ball from its position of rest, and merely release it, the
ball vibrates to and fro, the string continues in the same
plane, and the motion is that already discussed in the circular
pendulum. If at the same instant that I release the ball, I
impart to it a slight push in a direction not passing through
the position of rest, the ball describes a curved path, return-
ing to the point from which it started. This motion is that of
the conical pendulum, because the string supporting the
ball describes a cone.

649. In order to examine the nature of the motion, we
can make the ball depict its own path. At the opposite
point of the ball to that from which it is suspended, a hole
is drilled, and in this I have fitted a camel's hair paint-
brush filled with ink. I bring a sheet of paper on a
drawing-board under the vibrating ball; and you see
the brush traces an ellipse upon the paper, which I
quickly withdraw.

650. By starting the ball in different ways, I can make
it describe very different ellipses : here is one that is
extremely long and narrow, and here another almost cir-
cular. When the magnitude of the initial velocity is properly
adjusted, and its direction is perpendicular to the radius,
I can make the string describe a right cone, and the ball a
horizontal circle, but it requires some care and several
trials in order to succeed in this. The ellipse may also
become very narrow, so that we pass by insensible

xix. THE CONICAL PENDULUM. 311

gradations to the circular pendulum, in which the brush
traces a straight line.

651. When the ball is moving in a circle, its velocity

is uniform; when moving in an ellipse, its velocity is
greatest at the extremities of the least axes of this ellipse,
and least at the extremities of the greatest axes ; but, when
the ball is vibrating to and fro, as in the ordinary circular

312 EXPERIMENTAL MECHANICS. [LECT.

pendulum, the velocity is greatest at the middle of each
vibration, and vanishes of course each time the pendulum
attains the extremity of its swing. It is worthy of notice
that under all circumstances the brush traces an ellipse
upon the paper ; for the circle and the straight line are
only extreme cases, the one being a very round ellipse and
the other a very thin one. If, however, the arc of vibration
be large the movement is by no means so simple.

652. How are we to account for the elliptic movement?
To do so fully would require more calculation than can be
admitted here, but we may give a general account of
the phenomenon.

Let us suppose that the ellipse ACBD, Fig. 93, is the
path described by a particle when suspended by a string
from a point vertically above o, the centre of the ellipse.
To produce this motion I withdraw the particle from its
position of rest at o to A. If merely released, the particle
would swing over to B, and back again to A ; but I do not
simply release it, I impart a velocity impelling it in the
direction A T. Through o draw c D parallel to AT. If I
had taken the particle at o, and, without withdrawing it from
its position of rest, had started it off in the direction o D,
the particle would continue for ever to vibrate backwards
and forwards from c to D. Hence, when I release the
particle at A, and give it a velocity in the direction A T, the
particle commences to move under the action of two
distinct vibrations, one parallel to A B, the other parallel to
c D, and we have to find the effect of these two vibrations im-
pressed simultaneously upon the same particle. They are per-
formed in the same time, since all vibrations are isochronous.
We must conceive one motion starting from A towards o
at the same moment that the other commences to start from
o towards D. After the lapse of a short time, the body

THE CONICAL PENDULUM.

313

has moved through A Y in its oscillation towards o, and in
the same time through o z in its oscillation towards D ; it
is therefore found at x. Now, when the particle has moved
through a distance equal and parallel to A o, it must be
found at the point D, because the motion from o to D takes
the same time as from A to o. Similarly the body must pass
through B, because the time occupied by going from A to B,
would have been sufficient for the journey from o to D, and
back again. The particle is found at p, because, after the

FIG. 93.

vibration returning from B has arrived at Q, the movement
from D to o has travelled on to R. In this way the particle
may be traced completely round its path by the composition
of the two motions. It can be proved that for small motions
the path is an ellipse, by reasoning founded upon the fact
that the vibrations are isochronous.

653. Close examination reveals a very interesting circum-
stance connected with this experiment. It may be observed
that the ellipse described by the body is not quite fixed in
position, but that it gradually moves round in its plane.

3H EXPERIMENTAL MECHANICS. [LECT.

Thus, in Fig. 92, the ellipse which is being traced out by
the brush will gradually change its position to the dotted
line shown on the board. The axis of the ellipse revolves in
the same direction as that in which the ball is moving. This
phenomenon is more marked with an ellipse whose dimen-
sions are considerable in proportion to the length of the
string. In fact, if the ellipse be very small, the change of
position is imperceptible. The cause of this change is to be
found in the fact already mentioned (Art. 598), that though
the vibrations of a pendulum are very nearly isochronous,
yet they are not absolutely so ; the vibrations through a long
arc taking a minute portion of time longer than those
through a short arc.

This difference only becomes appreciable when the larger
arc is of considerable magnitude with reference to the length
of the pendulum.

654. How this causes displacement of the ellipse may be
explained by Fig. 94. The particle is describing the figure
A D c B in the direction shown by the arrows. This motion
may be conceived to be compounded of vibrations A c and
BD, if we imagine the particle to have been started from
A with the right velocity perpendicular to o A. At the
point A, the entire motion is for the instant perpendicu-
lar to o A; in fact, the motion is then exclusively due
to the vibration B D, and there is no movement
parallel to o A. We may define the extremity of the major
axis of the ellipse to be the position of the particle, when
the motion parallel to that axis vanishes. Of course this
applies equally to the other extremity of the axis c, and
similarly at the points B or D there is no motion of the
particle parallel to B D.

655. Let us follow the particle, starting from A until it
returns there again. The movement is compounded of

xix.] THE COMPOSITION OF VIBRATIONS. 315

two vibrations, one from A to c and back again, the other
along B D ; from o to D, then from D to B, then from p,
to o, taking exactly double the time of one vibration from
D to B. If the time of vibration
along A c were exactly equal to
that along B D, these two vibrations
would bring the particle back to
A precisely under the original
circumstances. But they do not
take place in the same time; the
motion along A c takes a shade
longer, so that, when the motion
parallel to A c has ceased, the
motion along D B has gone past o
to a point Q, very near o. Let
A P = O Q, and when the motion
parallel to A c has vanished, the

particle will be found at p; hence P must be the extre-
mity of the major axis of the ellipse. In the next revolution,
the extremity of the axis will advance a little more, and
thus the ellipse moves round gradually.

FIG. 94.

THE COMPOSITION OF VIBRATIONS.
656.~We have learned to regard the elliptic motion in the
conical pendulum as compounded of two vibrations. The
importance of the composition of vibrations justifies us in
examining this subject experimentally in another way.
The apparatus which we shall employ is represented in

Fig- 95-

A is a ball of cast iron weighing 25 Ibs., suspended from
the tripod by a cord : this ball itself forms the support of
another pendulum, B. The second pendulum is very light,
being merely a globe of glass filled with sand. Through a

EXPERIMENTAL MECHANICS.

[LECT.

hole at the bottom of the glass the sand runs out upon a
drawing-board placed underneath to receive it.

Thus the little stream of sand depicts its own journey upon

FIG. 96.

the drawing-board, and the curves traced out thus
indicate the path in which the bob of the second pendulum
has moved.

xix.] THE COMPOSITION OF VIBRATIONS. 317

657. If the lengths of the two pendulums be equal, and
their vibrations be in different planes, the curve described
is an ellipse, passing at one extreme into a circle, and at
the other into a straight line. This is what we might have
expected, for the two vibrations are each performed in the
same time, and therefore the case is analogous to that of
the conical pendulum of Art. 648.

658. But the curve is of a very different character when
the cords are unequal. Let us study in particular the case
in which the second pendulum is only one-fourth the length
of the cord supporting the iron ball. This is the experi-
ment represented in Fig. 95.

The form of the path delineated
by the sand is shown in Fig. 96.
The arrow-heads placed upon the
curve show the direction in which
it is traced. Let us suppose that
the formation of the figure com-
mences at A; it then goes
on to B, to o, to c, to D, and
back to A : this shows us that
the bob of the lower pendulum

must have performed two vibrations up and down, while
that of the upper has made one right and left. The motion
is thus compounded of two vibrations at right angles, and the
time of one is half that of the other.

The time of vibration is proportional to the square root
of the length ; and, since the lower pendulum is one-fourth
the length of the upper, its time of vibration is one-half
that of the upper. In this experiment, therefore, we have a
confirmation of the law of Art. 605.

FIG. 96.

LECTURE XX.
THE MECHANICAL PRINCIPLES OF A CLOCK.

Introduction. — The Compensating Pendulum. — The Escapement. —
The Train of Wheels.— The Hands.— The Striking Parts.

INTRODUCTION.

659. WE come now to the most important practical
application of the pendulum. The vibrations being always
isochronous, it follows that, if we count the number of
vibrations in a certain time, we shall ascertain the duration
of that time. It is simply the product of the number of
vibrations with the period of a single one. Let us take a
pendulum 39- 139 inches long; which will vibrate exactly
once a second in London, and is therefore called a
seconds pendulum (See Art. 607). If I set one of these
pendulums vibrating, and contrive mechanism by which the
number of its vibrations shall be recorded, I have a means
of measuring time. This is of course the principle of the
common clock : the pendulum vibrates once a second
and the number of vibrations made from one epoch to
another epoch is shown by the hands of the clock. For

LECT. XX.] THE COMPENSATING PENDULUM. 319

example, when the clock tells me that 15 minutes have
elapsed, what it really shows is that the pendulum has
made 60X15 = 900 vibrations, each of which has
occupied one second.

660. One duty of the clock is therefore to count and
record the number of vibrations, but the wheels and works
have another part to discharge, and that is to sustain the
motion of the pendulum. The friction of the air and the
resistance experienced at the point of suspension are forces
tending to bring the pendulum to rest ; and to counteract the
effect of these forces, the machine must be continually
invigorated with fresh energy. This supply is communicated
by the works of the clock, which will be sufficiently described
presently.

66 1.. When the weight driving the clock is wound up, a
store of energy is communicated which is doled out to the
pendulum in a very small impulse, at every vibration. The
clock-weight is just large enough to be able to counter-
balance the retarding forces when the pendulum has a proper
amplitude of vibration. In all machines there is some energy
lost in maintaining the parts in motion in opposition to friction
and other resistances ; in clocks this represents the whole
amount of the force, as there is no external work to be
performed.

THE COMPENSATING PENDULUM.
662. The actual length of the pendulum used, depends
upon the purposes for which the clock is intended, but it
is essential for correct performance that the pendulum
should vibrate at a constant rate; a small irregularity
in this respect may appreciably affect the indications
of the clock. If the pendulum vibrates in rooi
seconds instead of in one second, the clock loses one

320 EXPERIMENTAL MECHANICS. [I.ECT.

thousandth of a second at each beat ; and, since there are
86,400 seconds in a day, it follows that the pendulum
will make only 86,400 - 86-3 vibrations in a day, and
therefore the clock will lose 86 '3 seconds, or nearly a
minute and a half daily.

663. For accurate time-keeping it is therefore essential
that the time of vibration shall remain constant. Now the
time of vibration depends upon the length, and therefore
it is necessary that the length of the pendulum be absolutely
unalterable. If the length of the pendulum be changed
even by one-tenth of an inch, the clock will lose or gain nearly
two minutes daily, according to whether the pendulum has
been made longer or shorter. In general we may say that, if
the alteration in the length amount to k thousandths of an
inch, the number of seconds gained or lost per day is
1*103 x ^ with a seconds pendulum.

664. This explains the practice of raising the bob
of the pendulum when the clock is going too slow or
lowering it when going too fast. If the thread of the screw
used in doing this have twenty threads to the inch ; then one
complete revolution of the screw will raise the bob through
50 thousandths of an inch, and therefore the effect on the
rate will be i'io3 x 50 = 55 nearly. Thus, the rate of the
clock will be altered by about 55 seconds daily. Whatever
be the screw, its effect can be calculated by the simple rule
expressed as follows. Divide 1 103 by the number of threads
to the inch ; the quotient is the number of seconds that the
clock can be made to gain or lose daily by one revolution
of the screw on the bob of the pendulum.

665. Let us suppose that the length of the pendulum has
been properly adjusted so that the clock keeps accurate
time. It is necessary that the pendulum should not alter in
length. But there is an ever-present cause tending to

XX.]

THE COMPENSATING PENDULUM.

321

change it. That cause is the variation of temperature.
We shall first illustrate by actual experiment the well known
law that bodies expand under the action of heat ; then we
shall consider the irregularities thus introduced into the
motion of the pendulum ; and, finally, we shall point
out means by which these irregularities may be effectually
counteracted.

FIG. 97.

666. We have here a brass bar a yard long ; it is at present
at the temperature of the room. If we heat the bar over
a lamp, it becomes longer ; but upon cooling, it returns to
its original dimensions. These alterations of length are
very small, indeed too small to be perceived except by
careful measurement ; but we shall be able to demonstrate
in a simple way that elongation is the consequence of
increased temperature. I place the bar A D in the supports
shown in Fig. 97. It is firmly secured at B by means of a
binding screw, and passes quite freely through c ; if the bar

Y

322 EXPERIMENTAL MECHANICS. [LECT.

elongate when it is heated by the lamp, the point D must
approach nearer to E. At H is an electric battery, and G
is a bell rung by an electric current. One wire of the battery
connects H and G, another connects G with E, and a third
connects H with the end of the brass rod B. Until the electric
current becomes completed, the bell remains dumb, the
current is not closed until the point touches E : when this
is the case, the current rushes from the battery along the bar,
then from D to E, from that through the bell, and so back to
the battery. At present the point is not touching E, though
extremely close thereto. Indeed if I press E towards the
point, you hear the bell, showing that the circuit is complete ;
removing my finger, the bell again becomes silent, because
E springs back, and the current is interrupted.

667. I place the lamp under the bar: which begins to
heat and to elongate ; and as it is firmly held at B, the
point gradually approaches E : it has now touched E ; the
circuit is complete, and the bell rings. If I withdraw the
lamp, the bar cools. I can accelerate the cooling by
touching the bar with a damp sponge ; the bar contracts,
breaks the circuit, and the bell stops : heating the bar
again with the lamp, the bell again rings, to be again stopped
by an application of the sponge. Though you have not
been able to see the process, your ears have informed
you that heat must have elongated the bar, and that cold
has produced contraction.

668. What we have proved with respect to a bar of brass,
is true for a bar of any material ; and thus, whatever be the
substance of which a pendulum is made, a simple un-
compensated rod must be longer in hot weather than in
cold weather : hence a clock will generally have a tendency
to go faster in winter than in summer.

669. The amount of change thus produced is, it is true,

XX.] THE COMPENSATING PENDULUM. 323

very small. For a pendulum with a steel rod, the difference of
temperature between summer and winter would cause a
variation in the rate of five seconds daily, or about half a
minute in the week. The amount of error thus introduced
is of no great consequence in clocks which are only
intended for ordinary use ; but in astronomical clocks,
where seconds or even portions of a second are of import-
ance, inaccuracies of this magnitude would be quite
inadmissible.

670. There are, it is true, some substances — for example,
ordinary timber — in which the rate of expansion is less
than that of steel; consequently, the irregularities intro-
duced by employing a pendulum with a wooden rod
are less than those of the steel pendulum we have
mentioned ; but no substance is known which would not
originate greater variations than are admissible in the
performance of an astronomical clock.

We must, therefore, devise some means by which the
effect of temperature on the length of a pendulum can be
avoided. Various means have been proposed, and we
shall describe one of the best and simplest

671. The mercurial pendulum (Fig. 98) is frequently
used in clocks intended to serve as standard time-
keepers. The rod by which the pendulum is suspended is
made of steel ; and the bob consists of a glass jar of
mercury. The distance of the centre of gravity of the
mercury from the point of suspension may practically be
considered as the length of the pendulum. The rate of
expansion of mercury is about sixteen times that of steel :
hence, if the bob be formed of a column of mercury one-
eighth part of the length of the steel rod, the compensation
would be complete. For, suppose the temperature of the
pendulum be raised, the steel rod would be lengthened,

Y 2

3=4

EXPERIMENTAL MECHANICS.

[LECT.

and therefore the vase of mercury would be lowered ; on
the other hand, the column of mercury would expand by an
amount double that of the steel rod : thus the centre of the
column of mercury would be elevated as much as the steel
was elongated ; hence the centre of the mercury is raised
by its own expansion as much as it is
lowered by the expansion of the steel,
and therefore the effective length of
the pendulum remains unaltered. By
this contrivance the time of oscillation
of the pendulum is rendered indepen-
dent of the temperature. The bob of
the mercurial pendulum is shown in
Fig. 98. The screw is for the purpose
of raising or lowering the entire vessel
of mercury in order to make the rate
correct in the first instance.

THE ESCAPEMENT.
672. Practical skill as well as some
theoretical investigation has been
expended upon that part of a clock
which is called the escapement, the
excellence of which is essential to
the correct performance of a time-
piece. The pendulum must have its
FIG. 98- motion sustained by receiving an

impulse at every vibration : at the same time it is desirable
that the vibration should be hampered as little as possible
by mechanical connection. The isochronism on which the
time-keeping depends is in strictness only a characteristic of
oscillations performed with a total freedom from constraint of
every description ; hence we must endeavour to approximate

THE ESCAPEMENT.

325

the clock pendulum as nearly as possible to one which is
swinging quite freely. To effect this, and at the same time
to maintain the arc of vibration tolerably constant, is the
property of a good escapement.

FIG. 99.

673. A common form of escapement is shown in Fig. 99.
The arrangement is no doubt different from that actually
found in a clock ; but I have constructed the machine in

326 EXPERIMENTAL MECHANICS. [LECT.

this way in order to show clearly the action of the different
parts. G is called the escapement-wheel : it is surrounded
by thirty teeth, and turns round once when the pendulum
has performed sixty vibrations, — that is, once a minute, i
represents the escapement; it vibrates about an axis and
carries a fork at K which projects behind, and the rod
of the pendulum hangs between its prongs. The pendulum
is itself suspended from a point o. At N, H are a pair of
polished surfaces called the pallets : these fulfil a very
important function.

674. The escapement-wheel is constantly urged to turn
round by the action of the weight and train of wheels*
of which we shall speak presently ; but the action of the
pallets regulates the rate at which the wheel can revolve.
When a tooth of the wheel falls upon the pallet N, the
latter is gently pressed away : this pressure is transmitted
by the fork to the pendulum ; as N moves away from
the wheel, the other pallet H approaches the wheel ;
and by the time N has receded so far that the tooth
slips from it, H has advanced sufficiently far to catch
the tooth which immediately drops upon H. In fact,
the moment the tooth is free from N, the wheel begins
to revolve in conseq uence of the driving weight ; but it is
quickly stopped by another tooth falling on H : and the
noise of this collision is the well-known tick of the clock.
The pendulum is still swinging to the left when the tooth
falls on n. The pressure of the tooth then tends to push H
outwards, but the inertia of the pendulum in forcing H
inwards is at first sufficient to overcome the outward pressure
arising from the wheel ; the consequence is that, after the
tooth has dropped, the escapement-wheel moves back a
little, or "recoils," as it is called. If you look at any
ordinary clock, which has a second-hand, you will notice

XX.] THE ESCAPEMENT. 327

that after each second is completed the hand recoils before
starting for the next second. The reason of this is, that the
second-hand is turned directly by the escapement-wheel, and
that the inertia of the pendulum causes the escapement-
wheel to recoil But the constant pressure of the tooth
soon overcomes the inertia of the pendulum, and H is
gradually pushed out until the tooth is able to " escape " ;
the moment it does so the wheel begins to turn round,
but is quickly brought up by another tooth falling on N,
which has moved sufficiently inwards.

The process we have just described then recurs
over again. Each tooth escapes at each pallet, and the
escapements take place once a second ; hence the
escapement-wheel with thirty teeth will turn round once
in a minute.

675. When the tooth is pushing N, the pendulum is being
urged to the left ; the instant this tooth escapes, another
tooth falls on H, and the pendulum, ere it has accomplished
its swing to the left, has a force exerted upon it to bring it
to the right. When this force and gravity combined have
stopped the pendulum, and caused it to move to the right,
the tooth soon escapes at H, and another tooth falls on N,
then retarding the pendulum. Hence, except during the
very minute portion of time that the wheel turns after one
escapement, and before the next tick, the pendulum is never
free ; it is urged forwards when its velocity is great, but
before it comes to the end of its vibration it is urged
backwards ; this escapement does not therefore possess the
characteristics which we pointed out (Art. 672) as necessary
for a really good instrument. But for ordinary purposes
of time-keeping, the recoil escapement works sufficiently
well, as the force which acts upon the pendulum is in
reality extremely small. For the refined applications of the

328 EXPERIMENTAL MECHANICS. [LECT.

astronomical clock, the performance of a recoil-escapement
is inadequate.

The obvious defect in the recoil is that the pendulum
is retarded during a portion of its vibration ; the impulse
forward is of course necessary, but the retarding force is
useless and injurious.

676. The " dead-beat " escapement was devised by the
celebrated clockmaker Graham, in order to avoid this
difficulty. If you observe the second-hand of a clock, con-
trolled by this escapement, you will understand why it is
called the dead beat : there is no recoil ; the second-hand
moves quickly over each second, and remains there fixed
until it starts for the next second.

The wheel and escapement by which this effect is pro-
duced is shown in Fig. 100. A and B are the pallets, by the
action of the teeth on which the motion is given to the
crutch, which turns about the centre o ; from the axis
through this centre the fork descends, so that as the crutch
is made to vibrate to and fro by the wheel, the fork is also
made to vibrate, and thus sustain the motion of the pendulum.
The essential feature in which the dead-beat escapement
differs from the recoil escapement is that when the tooth
escapes from the pallet A, the wheel turns : but the tooth
which in the recoil escapement would have fallen on the
other pallet, now falls on a surface D, and not on the pallet B.
D is part of a circle with its-centre at o, the centre of motion ;
consequently, the tooth remains almost entirely inert so
long as it remains on the circular arc D.

677. There is thus no recoil, and the pendulum is allowed
to reach the extremity of its swing to the right unretarded ;
but when the pendulum is returning, the crutch moves until
the tooth passes from the circular arc D on to the pallet B :
instantly the tooth slides down the pallet, giving the crutch

XX.]

THE ESCAPEMENT.

329

an impulse, and escaping when the point has traversed B.
The next tooth that comes into action falls upon the
circular arc c, of which the centre is also at o ; this tooth
likewise remains at rest until the pendulum has finished its
swing, and has commenced its return ; then the tooth
slides down A, and the process recommences as before.

678. The operations are so timed that the pendulum
receives its impulse (which takes place when a tooth slides
down a pallet) precisely when the oscillation is at the point
of greatest velocity ; the pendulum is then unacted upon
till it reaches a similar position in the next vibration. This
impulse at the middle of the swing does not affect the time
of vibration.

330 EXPERIMENTAL MECHANICS. [LECT.

679. There is still a small frictional force acting to
retard the pendulum. This arises from the pressure of
the teeth upon the circular arcs, for there is a certain amount
of friction, no matter how carefully the surfaces may be
polished. It is not however found practically to be a
source of appreciable irregularity.

In a clock furnished with a dead-beat escapement and
a mercurial pendulum, we have a superb time-keeper.

THE TRAIN OF WHEELS.

680. We have next to consider the manner in which the
supply of energy is communicated to the escapement-
wheel, and also the mode in which the vibrations of the
pendulum are counted, A train of wheels for this purpose
is shown in Fig. 99. The same remark may be made about
this train that we have already made about the escapement,
— namely, that it is more designed to explain the principle
clearly than to show the actual construction of a clock.

68 1. The weight A which animates the whole machine
is attached to a rope, which is wound around a barrel B ;
the process of winding up the clock consists in raising this
weight. On the same axle as the barrel B is a large tooth-
wheel c ; this wheel contains 200 teeth. The wheel c
works into a pinion D, containing 20 teeth ; consequently,
when the wheel c has turned round once, the pinion D has
turned round ten times. The large wheel E is on the same
axle with the pinion D, and turns with D ; the wheel E
contains 180 teeth, and works into the pinion F, containing
30 teeth : consequently when E has gone round once, F will
have turned round six times ; and therefore, when the wheel
c and the barrel B have made one revolution, the pinion
F will have gone round sixty times ; but the wheel G is
on the same shaft as the pinion F, and therefore, for every

XX.] THE HANDS. 331

sixty revolutions of the escapement-wheel, the wheel c will
have gone round once. We have already shown that the
escapement- wheel goes round once a minute, and hence the
wheel c must go round once in an hour. If therefore
a hand be placed on the same axle with c, in front of a
clock dial, the hand will go completely round once an hour ;
that is, it will be the minute-hand of the clock.

682. The train of wheels serves to transmit the power
of the descending weight and thus supply energy to the
pendulum. In the clock model you see before you, the
weight sustaining the motion is 56 Ibs. The diameter of
the escapement-wheel is about double that of the barrel, and
the wheel turns round sixty times as fast as the barrel ;
therefore for every inch the weight descends, the circum-
ference of the escapement-wheel must move through 120
inches. From the principle of work it follows that the energy
applied at one end of a machine equals that obtained from
the other, friction being neglected. The force of 56 Ibs.
is therefore, reduced to the one hundred-and-twentieth
part of its amount at the circumference of the escape-
ment-wheel. And as the friction is considerable ; the actual
force with which each tooth acts upon the pallet is only a
few ounces.

683. In a good clock an extremely minute force need
only be supplied to the pendulum, so that, notwithstanding
86,400 vibrations have to be performed daily, one winding
of the clock will supply sufficient energy to sustainthe motion
for a week

THE HANDS.

684. We shall explain by the model shown in Fig. 101, how
the hour-hand and the minute-hand are made to revolve
\vith_different velocities about the same dial.

332 EXPERIMENTAL MECHANICS. [LECT.

G is a handle by which I can turn round the shaft which
carries the wheel F, and the hand B. There are 20 teeth
in F, and it gears into another wheel, E, containing 80
teeth; the shaft which is turned by E carries a third
wheel D, containing 25 teeth, and D works with a fourth c,
containing 75 teeth, c is capable of turning freely round
the shaft, so that the motion of the shaft does not
affect it, except through the intervention of the wheels E, F,
and D. To c another hand A is attached, which therefore
turns round simultaneously with c. Let us compare the

FIG. 101.

motion of the two hands A and B. We suppose that the handle
G is turned twelve times; then, of course, the hand B, since
it is on the shaft, will turn twelve times. The wheel F also
turns twelve times, but E has four times the number of
teeth that A has, and therefore, when F has gone round four
times, E will only have gone round once : hence, when F
has revolved twelve times, E will have gone round three
times. D turns with E, and therefore the twelve revolutions
of the handle will have turned D round three times, but since
c has 75 teeth and D 25 teeth, c will have only made one
revolution, while D has made three revolutions ; hence the

XX.] THE STRIKING PARTS. 333

hand A will have made only one revolution, while the hand
B has made twelve revolutions.

We have already seen (Art. 68 1) how, by a train of wheels,
one wheel can be made to revolve once in an hour. If
that wheel be upon the shaft instead of the handle G, the
hand B will be the minute-hand of the clock, and the hand
A the hour-hand.

685. The adjustment of the numbers of teeth is important,
and the choice of wheels which would answer is limited.
For since the shafts are parallel, the distance between the
centres of F and E must equal that between the centres
of c and of D. But it is evident that the distance
from the centre of F to the centre of E is equal to the
sum of the radii of the wheels F and E. Hence the
sum of the radii of the wheels F and E must be equal
to the sum of the radii of c and D. But the circumferences
of circles are proportional to their radii, and hence the sum
of the circumferences of F and E must equal that of c and
D ; it follows that the sum of the teeth in E and F must
be equal to the sum of the teeth in c and D. In the present
case each of these sums is one hundred.

686. Other arrangements of wheels might have been de-
vised, which would give the required motion ; for example, if
F weje 20, as before, and E 240, and if c and D were each
equal to 130, the sum of the teeth in each pair would be
260. E would only turn once for every twelve revolutions
of F, and c and D would turn with the same velocity as E ;
hence the motion of the hand A would be one-twelfth that of
B. This plan requires larger wheels than the train already
proposed.

THE STRIKING PARTS.

687. We have examined the essential features of the

334 EXPERIMENTAL MECHANICS. [LECT. xx.

going parts of the clock ; to complete our sketch of this
instrument we shall describe the beautiful mechanism by
which the striking is arranged. The model which we repre-
sent in Fig. 102 is, as usual, rather intended to illustrate
the principles of the contrivance than to be an exact counter-
part of the arrangement found in clocks. Some of the
details are not reproduced in the model ; but enough is
shown to explain the principle, and to enable the model
to work.

688. When the hour-hand reaches certain points on the
dial, the striking is to commence ; and a certain number of
strokes must be delivered, The striking apparatus has both
to initiate the striking and to control the number of strokes ;
the latter is by far the more difficult duty. Two contrivances
are in common use ; we shall describe that which is used in
the best clocks.

689. An essential feature of the striking mechanism in
the repeating clock is the snail, which is shown at B. This
piece must revolve once in twelve hours, and is, therefore,
attached to an axle which performs its revolution in exactly
the same time as the hour-hand of the clock. In the model,
the striking gear is shown detached from the going parts,
but it is easy to imagine how the snail can receive this
motion. The margin of the snail is marked with twelve
steps, numbered from one to twelve. The portions of the
margin between each pair of steps is a part of the circum-
ference of a circle, of which the axis of the snail is the centre.
The correct figuring of the snail is of the utmost importance
to the correct performance of the clock. Above the snail
is a portion of a toothed wheel, F, called the rack; this
contains about fourteen or fifteen teeth. When this
wheel is free, it falls down until a pin comes in contact
with the snail at B.

336 EXPERIMENTAL MECHANICS. [LECT.

690. The distance through which the rack falls depends
upon the position of the snail ; if the pin come in contact
with the part marked i., as it does in the figure, the rack will
descend but a small distance, while, if the pin fall on the
part marked VIL, the rack will have a longer fall : hence as
the snail changes its position with the successive hours, so
the distance through which the rack falls changes also. The
snail is so contrived that at each hour the rack falls on a
lower step than it does in the preceding hour ; for example,
during the hour of three o'clock, the rack would, if allowed
to fall, always drop upon the part of the snail marked m.,
but, when four o'clock has arrived, the rack would fall
on the part marked iv. ; it is to insure that this shall happen
correctly that such attention must be paid to the form of
the snail.

691. A is a small piece called the "gathering pallet " : it is
so placed with reference to the rack that, at each revolution
of A, the pallet raises the rack one tooth. Thus, after the
rack has fallen, the gathering pallet gradually raises it.

692. On the same axle as the gathering pallet, and
turning with it, is another piece c, the object of which
is to arrest the motion when the rack has been raised,
sufficiently. On the rack is a projecting pin ; the piece c
passes free of this pin until the rack has been lifted to its
original height, when c is caught by the pin, and the mecha-
nism is stopped. The magnitude of the teeth in the rack is
so arranged with reference to the snail, that the number
of lifts which the pallet must make in raising the rack is
equal to the number marked upon the step of the snail upon
which the rack had fallen ; hence the snail has the effect of
controlling the number of revolutions which the gathering
pallet can make. The rack is retained by a detent F, after
being raised each tooth.

xx. THE STRIKING PARTS. 337

693. The gathering pallet is turned by a small pinion of
27 teeth, and the pinion is worked by the wheel c, of 180
teeth. This wheel carries a barrel, to which a movement
of rotation is given by a weight, the arrangement of which is
evident : a second pinion of 27 teeth on the same axle with
D is also turned by the large wheel c. Since these pinions
are equal, they revolve with equal velocities. Over D the
bell i is placed ; its hammer E is so arranged that a pin
attached to D strikes the bell once in every revolution of D.
The action will now be easily understood. When the hour-
hand reaches the hour, a simple arrangement raises the detent
F ; the rack then drops ; the moment the rack drops, the
gathering pallet commences to revolve and raises up the
rack ; as each tooth is raised a stroke is given to the bell,
and thus the bell strikes until the piece c is brought to rest
against the pin.

694. The object of the fan H is to control the rapidity of
the motion : when its blades are placed more or less
obliquely, the velocity is lessened or increased.

APPENDIX I.

The formulas in the tables on p. 73 and after can be deduced
by two methods,— one that of graphical construction, the other
that of least squares. The first method is the more simple and
requires but little calculation ; though neatness and care are
necessary in constructing the diagrams. The second method
will be described for the benefit of those who possess the
requisite mathematical knowledge. The formulae used in the
preparation of the tables have been generally deduced from the
method of least squares, as the results are to a slight, though
insignificant, extent more accurate than those of the method of
graphical construction. This remark will explain why the figures
in some of the formulas are carried to a greater number of places
of decimals than could be obtained by the other method.

We shall confine the numerical examples to Tables III. and
IV., and show how the formulas of these tables have been
deduced by the two different methods.

Tables V., XIV., XVI., XXL, are to be found in the same
manner as Table III. ; and Tables VI., IX., X., XL, XV., XVIL,
XVIII., XIX., XX., XXL, XXIL, in the same manner as
Table IV.

THE METHOD OF GRAPHICAL CONSTRUCTION.

TABLE III.

A horizontal line APS, shown on a diminished scale in Fig.
103, is to be neatly drawn upon a piece of cardboard about
14" X 6". A scale which reads to the hundreth of an inch is to

z 2

340

APPENDIX.

be used in the construction of the figure. A pocket lens will be
found convenient in reading the small divisions. By means of
a pair of compasses and the scale, points are to be marked upon
the line APS, at distances i"'4, 2"-8, 4" -2, 5"'6, f'o, 8"'4, 9"'8, 1 1"'2
from the origin A. These distances correspond to the magni-
tudes of the loads placed upon the slide on the scale of o"'i to
i Ib. Perpendiculars to APS are to be erected at the points
marked, and distances F1} F2, F3, &c. set off upon these per-
pendiculars. These distances are to be equal, on the adopted
scale, to the frictions for the corresponding loads. For example,
we see from Table III., Experiment 3, that when the load upon
the slide is 42 Ibs., the friction is 12 '2 Ibs. ; hence the point F3
is found by measuring a distance 4"'2 from A, and erecting a

perpendicular i"22. Thus, for each of the loads a point is
determined. The positions of these points should be indicated
by making each of them the centre of a small circle o"'i
diameter. These circles, besides neatly defining the points,
will be useful in a subsequent part of the process.

It will be found that the points FJ, F2, &c. are very nearly in a
straight line. We assume that, if the apparatus and observations
were perfect, the points would lie exactly in a straight line. The
object of the construction is to determine the straight line,
which on the whole is most close to all the points. If it be
true that the friction is proportional to the pressure, this line

APPENDIX. 341

should pass through the origin A, for then the perpendicular
which represents the friction is proportional to the line cut off
from A, which represents the load. It will be found that a line
AT can be drawn through the origin A, so that all the points are
in the immediate vicinity of this line, if not actually upon it. A
string of fine black silk about 15" long, stretched by a bow of
wire or whalebone, is a convenient straight-edge for finding the
required line. The circles described about the points Fx, F2, &c.
will facilitate the placing of the silk line as nearly as possible
through all the points. It will not be found possible to draw a
line through A, which shall intersect all the circles ; the best line
passes below but very near to the circles round F1} F2, F3, F4, touches
the circle about F5, intersects the circles about F6 and F7, and
passes above the circle round F8. The line should be so placed
that its depth below the point which is most above it, is equal
to the height at which it passes above the point which is most
below it.

From A measure AS, a length of 10", and erect the perpendicular
s T. We find by measurement that ST is 2"7. If, then, we sup-
pose that the friction for any load is really represented by the
distance cut off by the line AT upon the perpendicular, it
follows that

F ; R : : 2*7 : 10".
or F = o'2j R.

This is the formula from which Table III. has been con-
structed.

TABLE IV.

By a careful application of the silk bow-string, x Y Q can be
drawn, which, itself in close proximity to A, passes more nearly
through FJ, F2, &c. than is possible for any line which passes
exactly through A. X Y Q will be found not only to intersect all
the small circles, but to cut off a considerable arc from each.
Measure off x P a distance of 10", and erect the perpendicular

342 APPENDIX.

P Q ; then, if 7i be the load, and f the corresponding friction,
we must have from similar triangles —

F-£ x i Ib. pQ

R PX

By measurement it is found that AY — o"'i4, and PQ
= 2"'53.
We have, therefore,

F = 1-4 -f- 0-253^.
This is practically the same formula as

F = 1-44 + 0-252 7?,

from which the table has been constructed. In fact, the column
of calculated values of the friction might have been computed
from the former, without appreciably differing from what is
found in the table.

THE METHOD OF LEAST SQUARES.

TABLE III.

Let k be the coefficient of friction. It is impossible to find
any value for k which will satisfy the equation,

F - k R =o,

for all the observed pairs of values of F and R. We have
then to find the value for k which, upon the whole, best repre-
sents the experiments. F - k R is^ to be as near zero as pos-
sible for each pair of values of F and R.

In accordance with the principle of least squares, it is well
known to mathematicians, the best value of k is that which
makes

(Fl -kfitf + (F, - kfij + &c. + (Fm - kRJ 2
a minimum where Fa and Rj, F2 and R2 &c. are the simultaneous
values of F and R in the several experiments.

APPENDIX. 343

In fact, it is easy to see that, if this quantity be small, each of
the essentially positive elements,

(Fa - k Rtf, &c.
of which it is composed, must be small also, and that therefore

F- kR
must always be nearly zero.

Differentiating the sum of squares and equating the differential
coefficeint to zero, we have according to the usual notation,

2 RI (F, - k A\) = o ;

whence k= 2 ^ FI.
1SS

The calculation of k becomes simplified when (as is generally
the case in the tables) the loads Rlt R2, &c., Rm are of the form,

N, 2 N, 3 N, &c., m N.
In this case,

2 R, F1 = N(F, + ^ + 3 F, + &c. + m Fm}.
= N* (i2 + 22 4- &c. + m2)

.

A m (m + J) (2 m +

In the case of Table III.

m = 8,N= 14,

Ft + 2F+ 3F, + mFM

whence k = 0*27.

Thus the formula F = 0*27 K is deduced both by the method
of least squares, and by the method of graphical construction.

TABLE IV.

The formula for this table is to be deduced from $he following
considerations.

No values exist for x and/, so that the equation
F=

344 APPENDIX.

shall be satisfied for all pairs of values of F and R, but the
best values for x and_y arc those which make

a minimum.

Differentiating with respect to x and y, and equating the
differential coefficients to zero, we have

2 (^ - x - y R^ = o,
2 R^ (F1 - x - y R^) = o.

This gives two equations for the determination of x and y.
Suppose, as is usually the case, the loads be of the form,

N, 2 N, 3 N, 4 N &c. m N,
and making

A = Fl + Fa + F3 + &c.+ Fm
B = Fl + 2 F2 + 3 F3 + &c. + m Fm,
we have the equations

A-mx-m(*m + ^ N y= o,

B _ m (m + i) _ m(m + l} (2 m +
2 6

Solving these, we find

— nt m* — m

12 B 6 A

m3 — m N m* - m N

In the present case, '

m = 8 N = 14, A = 138-4, B = 770. 9 ;

whence x = 1-44

_y = 0*252,
and we have the formula,

F = 1-44 + 0-252 /?.

APPENDIX II.

DETAILS OF THE WILLIS APPARATUS USED IN ILLUSTRATING
THE FOREGOING LECTURES.

THE ultimate parts of the various contrivances figured in this
volume are mainly those invented by the late Professor Willis
of Cambridge. They are minutely described and illustrated
in a work written by him for the purpose under the title
System of Apparatus for the use of Lecturers and Experi-
menters in Mechanical Philosophy, London, Weale & Co., 1851.
This work has long been out of print. It may therefore be
convenient if I give here a brief account of those parts of this
admirable apparatus that I have found especially useful. The
illustrations have been copied from the plates in Professor
Willis' book.1

The Willis system provides the means for putting versatile
framework together with or without revolving gear for the pur-
pose of mechanical illustration. Many parts which enter into
the construction of the machine used at the lecture to-day will
reappear to-morrow as essential parts of some totally different
contrivance. The parts are sufficiently substantial to work
thoroughly well. The scantlings and dimensions generally have

1 I ought to acknowledge the kindness with which Mr. J. Willis
Clark, of Cambridge, the literary executor of Professor Willis, has re-
sponded to my queries, while I am also under obligations to the courtesy
of Messrs. Crosby, Lockwood, & Co.

346 APPENDIX.

been so chosen as to produce models readily visible to a large
class.

It will of course be understood that every model contains
some one or more special parts such as the punch and die in
Fig- 73, or the spring balance in Fig. 17, or the pulley block
in Fig. 33. But for the due exhibition of the operation of the
machine a further quantity of ordinary framework and of
moving mechanism is usually necessary. This material, which
may be regarded as of a general type, it is the function of the
Willis system to provide.

THE BOLTS. — The system mainly owes its versatility and its
steadiness to the use of the iron screw bolt for all attachments.
The bolts used are § " diameter ; the shape of the head is
hemispherical and the shank must be square for a short distance
from the head so that the bolt cannot turn round when passed
through the slits of the brackets or rectangles. When the head
of the bolt bears on a slit in one of the wooden pieces a circular
iron washer 2" in diameter, or a square washer 2" on each side, is
necessary to protect the wood from crushing. There is to be a
square hole, in the washer to receive the square shank of the
bolt and the thickness of the washers should be & ". The nut is
square or hexagonal, and should always have a washer under-
neath when screwed home with a spanner or screw-wrench.
The most useful lengths are 2", 4", 6". The proper kind are
known commercially as coach-bolts, and they should be chosen
with easy screws, for facility in erecting or modifying apparatus.
At least two dozen of the intermediate size and a dozen of each
of the others are required. For elaborate contrivances many
more will be necessary.

THE BEDS. — The simplest as well as the longest parts of the
framework are called " beds" (Fig. 104). Each bed is made of
two wooden bars. These bars are united by strong screws
passing through small blocks of hard wood so as to keep the
bars full f" asunder, and thus allow the shanks of the bolts
to pass freely through the slit. The scantling of each bar
is 2i" X i$", and the beds are of various lengths from i' to 10'

APPENDIX.

347

or even longer. The beds can be attached together in any re-
quired position by bolts 6" long. The rectangles and the
brackets are attached to the beds by 4" bolts. In one con-

FIG. 104.

junction or another the beds will be found represented in almost
every figure in the book. We may specially refer to Figs. 20,
44, 48, 49, 50, 65, 83.

THE STOOL. — Most of the larger pieces of apparatus have
the stool as their foundation (see Figs. 11,39, Io2)- It is often

FIG. 105.

convenient as in Fig. 65 to employ a pair of stools, while one
stool superposed on another gives the convenient stand in

348 APPENDIX.

Fig. 80. The stool is a stout wooden frame, providing a choice
of slits to which beds or other pieces may be attached by bolts.
The structure of the frame is shown in Fig. 105. It is 2 ' 6 " high
and its extreme horizontal dimensions are 2' 6" X I ' 9" of
which the greater is A E. In other words, the longer sides of
the stool are those open at the top. Each top corner is
strengthened by an iron plate of which a separate sketch is
shown. The scantlings of the parts of the stool are as follows :—
The legs and horizontal top rails are 3" X 2^". Two of these
rails with the intervening f " slit make the top and legs to be
4| " wide. The bottom front rail I is 3 " wide and 4 " deep. The
double side rails D, H are 14 " wide and 2\" deep, being made
thinner than the legs into which they are mortised in order to
allow the washers of the bolts to pass behind them. The slits
are to be full f " wide throughout. Beech or birch are very
suitable materials, but softer woods will answer if large washers
are invariably used.

THE RECTANGLE. — The useful element of the Willis system
known by this name is of iron cast in one piece (Fig. 106). The
rectangles are used in the attachment of beds to each
other under special conditions, or they are often
attached to the stools or to brackets. Indeed their
uses are multifarious, see for examples Figs. 12, 58,
62, 89, 97, 102 and many others. The faces of the
rectangle are 2^" broad. The outside dimensions
are 6 " and 9 ", and the thickness of metal is \ ".
FIG. 106. Each side of the rectangle has the usual bolt slit f "
clear. Rectangles of a larger size are often found
useful, their weight makes them effective stands (see Figs. 35,
43, 52, 65).

THE TOOTHED WHEEL. — The most convenient type of
tqothed wheel for our present purpose is that known as the
cast-iron ten-pitch. In all such wheels the number of teeth
is simply ten times the number of inches in the diameter.
For example a wheel with 120 teeth is 12 inches in diameter.
A number of ten-pitch wheels large and small must be pro-

APPENDIX. 349

vided. The actual assortment that will be necessary depends
upon circumstances. For most purposes it will be sufficient
to have the multiples of 5 from 25 upwards to 120, and then
a few larger sizes such as 150, 180, 200. Duplicates of the
constantly recurring numbers such as 30, 60, 120 are convenient.
Arm wheels are always preferable to plate wheels in lightness
and appearance as well as in price. All wheels are to be I " thick
at the boss which is faced in the latter at each side, and bored
with a hole full i " diameter, in which a key groove is cut. A
pair of mitre wheels such as are used in Fig. 80 are sometimes
useful.

THE PULLEY. — We have frequent occasion to use the pulley
for conveying a cord, and a somewhat varied stock is con-
venient. Thus light brass pulleys are used in the apparatus
shown in Fig. 3, and a stout pulley in Fig. 71. A cast-iron
pulley about 10" in diameter is seen in Figs. 32 and 34. It is
bored i " in diameter with a key groove, and the boss is i " thick.
Some small pulley blocks similar to those used on yachts are
often very useful.

THE STUD-SOCKET. — For mounting toothed wheels on the
larger pulleys or for almost any rotating or oscil-
lating pieces the stud-socket is used (see Fig. 107).
The socket A B may be made of brass or of cast-
iron. It is i " in diameter so as to pass through
the bosses of the wheels that have been bored to
i " with this object : — The socket is provided with
a shoulder at one end (A) which is \\" diameter,
and with a strong screw B and octagonal nut at
the other end. The extreme length of the socket
is 3^", and the plain part of the I " cylinder is ij"
long. When two wheels are placed on the socket
each of which has a boss i " thick, the tightening
of the nut will secure the wheels against the
shoulder. A feather is screwed on the plain part fie. 107.
which enters the key grooves in the wheels, and thus
ensures that the wheels shall turn together. This feather should

350 APPENDIX.

be small enough to slip easily into the key groove. If only a single
wheel or if any peculiar piece such as a wooden cam or a disk
of sheet iron has to be mounted, then collars or large thick
washers must be placed on the socket so as permit the screw
to bind the whole together. The socket revolves upon a stout
iron stud C D, which is $ " in diameter. It bears a shoulder or
flange C at the back of the same diameter as the base of the
socket. The stud bears on the other side of the shoulder a
strong screw and nut which project l§" so as to allow the stud
to be secured in a hole I " deep in one of the brackets (to be
presently described). The plain part of this screw near the
shoulder must be f " diameter. The front end of the stud is
pierced with a hole to receive a spring pin to keep the socket
from sliding off the stud. Among the many applications of the
stud socket we may mention those shown in Figs. 30, 73, 74.

THE BRACKET. — There are six different forms of cast-iron
brackets represented in the adjoining figures (Figs. 108—113).

The brackets are primarily intended as the supports of the
stud-sockets. For this purpose each has a head I " thick bored

BRACKET No. I.

FIG. 108.

with a hole | " diameter, and thus fitted to receive the screw on
any of the studs. Each bracket stands on a base or sole with a
slit full | " wide for the bolts. The thickness of the sole is £ ".
The larger of the brackets I., II., and IV. have also slits in their

APPENDIX.

351

vertical faces. Brackets can be fastened either to the stool or
to the beds or rectangles, and the variety of their forms enables
the wheel-work carried on the stud sockets to be disposed in any

BRACKET No. IV

BRACKET No. \

BRACKET No. VI.

FIG. 113.

desired fashion. Brackets avail for many other purposes besides
those of supporting rotating mechanism. (Look at Figs, n,
12, 17, 20, 33, 38, 39, 73 and many others.)

THE SHAFTS AND TUBE-FITTINGS. — The stud sockets will
not provide for every case in which wheels have to be mounted
and driven. We must often employ shafts (see for instance
Figs. 30, 47, 101). The shafts we use are turned iron rods f " in
diameter, and of various lengths from 6 " up to 4 '. To support
the shafts we use for bearings the tube
fitting (Fig. 1 14). This is a brass cast-
ing which consists of a tube M N 2" long,
and ii" in external diameter, bored f "
so as to fit the shaft. The back of this
tube js a flat surface parallel to the
bore, and from it projects a screw f "
diameter, and \\" long with a nut FIG. n4.

which is however omitted in the draw-
ing. This screw may be of the same size as that of the
studs, and it is intended for the same purpose, namely to attach
the bearing to the hole in a bracket. The tube may of course
be fixed at any desired angle in the plane parallel to the face
of the bracket. To prevent the endlong motion of the shaft
cast-iron or brass rings are employed (Fig. 115). These are

352

APPENDIX.

THE PINNED
RING.

FIG. 115.

bored f ", and furnished with a binding screw by which they may
be tightened on the shaft in any position. To avoid injury to the
shaft it is well to have a narrow flat surface filed
along it to receive the end of the binding screw.
The use of the rings is shown in Fig. 47. If as
often happens (see for example Fig. 102) a
barrel has to be set in motion by a shaft the
required attachment can be made by simply
slipping on the barrel, and then putting at each
end of it two of the pinned rings (Fig. 115).
The pins enter holes bored into the barrel for
their reception so that when the rings are
bound to the shaft by their screws the barrel must revolve
with the shaft.

THE ADAPTER. — For the attachment of wheels or other
rotating pieces to the shaft an adapter is employed (Fig. 116).
It is bored with a $ " hole to fit the shaft, and
the external diameter is i ". At one end is a THE ADAPTER.
shoulder through which the binding screw is
tapped, and there is a nut and screw at the
opposite end. A feather will prevent the wheel
from turning round on the adapter which is
itself made to revolve with the shaft by screwing FIG. n6.
the binding screw down on the shaft. Some
adapters are only large enough for a single wheel i" thick in the
boss, but it is useful to have others
that will take two wheels. Adapters
are shown in use in Figs. 46 and

101.

THE LEVER ARM. - To give mo-
tion to the mechanism a lever arm
with a handle is frequently required
(Fig. 117). It is bored i" and has a
key groove, and the hole is i" long,
so that the lever arm can be fixed
on a stud socket like a wheel. By

THE LEVER ARM.

APPENDIX. 353

the aid of an adapter the lever arm is attached to a shaft. For
the use of the handle see Figs. 30 and 101. There are however
many other uses to which the lever arm is occasionally put It
can be used as a crank, and in linkage arrangements a pair
of lever arms are very convenient. Studs A or C can replace
the handle when necessary.

Such are the parts of the Willis apparatus which are adapted
for our present purpose. It remains to add that the fits should
be very easy, and the parts should be readily interchangeable.

A A

INDEX.

Accident, risk of, 32
Action, 6

Adapter, Willis apparatus, 352
Angle of friction, 78

of statical friction, 80
Apparatus for centre of gravity, 62

for equilibrium of three forces, 7

to show friction, 65, 78

the Willis, 345
Appendix!., 339
Attwood's machine, 232
Axes, permanent, 279

Balance, defective, 48

spring, 16

Bar, equilibrium of a, 38
Bat, cricket, 309
Beam, breadth of, 193

breaking load of, 193, 196

cast iron, 222

collapse of, 1 86

deflection of, 179

elasticity of, 184

load on, 197

placed edgewise, 193

strained, 178

strength of, 190

uniformly loaded, 198

with both ends secured, 200

with one end secured, 201
Beds in Willis apparatus, 346
Bob, raising or lowering the, 320
Bolts, use of, in Willis apparatus, 346
Bracket, Willis apparatus, 350
Brass, specific gravity of, 56
Breaking load, 177

Bridge, deflection of, 208

mechanics of, 218

Menai,?i8

suspension, 225

the Wye, 215

tubular, 223

with four struts, 210
two struts, 206
two ties, an
Brunei, Sir J., the Wye bridge, 215

Capstan, 151
Cast-iron beam, 222
Catenary, 226
Cathetometer, 180
Centre of gravity, 57
of a wheel, 61
position of, 59

oscillation, 304

percussion, 307
Circular motion, 267

action of, 271

applications of, 276

cause of, 270

in governor balls, 276

in sugar refining, 276

nature of, 267

on liquids, 271

on the earthy 276
Circular pendulum, 284
Clamps, 203
Clock pendulum, 299

principles of, 318

rate of, 322

Coefficient of friction, 74, 82
Collapse of a beam, 186
Compensating pendulum, 319

A A 2

356 INDEX.

Composition of forces, i, 9

parallel forces, 35, 37, 42

vibrations, 299, 315
Conical pendulum, 310
Couple, 44
Crane, 29, 162

friction in, 166

mechanical efficiency of, 163

Table XXI. 165
XXII. 166

velocity, ratio of, 163
Cricket bat, 309
Crowbar, 123
Cycloid, 295

D.

Dead-beat escapement, 328
Definition of force, 2
Deflection of a beam. Table XXIII. 182
Differential pulley, 112

Table XI. 114
Direction of a force, 5

Eade, Mr., epicycloidal pulley block
Easter Island, too
Elasticity of a beam, 184
Energy, 85, 94

storage of, 256, 258

unit of, 95

Engine, locomotive, 83
Epicycloidal pulley block, 80, 116

Equilibrium, neutral, 61

of a bar, 38, 41
three forces, 6
two forces, 6

stable, 59

unstable, 59
Escapement, 324

dead-beat, 328

recoil, 328

Expansion of bodies. 321
Experiment by M. Plateau, 273

F.

Fall in a second, 239
Falling body, motion of, 230
Feet, how represented, 7
Fibres in state of compression, 184

tension, 184
First law of motion, 230

, 116

Flywheel, 260

in steam-engine, 262
Foot pound, 95

Force, a small, and two larger, 12
definition of, 2
destroying motion, 3
direction of a. 5
magnitude of a, 4
measurement of, 4
of friction, 65
gravity, 50
one, resolved into three, 26

two, 17

representation of, 5
standard of, 4
Forces, composition of. 1,9
equilibrium of three. 6

two. 6

illustrations of. 3
in inclined plane, 136
parallel, 34
parallelogram of, 10
resolution of, 16

Formula for pulley block, 109, 114
Framework, 203, 345
Friction, 65

accurate law of, 75

a force, 66

and pressure. 72

angle of, 78

angle of statical, So

apparatus to show. 65, 68. 78

caused by roughness, 66

coefficient of. 74, 82

diminished, 66

excessive, 115

experimenting on, 66

in crane, 166

differential pulley block, 113
inclined plane, 132
lever, 123
pulleys, 89

law of, 91

rope and iron bar. 87
wheel and axle, 153
wheel and barrel, 158
laws of, 73, 8 1, 82
mean, 75

motion impeded by. 70
nature of, 65
overcoming, 93
Table I. 69
II. 71
III. 74
IV. 76
V. 78
VI. 81
VII. 81
VIII. 81
upon axle, 155
wheels, 93

INDEX.

357

Galileo and falling bodies, 235
kinetics. 230
the pendulum, 284
tower of Pisa, 233
Gathering pallet, 336
Girder, 219
as slight as possible, 221
Governor balls, 276
Graham, dead-beat escapement, 328

Large wheels, advantages of, 93
Law of falling bodies, 238
friction in pulleys, 91
lever of first order, 122
pressure, 37
Laws of friction, 73, 81, 82
Lead, specific gravity of, 56
Leaning tower of Pisa, 233
Level, 56

Graphical construction, 339
Gravity, 50
action of, 243

Lever, 119
and friction, 123
applications of, 123

and the pendulum, 292
and weight, 52
centre of, 57

arm, Willis apparatus, 352
laws of, 130
of first order, 119

defined, 246

law of, 122

different effects of, 53

of second order, 124

independent of motion, 241
in London, 292

of third order, 128
weight of, 121

specific, 53
Grindstone, treadle of, 128

Lifting crane, 29
Line and plummet, 56

Load, breaking, 177

H.

Locomotive engine, 83

Hammer, 252
theory of the, 252

M.

Hands of a clock, 331
Horse-power, 96

Machine, Attwood's, 232
punching, 263
Machines, pile-driving, 255

I.

Magnitude of a force, 4

Illustration ofparallelogram of forces, 10
Illustrations of forces, 10
resolution, 19

Margin of safety, 33
Mass, 236
Mean factions, 75
Measurement ot" force, 4

Inches, how represented, 7
Inclination of thread, 140
Inclined plane, 131

Mechanical powers, 85, 100
apparatus, Willis, 345
Menai Bridge, 218

forces on, 136

Method of least squares, 342

friction in, 132

Moment, 130

mechanical efficiency of, 139

Monkey, 257

Table XIII. 134 '
XIV. 137
XV. 138

Motion, first law of, 230
of falling body, 230

velocity, ratio of, 139

N.

Inertia, 250

inherent in matter, 252
Iron girders, 219
specific gravity of, 55
Isochronous simple pendulum, 303

Neutral equilibrium, 61
Newton and gravity, 289
Nut, 140

Ivory, specific gravity of, 56

O.

J-

Oscillation, centre of, 304

Jib, 29, 163

K.

P.
Pair of scales, 48

Kater, Captain, 305
Kinetics, 230

testing, 48
Parabola, 226

358

Parallel forces, 34

composition of, 35, 37, 42
opposite, 44
resultant of, 43
Parallelogram of forces, 10
Path of a projectile, 247
Pendulum and gravity, 292
circular, 284
compensating, 319
compound, 299, 301

INDEX.

Galileo and the, 286

isochronous simple, 303

length of the seconds, 292, 31!

motion of the, 283

of a clock, 299

simple, 284

time of oscillation, 286, 289
Percussion, centre of, 307, 309
Permanent axes, 279
Pile-driving machines, 255
Plateau, M., experiment by, 273
Plummet, 56
Powers, mechanical, 85
Pressure and friction, 72

law of, 37

of a loaded beam, 35, 37
Principles of framework, 203
Projectile, path of, 247
Pulley block, 99

differential, no

epicycloidal, 80

three sheave, 106

velocity, ratio of, 112
Pulley, ordinary form of, 86

single movable, 101
fixed, 86

use of, 88

velocity, ratio of, 103
Pulleys, friction in, 89

in windows, 86

in Willis apparatus, 349
Punching-machine, 263

force of, 265

R.

Rack, 334

Reaction, 6

Recoil escapement, 328

Rectangle in Willis apparatus, 348

Representation of a force, 5

Resistance to compression, 172, 175

extension, 172
Resolution of forces, 16

one force into three, 26

two, 17
Resultant, p

of parallel forces, 43

Rings in Willis apparatus, 352
Risk of accident, 32

s.

Safety, margin of, 33
Sailing, 21

against the wind, 24
Scales, 46
Screw, 139

and wheel and axle, 167

form of, 139

Table XVI. 142

velocity, ratio of, 143
Screw-bolt and nut, 148

Second, fall in a, 239
Seconds, pendulum, 318
Shafts, Willis apparatus, 351
Shears, 126
Simple pendulum, 284
Single movable pulley, Table IX. 104
Snail, 334
Specific gravity, 53
of brass, 56
iron, 55

Spirit level,

Spring balance, 16

Stable equilibrium, 59, 282

Standard of force, 4

Statical friction, angle of, 80

Stool in Willis apparatus, 347

Storage of energy, 256, 258

Stored-up energy exhibited, 261

Strength of a beam, 190

Striking parts, 333

Structures, 169

Strut, 28

Stud socket in Willis apparatus, 349

Sugar refining, 276

Suspension bridge, 225

mechanics of, 225

tension in, 228

Table I. 69
II. 71

III. 74

IV. 76
V. 78

VI. 78

VII. 81

VIII. 81

IX. 104

X. 108

INDEX.

359

XIII.

XVIII.

XIX.

XX.

XXI.

XXII.

XXIII.

9°

Tacking, as

Tension along a cord, 17
Three sheave pulley block, 106
Tie, 28, 175

rod, 29, 32
Timber, bending, 171

compression of, 172

properties of, 170

rings in, 171

seasoning, 171

warping, 171
Tin, 223

Toothed wheels, 160
Tower of Pisa, 233
Train of wheels, 330
Trans verse strain, 181
Treadle of a grindstone, 128
Tripod, 28

strength of, 28
Truss, simple form of, 212
Tube fittings, Willis apparatus, 351
Tubular bridge, 223

U.
Unstable equilibrium, 59,

Velocity, 231

ratio of inclined plane, 139

pulley, 103

pulley block, 112

screw, 143

wheel and axle, 152

wheel and pinion, 161
Vibrations, composition of, 299, 315

W.

Wedge, 139
Weighing machines, 123

scales, 46, ^8
Weight caused by gravity, 52

of water, 54
Wheel and axle, 149

and differential pulley, 167
screw, 167

experiments on, 152

formula for, 154

friction in, 153

Table XVIII. 154

velocity, ratio of, 152
Wheel and barrel, 158

formula for, 160

friction in, 158

Table XIX. 159
Wheel and pinion, 160

efficiency of, 161

Table XX. 162

velocity, ratio of, 161
Wheel, centre of gravity of, 6t
Wheels, 92

friction, 03

Wheels in Willis apparatus, 348
Willis system of apparatus, 345
Winch, 151
Wind, direction of, 22
Work, 85, 94
Wye Bridge, 215

RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNOAY.

A 000026014 1