THE INTERNATIONAL SCIENTIFIC SERIES.
VOLUME VII.
THE INTERNATIONAL SCIENTIFIC SERIES.
_Works already Published._
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III. FOODS. By EDWARD SMITH, M. D., LL. B., F. R. S. 1 vol. Cloth.
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OPTICS. By Dr. EUGENE LOMMEL, Professor of Physics in the University
of Erlangen. With 188 Illustrations and a Plate of Spectra in
Chromo-lithography. (_In press._)
THE INTERNATIONAL SCIENTIFIC SERIES.
THE
CONSERVATION OF ENERGY.
BY
BALFOUR STEWART, LL. D., F.R.S.,
PROFESSOR OF NATURAL PHILOSOPHY AT THE OWENS COLLEGE, MANCHESTER.
_WITH AN APPENDIX_,
TREATING OF THE VITAL AND MENTAL APPLICATIONS OF THE
DOCTRINE
NEW YORK:
D. APPLETON AND COMPANY,
549 & 551 BROADWAY.
1875.
ENTERED, according to Act of Congress, in the year 1874, by
D. APPLETON & COMPANY,
In the Office of the Librarian of Congress, at Washington.
NOTE TO THE AMERICAN EDITION.
The great prominence which the modern doctrine of the Conservation of
Energy or Correlation of Forces has lately assumed in the world of
thought, has made a simple and popular explanation of the subject very
desirable. The present work of Dr. Balfour Stewart, contributed to the
International Scientific Series, fully meets this requirement, as it
is probably the clearest and most elementary statement of the question
that has yet been attempted. Simple in language, copious and familiar
in illustration, and remarkably lucid in the presentation of facts and
principles, his little treatise forms just the introduction to the
great problem of the interaction of natural forces that is required by
general readers. But Professor Stewart having confined himself mainly
to the physical aspects of the subject, it was desirable that his views
should be supplemented by a statement of the operation of the principle
in the spheres of life and mind. An Appendix has, accordingly, been
added to the American edition of Dr. Stewart’s work, in which these
applications of the law are considered.
Professor Joseph Le Conte published a very able essay fourteen years
ago on the Correlation of the Physical and Vital Forces, which was
extensively reprinted abroad, and placed the name of the author among
the leading interpreters of the subject. His mode of presenting it was
regarded as peculiarly happy, and was widely adopted by other writers.
After further investigations and more mature reflection, he has
recently restated his views, and has kindly furnished the revised essay
for insertion in this volume.
Professor A. Bain, the celebrated Psychologist of Aberdeen, who
has done so much to advance the study of mind in its physiological
relations, prepared an interesting lecture not long ago on the
“Correlation of the Nervous and Mental Forces,” which was read with
much interest at the time of its publication, and is now reprinted as a
suitable exposition of that branch of the subject. These two essays, by
carrying out the principle in the field of vital and mental phenomena,
will serve to give completeness and much greater value to the present
volume.
NEW YORK, _December, 1873_.
PREFACE.
We may regard the Universe in the light of a vast physical machine, and
our knowledge of it may be conveniently divided into two branches.
The one of these embraces what we know regarding the structure of the
machine itself, and the other what we know regarding its method of
working.
It has appeared to the author that, in a treatise like this, these two
branches of knowledge ought as much as possible to be studied together,
and he has therefore endeavored to adopt this course in the following
pages. He has regarded a universe composed of atoms with some sort of
medium between them as the machine, and the laws of energy as the laws
of working of this machine.
The first chapter embraces what we know regarding atoms, and gives
also a definition of Energy. The various forces and energies of Nature
are thereafter enumerated, and the law of Conservation is stated. Then
follow the various transmutations of Energy, according to a list, for
which the author is indebted to Prof. Tait. The fifth chapter gives
a short historical sketch of the subject, ending with the law of
Dissipation; while the sixth and last chapter gives some account of the
position of living beings in this universe of Energy.
B. S.
_The Owens College, Manchester,
August, 1873._
CONTENTS.
NOTE TO THE AMERICAN EDITION, v
PREFACE, vii
CHAPTER
I.--WHAT IS ENERGY? 1
II.--MECHANICAL ENERGY AND ITS CHANGE INTO HEAT, 23
III.--THE FORCES AND ENERGIES OF NATURE: THE LAW OF CONSERVATION, 48
IV.--TRANSMUTATIONS OF ENERGY, 87
V.--HISTORICAL SKETCH: THE DISSIPATION OF ENERGY, 131
VI.--THE POSITION OF LIFE, 154
APPENDIX
I.--CORRELATION OF VITAL WITH CHEMICAL AND PHYSICAL FORCES.
By JOSEPH LE CONTE, Professor of Geology and Natural
History in the University of California, 171
II.--CORRELATION OF NERVOUS AND MENTAL FORCES. By ALEXANDER
BAIN, Professor of Logic and Mental Philosophy in
the University of Aberdeen, 205
THE CONSERVATION OF ENERGY.
CHAPTER I.
_WHAT IS ENERGY?_
_Our Ignorance of Individuals._
1. Very often we know little or nothing of individuals, while we yet
possess a definite knowledge of the laws which regulate communities.
The Registrar-General, for example, will tell us that the death-rate
in London varies with the temperature in such a manner that a very
low temperature is invariably accompanied by a very high death-rate.
But if we ask him to select some one individual, and explain to us in
what manner his death was caused by the low temperature, he will, most
probably, be unable to do so.
Again, we may be quite sure that after a bad harvest there will be a
large importation of wheat into the country, while, at the same time,
we are quite ignorant of the individual journeys of the various
particles of flour that go to make up a loaf of bread.
Or yet again, we know that there is a constant carriage of air from the
poles to the equator, as shown by the trade winds, and yet no man is
able to individualize a particle of this air, and describe its various
motions.
2. Nor is our knowledge of individuals greater in the domains of
physical science. We know nothing, or next to nothing, of the ultimate
structure and properties of matter, whether organic or inorganic.
No doubt there are certain cases where a large number of particles
are linked together, so as to act as one individual, and then we can
predict its action--as, for instance, in the solar system, where the
physical astronomer is able to foretell with great exactness the
positions of the various planets, or of the moon. And so, in human
affairs, we find a large number of individuals acting together as one
nation, and the sagacious statesman taking very much the place of the
sagacious astronomer, with regard to the action and reaction of various
nations upon one another.
But if we ask the astronomer or the statesman to select an individual
particle and an individual human being, and predict the motions of
each, we shall find that both will be completely at fault.
3. Nor have we far to look for the cause of their ignorance. A
continuous and restless, nay, a very complicated, activity is the order
of nature throughout all her individuals, whether these be living
beings or inanimate particles of matter. Existence is, in truth, one
continued fight, and a great battle is always and everywhere raging,
although the field in which it is fought is often completely shrouded
from our view.
4. Nevertheless, although we cannot trace the motions of individuals,
we may sometimes tell the result of the fight, and even predict how the
day will go, as well as specify the causes that contribute to bring
about the issue.
With great freedom of action and much complication of motion in the
individual, there are yet comparatively simple laws regulating the
joint result attainable by the community.
But, before proceeding to these, it may not be out of place to take a
very brief survey of the organic and inorganic worlds, in order that
our readers, as well as ourselves, may realize our common ignorance of
the ultimate structure and properties of matter.
5. Let us begin by referring to the causes which bring about disease.
It is only very recently that we have begun to suspect a large number
of our diseases to be caused by organic germs. Now, assuming that we
are right in this, it must nevertheless be confessed that our ignorance
about these germs is most complete. It is perhaps doubtful whether we
ever saw one of these organisms,[1] while it is certain that we are in
profound ignorance of their properties and habits.
We are told by some writers[2] that the very air we breathe is
absolutely teeming with germs, and that we are surrounded on all sides
by an innumerable array of minute organic beings. It has also been
conjectured that they are at incessant warfare among themselves, and
that we form the spoil of the stronger party. Be this as it may, we
are at any rate intimately bound up with, and, so to speak, at the
mercy of, a world of creatures, of which we know as little as of the
inhabitants of the planet Mars.
6. Yet, even here, with profound ignorance of the individual, we are
not altogether unacquainted with some of the habits of these powerful
predatory communities. Thus we know that cholera is eminently a low
level disease, and that during its ravages we ought to pay particular
attention to the water we drink. This is a general law of cholera,
which is of the more importance to us because we cannot study the
habits of the individual organisms that cause the disease.
Could we but see these, and experiment upon them, we should soon
acquire a much more extensive knowledge of their habits, and perhaps
find out the means of extirpating the disease, and of preventing its
recurrence.
Again, we know (thanks to Jenner) that vaccination will prevent the
ravages of small-pox, but in this instance we are no better off than
a band of captives who have found out in what manner to mutilate
themselves, so as to render them uninteresting to their victorious foe.
7. But if our knowledge of the nature and habits of organized molecules
be so small, our knowledge of the ultimate molecules of inorganic
matter is, if possible, still smaller. It is only very recently that
the leading men of science have come to consider their very existence
as a settled point.
In order to realize what is meant by an inorganic molecule, let us
take some sand and grind it into smaller and smaller particles, and
these again into still smaller. In point of fact we shall never
reach the superlative degree of smallness by this operation--yet in
our imagination we may suppose the sub-division to be carried on
continuously, always making the particles smaller and smaller. In
this case we should, at last, come to an ultimate molecule of sand or
oxide of silicon, or, in other words, we should arrive at the smallest
entity retaining all the properties of sand, so that were it possible
to divide the molecule further the only result would be to separate it
into its chemical constituents, consisting of silicon on the one side
and oxygen on the other.
We have, in truth, much reason to believe that sand, or any other
substance, is incapable of infinite sub-division, and that all we can
do in grinding down a solid lump of anything is to reduce it into lumps
similar to the original, but only less in size, each of these small
lumps containing probably a great number of individual molecules.
8. Now, a drop of water no less than a grain of sand is built up of a
very great number of molecules, attached to one another by the force of
cohesion--a force which is much stronger in the sand than in the water,
but which nevertheless exists in both. And, moreover, Sir William
Thomson, the distinguished physicist, has recently arrived at the
following conclusion with regard to the size of the molecules of water.
He imagines a single drop of water to be magnified until it becomes
as large as the earth, having a diameter of 8000 miles, and all the
molecules to be magnified in the same proportion; and he then concludes
that a single molecule will appear, under these circumstances, as
somewhat larger than a shot, and somewhat smaller than a cricket ball.
9. Whatever be the value of this conclusion, it enables us to realize
the exceedingly small size of the individual molecules of matter,
and renders it quite certain that we shall never, by means of the
most powerful microscope, succeed in making visible these ultimate
molecules. For our knowledge of the sizes, shapes, and properties
of such bodies, we must always, therefore, be indebted to indirect
evidence of a very complicated nature.
It thus appears that we know little or nothing about the shape or size
of molecules, or about the forces which actuate them; and, moreover,
the very largest masses of the universe share with the very smallest
this property of being beyond the direct scrutiny of the human
senses--the one set because they are so far away, and the other because
they are so small.
10. Again, these molecules are not at rest, but, on the contrary, they
display an intense and ceaseless energy in their motions. There is,
indeed, an uninterrupted warfare going on--a constant clashing together
of these minute bodies, which are continually maimed, and yet always
recover themselves, until, perhaps, some blow is struck sufficiently
powerful to dissever the two or more simple atoms that go to form a
compound molecule. A new state of things thenceforward is the result.
But a simple elementary atom is truly an immortal being, and enjoys the
privilege of remaining unaltered and essentially unaffected amid the
most powerful blows that can be dealt against it--it is probably in a
state of ceaseless activity and change of form, but it is nevertheless
always the same.
11. Now, a little reflection will convince us that we have in this
ceaseless activity another barrier to an intimate acquaintance with
molecules and atoms, for even if we could see them they would not
remain at rest sufficiently long to enable us to scrutinize them.
No doubt there are devices by means of which we can render visible, for
instance, the pattern of a quickly revolving coloured disc, for we may
illuminate it by a flash of electricity, and the disc may be supposed
to be stationary during the extremely short time of the flash. But we
cannot say the same about molecules and atoms, for, could we see an
atom, and could we illuminate it by a flash of electricity, the atom
would most probably have vibrated many times during the exceedingly
small time of the flash. In fine, the limits placed upon our senses,
with respect to space and time, equally preclude the possibility of our
ever becoming directly acquainted with these exceedingly minute bodies,
which are nevertheless the raw materials of which the whole universe is
built.
_Action and Reaction, Equal and Opposite._
12. But while an impenetrable veil is drawn over the individual in this
warfare of clashing atoms, yet we are not left in profound ignorance
of the laws which determine the ultimate result of all these motions,
taken together as a whole.
_In a Vessel of Goldfish._
Let us suppose, for instance, that we have a glass globe containing
numerous goldfish standing on the table, and delicately poised on
wheels, so that the slightest push, the one way or the other, would
make it move. These goldfish are in active and irregular motion, and he
would be a very bold man who should venture to predict the movements of
an individual fish. But of one thing we may be quite certain: we may
rest assured that, notwithstanding all the irregular motions of its
living inhabitants, the globe containing the goldfish will remain at
rest upon its wheels.
Even if the table were a lake of ice, and the wheels were extremely
delicate, we should find that the globe would remain at rest. Indeed,
we should be exceedingly surprised if we found the globe going away of
its own accord from the one side of the table to the other, or from
the one side of a sheet of ice to the other, in consequence of the
internal motions of its inhabitants. Whatever be the motions of these
individual units, yet we feel sure that the globe cannot move itself
_as a whole_. In such a system, therefore, and, indeed, in every system
left to itself, there may be strong internal forces acting between
the various parts, but these _actions and reactions are equal and
opposite_, so that while the small parts, whether visible or invisible,
are in violent commotion among themselves, yet the system as a whole
will remain at rest.
_In a Rifle._
13. Now it is quite a legitimate step to pass from this instance of the
goldfish to that of a rifle that has just been fired. In the former
case, we imagined the globe, together with its fishes, to form one
system; and in the latter, we must look upon the rifle, with its powder
and ball, as forming one system also.
Let us suppose that the explosion takes place through the application
of a spark. Although this spark is an external agent, yet if we reflect
a little we shall see that its only office in this case is to summon
up the internal forces already existing in the loaded rifle, and bring
them into vigorous action, and that in virtue of these internal forces
the explosion takes place.
The most prominent result of this explosion is the out-rush of the
rifle ball with a velocity that may, perhaps, carry it for the best
part of a mile before it comes to rest; and here it would seem to us,
at first sight, that the law of equal action and reaction is certainly
broken, for these internal forces present in the rifle have at least
propelled part of the system, namely, the rifle ball, with a most
enormous velocity in one direction.
14. But a little further reflection will bring to light another
phenomenon besides the out-rush of the ball. It is well known to all
sportsmen that when a fowling-piece is discharged, there is a kick or
recoil of the piece itself against the shoulder of the sportsman, which
he would rather get rid of, but which we most gladly welcome as the
solution of our difficulty. In plain terms, while the ball is projected
forwards, the rifle stock (if free to move) is at the same moment
projected backwards. To fix our ideas, let us suppose that the rifle
stock weighs 100 ounces, and the ball one ounce, and that the ball is
projected forwards with the velocity of 1000 feet per second; then it
is asserted, by the law of action and reaction, that the rifle stock is
at the same time projected backwards with the velocity of 10 feet per
second, so that the mass of the stock, multiplied by its velocity of
recoil, shall precisely equal the mass of the ball, multiplied by its
velocity of projection. The one product forms a measure of the action
in the one direction, and the other of the reaction in the opposite
direction, and thus we see that in the case of a rifle, as well as in
that of the globe of fish, action and reaction are equal and opposite.
_In a Falling Stone._
15. We may even extend the law to cases in which we do not perceive
the recoil or reaction at all. Thus, if I drop a stone from the
top of a precipice to the earth, the motion seems all to be in one
direction, while at the same time it is in truth the result of a mutual
attraction between the earth and the stone. Does not the earth move
also? We cannot see it move, but we are entitled to assert that it
does in reality move upwards to meet the stone, although quite to an
imperceptible extent, and that the law of action and reaction holds
here as truly as in a rifle, the only difference being that in the
one case the two objects are rushing together, while in the other
they are rushing apart. Inasmuch, however, as the mass of the earth
is very great compared with that of the stone, it follows that its
velocity must be extremely small, in order that the mass of the earth,
multiplied into its velocity upwards, shall equal the mass of the
stone, multiplied into its velocity downwards.
16. We have thus, in spite of our ignorance of the ultimate atoms and
molecules of matter, arrived at a general law which regulates the
action of internal forces. We see that these forces are always mutually
exerted, and that if A attracts or repels B, B in its turn attracts or
repels A. We have here, in fact, a very good instance of that kind of
generalization, which we may arrive at, even in spite of our ignorance
of individuals.
But having now arrived at this law of action and reaction, do we know
all that it is desirable to know? have we got a complete understanding
of what takes place in all such cases--for instance, in that of the
rifle which is just discharged? Let us consider this point a little
further.
_The Rifle further considered._
17. We define quantity of motion to mean the product of the mass by
the velocity; and since the velocity of recoil of the rifle stock,
multiplied by the mass of the stock, is equal to the velocity of
projection of the rifle ball, multiplied by the mass of the ball, we
conceive ourselves entitled to say that the quantity of motion, or
momentum, generated is equal in both directions, so that the law of
action and reaction holds here also. Nevertheless, it cannot but occur
to us that, _in some sense_, the motion of the rifle ball is a very
different thing from that of the stock, for it is one thing to allow
the stock to recoil against your shoulder and discharge the ball into
the air, and a very different thing to discharge the ball against your
shoulder and allow the stock to fly into the air. And if any man
should assert the absolute equality between the blow of the rifle stock
and that of the rifle ball, you might request him to put his assertion
to this practical test, with the absolute certainty that he would
decline. Equality between the two!--Impossible! Why, if this were the
case, a company of soldiers engaged in war would suffer much more than
the enemy against whom they fired, for the soldiers would certainly
feel each recoil, while the enemy would suffer from only a small
proportion of the bullets.
_The Rifle Ball possesses Energy._
18. Now, what is the meaning of this great difference between the two?
We have a vivid perception of a mighty difference, and it only remains
for us to clothe our naked impressions in a properly fitting scientific
garb.
_The something which the rifle ball possesses in contradistinction to
the rifle stock is clearly the power of overcoming resistance._ It
can penetrate through oak wood or through water, or (alas! that it
should be so often tried) through the human body, and this power of
penetration is the distinguishing characteristic of a substance moving
with very great velocity.
19. Let us define by the term _energy_ this power which the rifle
ball possesses of overcoming obstacles or of doing work. Of course
we use the word work without reference to the moral character of the
thing done, and conceive ourselves entitled to sum up, with perfect
propriety and innocence, the amount of work done in drilling a hole
through a deal board or through a man.
20. A body such as a rifle ball, moving with very great velocity,
has, therefore, energy, and it requires very little consideration
to perceive that this _energy will be proportional to its weight or
mass_, for a ball of two ounces moving with the velocity of 1000 feet
per second will be the same as two balls of one ounce moving with this
velocity, but the energy of two similarly moving ounce balls will
manifestly be double that of one, so that the energy is proportional
to the weight, if we imagine that, meanwhile, the velocity remains the
same.
21. But, on the other hand, the energy is not simply proportional to
the velocity, for, if it were, the energy of the rifle stock and of the
rifle ball would be the same, inasmuch as the rifle stock would gain as
much by its superior mass as it would lose by its inferior velocity.
Therefore, the energy of a moving body increases with the velocity more
quickly than a simple proportion, so that if the velocity be doubled,
the energy is more than doubled. Now, in what manner does the energy
increase with the velocity? That is the question we have now to answer,
and, in doing so, we must appeal to the familiar facts of everyday
observation and experience.
22. In the first place, it is well known to artillerymen, that if
a ball have a double velocity, its penetrating power or energy is
increased nearly fourfold, so that it will pierce through four, or
nearly four, times as many deal boards as the ball with only a single
velocity--in other words, they will tell us in mathematical language,
that the energy varies as the square of the velocity.
_Definition of Work._
23. And now, before proceeding further, it will be necessary to tell
our readers how to measure work in a strictly scientific manner. We
have defined energy to be the power of doing work, and although every
one has a general notion of what is meant by work, that notion may not
be sufficiently precise for the purpose of this volume. How, then, are
we to measure work? Fortunately, we have not far to go for a practical
means of doing this. Indeed, there is a force at hand which enables us
to accomplish this measurement with the greatest precision, and this
force is gravity. Now, the first operation in any kind of numerical
estimate is to fix upon our unit or standard. Thus we say a rod is
so many inches long, or a road so many miles long. Here an inch and
a mile are chosen as our standards. In like manner, we speak of so
many seconds, or minutes, or hours, or days, or years, choosing that
standard of time or duration which is most convenient for our purpose.
So in like manner we must choose our unit of work, but in order to
do so we must first of all choose our units of weight and of length,
and for these we will take the _kilogramme_ and the _metre_, these
being the units of the metrical system. The kilogramme corresponds
to about 15,432·35 English grains, being rather more than two pounds
avoirdupois, and the metre to about 39·371 English inches.
Now, if we raise a kilogramme weight one metre in vertical height,
we are conscious of putting forth an effort to do so, and of being
resisted in the act by the force of gravity. In other words, we spend
energy and do work in the process of raising this weight.
Let us agree to consider the energy spent, or the work done, in this
operation as one unit of work, and let us call it the _kilogrammetre_.
24. In the next place, it is very obvious that if we raise the
kilogramme two metres in height, we do two units of work--if three
metres, three units, and so on.
And again, it is equally obvious that if we raise a weight of two
kilogrammes one metre high, we likewise do two units of work, while if
we raise it two metres high, we do four units, and so on.
From these examples we are entitled to derive the following
rule:--_Multiply the weight raised (in kilogrammes) by the vertical
height (in metres) through which it is raised, and the result will be
the work done (in kilogrammetres)._
_Relation between Velocity and Energy._
25. Having thus laid a numerical foundation for our superstructure,
let us next proceed to investigate the relation between velocity and
energy. But first let us say a few words about velocity. This is one
of the few cases in which everyday experience will aid, rather than
hinder, us in our scientific conception. Indeed, we have constantly
before us the example of bodies moving with variable velocities.
Thus a railway train is approaching a station and is just beginning to
slacken its pace. When we begin to observe, it is moving at the rate of
forty miles an hour. A minute afterwards it is moving at the rate of
twenty miles only, and a minute after that it is at rest. For no two
consecutive moments has this train continued to move at the same rate,
and yet we may say, with perfect propriety, that at such a moment the
train was moving, say, at the rate of thirty miles an hour. We mean, of
course, that had it continued to move for an hour with the speed which
it had when we made the observation, it would have gone over thirty
miles. We know that, as a matter of fact, it did not move for two
seconds at that rate, but this is of no consequence, and hardly at all
interferes with our mental grasp of the problem, so accustomed are we
all to cases of variable velocity.
26. Let us now imagine a kilogramme weight to be shot vertically
upwards, with a certain initial velocity--let us say, with the velocity
of 9·8 metres in one second. Gravity will, of course, act against the
weight, and continually diminish its upward speed, just as in the
railway train the break was constantly reducing the velocity. But yet
it is very easy to see what is meant by an initial velocity of 9·8
metres per second; it means that if gravity did not interfere, and if
the air did not resist, and, in fine, if no external influence of any
kind were allowed to act upon the ascending mass, it would be found to
move over 9·8 metres in one second.
Now, it is well known to those who have studied the laws of motion,
that a body, shot upwards with the velocity of 9·8 metres in one
second, will be brought to rest when it has risen 4·9 metres in height.
If, therefore, it be a kilogramme, its upward velocity will have
enabled it to raise itself 4·9 metres in height against the force of
gravity, or, in other words, it will have done 4·9 units of work; and
we may imagine it, when at the top of its ascent, and just about to
turn, caught in the hand and lodged on the top of a house, instead of
being allowed to fall again to the ground. We are, therefore, entitled
to say that a kilogramme, shot upwards with the velocity of 9·8 metres
per second, has energy equal to 4·9, inasmuch as it can raise itself
4·9 metres in height.
27. Let us next suppose that the velocity with which the kilogramme
is shot upwards is that of 19·6 metres per second. It is known to all
who have studied dynamics that the kilogramme will now mount not only
twice, but four times as high as it did in the last instance--in other
words, it will now mount 19·6 metres in height.
Evidently, then, in accordance with our principles of measurement,
the kilogramme has now four times as much energy as it had in the
last instance, because it can raise itself four times as high, and
therefore do four times as much work, and thus we see that the energy
is increased four times by doubling the velocity.
Had the initial velocity been three times that of the first instance,
or 29·4 metres per second, it might in like manner be shown that the
height attained would have been 44·1 metres, so that by tripling the
velocity the energy is increased nine times.
28. We thus see that whether we measure the energy of a moving body by
the thickness of the planks through which it can pierce its way, or by
the height to which it can raise itself against gravity, the result
arrived at is the same. _We find the energy to be proportional to the
square of the velocity_, and we may formularize our conclusion as
follows:--
Let _v_ = the initial velocity expressed in metres per second, then
the energy in kilogrammetres = _v_²/19·6. Of course, if the body shot
upwards weighs two kilogrammes, then everything is doubled, if three
kilogrammes, tripled, and so on; so that finally, if we denote by
_m_ the mass of the body in kilogrammes, we shall have the energy in
kilogrammetres = _mv_²/19·6. To test the truth of this formula, we have
only to apply it to the cases described in Arts. 26 and 27.
29. We may further illustrate it by one or two examples. For instance,
let it be required to find the energy contained in a mass of five
kilogrammes, shot upwards with the velocity of 20 metres per second.
Here we have _m_ = 5 and _v_ = 20, hence--
Energy = 5(20)²/(19·6) = 2000/(19·6) = 102·04 nearly.
Again, let it be required to find the height to which the mass of the
last question will ascend before it stops. We know that its energy is
102·04, and that its mass is 5. Dividing 102·04 by 5, we obtain 20·408
as the height to which this mass of five kilogrammes must ascend in
order to do work equal to 102·04 kilogrammetres.
30. In what we have said we have taken no account either of the
resistance or of the buoyancy of the atmosphere; in fact, we have
supposed the experiments to be made in vacuo, or, if not in vacuo,
made by means of a heavy mass, like lead, which will be very little
influenced either by the resistance or buoyancy of the air.
We must not, however, forget that if a sheet of paper, or a feather,
be shot upwards with the velocities mentioned in our text, they will
certainly not rise in the air to nearly the height recorded, but
will be much sooner brought to a stop by the very great resistance
which they encounter from the air, on account of their great surface,
combined with their small mass.
On the other hand, if the substance we make use of be a large light bag
filled with hydrogen, it will find its way upwards without any effort
on our part, and we shall certainly be doing no work by carrying it
one or more metres in height--it will, in reality, help to pull us up,
instead of requiring help from us to cause it to ascend. In fine, what
we have said is meant to refer to the force of gravity alone, without
taking into account a resisting medium such as the atmosphere, the
existence of which need not be considered in our present calculations.
31. It should likewise be remembered, that while the energy of a moving
body depends upon its velocity, it is independent of the direction in
which the body is moving. We have supposed the body to be shot upwards
with a given velocity, but it might be shot horizontally with the same
velocity, when it would have precisely the same energy as before. A
cannon ball, if fired vertically upwards, may either be made to spend
its energy in raising itself, or in piercing through a series of deal
boards. Now, if the same ball be fired horizontally with the same
velocity it will pierce through the same number of deal boards.
In fine, direction of motion is of no consequence, and the only reason
why we have chosen vertical motion is that, in this case, there is
always the force of gravity steadily and constantly opposing the motion
of the body, and enabling us to obtain an accurate measure of the work
which it does by piercing its way upwards against this force.
32. But gravity is not the only force, and we might measure the energy
of a moving body by the extent to which it would bend a powerful
spring or resist the attraction of a powerful magnet, or, in fine,
we might make use of the force which best suits our purpose. If this
force be a constant one, we must measure the energy of the moving body
by the space which it is able to traverse against the action of the
force--just as, in the case of gravity, we measured the energy of the
body by the space through which it was able to raise itself against its
own weight.
33. We must, of course, bear in mind that if this force be more
powerful than gravity, a body moved a short distance against it will
represent the expenditure of as much energy as if it were moved a
greater distance against gravity. In fine, we must take account both of
the strength of the force and of the distance moved over by the body
against it before we can estimate in an accurate matter the work which
has been done.
FOOTNOTES:
[1] It is said that there are one or two instances where the microscope
has enlarged them into visibility.
[2] _See_ Dr. Angus Smith on Air and Rain.