Produced by deaurider, Paul Marshall and the Online
Distributed Proofreading Team at https://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)









Transcriber’s Notes:

  Underscores “_” before and after a word or phrase indicate _italics_
    in the original text.
  Small capitals have been converted to SOLID capitals.
  Illustrations have been moved so they do not break up paragraphs.
  Typographical errors have been silently corrected.




EXPERIMENTAL MECHANICS

[Illustration]

[Illustration: THE PATH OF A PROJECTILE IS A PARABOLA.]




                        EXPERIMENTAL MECHANICS

                         A COURSE OF LECTURES

              _DELIVERED AT THE ROYAL COLLEGE OF SCIENCE
                             FOR IRELAND_

                                  BY
                SIR ROBERT STAWELL BALL, LL.D., F.R.S.
                      ASTRONOMER ROYAL OF IRELAND

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

                         _WITH ILLUSTRATIONS_

                            SECOND EDITION

                                London
                           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 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 requirements 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. Culverwell.

                                              ROBERT S. BALL.

   OBSERVATORY, CO. DUBLIN,
        _3rd 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          1

                         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                       16

                         LECTURE III.
                      _PARALLEL FORCES._

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

                         LECTURE IV.
                   _THE FORCE OF GRAVITY._

    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                          50

                          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                          65

                         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                                85

                         LECTURE VII.
                      _THE PULLEY-BLOCK._

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

                          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                                    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
        Suspension Bridge                                    218

                         LECTURE XV.
              _THE MOTION OF A FALLING BODY._

    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 determined 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
        Composition of Vibrations                            299

                         LECTURE XX.
           _THE MECHANICAL PRINCIPLES OF A CLOCK._

    Introduction.—The Compensating Pendulum.—The
        Escapement.—The Train of Wheels.—The Hands.—The
        Striking Parts                                       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.




                          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.

1. I shall endeavour in this course of lectures to illustrate the
elementary laws of mechanics by means of experiments. 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 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 framework.
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
workshop. A vast deal of 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 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 understand 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 produce 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 release 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 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 measurements. 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.

[Illustration: FIG. 1.]

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 lbs. is applied at the point A, Fig.
1, tending to make A move in the direction AB. A standard line C of
certain length is to be taken. It is supposed that a line of this
length represents a force of 1 lb. 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 having an arrow-head
attached, we are able completely to represent a force.


EQUILIBRIUM OF TWO FORCES.

[Illustration: FIG. 2.]

7. In Fig. 2 we have represented two equal weights to which strings
are attached; these strings, after passing 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 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.

[Illustration: FIG. 3.]

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 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' 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., 1 lb., 2 lbs., &c., with rings at the
top, are used; one or more of these can easily be suspended from the
hooks as occasion may require.

[1] We shall often, in these lectures, represent feet or inches in
the manner usual among practical men—1' is one foot, 1" 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, 0"·5 is the mode of expressing a length of half an inch; 3'
1"·9 is to be read “three feet one inch and nine-tenths of an inch.”

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 positions we find 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.

[Illustration: FIG. 4.]

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 the others, experiment shows that the forces O P
and OQ 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 1 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 OR is capable of producing by itself the same effect as
the forces O P and OQ taken together. Therefore O R is equivalent to
O P and OQ. 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 P 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 parallelogram. Here we first perceive
the great law that the resultant of two forces acting at a point is the
diagonal of a parallelogram, of which they are the two sides.

14. This remarkable geometrical figure is called the _parallelogram of
forces_. Stated in its general form, the property 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.

[Illustration: FIG. 5.]

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 G 1·5 lb., and place 1 lb. 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 1 lb., while O S corresponds
to 1·5 lb. 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 O R 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 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.

[Illustration: FIG. 6.]

16. 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 counterbalanced 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 lbs., and weights
of 3 lbs. on the hook E and 4 lbs. 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 before, 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 the parallelogram on cardboard, as can be easily done
by forming 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.

18. It is easy, by the application of a set square, to prove that
in this case the cords attached to the 3 lb. and 4 lb. 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
applications. Attach 1 lb. 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 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 resultant of two
forces each of 1 lb. The apparent paradox is explained by noticing that
the forces of 1 lb. 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.

[Illustration: FIG. 7.]

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 impossible
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 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 lb. weight by means of a 2 lb. 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.

[Illustration: FIG. 8.]

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 16' distant. A
weight of 28 lbs. 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 lbs.)
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 lbs. 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 lbs., for, since two sides of a triangle, CB, 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 lb. weight than when it was
straight. But this can only have taken place by shortening the rope
between the pulley B and the 28 lb. 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 lb. 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 lb. 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 lbs. the current is
interrupted and the noise ceases.

23. I am sure you must all have noticed that the 2 lb. 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 therefore 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.

[Illustration: FIG. 9.]

24. As the last lecture was principally 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, provided it be the diagonal of the
parallelogram 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 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 lbs., corresponding to the scale which is seen. The 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 lbs., consequently for a
weight under 50 lbs. the side hook and ring are employed, as they give
a more accurate result than would be obtained by the top and bottom
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 lbs. 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 lbs. by the
balances. But the weight of 4 lbs. is the only weight acting; hence it
must be equivalent to two forces of very nearly 3 lbs. each along the
directions AE and BE. Here the two forces to which 4 lbs. is equivalent
are each of them less than 4 lbs., though taken together they exceed it.

[Illustration: FIG. 10.]

27. But remove the cords from AB and hang them on CD, the length CD
being 1' 10", then the forces shown along FC and D are each 5 lbs.;
here, therefore, one force of 4 lbs. is equivalent to two forces each
of 5 lbs. 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 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.

[Illustration: FIG. 11.]

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. 11 an arrangement
for this purpose is shown. A piece of stout twine AB, able to support
from 20 lbs. to 30 lbs., is fastened at one end A to a fixed support,
and at the other end B to the eye of a wire-strainer. A 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; 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 0"·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.
11, 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 siring, is magnified to upwards
of a hundredweight, for less than this would not break the wood. The
explanation is precisely the same as when the string was broken by the
thread.

[Illustration: FIG. 12.]


SAILING.

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

[Illustration: 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 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 O X 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
decompose 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 O A and O W,
thus forming the parallelogram O M R L. Hence, by the parallelogram of
forces, the force O R is equivalent 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 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 force 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.

[Illustration: FIG. 14.]

It will be noticed in this case that the force O L acting to leewards
exceeds O M 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 OW for the wind to act in the most efficient manner. Since, then, a
vessel can sail towards 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 manœuvre is called “tacking.” You will
understand that in a sailing-vessel the rudder has a more important
part 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.

[Illustration: FIG. 15.]

33. Up to the present we have only been considering forces which lie
in the same plane, but in nature we meet 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 (Fig. 15) three spring
balances are attached; A, B, C are not in the same straight line,
though they are at the same vertical 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 lbs. 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.

[Illustration: FIG. 16.]

34. Another mode of showing the decomposition of one force into three
forces not in the same plane is represented in Fig. 16. The tripod is
formed of three strips of pine, 4' × 0"·5 × 0"·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 lbs. 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
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
not 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 support very securely a considerable
weight. The pulleys by means of which gigantic weights are raised
are often supported by colossal tripods. They possess stability and
steadiness in addition to great strength.

36. An important point may be brought out by contrasting 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
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 either as a tie or as a strut, but a
cord or chain can only act as a tie. A pillar 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 of
wood B C 3' 6" long and 1" × 1" 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 A C 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 turning 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 lbs. 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 state of tension, as indeed
its name signifies, and the magnitude of that tension is shown to be
60 lbs. 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 as
great as the strain in the vertical wire by which it was produced.

[Illustration: FIG. 17.]

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

[Illustration: FIG. 18.]

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 lbs. 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 B P is equal to A C. Hence the
weight of 20 lbs. being represented 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 A B is 3' long, C B 3' 6", and A C 1', it
follows that the strain along the tie is 60 lbs., and along the strut
70 lbs., when the weight of 20 lbs. 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
importance to make every part strong enough to avoid the risk of
accident, not only under ordinary circumstances, 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
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.—Equilibrium of a Bar supported on
        a Knife-edge.—The Composition of 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.

[Illustration: FIG. 19.]

49. Fig. 19 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 lbs. 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 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 lbs., together with half the weight of the wooden
bar.

50. If the weight of 14 lbs. 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 lbs., 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 lbs. suspended from it by an S hook, without appreciable
deflection. Before the weight of 20 lbs. is suspended, the spring
balances each show a strain of 5 lbs. We would expect this, for it is
evident that the whole weight of the bar amounting to 10 lbs. 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 lbs. But 5 lbs. is due to the
weight of the bar. Hence the 20 lbs. is divided equally, as we have
already stated that it should be. But let the 20 lbs. be moved to any
other position, suppose 4 divisions from the right, and 16 from the
left; then the right-hand scale reads 21 lbs., and the left-hand reads
9 lbs. To get rid of the weight of the bar itself, we must subtract 5
lbs. from each. We learn therefore that the 20 lb. weight pulls the
right-hand spring balance with a strain of 16 lbs., and the left with a
strain of 4 lbs. 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 16 lbs. is shown on the right. At the same time the
weight is 4 divisions from the right, and 4 lbs. is the strain shown to
the left.

[Illustration: FIG. 20.]

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 lbs.; subtracting 5 lbs. for the
weight of the bar, the pressure due to the load is shown to be 18 lbs.,
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 lbs.; 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 commenced, 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 lbs.; if, then, the weight be at D, 10
divisions from one end A, and 4 from the other B, the pressure at the
corresponding ends will be 4 and 10. If the weight were 2·5 divisions
from one end, and therefore 11·5 from the other, the shares in which
this load would be supported at the ends are 11·5 lbs. and 2·5 lbs.
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 lbs. were in the middle, and
another of 14 lbs. at a distance of 11 divisions from one end, the
middle weight would produce 10 lbs. at each end and the 14 lbs. would
produce 3 lbs. and 11 lbs., and remembering the weight of the bar,
the total pressures produced would be 13 lbs. 6 oz. and 21 lbs. 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 KNIFE-EDGE.

60. The weight of the bar has hitherto somewhat complicated 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 overlook it altogether. By means of the
arrangement of Fig. 21, we can _counterpoise_ the weight of the bar.
To 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 1 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.

[Illustration: FIG. 21.]

61. A B is divided by lines drawn along it at distances of 1" 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 unloaded 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 horizontally. 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 lbs. 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 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 lb. weights and replace it by 2 lbs.,
leaving the other weight and the knife-edge unaltered, the bar
instantly descends on the 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 lbs., and therefore 36 inches
from the 2 lbs., 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 lbs., 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 lb. 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 this 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 the knife-edge are equal_.

65. We may further illustrate this law by suspending weights of 7 lbs.
and 5 lbs. respectively from the ends of 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 × 28 = 140, and
7 × 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 lbs. and 5 lbs. respectively.
These produce a pressure of 7 + 5 = 12 lbs. 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 lb. 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.

[Illustration: FIG. 22.]

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 lbs.—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 lbs.; thus the spring exerts in an upward pull a
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 lbs.,
which acts vertically 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 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 and BQ may be
considered as balanced by the force CR in the position shown in the
figure, but the force CR would be balanced by the equal and opposite
force CS, represented 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 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.


PARALLEL FORCES ACTING IN OPPOSITE DIRECTIONS.

70. Since the forces AP, BQ, CR (Fig. 22) are in equilibrium, 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 opposite 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 determined. 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 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.

[Illustration: FIG. 23.]

A wooden rod, A B 48" × 0"·5 × 0"·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 descends 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 lbs. to be placed at P, and 1 lb. at Q, we have
two parallel forces acting in opposite directions; and since their
difference is 1 lb., 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.

74. To study the case when the two forces are equal, a load of 2 lbs.
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 between
A and D. The forces evidently tend to raise OB 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 unsuccessful. But if
at the same time I press the rod downwards 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 of 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 consists of a slight frame of
wood A B C, 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.

[Illustration: FIG. 24.]

77. We suspend two small hooks from the points A and B: these are made
of fine wire, so that their weight may be 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 1 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 × 1, in the other
18 × 0·5, and these are obviously equal.

78. Let a weight of 1 lb. be placed on each of the hooks, the frame
will only be in equilibrium when the 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, however,
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 inaccurate result. The error
frequently 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 1 lb.; while if
the 1 lb. weight be placed at the long side, it will require more than
1 lb. 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 10 inches and 11 inches
long, then, if 1 lb. weight be placed in the pan of the 10-inch end,
its moment is 10; and if ¹⁰/₁₁ of 1 lb. be placed in the pan belonging
to the 11-inch end, its moment is also 10: hence 1 lb. at the short
end balances ¹⁰/₁₁ of 1 lb. at the long end; and therefore, if the
shopkeeper placed his weight in the short arm, his customers would
lose ¹/₁₁ part of each pound for which they paid; on the other hand,
if the shopkeeper placed his 1 lb. weight on the long arm, then not
less than ¹¹/₁₀ lb. would be required in the pan belonging to the short
arm. Hence in this case the customer would get ¹/₁₀ lb. 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 ¹/₁₁ lb. less than he ought, yet
the others get ¹/₁₀ lb. more than they have paid for.




                          LECTURE IV.
                     _THE FORCE OF GRAVITY._

    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.

81. 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 decomposed 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 properties of which
have led to the most sublime discoveries of human intellect. This is
the force of gravity.

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 experience 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.

[Illustration: FIG. 25.]

83. Let A B C D (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 D 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.

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; consequently 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 lbs. 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 weight, however its shape be altered: for
example, suppose I take a ball of putty which weighs 1 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 substances. 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
weighs 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 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.

[Illustration: FIG. 26.]

91. But the cubes may be 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 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 7¼ of the wooden cubes, which we may call cubes of
water, will balance it. We therefore say that the specific gravity of
iron is 7¼. 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 lbs., and that therefore one cubic inch weighs very nearly ¼ lb.

93. I have also cubes of brass, lead, and ivory; by 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                       8·1
      Lead                       11·3
      Ivory                       1·8

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
determining 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.

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.

[Illustration: FIG. 27.]

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 supported 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, 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, represented 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.

101. 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 suspension when the body is at
rest. If I attached a string to the plate and pulled it, the plate
would evidently place itself so 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.

[Illustration: FIG. 28.]

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 position 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
ascertaining the point about which it will balance. This may be done
by placing the bar on a knife-edge, and moving it to and fro until the
right position be secured; mark this position 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 suspension, 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 position 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.

[Illustration: FIG. 29.]

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.

106. 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
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.

[Illustration: FIG. 30.]

108. 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 particularly 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 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
disk D, which weighs 14 lbs, 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 as much as twelve times in a second. Still the stand remains 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 1 lb.; 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.

110. 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.

111. 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 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
understood 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.

112. 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.

[Illustration: FIG. 31.]

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

116. 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 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' × 11" × 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" × 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.

118. The mode of experimenting consists in placing a certain 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 1 lb., is always
included. A variety of small weights were provided for the hook B; they
consisted of 0·1, 0·5, 1, 2, 7, and 14 lbs. 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.

[Illustration: FIG. 32.]

119. An example of the experiments made is thus described. A weight of
56 lbs. is placed upon the slide, and it is found on trial that 29 lbs.
on B (including the weight of the hook itself) is sufficient to start
the slide; this weight is placed 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" × 11"; slide
          also of pine 9" × 9"; grain crosswise; slide is
          not started; force acting on slide is gradually
          increased until motion commences.
    +-----------+-------------+---------------+---------------+-------+
    | Number of |Load on slide|Force necessary|Force necessary| Mean  |
    |Experiment.|   in lbs.,  |to move slide. |to move slide. |values.|
    |           |  including  |               |               |       |
    |           |   weight    |  1st Series.  |  2nd Series.  |       |
    |           |  of slide.  |               |               |       |
    +-----------+-------------+---------------+---------------+-------+
    |    1      |     14      |       5       |       8       |  6·5  |
    |    2      |     28      |      15       |      16       | 15·5  |
    |    3      |     42      |      20       |      15       | 17·5  |
    |    4      |     56      |      29       |      24       | 26·5  |
    |    5      |     70      |      33       |      31       | 32·0  |
    |    6      |     84      |      43       |      33       | 38·0  |
    |    7      |     98      |      42       |      38       | 40·0  |
    |    8      |    112      |      50       |      33       | 41·5  |
    +-----------+-------------+---------------+---------------+-------+

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 lbs. In the third column is found the force necessary to
overcome the friction. The fourth column records a second series of
experiments performed 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 1st
series it would appear that the friction of 84 lbs. was 43 lbs., while
that of 98 lbs. was 42 lbs., 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
1st and the 2nd series do not agree, being generally greater in the
former than in the latter, the discordance being especially noticeable
in experiment 8, where the results were 50 lbs. and 33 lbs. 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 1st and the 2nd is
9 lbs., and that of the 2nd and the 3rd is only 2 lbs., 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 irregularities 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 16
inches per minute.

124. Indeed in no case can the slide _commence_ to move unless
the force _exceed_ the friction. The amount of this 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" × 11"; slide
          also of pine 9" × 9"; grain crosswise; slide
          started; force applied is sufficient to maintain
          uniform motion of the slide.
    +-----------+-------------+--------------+--------------+-------+
    |           |Load on slide|     Force    |   Force      |       |
    | Number of |    in lbs., |   necessary  |  necessary   | Mean  |
    |Experiment.|  including  | to maintain  | to  maintain |values.|
    |           |  weight of  |    motion.   |  motion.     |       |
    |           |    slide.   |  1st Series. | 2nd Series.  |       |
    +-----------+-------------+--------------+--------------+-------+
    |     1     |      14     |        4·9   |      4·9     |  4·9  |
    |     2     |      28     |        8·5   |      8·6     |  8·5  |
    |     3     |      42     |       12·6   |     12·4     | 12·5  |
    |     4     |      56     |       16·3   |     16·2     | 16·2  |
    |     5     |      70     |       19·7   |     20·0     | 19·8  |
    |     6     |      84     |       23·7   |     23·0     | 23·4  |
    |     7     |      98     |       26·5   |     26·1     | 26·3  |
    |     8     |     112     |       29·7   |     29·9     | 29·8  |
    +-----------+-------------+--------------+--------------+-------+

127. Two series of experiments to determine the power necessary to
maintain the motion have been recorded. Thus, in experiment 7, the load
on the slide being 98 lbs., it was found that 26·5 lbs. was sufficient
to sustain the motion, and a second trial being made independently,
the power found was 26·1 lbs.: a mean of the two values, 26·3 lbs., is
adopted as being near the truth. The greatest difference between the
two series, amounting to 0·7 lb., is found in experiment 6; a third
value was therefore obtained for the friction of 84 lbs.: this amounted
to 23·5 lbs., which is intermediate between the two former results, and
23·4 lbs., 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 corresponding loads upon the slide.

129. The mean frictions must, however, be slightly diminished 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 0·5 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 lbs. of friction in experiment 1 consists of 4·7,
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 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 (_R_)—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 experiments 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 determine a value for _k_ which, while it does
not represent any one of the experiments precisely, yet represents
them collectively better than it is possible for any other value to
do. The number thus found is 0·27. It is intermediate between 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 lbs. is 19·4 lbs., and the product of 70 and 0·27 is 18·9,
which is 0·5 lb. less than the true amount. In the last column of this
table the discrepancies between the observed and the calculated values
are recorded, for facility of comparison. It will be observed that the
greatest difference is under 1 lb.


                    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_ = 0·27 _R_.
    +-----------+----------+----------+----------+---------------+
    | Number of |    R.    |Corrected |    F.    | Discrepancies |
    |Experiment.|Total load|mean value|Calculated|  between the  |
    |           | on slide |    of    | value of |  observed and |
    |           |  in lbs. | friction.| friction.|   calculated  |
    |           |          |          |          |   frictions.  |
    +-----------+----------+----------+----------+---------------+
    |     1     |     14   |    4·7   |    3·8   |      -0·9     |
    |     2     |     28   |    8·2   |    7·6   |      -0·6     |
    |     3     |     42   |   12·2   |   11·3   |      -0·9     |
    |     4     |     56   |   15·8   |   15·1   |      -0·7     |
    |     5     |     70   |   19·4   |   18·9   |      -0·5     |
    |     6     |     84   |   23·0   |   22·7   |      -0·3     |
    |     7     |     98   |   25·8   |   26·5   |      +0·7     |
    |     8     |    112   |   29·3   |   30·2   |      +0·9     |
    +-----------+----------+----------+----------+---------------+

133. Hence the law _F_ = 0·27 _R_ represents the experiments with
tolerable accuracy; and the numerical ratio O·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 lbs. and 112 lbs.;
for example, if the load be 63 lbs., the friction is 63 × 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 between the extreme
weights employed in calculating the value of the coefficient which is
employed.


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 1 lb. and it is hardly possible
that any of the _mean frictions_ we have found should be in error to
so great an extent as 0·5 lb. But with the value of the coefficient
of friction which is used in Table III., the discrepancies amount
sometimes to 0·9 lbs. 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 _F_ and _R_ 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 reconciliation
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 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_ = 1·44 + 0·252 _R_.

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_ = 1·44 + 0·252 _R_.
    +-----------+----------+-------------+----------+-------------+
    |           |          |             |          |Discrepancies|
    | Number of |    R.    |  Corrected  |    F.    | between the |
    |Experiment.|Total load|    mean     |Calculated|observed and |
    |           | on slide |  value of   | value of | calculated  |
    |           |  in lbs. |  friction.  | friction.|  frictions. |
    +-----------+----------+-------------+----------+-------------+
    |    1      |     14   |      4·7    |    5·0   |    +0·3     |
    |    2      |     28   |      8·2    |    8·5   |    +0·3     |
    |    3      |     42   |     12·2    |   12·0   |    -0·2     |
    |    4      |     56   |     15·8    |   15·6   |    -0·2     |
    |    5      |     70   |     19·4    |   19·1   |    -0·3     |
    |    6      |     84   |     23·0    |   22·6   |    -0·4     |
    |    7      |     98   |     25·8    |   26·1   |    +0·3     |
    |    8      |    112   |     29·3    |   29·7   |    +0·4     |
    +-----------+----------+-------------+----------+-------------+

The fourth column contains the calculated values: thus, for example,
in experiment 4, where the load is 56 lbs., the calculated value is
1·44 + 0·252 × 56 = 15·6; the difference 0·2 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 lbs., and that therefore the formula represents the experiments
with considerable accuracy. It is undoubtedly nearer the truth than the
former law (Art. 132); in fact, the differences are now such as might
really belong to errors unavoidable in making the experiments.

139. This formula may be used for calculating the friction for any load
between 14 lbs. and 112 lbs. Thus, if the load be 63 lbs., the friction
is 1·44 + 0·252 × 63 = 17·3 lbs., which does not differ much from 17·0
lbs., 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 formula have been determined;
for example, to take an extreme case, if _R_ = 0, the formula would
indicate that the friction was 1·44, 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 lbs. to 56 lbs. 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 lbs. and 112 lbs. Suppose we only
consider loads up to 56 lbs., we find that the coefficient 0·288 will
best represent the experiments within this range, though for 112 lbs.
it would give an error of nearly 3 lbs. The results calculated by the
formula _F_ = 0·288 _R_ are shown in Table V., where the greatest
difference is 0·7 lb.


                     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_
    +-----------+----------+---------+----------+-------------+
    |           |          |         |          |Discrepancies|
    | Number of |    R.    |Corrected|    F.    | between the |
    |Experiment.|Total load|  mean   |Calculated| observed and|
    |           |on slide  |value of |value of  |  calculated |
    |           | in lbs.  |friction.| friction.|  frictions. |
    +-----------+----------+---------+----------+-------------+
    |     1     |    14    |   4·7   |    4·0   |     -0·7    |
    |     2     |    28    |   8·2   |    8·1   |     -0·1    |
    |     3     |    42    |  12·2   |   12·1   |     -0·1    |
    |     4     |    56    |  15·8   |   16·1   |     +0·3    |
    +-----------+----------+---------+----------+-------------+

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 lbs., disregarding the higher loads, is
_F_ = 0·9 + 0·266 _R_. This 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 0·1.


                     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_.
    +-----------+----------+---------+----------+-------------+
    |           |          |         |          |Discrepancies|
    | Number of |    R.    |Corrected|    F.    | between the |
    |Experiment.|Total load|  mean   |Calculated| observed and|
    |           |on slide  |value of |value of  |  calculated |
    |           | in lbs.  |friction.| friction.|  frictions. |
    +-----------+----------+---------+----------+-------------+
    |     1     |    14    |   4·7   |   4·6    |     -0·1    |
    |     2     |    28    |   8·2   |   8·3    |     +0·1    |
    |     3     |    42    |  12·2   |  12·1    |     -0·1    |
    |     4     |    56    |  15·8   |  15·8    |      0·0    |
    +-----------+----------+---------+----------+-------------+

[Illustration: FIG. 33.]

THE ANGLE OF FRICTION.

142. There is another mode of examining the action of friction besides
that we have been considering. The apparatus for this purpose is
shown in Fig. 33, in which B C represents the plank of pine we have
already used. It is 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 convenient 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 framework 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 lbs., 56 lbs., and 112 lbs., and the results are given in Table
VII.

We see that a load of 56 lbs. started when the plane reached an
inclination of 20°·1 in the first series, and of 17°·2 in the second,
the mean value 18°·6 being given in the fifth column. These means
for the three different weights 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" × 11" carries a loaded
          slide of pine 9" × 9"; one end of the plane is
          gradually elevated until the slide starts off.
    +-----------+-------------+-----------+-----------+-----------+
    | Number of |Total load on| Angle of  | Angle of  |Mean values|
    |Experiment.|  the slide  |elevation. |elevation. |  of the   |
    |           |   in lbs.   |1st Series.|2nd Series.|  angles.  |
    +-----------+-------------+-----------+-----------+-----------+
    |    1      |      14     |  19°·5    |    ——     |   19°·5   |
    |    2      |      56     |  20°·1    |   17°·2   |   18°·6   |
    |    3      |     112     |  20°·3    |   18°·9   |   19°·6   |
    +-----------+-------------+-----------+-----------+-----------+

145. We might, however, proceed differently in determining 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" × 11" carries a loaded
          slide of pine 9" × 9"; one end of the plane
          is gradually elevated until the slide, having
          received a start, moves off uniformly.
          +-----------+-----------------+------------+
          |Number of  |  Total load on  |  Angle of  |
          |Experiment.|the slide in lbs.|inclination.|
          +-----------+-----------------+------------+
          |    1      |        14       |    14°·3   |
          |    2      |        56       |    13°·0   |
          |    3      |       112       |    13°·0   |
          +-----------+-----------------+------------+

We 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.

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 13°·4. The tangent of this angle is 0·24: this is about 11 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 19°·2, and its tangent is 0·35. The experiments of
Table I. are, as already pointed out, rather unsatisfactory, 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 corresponding coefficient of
friction. When the slide is started, the tangent is about 11 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
differences 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" × 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 the approximate law that
the friction is proportional to the pressure. Suppose that the weight
were 100 lbs., and the area of the slide 100 inches, there would
then be a pressure of 1 lb. per square inch over the surface of the
slide, and therefore the friction to be overcome on each square inch
would be 0·27 lb., or for the whole slide 27 lbs. If, however, the
slide had only an area of 50 square inches, the load would produce a
pressure of 2 lbs., per square inch; the friction would therefore be
2 × 0·27 = 0·54 lb. for each square inch, and the total friction would
be 50 × 0·54 = 27 lbs., 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 lbs. for every ton 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 disappear, 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 important 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 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 stories 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 complicated 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 two iron studs, G, H, 0"·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 lbs. 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 lbs. 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 lbs. on A no less than 47 lbs. is necessary on B,
consequently there is an enormous loss of force: more than two-thirds
of the force which is exerted is expended uselessly. If instead of the
14 lbs. 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 presses 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 lbs. 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
lbs. on one hook would exactly balance 14 lbs. 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 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-shaped groove in its circumference to receive the
rope: this wheel turns on a ⅝ inch wrought iron axle, which is well
oiled. The rope used is about 0"·25 in diameter.

159. From the hooks E, F at each end of the rope a 14 lb. 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 lb. placed on either hook causes
it to descend and make the other ascend. This is a great improvement;
0·5 lb. now accomplishes what 33 lbs. was before required for. We
have avoided a great deal of friction, but we have not got rid of it
altogether, for 0·25 lb. 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.

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 in 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 lb.
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 lb. on E, we may consider the friction due to the weight
of the pulley, rope, and hooks as neutralized.

[Illustration: FIG. 34.]

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 lb.
This does not of course include the weight of 0·16 lb. 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 lb.

164. Let us now perform precisely the same experiment with the small
pulley. I transfer the same rope and hooks to K, and I find that 0·16
lb. is not now sufficient to overcome the friction of the pulley, but
I add on weights until C will just descend, which occurs when the load
reaches 0·95 lb. 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 lbs.; 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"·0 and 9"·5. The ratio of
these is 4·5; the corresponding resistances from friction we have seen
to be 0·28 lb. and 1·35 lbs. 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 ¹/₁₆th 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_.

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

167. We may deduce from these considerations the practical 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 lbs. is placed on each hook, 0·5 lb.
added to either will make it descend and raise up the other. Let 14
lbs. be placed on each hook, 0·5 lb. is no longer sufficient; 1 lb.
is required: hence when the weight is doubled the friction is also
doubled. Repeating the experiment with 21 lbs. and 28 lbs. on each
side, the corresponding weights necessary to overcome friction are 1·5
lbs. and 2 lbs. In the four experiments the weights used are in the
proportion 1, 2, 3, 4; and the forces necessary to overcome friction,
0·5 lb., 1 lb., 1·5 lbs., and 2 lbs., 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 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 circumference 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 depressions 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 bearings, 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 transferred 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, and 1 lb. weight is attached to each of its ends: these
of course balance. A number of fine wire hooks, each weighing 0·001
lb., are prepared, and it is found that when a weight of 0·004 lb. 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 depends 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 exertion that will be required until we know
the height through which it is to be raised; and if we know the height,
we cannot 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 1 lb.
avoirdupois through one British foot is taken as a standard, compared
with which all other quantities of energy are estimated. This quantity
of exertion is called in mechanics the unit of energy, and sometimes
also the “foot-pound.”

176. If a weight of 1 lb. has to be raised through a height of 2 feet,
or a weight of 2 lbs. through a height of 1 foot, it will be necessary
to expend twice as much energy as would have raised a weight of 1 lb.
through 1 foot, that is, 2 foot-pounds.

If a weight of 5 lbs. had to be raised from the floor up to a stool 3
feet high, how many units of energy would be required? To raise 5 lbs.
through 1 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 necessary.

If 100 lbs. be raised through 20 feet, 100 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 lbs. to a height of 20 feet, or a trunk
weighing 50 lbs. to a height of 15 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 lbs.
Hence it is less exertion to carry 50 lbs. up 15 feet than 40 lbs. 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 be measured by the number of foot-pounds
produced in the unit of time.

The power of a steam-engine is defined 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 uncertain 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 × 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

                         50 × 33,000
                         ———————————— = 1,650.
                            1,000

179. We shall apply the principle of work to the consideration of the
pulley already described (p. 90). In order to raise the weight of 14
lbs., it is necessary that the rope to which the power is applied
should be pulled downwards by a force of 15 lbs., the extra pound
being on account of the friction. To fix our ideas, we shall suppose
the 14 lbs. to be raised 1 foot; to lift this load directly, without
the intervention of the pulley, 14 foot-pounds would be necessary,
but when it is raised by means of the pulley, 15, foot-pounds are
necessary. Hence there is an absolute loss of ¹/₁₅th of the energy when
the pulley is used. If a steam-engine of 1 horse-power were employed
in raising weights by a rope passing over a pulley similar to that on
which we have experimented, only ¹⁴/₁₅ths of the work would be usefully
employed; but we find

                        33,000 × 14 = 30,800.
                                ————
                                 15

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

180. 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 1 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 rubbing in
its bearings, and, if it be not properly oiled, it becomes heated. The
amount of energy that seems to disappear 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 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 purpose 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 Moveable 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 importance. 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 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 corresponding 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 which 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 multitude of
men, it is difficult to 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 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 with 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 moveable 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.

188. 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 concerned, 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 lbs. So that when a load is being raised we
must reduce the apparent power by 1·5 lbs. to obtain the power really
effective.

189. Let us suspend 14 lbs. from the load hook at B, and ascertain
what power will raise the load. We leave the weight of the moveable
pulley and 1·5 lbs. of the power at C out of consideration. I then
find by experiment that 7 lbs. of effective power is not sufficient to
raise the load, but if one pound more be added, the power descends, and
the load is raised. Here, then, is a remarkable result; a weight of 8
lbs. has overcome 14 lbs. In this we have the first application of the
mechanical powers to increase our available forces.

[Illustration: FIG. 35.]

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 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 lbs. 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 lbs. one foot requires 14 foot-pounds of energy.
Hence, were there no such thing as friction, 7 lbs. on the power hook
would be sufficient to raise the load; because 7 lbs. descending
through two feet yields 14 foot-pounds. But there is a loss of energy
on account of friction, and a power of 7 lbs. is not sufficient: 8
lbs. are necessary. Eight lbs. in descending two feet performs 16
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 lbs. be placed on the load hook, a few trials
assure us that a power of 16 lbs. (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 directly. From a series of
careful experiments it has been found that when a man is employed in
the particular exertion 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 lbs. 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 lbs. than when it is larger or smaller. If
therefore the weights to be lifted amount to about 80 lbs., energy may
really be economized 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
          0"·6 wide, wrought iron axle 0"·6 diameter;
          fixed pulley of cast iron 5" diameter, groove
          0"·4 wide, wrought iron axle 0"·6 diameter,
          axles oiled; flexible plaited rope 0"·25
          diameter; velocity ratio 2, mechanical efficiency
          1·8, useful effect 90 per cent.; formula
          _P_ = 2·21 + 0·5453 _R_.
    +-----------+---------+----------+----------+------------------+
    |           |         |          |    P.    |  Discrepancies   |
    | Number of |   R.    | Observed |Calculated| between observed |
    |Experiment.|  Load   |   power  |   power  |  and calculated  |
    |           | in lbs. |  in lbs. |  in lbs. |      powers.     |
    +-----------+---------+----------+----------+------------------+
    |     1     |    28   |   17·5   |   17·5   |        0·0       |
    |     2     |    57   |   33·5   |   33·3   |      - 0·2       |
    |     3     |    85   |   48·5   |   48·6   |      + 0·1       |
    |     4     |   113   |   64·0   |   63·8   |      - 0·2       |
    |     5     |   142   |   80·0   |   79·6   |      - 0·4       |
    |     6     |   170   |   94·5   |   94·9   |      + 0·4       |
    |     7     |   198   |  110·5   |   110·2  |      - 0·3       |
    |     8     |   226   |  125·5   |   125·5  |        0·0       |
    +-----------+---------+----------+----------+------------------+

The dimensions of the pulleys are precisely stated because, for
pulleys of different construction, the numerical 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 _R_, 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 lbs. will be sufficient to raise a load of 198 lbs. 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 lbs.)
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
lower block were without weight, a power of 99 lbs. would have been
sufficient; but, owing to the presence of these disturbing causes,
110·5 lbs. are necessary: of this amount 1·5 lbs. is due to the weight
of the pulley, 10 lbs. is the force of friction, and the remaining 99
lbs. 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 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_ = 2·21 + 0·5453 _R_. For example,
in experiment 5, the product of 142 and 0·5453 is 77·43, to which, when
2·21 is added, we find for _P_ 79·64, very nearly the same as 80 lbs.,
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 lbs.
be attached to the moveable pulley, we find that 111 lbs. must be
applied as the power. But in order to raise 200 lbs. one foot, the
power exerted must act over two feet; hence the number of foot-pounds
required is 2 × 111 = 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 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 _velocity 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 principles already explained.
Suppose the load be 60 lbs., 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 lbs. 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 of the lower block out of consideration.

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

202. We are thus enabled to see how much of our exertion in raising
weights must be expended in merely overcoming friction, and how much
may be utilized. Suppose for example that we have to raise a weight of
100 lbs. one foot by means of the pulley-block; the power we must apply
is 25 lbs., 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 Table X.


            TABLE X.—THREE-SHEAVE PULLEY-BLOCKS.

        Sheaves cast iron 2"·5 diameter; plaited rope
          0"·25 diameter; velocity ratio 6; mechanical
          advantage 4; useful effect 67 per cent.; formula
          _P_ = 2·36 + 0·238 _R_.
    +-----------+---------+----------+----------+------------------+
    |           |         |          |    P.    |  Discrepancies   |
    | Number of |   R.    | Observed |Calculated| between observed |
    |Experiment.|  Load   |   power  |   power  |  and calculated  |
    |           | in lbs. |  in lbs. |  in lbs. |      powers.     |
    +-----------+---------+----------+----------+------------------+
    |     1     |    57   |   15·5   |   15·9   |      + 0·4       |
    |     2     |   114   |   29·5   |   29·5   |        0·0       |
    |     3     |   171   |   43·5   |   43·1   |      - 0·4       |
    |     4     |   228   |   56·0   |   56·6   |      + 0·6       |
    |     5     |   281   |   70·0   |   69·2   |      - 0·8       |
    |     6     |   338   |   83·0   |   82·8   |      - 0·2       |
    |     7     |   395   |   97·0   |   96·4   |      - 0·6       |
    |     8     |   452   |  109·0   |  109·9   |      + 0·9       |
    +-----------+---------+----------+----------+------------------+

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 lbs. by 0·238, and add 2·36 lbs. 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 lbs.; the product of
228 and 0·238 is 54·26; adding 2·36, we find 56·6 lbs. for the power
required; the actual observed power is 56 lbs., 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 difference
in no case reaches 1 lb. 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 lbs. and 452 lbs., 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 1·6 lbs. is sufficient for this
purpose. I attach a 56 lb. weight as a load, and find that 13·1 lbs.
is sufficient power for motion. This amount is partly composed 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 lbs. (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 lbs. or greater raises 56 lbs., that any power between
13·1 lbs. and 5·5 lbs. balances 56 lbs., and that any power less than
5·5 lbs. is raised by 56 lbs.

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 lbs. in a perfectly frictionless
machine, and _Y_ the force of friction. We shall be able to determine
_X_ and _Y_ by the experiments just performed. When the load is
raised a power equal to _X_ + _Y_ must be applied, and therefore
_X_ + _Y_ = 13·1. 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 _Y_ = 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 lbs. 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 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.

[Illustration: FIG. 36.]

It consists of three parts,—an upper pulley-block, a 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; it carries a hook, to 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 11"·84, while that of the small groove is 10"·36. When the upper
pulley has made one revolution, the large groove must have drawn up
11"·84 of chain, since the chain cannot slip on account of the ridges;
but in the same time the small groove has lowered 10"·36 of chain:
hence when the upper pulley has revolved once, the chain between the
two must have been shortened by the difference between 11"·84 and
10"·36, that is by 1"·48; but this can only have taken place by raising
the moveable pulley through half 1"·48, that is, through a space 0"·74.
The power has then acted through 11"·84, and has raised the resistance
0"·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 1 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 pulley descends. The load is
thus raised or lowered by simply pulling one chain A or the other D.

[Illustration: FIG. 37.]

213. We shall next consider the mechanical 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
sixteen 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 importance.

214. Placing upon the load hook a weight of 200 lbs., I find that 38
lbs. 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 lbs. I find the requisite power to be
64 lbs., which is only about 3 lbs. less than one-sixth of 400 lbs.
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.


      TABLE XI.—THE DIFFERENTIAL PULLEY-BLOCK.

        Circumference of large groove 11"·84, of small
          groove 10"·36; velocity ratio 16; mechanical
          efficiency 6·07; useful effect 38 per cent.;
          formula _P_ = 3·87 + 0·1508 _R_.
    +-----------+---------+----------+----------+---------------+
    |           |         |          |    P.    | Difference    |
    | Number of |   R.    | Observed |Calculated|of the observed|
    |Experiment.|  Load   |   power  |   power  | and calculated|
    |           | in lbs. |  in lbs. |  in lbs. |     values.   |
    +-----------+---------+----------+----------+---------------+
    |     1     |    56   |    10    |   12·3   |      + 2·3    |
    |     2     |   112   |    20    |   20·8   |      + 0·8    |
    |     3     |   168   |    31    |   29·2   |      - 1·8    |
    |     4     |   224   |    38    |   37·7   |      - 0·3    |
    |     5     |   280   |    48    |   46·1   |      - 1·9    |
    |     6     |   336   |    54    |   54·6   |      + 0·6    |
    |     7     |   392   |    64    |   63·1   |      - 0·9    |
    |     8     |   448   |    72    |   71·5   |      - 0·5    |
    |     9     |   504   |    80    |   80·0   |        0·0    |
    |    10     |   560   |    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
lbs. to the product; this rule may be expressed by the formula
_P_ = 3·87 + 0·1508 _R_. (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
lbs., 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 lbs. is to be raised; the product of 280
and 0·1508 is 42·22, to which, when 3·87 is 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 lbs. one foot 280 foot-pounds of energy would be
necessary, but in the differential pulley-block 46·09 lbs. must
be exerted for a distance of 16 feet in order to accomplish this
object. The product of 46·09 and 16 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 lbs. only 38 per cent. of the
energy applied to a differential 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 lbs.
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 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 ¹/₆th of the load, the pressure on
the axle when the power-weight has been removed is only ⁶/₇ths 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
⁶/₇ths of its previous value, while the friction on the lower pulley
remains unaltered.

We may therefore assume that the total friction is at least ⁶/₇ths 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 ⁶/₇ × 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 experiments upon a
useful mechanical power introduced by Mr. Eade under the name of the
epicycloidal pulley-block. It 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 lbs., and insert a few small
hooks into the links of the power chain in order to receive weights: 56
lbs. is sufficient to produce motion, hence the mechanical efficiency
is 5. Had there been no friction a power of 56 lbs. would have been
capable of overcoming a load of 12·5 × 56 = 700 lbs. 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.


          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 _R_.
    +-----------+---------+----------+----------+---------------+
    |           |         |          |    P.    | Difference    |
    | Number of |   R.    | Observed |Calculated|of the observed|
    |Experiment.|  Load   |   power  |   power  | and calculated|
    |           | in lbs. |  in lbs. |  in lbs. |     values.   |
    +-----------+---------+----------+----------+---------------+
    |     1     |    56   |    15    |   16·2   |    + 1·2      |
    |     2     |   112   |    27    |   26·5   |    - 0·5      |
    |     3     |   168   |    40    |   36·9   |    - 3·1      |
    |     4     |   224   |    47    |   47·2   |    + 0·2      |
    |     5     |   280   |    56    |   57·6   |    + 1·6      |
    |     6     |   336   |    66    |   68·0   |    + 2·0      |
    |     7     |   392   |    78    |   78·3   |    + 0·3      |
    |     8     |   448   |    88    |   88·6   |    + 0·6      |
    |     9     |   504   |   100    |   99·0   |    - 1·0      |
    |    10     |   560   |   110    |  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 epicycloidal 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 overcoming 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, 1" × 1" in
section and 6' long; it weighs 19 lbs. The power is produced by a 56
lb. weight: the fulcrum consists of a moderately sharp steel edge
firmly secured to the framework. 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
records the magnitude of the force produced at the short end. This is
the lever with which we shall commence our experiments.

[Illustration: FIG. 38.]

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
lbs.), 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 lbs. 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 1'; A is at one of
these divisions, 1' distant from B, and C is 5' distant from 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 lbs. being 4' from the fulcrum, while the load
is 1' from the fulcrum, it is found that the strain indicated by the
balance is 224 lbs.; 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 lbs.; 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 _the power 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 lbs.; 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.

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 allowance for friction. In this we notice a marked
difference between the lever and the pulley-blocks already described.

239. In the lever of the first order we find an excellent machine
for augmenting power. A power of 14 lbs. 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 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" × 1" × 1" 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
horizontally. The part of the bar between the fulcrum A and the power
C is divided into five portions, each 1' long, and the points A and C
are each 6" distant from the extremities of the bar. The load employed
is 56 lbs.; through the ring of this weight the bar passes, and thus
the bar supports the load. The bar is counterpoised by the weight of 19
lbs. at G, in the manner already explained (Art. 231).

243. The mode of experimenting is as follows:—Let the weight B be
placed 1' from the fulcrum; the strain shown by the spring balance
is about 11 lbs. 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 ÷ 5 = 11·2.

[Illustration: FIG. 39.]

244. If the weight be placed 2' from the fulcrum the strain is about
22·5 lbs. and it is easy to ascertain that this is the same amount as
would have been found by the application of the rule. A similar result
would have been obtained if the 56 lb. 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 lever of the first order:
that, the power multiplied by its distance from the fulcrum is equal to
the load multiplied by its distance from the fulcrum. In the present
case the load is uniformly 56 lbs., while the power by which it is
sustained is always less than 56 lbs.

[Illustration: 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 square.
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. At a point B on this lever, 1"·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 0"·25 square between the jaws of the shears
in the position E, the distance D E being 3"·5, 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 lbs., 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 × 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 lbs. at F produces a pressure of 279
lbs. 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 lbs. is not sufficient to shear across the bar
of iron, even though it be magnified to 279 lbs. I therefore suspend
weights from F, and gradually increase the load until the bar is cut. I
find at the first trial that 112 lbs. is sufficient, and a second trial
with the same bar gives 114 lbs.; 113 lbs., the mean between these
results, may be considered an adequate force. This is the load on F;
the real pressure on the bar is 113 × 27·9 = 3153 lbs.: 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 lbs. 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 16 × 3153 lbs., 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 understood 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 lb. weight, while the load is indicated by the
spring balance. The power always exceeds the load, and consequently
this lever is to be used where speed is to be gained instead of power.
Thus, for example; when the power, 56 lbs., is 2' distant from the
fulcrum, the load indicated by the spring balance is about 23 lbs.

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.

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 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 12 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 1 lb. 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 ¹/₁₂ of a
foot, and therefore a load of 12 lbs. 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.

[Illustration: FIG. 41.]

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 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 1
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 0·1 lb. and 0·01
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, and 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
O·45 lb. R is just pulled up; and on the other hand, when P is only
0·40 lb. 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 lb., and the friction 0·025 lb. For
when P raises R, it must overcome friction 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 = 0·40; and R is thus 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 0·66 and 0·71; therefore R is 0·685 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 constant at these inclinations. (Were
the friction large, other methods are necessary, see Art. 265.) As in
the experiments 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 1 lb. 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', 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 lb. acting
over 4' produces two units of work. In general, if _l_ 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 _P l_: hence
_R h_ = _P l_, and consequently _P_ : _h_ :: _R_ : _l_; that is, the
power is to the height of the plane as the load is to its length. In
the present case _R_ = 1 lb., _l_ = 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 1 lb. in
               weight, friction counterpoised; formula
               _P_ = 0·0208 × _h_".
    +-----------+------+--------+-------------+-----------------------+
    | Number of |Height|Observed|     P.      |   Difference of the   |
    |Experiment.|  of  |  power | Calculated  |observed and calculated|
    |           |plane.| in lbs.|power in lbs.|       powers          |
    +-----------+------+--------+-------------+-----------------------+
    |     1     |  2"  |  0·04  |     0·04    |          0·00         |
    |     2     |  4"  |  0·08  |     0·08    |          0·00         |
    |     3     |  6"  |  0·13  |     0·12    |         -0·01         |
    |     4     |  8"  |  0·16  |     0·17    |         +0·01         |
    |     5     | 10"  |  0·21  |     0·21    |          0·00         |
    |     6     | 15"  |  0·31  |     0·31    |          0·00         |
    |     7     | 20"  |  0·42  |     0·42    |          0·00         |
    |     8     | 33"  |  0·71  |     0·69    |         -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 0·31 lb. The
truck is placed in the middle of the plane, and the power is adjusted
so as to be sufficient to draw the truck to the top with certainty; the
necessary power calculated by the formula is also 0·31 lbs., 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 variation 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. 117. 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 17°·2, so that its length, base, and height are in
the proportions of the numbers 1, 0·955, and 0·296. Weights ranging
from 7 lbs. to 56 lbs. 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 overcome friction.

[Illustration: FIG. 42.]

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 component acting down
the plane, and R M the pressure against the plane; the triangle G L R
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 proportional, 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 lbs., the best value for the coefficient of friction is 0·288 (Art.
141): hence the amount of friction upon the plane is

    0·288 × 0·955 _R_ = 0·275 _R_.

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_.

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" × 11"; angle of
             inclination 17°·2; slide of pine, grain
             crosswise; slide started; formula
             _P_ = 0·571 _R_.
    +-----------+----------+----------+----------+--------------+
    |           |    R.    |Power in  |     P.   |Difference of |
    |Number of  |Total load|lbs. which|Calculated| the observed |
    |Experiment.| on slide |just draws| value of |and calculated|
    |           | in lbs.  | up slide.|the power.|    powers.   |
    +-----------+----------+----------+----------+--------------+
    |     1     |    7     |    4·6   |   4·0    |    -0·6      |
    |     2     |   14     |    8·3   |   8·0    |    -0·3      |
    |     3     |   21     |   12·3   |  12·0    |    -0·3      |
    |     4     |   28     |   16·5   |  16·0    |    -0·5      |
    |     5     |   35     |   20·0   |  20·0    |     0·0      |
    |     6     |   42     |   24·2   |  24·0    |    -0·2      |
    |     7     |   49     |   28·0   |  28·0    |     0·0      |
    |     8     |   56     |   31·8   |  32·0    |    +0·2      |
    +-----------+----------+----------+----------+--------------+

271. Thus for example, in experiment 6, a load of 42 lbs. was raised
by a force of 24·2 lbs., while the calculated value is 24·0 lbs.; the
difference, 0·2 lbs., 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. 1 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 lbs. is expressed by the formula _F_ = 0·9 + 0·266 × pressure, but
the pressure is in this case = 0·955 _R_, and hence the friction is

    0·9 + 0·254 _R_.

To this must be added 0·296 _R_, the component of the force of gravity
which must be overcome, and hence the total force necessary is

    0·9 + 0·55 _R_.

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


                  TABLE XV.—INCLINED PLANE.

        Smooth plane of pine 72"× 11"; angle of inclination
          17°·2; slide of pine, grain crosswise; slide
          started; formula _P_ = 0·9 + 0·55 _R_.
    +-----------+----------+----------+----------+--------------+
    |           |    R.    |Power in  |     P.   |Difference of |
    |Number of  |Total load|lbs. which|Calculated| the observed |
    |Experiment.| on slide |just draws| value of |and calculated|
    |           | in lbs.  | up slide.|the power.|    powers.   |
    +-----------+----------+----------+----------+--------------+
    |     1     |     7    |    4·6   |    4·7   |    +0·1      |
    |     2     |    14    |    8·3   |    8·6   |    +0·3      |
    |     3     |    21    |   12·3   |   12·5   |    +0·2      |
    |     4     |    28    |   16·5   |   16·3   |    -0·2      |
    |     5     |    35    |   20·0   |   20·1   |    +0·1      |
    |     6     |    42    |   24·2   |   24·0   |    -0·2      |
    |     7     |    49    |   28·0   |   27·8   |    -0·2      |
    |     8     |    56    |   31·8   |   31·7   |    -0·1      |
    +-----------+----------+----------+----------+--------------+

For example: in experiment 5, a load of 35 lbs. is found to be
raised by a power of 20·0 lbs., while the calculated power is
0·9 + 0·55 × 35 = 20·1 lbs.

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 O·3 lb. 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 lbs., we calculate by the formula that 17·4 lbs.
would be sufficient to raise it, so that, 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 30 ÷ 17·4 = 1·72.

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 1 ÷ 0·296 = 3·38. To raise 30 lbs. one foot,
a force of 17·4 lbs. must therefore be exerted through 3·38 feet. The
number of units of work expended is thus 17·4 × 3·38 = 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 13°·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 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 backwards
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 described 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 experiments 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 lbs.; 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.

[Illustration: FIG. 43.]

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 1"
the power moves 3 × 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 efficiency. 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, the arm is steadily drawn; but the power will only move the arm
a few inches, for when the cord ceases to be perpendicular 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
          1"·25, with 3 threads to the inch, length of
          arm 10"·25; nut of cast iron, bearing surfaces
          oiled, velocity ratio 193, useful effect 36
          per cent., mechanical efficiency 70; formula
          _P_ = 0·0143 _R_.
    +-----------+-------+-------------+-------------+----------------+
    |           |       |             |     P.      | Difference of  |
    | Number of |  R.   |  Observed   | Calculated  |the observed and|
    |Experiment.| Load  |    power    |   power     |   calculated   |
    |           |in lbs.|    in lbs.  |  in lbs.    |     powers.    |
    +-----------+-------+-------------+-------------+----------------+
    |    1      |    28 |     0·4     |     0·4     |      0·0       |
    |    2      |    56 |     0·8     |     0·8     |      0·0       |
    |    3      |    84 |     1·2     |     1·2     |      0·0       |
    |    4      |   112 |     1·6     |     1·6     |      0·0       |
    |    5      |   140 |     2·0     |     2·0     |      0·0       |
    |    6      |   168 |     2·4     |     2·4     |      0·0       |
    |    7      |   196 |     2·7     |     2·8     |     +0·1       |
    |    8      |   224 |     3·3     |     3·2     |     -0·1       |
    +-----------+-------+-------------+-------------+----------------+

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 lbs. is the power with a start, when
without a start 3·2 lbs. 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 lb. The fourth
column contains the values of the powers computed by the formula
_P_ = 0·0143 _R_. This formula has been deduced from the observations
in the 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 0·1 lb., but this is quite inconsiderable in
comparison with the weights we are employing.

284. In order to lift 100 lbs. the expression 0·0143 _R_ shows that
1·43 lbs. would be necessary: hence the mechanical efficiency of the
screw is 100 ÷ 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 compact, 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 of 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; therefore, to raise
100 lbs. 1 foot, we find that 1·43 × 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 weights through considerable distances, and the
former for exerting pressures through short distances.

[Illustration: FIG. 44.]


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 pressures, 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 pressure of 1 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 consideration. 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.

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_ = 0·66 + 0·0075 _R_.
    +-----------+--------+---------+----------+-----------------+
    |           |        |         |   P.     |Difference of the|
    | Number of |  R.    |Observed |Calculated|   observed and  |
    |Experiment.| Load   |  power  |   power  |    calculated   |
    |           |in lbs. | in lbs. |  in lbs. |       powers.   |
    +-----------+--------+---------+----------+-----------------+
    |     1     |   112  |   1·4   |    1·5   |       +0·1      |
    |     2     |   224  |   2·2   |    2·3   |       +0·1      |
    |     3     |   336  |   3·3   |    3·2   |       -0·1      |
    |     4     |   448  |   4·1   |    4·0   |       -0·1      |
    |     5     |   560  |   5·0   |    4·9   |       -0·1      |
    |     6     |   672  |   5·7   |    5·7   |        0·0      |
    |     7     |   784  |   6·5   |    6·5   |        0·0      |
    |     8     |   896  |   7·4   |    7·4   |        0·0      |
    |     9     |  1008  |   8·1   |    8·2   |       +0·1      |
    |    10     |  1120  |   9·0   |    9·1   |       +0·1      |
    +-----------+--------+---------+----------+-----------------+

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 lbs., may be calculated by
this formula; it is 0·66 + 0·0075 × 600 = 5·16. Hence the mechanical
efficiency of the screw-jack is 600 ÷ 5·16 = 116. Thus the screw
is very powerful, increasing the force applied to it more than a
hundredfold. In order to raise 600 lbs. one foot, a quantity of work
represented by 5·16 × 414 = 2136 units must be expended; 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 lbs. suffice to raise 1,120 lbs.; 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 lbs. must
be applied. Hence, even with the assistance of the load, a force of
3·4 lbs. 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 overcome both
X and Y, and therefore we have X + Y = 9·0. 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
lbs., and we have therefore X + 3·4 = Y, and hence X = 2·8, Y = 6·2.

That is, 2·8 is the amount of force which with a frictionless 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 = 2·7, which is within 0·1 lb. of the value of X.
The agreement of these results is satisfactory.


THE SCREW BOLT AND NUT.

[Illustration: FIG. 45.]

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.




                           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 lectures 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, 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.

[Illustration: FIG. 46.]

299. The wheel and axle assumes very many forms corresponding to
the various purposes to which it is applied. 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 arrangement 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 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 efficiency 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 circumference of the wheel. The axle is 0"·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 circumference 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 2"·87.

306. We can now ascertain the velocity ratio in this machine. When the
wheel and axle have made one complete revolution the power has been
lowered through a distance of 88"·5, and the load has been raised
through 2"·87. This is evident because the wheel and axle are attached
together, and therefore each completes one revolution 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 ÷ 2"·87 = 31 very nearly. We
shall therefore 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 lbs. 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 lbs. applied
at E is just sufficient to raise D. We infer from this result that
the mechanical efficiency of this machine is 56 ÷ 2·6 = 21·5. I add a
second 56 lb. weight to the load, and I find that a power of 5·0 lbs.
raises the load of 112 lbs. The mechanical efficiency in this case is
112 ÷ 5·5 = 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 applied. Let us suppose a
load of 100 lbs. is to be raised one foot; a force of 100 ÷ 22 = 4·6
lbs. will suffice to raise this load. This force must be exerted
through a space of 31', and consequently 31 × 4·6 = 143 units of energy
must be expended; of this amount 100 units are usefully employed, and
therefore the percentage of energy utilized is 100 ÷ 143 × 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 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 lbs.; 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_.
    +-----------+---------+----------+----------+--------------+
    |           |         |          |    P.    |Difference of |
    | Number of |    R.   | Observed |Calculated| the observed |
    |Experiment.|  Load   |  power   |  power   |and calculated|
    |           | in lbs. |  in lbs. |  in lbs. |    values.   |
    +-----------+---------+----------+----------+--------------+
    |     1     |    28   |    1·4   |    1·4   |      0·0     |
    |     2     |    42   |    2·0   |    2·0   |      0·0     |
    |     3     |    56   |    2·6   |    2·6   |      0·0     |
    |     4     |    70   |    3·2   |    3·2   |      0·0     |
    |     5     |    84   |    3·7   |    3·8   |    + 0·1     |
    |     6     |    98   |    4·4   |    4·4   |      0·0     |
    |     7     |   112   |    5·0   |    5·0   |      0·0     |
    +-----------+---------+----------+----------+--------------+

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

    _P_ = 0·204 + 0·0426 _R_.

311. Thus for example in experiment 5 a load of 84 lbs. was found to be
raised by a power of 3·7 lbs. The value calculated by the formula is
0·204 + 0·0426 × 84 = 3·8. The calculated value only differs from the
observed value by 0·1 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 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 axle 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 proportion 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 artificial; 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 transmission 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 feet 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 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.

[Illustration: FIG. 47.]

316. We can examine experimentally the effect of friction on the axle
by the apparatus of Fig. 47. B is a shaft 0"·75 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 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 lbs.—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 lbs., when E
descends. We thus see that to raise 56 lbs. an excess of 29 lbs. 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 lbs. at
F, when it is found that a power of 43 lbs. at E is required to lift
it. Had the power been 42 lbs., 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 lbs. 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 circumference 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 times
about the barrel, and let the ends hang down as before. I then attach
to each end 56 lbs. weight, and I find that 10 lbs. 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 lbs. is
sufficient to overcome the friction which required 29 lbs. 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 circumference of the wheel is 88"·5;
the circumference of the barrel is 14"·9. The proper mode of finding
the circumference 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 determined by
experiment. I suspend a weight of 56 lbs. from the hook, and apply
power to the wheel. I find that 10·1 lbs. is just sufficient to raise
the load.

322. The mechanical efficiency is to be found by dividing 10·1 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 percentage of applied
energy which is utilized. Let us suppose a weight of 100 lbs. has to
be raised one foot: for this purpose a force of 100 ÷ 5·54 = 18·1
lbs. 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 18·1, 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 × 100 = 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 14"·9
          circumference; axle is of wrought iron, 0"·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_.
    +-----------+--------+----------+------------+----------------+
    |           |        |          |     P.     | Difference of  |
    | Number of |   R.   | Observed | Calculated |  the observed  |
    |Experiment.|  Load  |  power   |    power   | and calculated |
    |           | in lbs.|  in lbs. |   in lbs.  |     values.    |
    +-----------+--------+----------+------------+----------------+
    |     1     |   14   |    2·7   |     2·9    |     + 0·2      |
    |     2     |   28   |    5·3   |     5·2    |     - 0·1      |
    |     3     |   42   |    7·7   |     7·6    |     - 0·1      |
    |     4     |   56   |   10·1   |    10·0    |     - 0·1      |
    |     5     |   70   |   12·4   |    12·4    |       0·0      |
    |     6     |   84   |   14·7   |    14·7    |       0·0      |
    |     7     |   98   |   17·1   |    17·1    |       0·0      |
    |     8     |  112   |   19·4   |    19·5    |     + 0·1      |
    +-----------+--------+----------+------------+----------------+

The formula which represents the experiments with the greatest amount
of accuracy is _P_ = 0·5 + 0·169 _R_. 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
10-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 wheels upon two parallel
shafts, so that they gear one into the other 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 0"·75 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 lbs. to the axle of the large wheel, it is observed that a
power of 13·7 lbs. 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 will 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 lbs. from the load hook, 26 lbs. is just enough to
raise it; the mechanical efficiency that would be deduced from this
result is 112 ÷ 26 = 4·3, which is slightly in excess of the amount
obtained by the former experiment. It is often found to be a property
of the mechanical powers, that as the load increases the mechanical
efficiency slightly improves.

329. In Table XX. will be found a record of experiments upon 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 (10-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 + 0·21 _R_.
    +-----------+---------+---------+-----------+---------------+
    |           |         |         |    P.     | Difference of |
    | Number of |    R.   |Observed |Calculated |  the observed |
    |Experiment.|  Load   | power   |   power   |and calculated |
    |           | in lbs. | in lbs. |  in lbs.  |     powers.   |
    +-----------+---------+---------+-----------+---------------+
    |     1     |    14   |   5·4   |    5·4    |       0·0     |
    |     2     |    28   |   8·7   |    8·3    |     - 0·4     |
    |     3     |    42   |  11·0   |   11·3    |     + 0·3     |
    |     4     |    56   |  13·7   |   14·2    |     + 0·5     |
    |     5     |    70   |  17·5   |   17·2    |     - 0·3     |
    |     6     |    84   |  20·0   |   20·1    |     + 0·1     |
    |     7     |    98   |  23·0   |   23·0    |       0·0     |
    |     8     |   112   |  26·0   |   26·0    |       0·0     |
    +-----------+---------+---------+-----------+---------------+

330. The large amount of friction present in this contrivance is the
consequence of winding the rope directly upon the axle instead of upon
a barrel, as already pointed out in Art. 319. 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 (Art. 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 represented 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 14"·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 × 43" = 344", in order to raise the load 15"·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 1', when it appears
that the number of revolutions of the wheel B is such that the power
must have moved 23'.

[Illustration: FIG. 48.]

334. The mechanical efficiency is to be found as usual by trial. 56
lbs. placed at F is raised by 3·1 lbs. at G; hence the mechanical
efficiency deduced from this experiment is 56 ÷ 3·1 = 18. 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 ÷ 200;
          circumference of drum on which rope is wound,
          14"·9; velocity ratio, 23; mechanical
          efficiency, 18; useful effect, 78 per cent.;
          formula, _P_ = 0·0556 _R_.
    +-----------+--------+--------+----------+--------------+
    |           |        |        |   P.     |Difference of |
    | Number of |   R.   |Observed|Calculated| the observed |
    |Experiment.|  Load  | power  |  power   |and calculated|
    |           | in lbs.| in lbs.| in lbs.  |    values.   |
    +-----------+--------+--------+----------+--------------+
    |     1     |   14   |   0·9  |    0·8   |     -0·1     |
    |     2     |   28   |   1·6  |    1·6   |      0·0     |
    |     3     |   42   |   2·4  |    2·3   |     -0·1     |
    |     4     |   56   |   3·1  |    3·1   |      0·0     |
    |     5     |   70   |   3·8  |    3·9   |     +0·1     |
    |     6     |   84   |   4·5  |    4·7   |     +0·2     |
    |     7     |   98   |   5·3  |    5·5   |     +0·2     |
    |     8     |  112   |   6·2  |    6·2   |     +0·0     |
    +-----------+--------+--------+----------+--------------+

335. A series of experiments made with this crane is recorded in Table
XXI., and a comparison of the calculated and observed values will show
that the formula _P_ = 0·0556 _R_ represents the experiments with
considerable accuracy.

336. It may be noticed that in this formula the term independent of
_R_, which we frequently meet with in the expression 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 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 ÷ 200 × 30 ÷ 180;
          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_.
    +-----------+--------+--------+----------+--------------+
    |           |        |        |   P.     |Difference of |
    | Number of |   R.   |Observed|Calculated| the observed |
    |Experiment.|  Load  | power  |  power   |and calculated|
    |           | in lbs.| in lbs.| in lbs.  |    values.   |
    +-----------+--------+--------+----------+--------------+
    |     1     |   14   |  0·30  |   0·29   |    -0·01     |
    |     2     |   28   |  0·40  |   0·40   |     0·00     |
    |     3     |   42   |  0·50  |   0·51   |    +0·01     |
    |     4     |   56   |  0·60  |   0·62   |    +0·02     |
    |     5     |   70   |  0·75  |   0·73   |    -0·02     |
    |     6     |   84   |  0·85  |   0·84   |    -0·01     |
    |     7     |   98   |  0·95  |   0·95   |     0·00     |
    |     8     |  112   |  1·05  |   1·06   |    +0·01     |
    +-----------+--------+--------+----------+--------------+

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
lbs. 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 _combination_ 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.

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 inconvenient, and indeed
impracticable, and the additional circumstance 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 application 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 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 elementary 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 remembered 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 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 composed.

349. Timber may be softened by steaming. I have here a rod of pine,
24" × 0"·5 × 0"·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 together and twisted, but
the fibres are not coherent as they are in wood. Hence we find that a
rod of wood is stiff, while 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" × 1" × 1". One of
them I easily snap across with a blow, while my blows are unable to
break the other. The difference 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 extent 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 resistance 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
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, 1" × 0"·5 and 0"·5 × 0"·5 being generally used,
but sometimes 1" × 1" is employed.

[Illustration: FIG. 49.]

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" × 0".5 × 0"·5, Fig. 49. Each end of this
rod is firmly secured between two cheeks of iron, which are bolted
together: the rod is suspended by its upper extremity from the hook of
the epicycloidal pulley-block (Art. 213), which is itself supported 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 insignificant extent.

354. Let us contrast the resistance of a rod of timber to extension
with the effect upon a rope under the same circumstances. I have here
a rope about 0".25 diameter; it is suspended from a point, and bears a
14 lb. 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 1
cwt., and on measurement I find that the two points which before were
2' apart, are now 2' 2"; thus the rope has stretched at the rate of an
inch per foot for a strain of 1 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
“tie.” 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 experiment, 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 advantage 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 lbs. on the pillar at D. The fulcrum is at A and guides at
G. The lever and the tray would somewhat complicate our 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 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

[Illustration: FIG. 50.]

358. The first experiment we shall make with this apparatus is upon a
pine rod 40" long and 0"·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 lbs. in the tray: this produces a pressure of 84 lbs. upon
the rod, but the rod still remains perfectly straight, so that it bears
this pressure easily. When the pressure is increased to 96 lbs. a very
slight amount of deflection may be seen. When the strain reaches 114
lbs. 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
0"·25. Gradually augmenting the pressure, I find that when it reaches
132 lbs. the deviation has reached 0"·5; and finally, when 48 lbs. is
placed in the tray, that is, when the rod is subjected to 144 lbs., it
breaks across the middle. Hence we see that this rod sustained a load
of 96 lbs. without sensibly bending, but that fracture ensued when the
load was increased about half as much again. Another experiment with
a similar rod gave a slightly less value (132 lbs.) for the breaking
load. If I add these results together, and divide the sum by 2, I
find 138 lbs. 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 0"·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 126
lbs.; a very slight amount of bending is noticeable with 186 lbs.;
with 228 lbs., the amount by which the centre of the rod has deviated
laterally from its original position is about 0"·2; and finally, when
the load reaches 294 lbs., 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 1" 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" × 1" × 1" 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 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.

[Illustration: FIG. 51.]

For convenience in recording the experiments the tray chain and hooks
have been adjusted to weigh exactly 14 lbs. (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 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 position can be accurately determined.
The cathetometer is levelled by means of the screws H H, so that the
triangular 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.

[Illustration: FIG. 52.]

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
distinctly. 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 0"·19, 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.


     TABLE XXIII.—DEFLECTION OF A BEAM.

    A rod of pine 48" × 1" × 1"; resting freely on
       supports 40" apart; and laden in the middle.
    +-------------+-----------+-------------+
    | Number of   | Magnitude | Deflection. |
    | Experiment. | of load.  |             |
    +-------------+-----------+-------------+
    |    1        |    14     |    0"·19    |
    |    2        |    28     |    0"·37    |
    |    3        |    42     |    0"·55    |
    |    4        |    56     |    0"·74    |
    |    5        |    70     |    0"·94    |
    |    6        |    84     |    1"·13    |
    |    7        |    98     |    1"·35    |
    |    8        |   112     |    1"·61    |
    |    9        |   126     |    1"·95    |
    |   10        |   140     |    2"·37    |
    +-------------+-----------+-------------+

365. The first column records the number of the experiment. The
second represents the load, and the third contains the corresponding
deflections. It will be seen that up to 98 lbs. the deflection is about
0"·2 for every stone weight, but afterwards the deflection increases
more rapidly. When the weight reaches 140 lbs. 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; 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" × 1" × 1" 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 lbs. 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 lbs.; the beam was, of course, considerably deflected, but when
the weight was removed the cross again returned,—at all events, to
within 0"·01 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 borne a strain which,
including the tray, amounted to 70 lbs. But when the beam has been
made to carry 84 lbs. 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 0"·03. This is still
more apparent after the beam has carried 98 lbs., for when this load is
removed the centre of the beam is permanently deflected by 0"·13. Here,
then, we may infer that the fibres of the beam are beginning to be
strained beyond their powers of resistance, and this is 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" × 1" × 1", 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 81 lbs.

If I were to place the second beam on the same supports with the cut
upwards, then there can be no doubt that it 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 31 lbs. 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" × 0"·5 × 1", placed with its
shortest dimension vertical. If we remember that an entire beam of the
same size required about 140 lbs. 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 0"·5 deep in the middle of
a pine rod 48" × 1" × 1". These cuts are 0"·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 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 lbs. 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 lbs. 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 consequently 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" × 1" × 1" have been broken by a
strain of 140 lbs. (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
140 lbs. 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 lbs. is changed into two other forces,
each of them enormously greater and sufficiently strong to rupture
the fibres. We need not suppose that 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 rectangular, as the material can be more
effectively disposed 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 1' in diameter is 6" broad and 10"·5 deep, while the
strongest beam is 7" broad and 9"·75 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 1' to 4', and of three
different sections, viz. 1" × 1", 1" × 0"·5, and 0"·5 × 0"·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 1' 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 are 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.


               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;

                             area of section × depth
         Formula, _P_ = 6080 ————————————————————————
                                     span

    +-------+---------------------+------------+----------+------------+
    |       |                     |Mean of the |    P.    |Difference  |
    |       |      Dimensions.    |observations|Calculated|  of the    |
    |No. of +-----+--------+------+  of the    | breaking |observed and|
    |Experi-|     |        |      |  breaking  |   load   | calculated |
    |ment.  |Span.|Breadth.|Depth.|load in lbs.| in lbs.  |   values.  |
    +-------+-----+--------+------+------------+----------+------------+
    |   1   |40"·0|  1"·0  | 1"·0 |    152     |   152    |    0·0     |
    |   2   |40"·0|  0"·5  | 1"·0 |     77     |    76    |   -1·0     |
    |   3   |40"·0|  1"·0  | 0"·5 |     38     |    38    |    0·0     |
    |   4   |40"·0|  0"·5  | 0"·5 |     19     |    19    |    0·0     |
    |   5   |30"·0|  1"·0  | 0"·5 |     59     |    51    |   -8·0     |
    |   6   |30"·0|  0"·5  | 0"·5 |     25     |    25    |    0·0     |
    |   7   |20"·0|  1"·0  | 0"·5 |     74     |    76    |   +2·0     |
    |   8   |20"·0|  0"·5  | 0"·5 |     36     |    38    |   +2·0     |
    |   9   |10"·0|  1"·0  | 0"·5 |    154     |   152    |   -2·0     |
    |  10   |10"·0|  0"·5  | 0"·5 |     68     |    76    |   +8·0     |
    +-------+-----+--------+------+------------+----------+------------+

381. In the first column is a series of figures for convenience 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 two
observations of the breaking load. Thus for example, in experiment
No. 5 the two beams used were each 36" × 1" × 0"·5, they were placed
on points of support 30" distant, so the span recorded is 30": one of
the beams was broken by a load of 58 lbs., and the second by a load
of 60 lbs.; the mean between the two, 59 lbs., 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 0"·5 × 0"·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
lbs. and 36 lbs.; the former of these numbers is rather more than half
of the latter. In fact, had the breaking load of 40" been ¾ lb. less,
18·25 lbs., and had that of 20" been ½ lb. more, 36·5 lbs., 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 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 lbs.; on reference to the table we find 25 lbs. 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
lbs. less; this does not agree very well with the theory, but still
the difference, though 8 lbs., is only about 11 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, 1" broad, and 0"·5 deep, broke with 38 lbs.; and in
experiment No. 7 a 20" bar of the same section broke with 74 lbs.; 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 40" × 1" × 0"·5 is broken by a load of 38 lbs.,
while a bar just half the breadth is broken by 19 lbs. 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
lbs., is more than 25, the observed breaking weight of No. 6: a similar
remark may be made about Nos. 9 and 10.

391. Supposing we had a beam of 40" span, 2" broad, and 0"·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 two
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 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
1"·0 × 0"·5, but the first bar is broken edgewise, and the second
flatwise. The first breaks with 77 lbs., and the second with 38 lbs.;
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 dimension of the section is to the
lesser dimension_.

394. The strength of a beam 40" × 0"·5 × 1" is four times as great as
the strength of 40" × 0"·5 × 0"·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 lbs., whereas if
the same rods were in one piece, the breaking load would be nearly 80
lbs.

395. This may be illustrated in a different manner. I have here two
beams of 40" × 1" × 0"·5 superposed; they form one beam, equivalent to
that of No. 1 in bulk, but I find that they break with 80 lbs., 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 clamps like those
represented in Fig. 56. I now find that the bars thus fastened together
require 104 lbs. 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 lbs. would be required to
produce fracture.

397. We shall be able by a little consideration to understand 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 1" 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 1" broad and 7" deep, is 7 times as strong as a beam 7"
broad, 1" deep (Art. 393). Hence the original beam must be 35 times as
strong as a beam 7" broad, 1" deep; but the beam 7" broad and 1" deep
is seven times stronger than a beam the section of which is 1" × 1",
hence the original beam is 245 times as strong as a beam 12' long and
1" × 1" in section; of which we can calculate the strength, by Art.
388, from the proportion—

    144" : 40" :: 152 : Answer.

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

399. It will be useful to deduce the _general_ expression for the
breaking load of a beam _l"_ span, _b"_ broad, and _d"_ deep, supported
freely at the ends and laden in the centre.

Let us suppose a bar _l"_ long, and 1" × 1" in section. The breaking
load is found by the proportion—

    _l_ : 40 :: 152 : Answer;

and the result obtained is 6080/_l_. A beam which is _d"_ broad,
_l"_ span, and 1" deep, would be just as strong as _d_ of the beams
_l"_ × 1" × 1" placed side by side; of which the collective strength
would be—

     6080
   ———————— × _d_.
      _l_

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

     6080
   ————————— × _d_².
      _l_

We thus learn the strength of a beam 1" broad, _d"_ deep, and _l"_
span. The strength of _b_ of these beams placed side by side, would be
the same as the strength of one beam _b"_ broad, _d"_ deep, and _l"_
span, and thus we finally obtain

     6080
   ———————— _d_² × _b_.
      _l_

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

                      area of section × depth;
              6080 ×  —————————————————————————
                                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,
1", depth 0"·5; the formula gives, since the area is 0"·5,

                             0·5 × 0·5
                 _P_ = 6080 —————————— = 76
                                 20

This agrees sufficiently with 74 lbs., 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 which is part of a permanent structure
with more than about a third or a fourth of the breaking weight. Hence
if we wanted to calculate a fair working load in lbs. for a beam of
pine, we might obtain it from the formula.

           area of section × depth.
    1500 × —————————————————————————
                   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 imperfections 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 warehouse 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.

406. We shall employ, in the first place, a beam 40" span, 0"·5 broad,
and 1" deep; and we shall break it by applying a load simultaneously at
two points, as may be most conveniently done by the contrivance shown
in the diagram, Fig. 53. A B is the beam resting on 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 divided equally between the two points C
and D, and we may regard A B as a beam loaded at its two points of
trisection. The tray and weights are employed which we have used in the
apparatus represented in Fig. 58.

[Illustration: FIG. 53.]

407. We proceed to break this beam. Adding weights to the tray, we see
that it yields with 117 lbs., 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 77 lbs. at the centre; now ³/₂ × 77 = 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 fracture is equally likely to occur at any
point between C and D.

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.

[Illustration: FIG. 54.]

In an experiment actually tried, a beam 40" × 0"·5 × 1" placed edgewise
was found to support ten 14 lb. 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 considerably stronger than would
be indicated by the results in Table XXIV.


EFFECT OF SECURING THE ENDS OF 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" × 1" × 1". We clamp each of the
ends and then break the beam by a weight suspended from the centre. It
requires 238 lbs. to accomplish fracture. This is a little more than
half as much again as 152 lbs., 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 circumstances, 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" × 0"·5 × 0"·5, and we find that, when W reaches 10 lbs., the beam
breaks. In experiment No. 8, Table XXIV., a similar beam required 36
lbs.; hence we see that 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.

[Illustration: 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 0"·5 × 0"·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 produced may be seen from Figs. 58 and 62.

[Illustration: FIG. 56.]

416. The following experiment shows the tenacity with which these
clamps hold. Two slips of pine, each 12" × 0"·5 × 0"·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 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
constructions. 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 20" 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 diagrams used in this lecture. Were A B unsupported except
at its end A, it would of course break when a weight of 10 lbs. 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 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 lbs. 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.

[Illustration: FIG. 57.]

420. The actual magnitudes of these can be computed. Draw the
parallelogram C D E B; if B D represent 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, A B C.
In the present case

    A B = 20", A C = 18", B C = 27";

therefore, when W is 112 lbs., we calculate that the force on A B is
124 lbs., and on C B 168 lbs. A B would require about 300 lbs. to
crush it, and C B about 2,000 lbs. to tear it asunder, consequently
the tie and strut can support 1 cwt. with ease. If, however, W were
increased to about 270 lbs., the force on A B would become too great,
and fracture would arise from the collapse of 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 lbs. When W is 270 lbs. the compression
on A B is 300 lbs., and the tension on B C is 405 lbs., 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 inducement 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.

[Illustration: FIG. 58.]

It consists of two beams, A B, 4' long, placed parallel to each other
at a distance of 3"·5, and supported at each end; they are firmly
clamped to the supports, and a roadway of short pieces is laid upon
them. At the points of 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 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 explain 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 11 stone weights
along the bridge, and the cathetometer shows that the deflection is
only 0"·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. 1 cwt.
being the load, I see that the deflection amounts to 0"·2; with 2 cwt.
the deflection reaches 0·43"; and the bridge breaks with 238 lbs.

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" × 0"·5 × 0"·5 is broken by a load of 19 lbs.: hence the beams of
the bridge would have been broken by a load of 38 lbs. 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. We
may, however, be confident that about 80 lbs. would have broken the
unsupported bridge. The strength is, therefore, increased threefold by
the struts, for a load of 238 lbs. 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 lbs. applied near the centre is necessary to
break this bridge, it follows from the principle 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 11 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, 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
determined 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.

[Illustration: FIG. 59.]

We have here two horizontal rods, 48" × 0"·5 × 0"·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 to break a
piece C C´, which is secured at the ends; the rod A A´ would have been
broken with 38 lbs., hence 190 lbs. 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 lbs. 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
0"·1, 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 0"·15; with 4 cwt. the deflection
amounts to 0'·72. We therefore infer that the bridge is beginning to
yield, and the clamps give way when the load is increased to 500 lbs.


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 40" × 0"·5 × 0"·5; it is trisected in
the points B and C, from which points the ties B E and C F are secured
to the upper parts of the framework. 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 that the bridge
has two beams we know it would require about 70 lbs. or 80 lbs. to
produce fracture without the ties, and therefore we might expect that
over 200 lbs. 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 lbs.: this is somewhat less than the amount we had
calculated; the reason being, I think, that one of the clamps slipped
before fracture.

[Illustration: FIG. 60.]


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 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.

[Illustration: FIG. 61.]

434. Let A B (Fig. 61) be a rod of pine 40" × 0"·5" × 0"·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 lbs. is sufficient; much less would
have done had not the ends been clamped. We have already applied a load
in this manner in Art. 406.

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 112 lbs. 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.

[Illustration: FIG. 62.]

436. Two frames of this kind, with a roadway laid between them, would
form a bridge, or if the frames were turned upside 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.


THE WYE BRIDGE.

437. An instructive bridge was erected by the late Sir I. Brunel over
the Wye, for the purpose of carrying a railway. 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" × 0"·5 × 0"·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.

[Illustration: FIG. 63.]

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. Even if there were actual joints at the corners, it would be
almost impossible 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 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 is divided into 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 extend 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 versâ_. 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 framework should be composed of triangles.

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 represented
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.


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 proportional 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.

451. I take here a piece of cast iron, which is 2' long, and
0"·5 × 0"·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 tray, I find that with 280 lbs. the crash
comes.

452. Let us compare this result with No. 8 of Table XXIV. (p. 190).
There we find that a piece of pine, the same size as the cast iron,
was broken with 36 lbs.: 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 1" × 1" 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 0"·5 × 0"·5 of section,
would require 20 × 280 ÷ 12 = 467 lbs. by the law that the strength
is inversely as the length. We also know that one beam 12" × 1" × 1"
is just as strong as two beams 12" × 1" × 0"·5, each placed edgewise;
each of these latter beams is twice as strong as 12" × 1" × 0"·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 1: hence the original beam is four times as strong as one beam
12" × 1" × 0"·5 placed flatwise: but this last beam is twice as strong
as a beam 12" × 0"·5 × 0"·5, and hence we see that a beam 12" × 1" × 1"
must be 8 times as strong as a beam of 12" × 0"·5 × 0"·5, but this last
beam would require a load of 467 lbs. to break it, and hence the beam
of 12" × 1" × 1" would require 467 × 8 = 3736 lbs. to produce fracture.
This amounts to more than a ton and a half.

455. It is a rule sometimes useful to practical men that 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
employing 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 deduced 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 weight, the fibres at the bottom of the beam are
extended, the tendency being to tear them (Art. 376). 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.

[Illustration: FIG. 64.]

458. E F (Fig. 64) is the thin iron beam along the bottom of which is
the stout flange shown at C D; 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 the lower one, in
consequence of a property of cast iron. This metal is more capable of
resisting forces of compression 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 strengthening 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, 1" × 1" 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 experiment 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 lbs. produce a deflection
of 0"·51, and 42 lbs. bends down the rod through 3"·18. This is a
large deflection; and when I remove the load, the rod only returns
through 1"·78, thus showing that a permanent deflection of 1"·40 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 lbs. only produces a deflection of 0"·09,
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 lbs. 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 1 cwt. in the tray, the deflection only
reaches 0"·26, and, when the load is removed, the tube is found to be
permanently deflected by a quantity, at all events not greater than
0"·004; 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, containing more iron than the tube. We load the
tube still further by placing additional weights in the tray, and with
140 lbs. 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, however, 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, 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 extension, 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.

471. 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 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 distance 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 suspension bridge has to carry. The series of
weights thus arranged produces substantially the same effect as if
it were actually 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 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.

[Illustration: FIG. 65.]

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 exceeds 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 lbs.; 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 lbs.

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 lbs.
gives 87 lbs. 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 lbs.: a
sufficiently close approximation to the calculated tension of 87 lbs.

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 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 produced 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 lbs.; of course the
tension in the other chain is the same: and thus we find a weight of
14 lbs. has produced additional tensions of 10 lbs. each in the two
chains. With a weight of 28 lbs. at the centre we find a strain of 110
lbs. on the chain.

482. These additional weights may be regarded as analogous 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 BODY._

    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 convey a notion
of rapid motion. Such is not precisely the 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.

[Illustration: FIG. 66.]

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 cannot 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
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 lecture 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 enable 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 pulley; each of the weights is
counterbalanced by the other: so that when the two are in motion, we
may consider either as a body 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 consequences 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 themselves 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 dropping 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 experiment on a scale
sufficiently reduced to correspond with the dimensions of the lecture
room.

[Illustration: FIG. 67.]

489. The apparatus used is shown in Fig. 67. It consists 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 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 lb. and 1 lb. respectively.

490. We are now ready to perform the experiment. I break the circuit;
the two balls are disengaged simultaneously; 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 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 difference 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 two 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 not correct to define _mass_ by the introduction of the idea
of weight, because the _mass_ of a body is something 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 1 lb. at London; it is then evident that the mass of any
other piece of platinum should be expressed 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 the first law of motion, by
indirect evidence.

497. Let us apply these principles to explain the experiment 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
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 ascertain 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 pendulum. 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 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 1, 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 continue 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 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 uninterruptedly. 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.

506. We have previously shown that the space is proportional 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 × 16 = 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 OF GRAVITY IS INDEPENDENT OF THE MOTION OF THE BODY.

508. 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 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 16' 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 apparatus. 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.

[Illustration: 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 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 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 sustain 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 consequences 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 but gravity act upon the particle
for _t_ seconds, it will then be 16_t_² feet nearer the earth than it
would have been had gravity not acted.

[Illustration: FIG. 69.]

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.


HOW THE FORCE OF GRAVITY IS DEFINED.

516. From the formula

    Distance = 16_t_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, according 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 altogether 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' - 16' = 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 end of _t_ seconds a velocity
of 32_t_. 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_ = 32_t_ and _s_ = 16_t_² it is easy to
infer another very well known formula, namely, _v_² = 64_s_.


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 therefore 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 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 which the ball moves;
this curve is a parabola, of which F is the focus and the line A A the
directrix.

[Illustration: FIG. 70.]

It is a property of the parabola that the distance of any point on the
curve from the focus is equal to its perpendicular distance from the
directrix. This is shown in the figure. For example, the dotted line F
D, drawn from F to 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 projection, the path is
a parabola. This can be proved mathematically 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 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 lbs., 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.

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 locomotive 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
considerable 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.

[Illustration: FIG. 71.]

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 14 lbs.
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 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 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 without being acted upon by a force. This force
must have been 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 lb. weight from B, and attach there a weight
of 28 lbs., A remaining the same as before (14 lbs.). 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 lbs.

532. Finally, let us remove the 28 lbs. from B, put on 56 lbs., and
perform the experiment again; you see that even the 56 lbs. 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 lbs. was raised was
considerably greater than that through which the 56 lbs. 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 lbs., 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 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 overcome 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 the hammer. If the wood be
lying on the ground, the reaction 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 force will make
the nail penetrate than would suffice to move the wood quickly enough.
If the wood be thin and unsupported, 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 A be 14 lbs., and B, on the ground, be 56 lbs. 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 × 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 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 × (6 + X). But the work done by A must be equal to
that done upon B, and therefore

    14(6 + X) = 56 X,

whence X = 2. If there were no loss by friction, B would therefore be
raised 2'; but owing to friction, and doubtless also to the imperfect
flexibility of the rope, the effect 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 lbs. (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 × 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 reaction 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 × 12 = 100,800 lbs. If the reaction did not
reach this amount, the monkey could not be brought to rest in so short
a distance. The reaction 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.

540. 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 224,000 lbs.: this exerted over a
space of 0"·45 would be 8,400 units of energy; hence the pile would
be driven 0"·45. The more the resistance, the less the penetration
produced by each blow. 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 accumulated 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 reproduced 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 principles.
In the clock or watch the demand for energy to sustain the motion
is constant, while the supply is only occasional; 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 powerful 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 arranged 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 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 referred to in the case of the
cannon-ball. In order to render this method practically available for
the purposes of machinery, 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 18", and its
weight is 26 lbs.; the fly is carried upon a shaft (A) of wrought iron
¾" 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 backwards
as quickly as is convenient, pulling the rope all the time, the scale
shows a tension of about 50 lbs. 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 × 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.

[Illustration: FIG. 72.]

549. Let us see how the stored-up energy can be exhibited. A piece of
pine 24" × 1" × 1" of which both ends are supported, requires a force
of 300 lbs. 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 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 lbs. in setting the wheel in motion. The wheel stored
up and modified my energy so as to produce a force of 300 lbs., 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 lb. 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 lbs. 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 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 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.

[Illustration: FIG. 73.]

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, because, before this could happen, an
infinite force 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 required 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 0"·5 in diameter through
a plate 0"·8 thick, the area of iron that has to be cut across is
²²/₇ × ½ × ⁴/₅ = 1·26 square inches: and as 22·5 tons per square inch
are required for shearing, this hole will require 22·5 × 1·26 = 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
0"·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 0"·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 ¹/₆₀ × 2240 × 28·4 = 1062.

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 1 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 velocity.
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 motion 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 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; 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.

[Illustration: FIG. 74.]

A small toothed wheel is mounted on the same socket as C; this is
behind P Q, and is therefore not seen in the 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 0·1 lb. I find that I must pull it with a force of 3
lbs. 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 lbs.;
that is, it must be nearly thirty times as great as gravity.

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.

[Illustration: 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 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 rectilinear 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 revolving in two
circles are equal, the central force in the smaller 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 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.

[Illustration: 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 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 inside 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 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 composition 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.

[Illustration: FIG. 77.]

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.

[Illustration: FIG. 78.]

574. The explanation of the development 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 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 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 factory should work
uniformly. Irregularities of motion may be productive of loss and
various inconveniences. An engine would work irregularly either from
variation in the production 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 occasionally 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 arrangement which shall
admit more steam to the cylinder when 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 depends upon its velocity. Such
was the train of reasoning which led to the happy invention of the
governor-balls: these are shown in Fig. 79.

[Illustration: FIG. 79.]

A B is a vertical spindle which is turned by the engine. P P 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 P P. 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 outwards 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 the piece R R, which diminishes
the supply of steam, 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 charcoal. The syrup is
then boiled. In order to preserve the colour of the sugar, and to
prevent loss, this boiling is conducted _in vacuo_, 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 perforated 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 layer of perfectly pure white sugar, some inches in
thickness, ready for the market.

579. Circular motion is peculiarly fitted for this purpose; 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 rope is fastened, by means
of which the disk is suspended from the spindle. The disk hangs of
course in a vertical plane.

[Illustration: FIG. 80.]

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 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.

[Illustration: FIG. 81.]

585. The principles already enunciated will explain these 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 outwards
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 O P
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 P and Q 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 horizontal plane. Once
in the horizontal plane, the motion there is 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 permanently 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.

[Illustration: FIG. 82.]

590. We first experiment with a pendulum on a large scale. Our lecture
theatre is 32 feet high, and there is a wire suspended from the ceiling
27' long; to the end of this a ball of cast iron weighing 25 lbs. 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 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 A D and B C, each
12' long, and suspended from two points A B, about 1' apart, in the
same horizontal line. Each of these pendulums carries a weight of the
same size: they are in fact identical.

[Illustration: FIG. 83.]

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
1', 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
1' 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 12 lbs. on one hook and
one of 6 lbs. on the other. I withdraw one in each hand; I release
them; they return to my hand at the same moment. Whether I withdraw
the weights through long arcs or short arcs, equal or unequal, they
invariably return together, and both therefore 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
substances. 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.

[Illustration: FIG. 84.]

603. I have here (Fig. 84) two pendulums A D, B C, one 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 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 oscillation. 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 pendulum 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, 1; and the
times of the oscillations are proportional to 3, 2, 1: hence we learn
that _the period of oscillation 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.

It is found by calculation, and the result is confirmed by experiment,
that the time of vibration is represented by the expression,

            _____________________________
    3·1416 √ ( Length / Force of gravity).

607. The accurate value of the force of gravity in London is 32·1908, so
                                                          ________
that the time of vibration of a pendulum there is 0·5537 √ length: the
length of the seconds pendulum is 3'·2616.


THE FORCE OF GRAVITY DETERMINED BY THE PENDULUM.

608. The pendulum affords the proper means of measuring 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 × (3·1416 / Time)².

609. It is impossible to observe the time of a single vibration with
the necessary degree of accuracy; but supposing we consider a large
number of vibrations, say 100, 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 in the whole
time, there cannot be an error of more than 0·01 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.

610. But the length of the pendulum has also to be ascertained, 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 pendulum is embarrassed by considerable difficulties, which have
however been overcome by ingenious contrivances to be described in the
next chapter.

611. But we shall perform, in a very simple way, an experiment 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 0"·03 greater than the distance
from the point of suspension to the centre of the sphere.

612. The length having been ascertained, the next element to be
determined is the time of vibration. For this purpose I use a
stop-watch, which can be started or stopped instantaneously 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 silken 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 300th 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 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 Dublin is 2'·114 × (3·1416 / 0.805)² = 32·196. This result
agrees with one which has been determined by measurement made with
every precaution.

Another method of measuring gravity by the pendulum will be described
in the next lecture (Art. 637).


THE CYCLOID.

616. 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 circumference 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.

618. 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
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 O B. 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 T, the part T P only being
free. This arrangement will always force the point P to move in the
cycloidal arc.

[Illustration: FIG. 85.]

620. We are now in a condition to ascertain experimentally, 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 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 down to C, I shall have proved that the vibrations
are isochronous.

[Illustration: FIG. 86.]

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, 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.

[Illustration: FIG. 87.]

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 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 amplitude 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 suspension 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 exclusively in the neighbourhood
of the bob, inasmuch 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, as
otherwise we cannot have a definite point from which to measure. To
illustrate the mode that is to be adopted, I take here an iron bar 6'
long and 1" square, which weighs 19 lbs. 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 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 upper surface; this piece
of steel is firmly secured to the framework. 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.

[Illustration: FIG. 88.]

626. The general appearance of the apparatus, when mounted, 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 deduced by finding the time of vibration,
and then calculating 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 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.

[Illustration: FIG. 89.]

628. We first try with 6' of cord, so that the simple pendulum 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 necessary 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" × 0"·5 × 0"·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 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" × ⅔ = 16".

A bar 72" long will vibrate in a shorter time when the edge is 15"·2
from one end than when it has any other position. The length of the
corresponding simple pendulum is 41"·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 the 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 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 comparison 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 oscillation 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 oscillation 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
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 distance 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 suspension, 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. Reversing the bar, so
that it vibrates about its centre of oscillation B, I now find that
110·0 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 ¹/₂₅₀th part of a second.

639. It would be difficult to render the times of oscillation 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
1·102 seconds. The distance of the knife-edges is about 3'·992; hence
gravity may be found from the expression (Art. 608)

    3'·992 × (3·1416 / 1.102)².

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.

[Illustration: FIG. 90.]

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 which the
rod can rotate. A B, Fig. 90, is a pine rod 48" × 1" × 1", 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 communicated 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 circumstances already
stated can be verified.

[Illustration: FIG. 91.]

644. A more visible way of investigating the subject is shown in Fig.
91. F B is a rod of wood, suspended from a beam by the string F G. 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 similarly clamped at P and Q.

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. 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 general, whatever be
the shape of the body, that the centre of oscillation is identical with
the centre of percussion.

647. The principle embodied in the 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 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, returning 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 circular. 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 gradations to the circular pendulum, in
which the brush traces a straight line.

[Illustration: FIG. 92.]

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 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 A C B D, 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 A T. 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 impressed simultaneously upon the same
particle. They are performed in the same time, since all vibrations
are isochronous. We must conceive one motion starting from A 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 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 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.

[Illustration: FIG. 93.]

653. Close examination reveals a very interesting circumstance
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. 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 dimensions 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.

[Illustration: FIG. 94.]

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 B D, 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 perpendicular 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 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 B 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 extremity 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.


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 lbs., 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 hole at the bottom of the glass the sand runs out
upon a drawing-board placed underneath to receive it.

[Illustration: FIG. 95.]

Thus the little stream of sand depicts its own journey upon the
drawing-board, and the curves traced out thus indicate the path in
which the bob of the second pendulum has moved.

[Illustration: FIG. 96.]

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 experiment represented in Fig. 95. The form of
the path delineated by the sand is shown in Fig. 96. The arrowheads
placed upon the curve show the direction in which it is traced. Let us
suppose that the formation of the figure commences 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.




                        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 example, when the clock tells me that 15 minutes have elapsed, what
it really shows is that the pendulum has made 60 × 15 = 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.

661. 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 counterbalance 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 1·001 seconds
instead of in one second, the clock loses one 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 × _k_ 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 1·103 × 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 1103 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 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.

[Illustration: 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 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 uncompensated 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, 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 importance, inaccuracies of this magnitude
would be quite inadmissible.

[Illustration: FIG. 98.]

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 introduced 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, 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
independent 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
timepiece. The pendulum must have its 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 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.

[Illustration: 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 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 consequence 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 H. 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 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 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, controlled 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 produced 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 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.

[Illustration: FIG. 100.]

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.

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.

681. 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 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 lbs.
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 circumference 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 lbs. is therefore, reduced to the one hundred-and-twentieth part of
its amount at the circumference of the escapement-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 sustain the 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 with different velocities about
the same dial.

[Illustration: FIG. 101.]

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 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 hand A will have made only one
revolution, while the hand B has made twelve revolutions.

We have already seen (Art. 681) 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 devised, which would
give the required motion; for example, if F were 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 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 represent 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 circumference 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.

[Illustration: FIG. 102.]

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 VII., 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 III., 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 mechanism 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.

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 formulæ 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 formulæ
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 formulæ 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 formulæ of these tables have been deduced by the two
different methods.

Tables V., XIV., XVI., XXI., are to be found in the same manner as
Table III.; and Tables VI., IX., X., XI., XV., XVII., XVIII., XIX.,
XX., XXI., XXII., 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" × 6". A
scale which reads to the hundredth of an inch is to 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
1"·4, 2"·8, 4"·2, 5"·6, 7"·0, 8"·4, 9"·8, 11"·2 from the origin A.
These distances correspond to the magnitudes of the loads placed upon
the slide on the scale of 0"·1 to 1 lb. Perpendiculars to APS are to
be erected at the points marked, and distances F₁, F₂, F₃, &c. set
off upon these perpendiculars. 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 lbs., the friction is 12·2 lbs.; hence the point F₃ is
found by measuring a distance 4"·2 from A, and erecting a perpendicular
1"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 0"·1 diameter. These circles, besides neatly defining
the points, will be useful in a subsequent part of the process.

[Illustration: FIG. 103.]

It will be found that the points F₁, 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 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
F₁, 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 ST.
We find by measurement that ST is 2"·7. If, then, we suppose 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_ = 0·27 _R_.

This is the formula from which Table III. has been constructed.


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 F₁,
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 P Q; then, if _R_ be the load, and
_F_ the corresponding friction, we must have from similar triangles—

          _AY_
    _F_ - ———— × 1 lb.
          0"·1           _PQ_
   ————————————————— =  ——————.
              _R_        _PX_

By measurement it is found that _AY_ = 0"·14, and _PQ_ = 2"·53.

We have, therefore,

    _F_ = 1·4 + 0·253 _R_.

This is practically the same formula as

    _F_ = 1·44 + 0·252 _R_,

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_ = 0,

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 represents the
experiments. _F_ - _k R_ is to be as near zero as possible 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

    (_F₁_ - _k R₁_)² + (_F₂_ - _k R₂_)² + &c. + (_Fₘ_ - _k Rₘ_)²

a minimum where F₁ and R₁, F₂ and R₂ &c. are the simultaneous values of
F and R in the several experiments.

In fact, it is easy to see that, if this quantity be small, each of the
essentially positive elements,

    (_F₁_ - _k R₁_)², &c.

of which it is composed, must be small also, and that therefore

    _F_ - _k R_

must always be nearly zero.

Differentiating the sum of squares and equating the differential
coefficient to zero, we have according to the usual notation,

    Σ _R₁_ (_F₁_ - _k R₁_) = 0;

                    Σ _R₁ F₁_
    whence _k_ =  ————————————————.
                      Σ _R₁_²

The calculation of _k_ becomes simplified when (as is generally the
case in the tables) the loads _R₁_, _R₂_, &c., _Rₘ_ are of the form,

    _N_, 2_N_, 3_N_ &c., _m N_.

In this case,

    Σ _R₁_ _F₁_ = _N_ (_F₁_ + 2_F₂_ + 3_F₃_ + &c. + _mFₘ_).

    Σ _R₁_² = _N_² (1² + 2² + &c. + _m_²)

                _m_(_m_ + 1) (2_m_ +1)
         = _N_² ———————————————————————
                           6

               (_F₁_ + 2_F₂_ + &c. + _mFₘ_)
    ∴ _k_ = 6 ————————————————————————————— .
                _Nm_(_m_ + 1) (2_m_ + 1)

In the case of Table III.

    _m_ = 8, _N_ = 14,

    _F₁_ + 2_F_ + 3_F₃_ + _mFₘ_ = 770·9;

whence _k_ = 0·27.

Thus the formula _F_ = 0·27 _R_ 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 the following
considerations.

No values exist for _x_ and _y_, so that the equation

    _F_ = _x_ + _y R_

shall be satisfied for all pairs of values of _F_ and _R_, but the best
values for _x_ and _y_ are those which make

    (_F₁_ - _x_ - _y R₁_)² + (_F₂_ - _x_ - _y R₂_)² + &c.

           + (_Fₘ_ - _x_ - _y Rₘ_)²

a minimum.

Differentiating with respect to _x_ and _y_, and equating the
differential coefficients to zero, we have

    Σ (_F₁_ - _x_ - _y R₁_) = 0,

    Σ _R₁_(_F₁_ - _x_ - _y R₁_) = 0.

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. _mN_,

and making

    _A_ = _F₁_ + _F₂_ + _F₃_ + &c. + _Fₘ_

    _B_ = _F₁_ +2_F₂_ + 3_F₃_ + &c. + _mFₘ_,

we have the equations

                 _m_(_m_ + 1)
    _A_ - _mx_ - ————————————— _N y_ = 0,
                       2

          _m_(_m_ + 1)        _m_(_m_ + 1) (2_m_ + 1)
    _B_ - ————————————— _x_ - ——————————————————————— _N y_ = 0.
                 2                      6

Solving these, we find

           2 + 4_m_            6
    _x_ = ——————————— _A_ - ————————— _B_,
          _m_² - _m_       _m_² - _m_

              12     _B_       6        _A_
    _y_ = ————————— ————— - -————————— ————— .
          _m_³ - _m   N_   _m_² - _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 _R_.




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 Experimenters 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.[2]

[2] I ought to acknowledge the kindness with which Mr. J. Willis Clark,
of Cambridge, the literary executor of Professor Willis, has responded
to my queries, while I am also under obligations to the courtesy of
Messrs. Crosby, Lockwood, & Co.

The Willis system provides the means for putting versatile framework
together with or without revolving gear for the purpose 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 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 ⅜ 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
⅛". The nut is square or hexagonal, and should _always_ have a washer
underneath 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.

[Illustration: THE BED. FIG. 104.]

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 ⅜" asunder, and thus allow the
shanks of the bolts to pass freely through the slit The scantling of
each bar is 2½" × 1½", and the beds are of various lengths from 1' to
10' or even longer. The beds can be attached together in any required
position by bolts 6" long. The rectangles and the brackets are attached
to the beds by 4" bolts. In one conjunction 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.

[Illustration: THE STOOL. FIG. 105.]

THE STOOL.—Most of the larger pieces of apparatus have the _stool_ as
their foundation (see Figs. 11, 39, 102). It is often convenient as
in Fig. 65 to employ a pair of stools, while one stool superposed on
another gives the convenient stand in 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" × 1' 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" × 2⅛". Two of these rails with the
intervening ⅜" slit make the top and legs to be 4⅝" wide. The bottom
front rail I is 3" wide and 4" deep. The double side rails D, H are 1¾"
wide and 2½" 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 ⅜" wide throughout. Beech or birch are
very suitable materials, but softer woods will answer if large washers
are invariably used.

[Illustration: THE RECTANGLE. FIG. 106.]

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 2½" broad. The outside
dimensions are 6" and 9", and the thickness of metal is ⅝". Each side
of the rectangle has the usual bolt slit ⅜" 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 toothed 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
provided. 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 1" thick at the boss which is faced in the latter
at each side, and bored with a hole full 1" diameter, in which a key
groove is cut. A pair of mitre wheels such as are used in Fig. 80 are
sometimes useful.

[Illustration: THE STUD SOCKET. FIG. 107.]

THE PULLEY.—We have frequent occasion to use the pulley for conveying
a cord, and a somewhat varied stock is convenient 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 1" in diameter with a key groove, and the boss
is 1" 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 oscillating pieces the stud socket is used
(see Fig. 107). The socket A B may be made of brass or of cast iron. It
is 1" in diameter so as to pass through the bosses of the wheels that
have been bored to 1" with this object:—The socket is provided with a
shoulder at one end (A) which is 1½" diameter, and with a strong screw
B and octagonal nut at the other end. The extreme length of the socket
is 3½", and the plain part of the 1" cylinder is 1¾" long. When two
wheels are placed on the socket each of which has a boss 1" thick, the
tightening of the nut will secure the wheels against the shoulder. A
feather is screwed on the plain part which enters the key grooves in
the wheels, and thus ensures that the wheels shall turn together. This
feather should 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 ⅝" 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 1⅝" so as to
allow the stud to be secured in a hole 1" deep in one of the brackets
(to be presently described). The plain part of this screw near the
shoulder must be ⅝" 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.

[Illustration: BRACKET NO. I. FIG. 108.]

[Illustration: BRACKET NO. II. FIG. 109.]

[Illustration: BRACKET NO. III. FIG. 110.]

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 1" thick bored with
a hole ⅝" 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
⅝" wide for the bolts. The thickness of the sole is ⅝". The larger
of the brackets I., II., and IV. have also slits in their 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 desired fashion.
Brackets avail for many other purposes besides those of supporting
rotating mechanism. (Look at Figs. 11, 12, 17, 20, 33, 38, 39, 73 and
many others.)

[Illustration: BRACKET NO. IV. FIG. 111.]

[Illustration: BRACKET NO. V. FIG. 112.]

[Illustration: BRACKET NO. VI. FIG. 113.]

[Illustration: THE TUBE FITTING. FIG. 114.]

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 ¾" in diameter, and of various lengths
from 6" up to 4'. To support the shafts we use for bearings the _tube
fitting_ (Fig. 114). This is a brass casting which consists of a tube
M N 2" long, and 1¼" in external diameter, bored ¾" so as to fit the
shaft. The back of this tube is a flat surface parallel to the bore,
and from it projects a screw ⅝" diameter, and 1⅝" long with a nut which
is however omitted in the drawing. 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 bored ¾", 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.

[Illustration: THE PINNED RING. FIG. 115.]

[Illustration: THE ADAPTER. FIG. 116.]

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 ¾" hole
to fit the shaft, and the external diameter is 1". At one end is a
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 the binding screw down on the shaft. Some adapters
are only large enough for a single wheel 1" 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.

[Illustration: THE LEVER ARM. Fig. 117.]

THE LEVER ARM.—To give motion to the mechanism a lever arm with a
handle is frequently required (Fig. 117). It is bored 1" and has a
key groove, and the hole is 1" long, so that the lever arm can be
fixed on a stud socket like a wheel. By 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.




INDEX.


            A.
    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 I., 339
    Atwood’s machine, 232
    Axes, permanent, 279

            B.
    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, 186
      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, 218
      suspension, 225
      the Wye, 215
      tubular, 223
      with four struts, 210
        two struts, 206
        two ties, 211
    Brunei, Sir J., the Wye bridge, 215

            C.
    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 earth, 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
    Composition of forces, 1, 9
      parallel forces, 35, 37, 42
      vibrations, 299, 315
    Conical pendulum, 310
    Couple, 44
    Crane, 29, 162
      friction in, 166
      mechanical efficiency of, 165
      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

            E.
    Eade, Mr., epicycloidal pulley-block, 116
    Easter Island, 100
    Elasticity of beam, 184
    Energy, 85, 94
      storage of, 256, 258
      unit of, 95
    Engine, locomotive, 83
    Epicycloidal pulley-block, 80, 116
      Table XII. 118
    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
    Fly-wheel, 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, 80
      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, 81, 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

            G.
    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
    Graphical construction, 339
    Gravity, 50
      action of, 243
      and the pendulum, 292
      and weight, 52
      centre of, 57
      defined, 246
      different effects of, 53
      independent of motion, 241
      in London, 292
      specific, 53
    Grindstone, treadle of, 128

            H.
    Hammer, 252
      theory of the, 252
    Hands of a clock, 331
    Horse-power, 96

            I.
    Illustration of parallelogram of forces, 10
    Illustrations of forces, 10
      resolution, 19
    Inches, how represented, 7
    Inclination of thread, 140
    Inclined plane, 131
      forces on, 136
      friction in, 132
      mechanical efficiency of, 139
      Table XIII. 134
            XIV. 137
            XV. 138
      velocity, ratio of, 139
    Inertia, 250
      inherent in matter, 252
    Iron girders, 219
      specific gravity of, 55
    Isochronous simple pendulum, 303
    Ivory, specific gravity of, 56

            J.
    Jib, 29, 163

            K.
    Kater, Captain, 305
    Kinetics, 230

            L.
    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
    Lever, 119
      and friction, 123
      applications of, 123
      arm, Willis apparatus, 352
      laws of, 130
      of first order, 119
        law of, 122
      of second order, 124
      of third order, 128
      weight of, 121
    Lifting crane, 29
    Line and plummet, 56
    Load, breaking, 177
    Locomotive engine, 83

            M.
    Machine, Atwood’s, 232
      punching, 263
    Machines, pile-driving, 255
    Magnitude of a force, 4
    Margin of safety, 33
    Mass, 236
    Mean frictions, 75
    Measurement of force, 4
    Mechanical powers, 85, 100
      apparatus, Willis, 345
    Menai Bridge, 218
    Method of least squares, 342
    Moment, 130
    Monkey, 257
    Motion, first law of, 230
      of falling body, 230

            N.
    Neutral equilibrium, 61
    Newton and gravity, 289
    Nut, 140

            O.
    Oscillation, centre of, 304

            P.
    Pair of scales, 48
      testing, 48
    Parabola, 226
    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
      conical, 310
      formula for, 292
      Galileo and the, 286
      isochronous simple, 303
      length of the seconds, 292, 318
      motion of the, 285
      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, 110
      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, 9
      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
      jack, 131, 145
        Table XVII. 146
    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
        ivory, 56
        lead, 56
    Spirit-level, 56
    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

            T.
    Table I. 69
          II. 71
          III. 74
          IV. 76
          V. 78
          VI. 78
          VII. 81
          VIII. 81
          IX. 104
          X. 108
          XI. 114
          XII. 118
          XIII. 134
          XIV. 137
          XV. 138
          XVI. 142
          XVII. 146
          XVIII. 154
          XIX. 159
          XX. 162
          XXI. 165
          XXII. 166
          XXIII. 182
          XXIV. 190
    Tacking, 25
    Tension along a cord, 17
    Three-sheave pulley-block, 106
    Tie, 28, 175
      rod, 29, 32
    Timber, bending, 171
      compression of, 172
      extension 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
    Transverse strain, 181
    Treadle of a grindstone, 128
    Tripod, 28
      strength of, 28
    Truss, simple form of, 212
    Tube fitting, Willis apparatus, 351
    Tubular bridge, 223

            U.
    Unstable equilibrium, 59, 282

            V.
    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, 48
    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, 61
    Wheels, 92
      friction, 93
    Wheels in Willis apparatus, 348
    Willis system of apparatus, 345
    Winch, 151
    Wind, direction of, 22
    Work, 85, 94
    Wye Bridge, 215


THE END.

RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.





End of Project Gutenberg's Experimental Mechanics, by Robert Stawell Ball