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* * * * *
THE EARL OF PEMBROKE TO THE ABBESS OF WILTON.
"Go spin, you jade! go spin!"
[Illustration: MAGNETISM, LIGHT, AND MOLECULAR SPINNING TOPS.
_Page 122._
_THE ROMANCE OF SCIENCE._
SPINNING TOPS.
_THE "OPERATIVES' LECTURE"_
OF THE BRITISH ASSOCIATION MEETING AT LEEDS,
6th SEPTEMBER, 1890.
BY
PROFESSOR JOHN PERRY,
M.E., D.Sc, LL.D., F.R.S.
With Numerous Illustrations.
_REPRINT OF NEW AND REVISED EDITION,_
_With an Illustrated Appendix on the Use of Gyrostats._
LONDON
SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE,
Northumberland Avenue, W.C.; 43, Queen Victoria Street, E.C.
BRIGHTON: 129, North Street.
NEW YORK: E. S. GORHAM.
1910
PUBLISHED UNDER THE DIRECTION OF THE GENERAL LITERATURE COMMITTEE
[_Date of last impression, April 1908_]
This Report of an Experimental Lecture
WAS INSCRIBED TO
THE LATE
LORD KELVIN,
BY HIS AFFECTIONATE PUPIL, THE LECTURER, WHO
HEREBY TOOK A CONVENIENT METHOD OF
ACKNOWLEDGING THE REAL AUTHOR OF
WHATEVER IS WORTH PUBLICATION
IN THE FOLLOWING
PAGES.
* * * * *
PREFACE.
This is not the lecture as it was delivered. Instead of two pages of
letterpress and a woodcut, the reader may imagine that for half a minute
the lecturer played with a spinning top or gyrostat, and occasionally
ejaculated words of warning, admonition, and explanation towards his
audience. A verbatim report would make rather uninteresting reading, and I
have taken the liberty of trying, by greater fullness of explanation, to
make up to the reader for his not having seen the moving apparatus. It has
also been necessary in a treatise intended for general readers to simplify
the reasoning, the lecture having been delivered to persons whose life
experiences peculiarly fitted them for understanding scientific things. An
"argument" has been added at the end to make the steps of the reasoning
clearer.
JOHN PERRY.
* * * * *
{9}
SPINNING TOPS.
--------
At a Leeds Board School last week, the master said to his class, "There is
to be a meeting of the British Association in Leeds. What is it all about?
Who are the members of the British Association? What do they do?" There was
a long pause. At length it was broken by an intelligent shy boy: "Please,
sir, I know--they spin tops!"[1]
Now I am sorry to say that this answer was wrong. The members of the
British Association and the Operatives of Leeds have neglected top-spinning
since they were ten years of age. If more attention were paid to the
intelligent examination of the behaviour of tops, there would be greater
advances in mechanical engineering and a great many industries. There would
be a better general knowledge of astronomy. Geologists would not make
mistakes by millions of years, and our knowledge of Light, and Radiant
Heat, and other {10} Electro-magnetic Phenomena would extend much more
rapidly than it does.
I shall try to show you towards the end of the lecture that the fact of our
earth's being a spinning body is one which would make itself known to us
even if we lived in subterranean regions like the coming race of an
ingenious novelist.[2] It is the greatest and most persistent cause of many
of the phenomena which occur around us and beneath us, and it is probable
that even Terrestrial Magnetism is almost altogether due to it. Indeed
there is only one possible explanation of the _Vril-ya_ ignorance about the
earth's rotation. Their knowledge of mechanics and dynamics was immense; no
member attending the meeting of the British Association can approach them
in their knowledge of, I will not say, _Vril_, but even of quite vulgar
electricity and magnetism; and yet this great race which expresses so
strongly its contempt for Anglo-Saxon _Koom-Poshery_ was actually ignorant
of the fact that it had existed for untold generations inside an object
that spins about an axis.
Can we imagine for one instant that the children of that race had never
spun a top or trundled a hoop, and so had had no chance of being led to the
greatest study of nature? No; the only possible explanation lies in the
great novelist's never {11} having done these things himself. He had
probably as a child a contempt for the study of nature, he was a baby
Pelham, and as a man he was condemned to remain in ignorance even of the
powers of the new race that he had created.
The _Vril-ya_ ignorance of the behaviour of spinning bodies existing as it
does side by side with their deep knowledge of magnetism, becomes even more
remarkable when it comes home to us that the phenomena of magnetism and of
light are certainly closely connected with the behaviour of spinning
bodies, and indeed that a familiar knowledge of the behaviour of such
bodies is absolutely necessary for a proper comprehension of most of the
phenomena occurring in nature. The instinctive craving to investigate these
phenomena seems to manifest itself soon after we are able to talk, and who
knows how much of the intellectual inferiority of woman is due to her
neglect of the study of spinning tops; but alas, even for boys in the
pursuit of top-spinning, the youthful mind and muscle are left with no
other guidance than that which is supplied by the experience of young and
not very scientific companions. I remember distinctly that there were many
puzzling problems presented to me every day. There were tops which nobody
seemed able to spin, and there were others, well {12} prized objects, often
studied in their behaviour and coveted as supremely valuable, that behaved
well under the most unscientific treatment. And yet nobody, even the
makers, seemed to know why one behaved badly and the other well.
I do not disguise from myself the fact that it is rather a difficult task
to talk of spinning tops to men who have long lost that skill which they
wonder at in their children; that knowingness of touch and handling which
gave them once so much power over what I fear to call inanimate nature. A
problem which the child gives up as hopeless of solution, is seldom
attacked again in maturer years; he drives his desire for knowledge into
the obscure lumber-closets of his mind, and there it lies, with the
accumulating dust of his life, a neglected and almost forgotten instinct.
Some of you may think that this instinct only remains with those minds so
many of which are childish even to the limit of life's span; and probably
none of you have had the opportunity of seeing how the old dust rubs off
from the life of the ordinary man, and the old desire comes back to him to
understand the mysteries that surround him.
But I have not only felt this desire myself, I have seen it in the excited
eyes of the crowd of people who stand by the hour under the dropping
cherry-blossoms beside the red-pillared temple of {13} Asakusa in the
Eastern capital of Japan, watching the _tedzu-mashi_ directing the
evolutions of his heavily rimmed _Koma_. First he throws away from him his
great top obliquely into the air and catches it spinning on the end of a
stick, or the point of a sword, or any other convenient implement; he now
sends it about quite carelessly, catching it as it comes back to him from
all sorts of directions; he makes it run up the hand-rail of a staircase
into a house by the door and out again by the window; he makes it travel up
a great corkscrew. Now he seizes it in his hands, and with a few dexterous
twists gives it a new stock of spinning energy. He makes it travel along a
stretched string or the edge of a sword; he does all sorts of other curious
things with his tops, and suddenly sinks from his masterful position to beg
for a few coppers at the end of his performance.
How tame all this must seem to you who more than half forget your childish
initiation into the mysteries of nature; but trust me, if I could only make
that old top-spinner perform those magical operations of his on this
platform, the delight of the enjoyment of beautiful motion would come back.
Perhaps it is only in Japan that such an exhibition is possible; the land
where the waving bamboo, and the circling hawk, and the undulating summer
sea, and every beautiful motion of nature {14} are looked upon with
tenderness; and perhaps it is from Japan that we shall learn the
development of our childish enthusiasm.
The devotees of the new emotional art of beautiful motion and changing
colour are still in the main beggars like Homer, and they live in garrets
like Johnson and Savage; but the dawn of a new era is heralded, or rather
the dawn has already come, for Sir William Thomson's achievements in the
study of spinning tops rank already as by no means the meanest of his great
career.
If you will only think of it, the behaviour of the commonest spinning top
is very wonderful. When not spinning you see that it falls down at once, I
find it impossible to balance it on its peg; but what a very different
object it is when spinning; you see that it not only does not fall down, it
offers a strange resistance when I strike it, and actually lifts itself
more and more to an upright position. Once started on scientific
observation, nature gives us facts of an analogous kind in great plenty.
Those of you who have observed a rapidly moving heavy belt or rope, know
that rapid motion gives a peculiar quasi-rigidity to flexible and even to
fluid things.
Here, for example, is a disc of quite thin paper (Fig. 1), and when I set
it in rapid rotation you observe that it resists the force exerted by my
{15} hand, the blow of my fist, as if it were a disc of steel. Hear how it
resounds when I strike it with a stick. Where has its flexibility gone?
[Illustration: FIG. 1.]
Here again is a ring of chain which is quite flexible. It seems ridiculous
to imagine that this {16} could be made to stand up like a stiff hoop, and
yet you observe that when I give it a rapid rotation on this mandril and
let it slide off upon the table, it runs over the table just as if it were
a rigid ring, and when it drops on the floor it rebounds like a boy's hoop
(Fig. 2).
[Illustration: FIG. 2.]
Here again is a very soft hat, specially made for this sort of experiment.
You will note that it collapses to the table in a shapeless mass when I lay
it down, and seems quite incapable of resisting forces which tend to alter
its shape. In fact, there is almost a complete absence of rigidity; but
when this is spun on the end of a stick, first note {17} how it has taken a
very easily defined shape; secondly, note how it runs along the table as if
it were made of steel; thirdly, note how all at once it collapses again
into a shapeless heap of soft material when its rapid motion has ceased.
Even so you will see that when a drunken man is not leaning against a wall
or lamp-post, he feels that his only chance of escape from ignominious
collapse is to get up a decent rate of speed, to obtain a quasi-sobriety of
demeanour by rapidity of motion.
The water inside this glass vessel (Fig. 3) is in a state of rapid motion,
revolving with the vessel itself. Now observe the piece of paraffin wax A
immersed in the water, and you will see when I push at it with a rod that
it vibrates just as if it were surrounded with a thick jelly. Let us now
apply Prof. Fitzgerald's improvement on this experiment of Sir William
Thomson's. Here is a disc B stuck on the end of the rod; observe that when
I introduce it, although it does not touch A, A is repelled from the disc.
Now observe that when I twirl the disc it seems to attract A.
[Illustration: FIG. 3.[3]]
At the round hole in front of this box a rapid motion is given to a small
quantity of air which is mixed with smoke that you may see it. That
smoke-ring moves through the air almost like a solid body for a
considerable distance unchanged, and I am not sure that it may not be
possible yet {18} to send as a projectile a huge poisoned smoke-ring, so
that it may destroy or stupefy an army miles away. Remember that it is
really the same air all the time. You will observe that two smoke rings
sent from two boxes have curious actions {19} upon one another, and the
study of these actions has given rise to Thomson's smoke-ring or vortex
theory of the constitution of matter (Fig. 4).
[Illustration: FIG. 4.]
It was Rankine, the great guide of all engineers, who first suggested the
idea of molecular vortices in his explanations of heat phenomena and the
phenomena of elasticity--the idea that every particle of matter is like a
little spinning top; but I am now speaking of Thomson's theory. To imagine
that an atom of matter is merely a {20} curiously shaped smoke-ring formed
miraculously in a perfect fluid, and which can never undergo permanent
alteration, looks to be a very curious and far-fetched hypothesis. But in
spite of certain difficulties, it is the foundation of the theory which
will best explain most of the molecular phenomena observed by philosophers.
Whatever be the value of the theory, you see from these experiments that
motion does give to small quantities of fluid curious properties of
elasticity, attraction and repulsion; that each of these entities refuses
to be cut in two; that you cannot bring a knife even near the smoke-ring;
and that what may be called a collision between two of them is not very
different in any way from the collision between two rings of india-rubber.
Another example of the rigidity given to a fluid by rapid motion, is the
feeling of utter helplessness which even the strongest swimmers sometimes
experience when they get caught in an eddy underneath the water.
I could, if I liked, multiply these instances of the quasi-rigidity which
mere motion gives to flexible or fluid bodies. In Nevada a jet of water
like the jet from a fireman's hose, except that it is much more rapid,
which is nearly as easily projected in different directions, is used in
mining, and huge masses of earth and rock are rapidly disintegrated {21} by
the running water, which seems to be rather like a bar of steel than a jet
of water in its rigidity.
It is, however, probable that you will take more interest in this box of
brass which I hold in my hands. You see nothing moving, but really, inside
this case there is a fly-wheel revolving rapidly. Observe that I rest this
case on the table on its sharp edge, a sort of skate, and it does not
tumble down as an ordinary box would do, or as this box will do after a
while, when its contents come to rest. Observe that I can strike it violent
blows, and it does not seem to budge from its vertical position; it turns
itself just a little round, but does not get tilted, however hard I strike
it. Observe that if I do get it tilted a little it does not fall down, but
slowly turns with what is called a precessional motion (Fig. 5).
You will, I hope, allow me, all through this lecture, to use the term
_precessional_ for any motion of this kind. Probably you will object more
strongly to the great liberty I shall take presently, of saying that the
case _precesses_ when it has this kind of motion; but I really have almost
no option in the matter, as I must use some verb, and I have no time to
invent a less barbarous one.
[Illustration: FIG. 5.]
When I hold this box in my hands (Fig. 6), I find that if I move it with a
motion of mere translation in any direction, it feels just as it would do
{22} if its contents were at rest, but if I try to turn it in my hands I
find the most curious great resistance to such a motion. The result is that
when you hold this in your hands, its readiness to move so long as it is
not turned round, and its great resistance to turning round, and its
unexpected tendency to turn in a different way from that in which you try
to turn it, give one the most uncanny sensations. It seems almost as if an
invisible being had hold of the box and exercised forces capriciously. And
{23} indeed there is a spiritual being inside, what the algebraic people
call an impossible quantity, what other mathematicians call "an operator."
[Illustration: FIG. 6.]
Nearly all the experiments, even the tops and other apparatus you have seen
or will see to-night, have been arranged and made by my enthusiastic
assistant, Mr. Shepherd. The following experiment is not only his in
arrangement; even the idea of it is his. He said, you may grin and contort
your body with that large gyrostat in your hands, but many of your audience
will simply say to {24} themselves that you only _pretend_ to find a
difficulty in turning the gyrostat. So he arranged this pivoted table for
me to stand upon, and you will observe that when I now try to turn the
gyrostat, it will not turn; however I may exert myself, it keeps pointing
to that particular corner of the room, and all my efforts only result in
turning round my own body and the table, but not the gyrostat.
Now you will find that in every case this box only resists having the axis
of revolution of its hidden flywheel turned round, and if you are
interested in the matter and make a few observations, you will soon see
that every spinning body like the fly-wheel inside this case resists more
or less the change of direction of its spinning axis. When the fly-wheels
of steam-engines and dynamo machines and other quick speed machines are
rotating on board ship, you may be quite sure that they offer a greater
resistance to the pitching or rolling or turning of the ship, or any other
motion which tends to turn their axes in direction, than when they are not
rotating.
Here is a top lying on a plate, and I throw it up into the air; you will
observe that its motion is very difficult to follow, and nobody could
predict, before it falls, exactly how it will alight on the plate; it may
come down peg-end foremost, or hindmost, or sideways. But when I spin it
(Fig. 7), and now throw it up into the air, there is no doubt whatever {25}
as to how it will come down. The spinning axis keeps parallel to itself,
and I can throw the top up time after time, without disturbing much the
spinning motion.
[Illustration: FIG. 7.]
[Illustration: FIG. 8.]
If I pitch up this biscuit, you will observe that I can have no certainty
as to how it will come down, but if I give it a spin before it leaves my
hand there is no doubt whatever (Fig. 8). Here is a hat. I throw it up, and
I cannot be sure as to how it will move, but if I give it a spin, you see
that, as {26} with the top and the biscuit, the axis about which the
spinning takes place keeps parallel to itself, and we have perfect
certainty as to the hat's alighting on the ground brim downwards (Fig. 9).
[Illustration: FIG. 9.]
I need not again bring before you the very soft hat to which we gave a
quasi-rigidity a few minutes ago; but you will remember that my assistant
sent that off like a projectile through the air when it was spinning, and
that it kept its spinning axis parallel to itself just like this more rigid
hat and the biscuit.
[Illustration: FIG. 10.]
[Illustration: FIG. 11.]
I once showed some experiments on spinning tops to a coffee-drinking,
tobacco-smoking audience in that most excellent institution, the Victoria
Music Hall in London. In that music hall, things are not very different
from what they are at any other {27} music hall except in beer, wine, and
spirits being unobtainable, and in short scientific addresses being
occasionally given. Now, I impressed my audience as strongly as I could
with the above fact, that if one wants to throw a quoit with certainty as
to how it will alight, one gives it a spin; if one wants to throw a hoop or
a hat to somebody to catch upon a stick, one gives the hoop or hat a spin;
the disinclination of a spinning body to let its axis get altered in
direction can always be depended upon. I told them that this was why
smooth-bore guns cannot be depended upon for accuracy;[4] that the spin
which an ordinary bullet took depended greatly on how it chanced to touch
the muzzle as it just left the gun, whereas barrels are now rifled, that
is, spiral grooves are now cut inside the barrel of a gun, and excrescences
from the bullet or projectile fit into these grooves, so that as it is
forced along the barrel of the gun by the explosive force of the powder, it
must also spin about its axis. Hence it leaves the gun with a perfectly
well-known spinning motion about which there can be no doubt, and we know
too that Fig. 10 shows the {28} kind of motion which it has afterwards,
for, just like the hat or the biscuit, its spinning axis keeps nearly
parallel to itself. Well, this was all I could do, for I am not skilful in
throwing hats or quoits. But after my address was finished, and after a
young lady in a spangled dress had sung a comic song, two jugglers came
upon the stage, and I could not have had better illustrations of the above
principle than were given in almost every trick performed by this lady and
gentleman. They sent hats, and hoops, and plates, and umbrellas spinning
from one to the other. One of them threw a stream of knives into the air,
catching them and throwing them up again with perfect precision and my now
educated audience shouted with delight, and showed in other unmistakable
{29} ways that they observed the spin which that juggler gave to every
knife as it left his hand, so that he might have a perfect knowledge as to
how it would come back to him again (Fig. 11). {30} It struck me with
astonishment at the time that, almost without exception, every juggling
trick performed that evening was an illustration of the above principle.
And now, if you doubt my statement, just ask a child whether its hoop is
more likely to tumble down when it is rapidly rolling along, or when it is
going very slowly; ask a man on a bicycle to go more and more slowly to see
if he keeps his balance better; ask a ballet-dancer how long she could
stand on one toe without balancing herself with her arms or a pole, if she
were not spinning; ask astronomers how many months would elapse before the
earth would point ever so far away from the pole star if it were not
spinning; and above all, ask a boy whether his top is as likely to stand
upright upon its peg when it is not spinning as when it is spinning.
[Illustration: FIG. 12.]
We will now examine more carefully the behaviour of this common top (Fig.
12). It is not {31} spinning, and you observe that it tumbles down at once;
it is quite unstable if I leave it resting upright on its peg. But now note
that when it is spinning, it not only will remain upright resting on its
peg, but if I give it a blow and so disturb its state, it goes circling
round with a precessional motion which grows gradually less and less as
time goes on, and the top lifts itself to the upright position again. I
hope you do not think that time spent in careful observation of a
phenomenon of this kind is wasted. Educated observation of the commonest
phenomena occurring in our everyday life is never wasted, and I often feel
that if workmen, who are the persons most familiar with inorganic nature,
could only observe and apply simple scientific laws to their observations,
instead of a great discovery every century we should have a great discovery
every year. Well, to return to our top; there are two very curious
observations to make. Please neglect for a short time the slight wobbling
motions that occur. One observation we make is, that the top does not at
first bow down in the direction of the blow. If I strike towards the south,
the top bows towards the west; if I strike towards the west, the top bows
down towards the north. Now the reason of this is known to all scientific
men, and the principle underlying the top's behaviour is of very great {32}
importance in many ways, and I hope to make it clear to you. The second
fact, that the top gradually reaches its upright position again, is one
known to everybody, but the reason for it is not by any means well known,
although I think that you will have no great difficulty in understanding
it.
The first phenomenon will be observed in this case which I have already
shown you. This case (Fig. 5), {33} with the fly-wheel inside it, is called
a _gyrostat_. When I push the case it does not bow down, but slowly turns
round. This gyrostat will not exhibit the second phenomenon; it will not
rise up again if I manage to get it out of its upright position, but, on
the contrary, will go precessing in wider and wider circles, getting
further and further away from its upright position.
[Illustration: FIG. 13.]
[Illustration: FIG. 14.]
The first phenomenon is most easily studied in this balanced gyrostat (Fig.
13). You here see the fly-wheel G in a strong brass frame F, which is
supported so that it is free to move about the vertical axis A B, or about
the horizontal axis C D. The gyrostat is balanced by a weight W. Observe
that I can increase the leverage of W or diminish it by shifting the
position of the sleeve at A so that it will tend to either lift or lower
the gyrostat, or exactly balance it as it does now. You must observe
exactly what it is that we wish to study. If I endeavour to push F
downwards, with the end of this stick (Fig. 14), it really moves
horizontally to the right; now I push it to the right (Fig. 15), and it
only rises; now push it up, and you see that it goes to the left; push it
to the left, and it only goes downwards. You will notice that if I clamp
the instrument so that it cannot move vertically, it moves at once
horizontally; if I prevent mere horizontal motion it readily moves
vertically when I push it. Leaving it free as {34} before, I will now shift
the position of the weight W, so that it tends continually to lift the
gyrostat, and of course the instrument does not lift, it moves horizontally
with a slow precessional motion. I now again shift the weight W, so that
the gyrostat would fall if it were not spinning (Fig. 16), and it now moves
horizontally with a slow precessional motion which is in a direction
opposed to the last. These phenomena are easily explained, but, {35} as I
said before, it is necessary first to observe them carefully. You all know
now, vaguely, the fundamental fact. It is that if I try to make a very
quickly spinning body change the direction of its axis, the direction of
the axis will change, but not in the way I intended. It is even more
curious than my countryman's pig, for when he wanted the pig to go to Cork,
he had to pretend that he was driving the pig home. His rule was a very
{36} simple one, and we must find a rule for our spinning body, which is
rather like a crab, that will only go along the road when you push it
sidewise.
[Illustration: FIG. 15.]
[Illustration: FIG. 16.[5]]
[Illustration: FIG. 10.]
As an illustration of this, consider the spinning projectile of Fig. 10.
The spin tends to keep its axis always in the same direction. But there is
a defect in the arrangement, which you are now in a {37} position to
understand. You see that at A the air must be pressing upon the
undersurface A A, and I have to explain that this pressure tends to make
the projectile turn itself broadside on to the air. A boat in a current not
allowed to move as a whole, but tied at its middle, sets itself broadside
on to the current. Observe this disc of cardboard which I drop through the
air edgewise, and note how quickly it sets itself broadside on and falls
more slowly; and some of you may have thrown over into the water at Aden
small pieces of silver for the diving boys, and you are aware that if it
were not for this slow falling of the coins with a wobbling motion
broadside on, it would be nearly impossible for any diving boy to get
possession of them. Now all this is a parenthesis. The {38} pressure of the
air tends to make the projectile turn broadside on, but as the projectile
is spinning it does not tilt up, no more than this gyrostat does when I try
to tilt it up, it really tilts out of the plane of the diagram, out of the
plane of its flight; and only that artillerymen know exactly what it will
do, this kind of _windage_ of the projectile would give them great trouble.
You will notice that an experienced child when it wants to change the
direction of a hoop, just exerts a tilting pressure with its hoop-stick. A
man on a bicycle changes his direction by leaning over so as to be out of
balance. It is well to remind you, however, that the motion of a bicycle
and its rider is not all rotational, so that it is not altogether the
analogue of a top or gyrostat. The explanation of the swerving from a
straight path when the rider tilts his body, ultimately comes to the same
simple principle, Newton's second law of motion, but it is arrived at more
readily. It is for the same reason--put briefly, the exercise of a
centripetal force--that when one is riding he can materially assist his
horse to turn a corner quickly, if he does not mind appearances, by
inclining his body towards the side to which he wants to turn; and the more
slowly the horse is going the greater is the tendency to turn for a given
amount of tilting of one's body. Circus-riders, when galloping in a circle,
assist their horses greatly by the position of their bodies; it is {39} not
to save themselves from falling by centrifugal force that they take a
position on a horse's back which no riding-master would allow his pupil to
imitate; and the respectable riders of this country would not scorn to help
their horses in this way to quick turning movements, if they had to chase
and collect cattle like American cowboys.
Very good illustrations of change of direction are obtained in playing
_bowls_. You know that a bowl, if it had no _bias_, that is, if it had no
little weight inside it tending to tilt it, would roll along the level
bowling-green in a straight path, its speed getting less and less till it
stopped. As a matter of fact, however, you know that at the beginning, when
it is moving fast, its path is pretty straight, but because it always has
bias the path is never quite straight, and it bends more and more rapidly
as the speed diminishes. In all our examples the slower the spin the
quicker is the precession produced by given tilting forces.
Now close observation will give you a simple rule about the behaviour of a
gyrostat. As a matter of fact, all that has been incomprehensible or
curious disappears at once, if instead of speaking of this gyrostat as
moving up or down, or to the right or left, I speak of its motions about
its various axes. It offers no resistance to mere motion of translation.
But when I spoke of its moving {40} horizontally, I ought to have said that
it moved about the vertical axis A B (Fig. 13). Again, what I referred to
as up and down motion of F is really motion in a vertical plane about the
horizontal axis C D. In future, when I speak of trying to give motion to F,
think only of the axis about which I try to turn it, and then a little
observation will clear the ground.
[Illustration: FIG. 17.]
[Illustration: FIG. 18.]
Here is a gyrostat (Fig. 17), suspended in gymbals so carefully that
neither gravity nor any frictional forces at the pivots constrain it;
nothing that I can do to this frame which I hold in my hand will affect the
direction of the axis E F of the gyrostat. Observe that I whirl round on my
toes like a ballet-dancer while this is in my hand. I move it about in all
sorts of ways, but if it was pointing to the pole star at the beginning it
remains pointing to the pole star; if it pointed towards the moon at the
beginning it still points {41} towards the moon. The fact is, that as there
is almost no frictional constraint at the pivots there are almost no forces
tending to turn the axis of rotation of the gyrostat, and I can only give
it motions of translation. But now I will clamp this vertical spindle by
means of a screw and repeat my ballet-dance whirl; you will note that I
need not whirl round, a very small portion of a whirl is enough to cause
this gyrostat (Fig. 18) to set its spinning axis vertical, to set its axis
parallel to the vertical axis of rotation which I give it. Now I whirl in
the opposite direction, the gyrostat at once turns a somersault, turns
completely round and remains again with its axis vertical, and if you were
to carefully note the direction of the spinning of the {42} gyrostat, you
would find the following rule to be generally true:--Pay no attention to
mere translational motion, think only of rotation about axes, and just
remember that when you constrain the axis of a spinning body to rotate, it
will endeavour to set its own axis parallel to the new axis about which you
rotate it; and not only is this the case, but it will endeavour to have the
direction of its own spin the same as the direction of the new rotation. I
again twirl on my toes, holding this frame, and now I know that to a person
looking down upon the gyrostat and me from the ceiling, as I revolved in
the direction of the hands of a clock, the gyrostat is spinning in the
direction of the hands of a clock; but if I revolve against the clock
direction (Fig. 19) the gyrostat tumbles over so as again to be revolving
in the same direction as that in which I revolve.
[Illustration: FIG. 19.]
This then is the simple rule which will enable you to tell beforehand how a
gyrostat will move {43} when you try to turn it in any particular
direction. You have only to remember that if you continued your effort long
enough, the spinning axis would become parallel to your new axis of motion,
and the direction of spinning would be the same as the direction of your
new turning motion.
Now let me apply my rule to this balanced gyrostat. I shove it, or give it
an impulse downwards, but observe that this really means a rotation about
the horizontal axis C D (Fig. 13), and hence the gyrostat turns its axis as
if it wanted to become parallel to C D. Thus, looking down from above (as
shown by Fig. 20), O E was the direction of the spinning axis, O D was the
axis about which I endeavoured to move it, and the instantaneous effect was
that O E altered to the position O G. A greater impulse of the same kind
would have caused the spinning axis instantly to go to O H or O J, whereas
an upward opposite impulse would have instantly made the spinning axis
point in the direction O K, O L or O M, depending on how great the impulse
was and the rate of spinning. When one observes these phenomena for the
first time, one says, "I shoved it down, and it moved to the right; I
shoved it up, and it moved to the left;" but if the direction of the spin
were opposite to what it is, one would say, "I shoved it down, and it moved
to the left; I shoved it up, and it moved to the right." The simple {44}
statement in all cases ought to be, "I wanted to rotate it about a new
axis, and the effect was to send its spinning axis towards the direction of
the new axis." And now if you play with this balanced gyrostat as I am
doing, shoving it about in all sorts of ways, you will find the rule to be
a correct one, and there is no difficulty in predicting what will happen.
[Illustration: FIG. 20.]
{45}
If this rule is right, we see at once why precession takes place. I put
this gyrostat (Fig. 13) out of balance, and if it were not rotating it
would fall downwards; but a force acting downwards really causes the
gyrostat to move to the right, and so you see that it is continually moving
in this way, for the force is always acting downwards, and the spinning
axis is continually chasing the new axes about which gravity tends
continually to make it revolve. We see also why it is that if the want of
balance is the other way, if gravity tends to lift the gyrostat, the
precession is in the opposite direction. And in playing with this gyrostat
as I do now, giving it all sorts of pushes, one makes other observations
and sees that the above rule simplifies them all; that is, it enables us to
remember them. For example, if I use this stick to hurry on the precession,
the gyrostat moves in opposition to the force which causes the precession.
I am particularly anxious that you should remember this. At present the
balance-weight is so placed that the gyrostat would fall if it were not
spinning. But it is spinning, and so it precesses. If gravity were greater
it would precess faster, and it comes home to us that it is this precession
which enables the force of gravity to be inoperative in mere downward
motion. You see that if the precession is hurried, it is more than
sufficient to balance gravity, {46} and the gyrostat rises. If I retard the
precession, it is unable to balance gravity, and the gyrostat falls. If I
clamp this vertical axis so that precession is impossible, you will notice
that the gyrostat falls just as if it were not spinning. If I clamp the
instrument so that it cannot move vertically, you notice how readily I can
make it move horizontally; I can set it rotating horizontally like any
ordinary body.
In applying our rule to this top, observe that the axis of spinning is the
axis E F of the top (Fig. 12). As seen in the figure, gravity is tending to
make the top rotate about the axis F D, and the spinning axis in its chase
of the axis F D describes a cone in space as it precesses. This gyrostat,
which is top-heavy, rotates and precesses in much the same way as the top;
that is, if you apply our rule, or use your observation, you will find that
to an observer above the table the spinning and precession occur in the
same direction, that is, either both with the hands of a watch, or both
against the hands of a watch. Whereas, a top like this before you (Fig.
21), supported above its centre of gravity, or the gyrostat here (Fig. 22),
which is also supported above its centre of gravity, or the gyrostat shown
in Fig. 56, or any other gyrostat supported in such a way that it would be
in stable equilibrium if it were not spinning; in all these {47} cases, to
an observer placed above the table, the precession is in a direction
opposite to that of the spinning.
[Illustration: FIG. 21.]
[Illustration: FIG. 22.]
{48}
If an impulse be given to a top or gyrostat in the direction of the
precession, it will rise in opposition to the force of gravity, and should
at any instant the precessional velocity be greater than what it ought to
be for the balance of the force of gravity, the top or gyrostat will rise,
its precessional velocity diminishing. If the precessional velocity is too
small, the top will fall, and as it falls the precessional velocity
increases.
Now I say that all these facts, which are mere facts of observation, agree
with our rule. I wish I dare ask you to remember them all. You will observe
that in this wall sheet I have made a list of them. I speak of gravity as
causing the precession, but the forces may be any others than such as are
due to gravity.
WALL SHEET.
I. RULE. When forces act upon a spinning body, tending to cause rotation
about any other axis than the spinning axis, the spinning axis sets itself
in better agreement with the new axis of rotation. Perfect agreement would
mean perfect parallelism, the directions of rotation being the same.
II. Hurry on the precession, and the body rises in opposition to gravity.
{49}
III. Delay the precession and the body falls, as gravity would make it do
if it were not spinning.
IV. A common top precesses in the same direction as that in which it spins.
V. A top supported above its centre of gravity, or a body which would be in
stable equilibrium if not spinning, precesses in the opposite direction to
that of its spinning.
VI. The last two statements come to this:--When the forces acting on a
spinning body tend to make the _angle_ of precession greater, the
precession is in the same direction as the spinning, and _vice versa_.
Having by observation obtained a rule, every natural philosopher tries to
make his rule a rational one; tries to explain it. I hope you know what we
mean when we say that we explain a phenomenon; we really mean that we show
the phenomenon to be consistent with other better known phenomena. Thus
when you unmask a spiritualist and show that the phenomena exhibited by him
are due to mere sleight-of-hand and trickery, you explain the phenomena.
When you show that they are all consistent with well-observed and
established mesmeric influences, you are also said to explain the
phenomena. When you show that they can be effected by means of telegraphic
messages, or by reflection of light from mirrors, you explain the {50}
phenomena, although in all these cases you do not really know the nature of
mesmerism, electricity, light, or moral obliquity.
The meanest kind of criticism is that of the man who cheapens a scientific
explanation by saying that the very simplest facts of nature are
unexplainable. Such a man prefers the chaotic and indiscriminate wonder of
the savage to the reverence of a Sir Isaac Newton.
[Illustration: FIG. 23.]
The explanation of our rule is easy. Here is a gyrostat (Fig. 23) something
like the earth in shape, and it is at rest. I am sorry to say that I am
compelled to support this globe in a very visible manner by gymbal rings.
If this globe were just floating in the air, if it had no tendency to fall,
my explanation would be easier to understand, and I could illustrate it
better experimentally. Observe the point P. If I move the globe slightly
about the axis A, the point P moves to Q. But suppose instead of this that
the globe and inner gymbal {51} ring had been moved about the axis B; the
point P would have moved to R. Well, suppose both those rotations took
place simultaneously. You all know that the point P would move neither to Q
nor to R, but it would move to S; P S being the diagonal of the little
parallelogram. The resultant motion then is neither about the axis O A in
space, nor about the axis O B, but it is about some such axis as O C.
To this globe I have given two rotations simultaneously. Suppose a little
being to exist on this globe which could not see the gymbals, but was able
to observe other objects in the room. It would say that the direction of
rotation is neither about O A nor about O B, but that the real axis of its
earth is some line intermediate, O C in fact.
If then a ball is suddenly struck in two different directions at the same
instant, to understand how it will spin we must first find how much spin
each blow would produce if it acted alone, and about what axis. A spin of
three turns per second about the axis O A (Fig. 24), and a spin of two
turns per second about the axis O B, really mean that the ball will spin
about the axis O C with a spin of three and a half turns per second. To
arrive at this result, I made O A, 3 feet long (any other scale of
representation would have been right) {52} and O B, 2 feet long, and I
found the diagonal O C of the parallelogram shown on the figure to be 3-1/2
feet long.
Observe that if the rotation about the axis O A is _with_ the hands of a
watch looking from O to A, the rotation about the axis O B looking from O
to B, must also be with the hands of a watch, and the resultant rotation
about the axis O C is also in a direction with the hands of a watch looking
from O to C. Fig. 25 shows in two diagrams how necessary it is that on
looking from O along either O A or O B, the rotation should be in the same
direction as regards the hands of a watch. These constructions are well
known to all who have studied elementary mechanical principles. Obviously
if the rotation about O A is very much greater than the rotation about O B,
then the position of the new axis O C must be much nearer O A than O B.
[Illustration: FIG. 24.]
[Illustration: FIG. 25.]
We see then that if a body is spinning about an axis O A, and we apply
forces to it which {53} would, if it were at rest, turn it about the axis O
B; the effect is to cause the spinning axis to be altered to O C; that is,
the spinning axis sets itself in better agreement with the new axis of
rotation. This is the first statement on our wall sheet, the rule from
which all our other statements are derived, assuming that they were not
really derived from observation. Now I do not say that I have here given a
complete proof for all cases, for the fly-wheels in these gyrostats are
running in bearings, and the bearings constrain the axes to take the new
positions, whereas there is no such {54} constraint in this top; but in the
limited time of a popular lecture like this it is not possible, even if it
were desirable, to give an exhaustive proof of such a universal rule as
ours is. That I have not exhausted all that might be said on this subject
will be evident from what follows.
If we have a spinning ball and we give to it a new kind of rotation, what
will happen? Suppose, for example, that the earth were a homogeneous
sphere, and that there were suddenly impressed upon it a new rotatory
motion tending to send Africa southwards; the axis of this new spin would
have its pole at Java, and this spin combined with the old one would cause
the earth to have its true pole somewhere between the present pole and
Java. It would no longer rotate about its present axis. In fact the axis of
rotation would be altered, and there would be no tendency for anything
further to occur, because a homogeneous sphere will as readily rotate about
one axis as another. But if such a thing were to happen to this earth of
ours, which is not a sphere but a flattened spheroid like an orange, its
polar diameter being the one-third of one per cent. shorter than the
equatorial diameter; then as soon as the new axis was established, the axis
of symmetry would resent the change and would try to become again the axis
of rotation, and a great wobbling motion would ensue. {55} I put the matter
in popular language when I speak of the resentment of an axis; perhaps it
is better to explain more exactly what I mean. I am going to use the
expression Centrifugal Force. Now there are captious critics who object to
this term, but all engineers use it, and I like to use it, and our captious
critics submit to all sorts of ignominious involution of language in
evading the use of it. It means the force with which any body acts upon its
constraints when it is constrained to move in a curved path. The force is
always directed away from the centre of the curve. When a ball is whirled
round in a curve at the end of a string its centrifugal force tends to
break the string. When any body keyed to a shaft is revolving with the
shaft, it may be that the centrifugal forces of all the parts just balance
one another; but sometimes they do not, and then we say that the shaft is
out of balance. Here, for example, is a disc of wood rotating. It is in
balance. But I stop its motion and fix this piece of lead, A, to it, and
you observe when it rotates that it is so much out of balance that the
bearings of the shaft and the frame that holds them, and even the
lecture-table, are shaking. Now I will put things in balance again by
placing another piece of lead, B, on the side of the spindle remote from A,
and when I again rotate the disc (Fig. 26) there {56} is no longer any
shaking of the framework. When the crank-shaft of a locomotive has not been
put in balance by means of weights suitably placed on the driving-wheels,
there is nobody in the train who does not feel the effects. Yes, and the
coal-bill shows the effects, for an unbalanced engine tugs the train
spasmodically instead of exerting an efficient steady pull. My friend
Professor Milne, of Japan, places earthquake measuring instruments on
engines and in trains for measuring this and other wants of balance, and he
has shown unmistakably that two engines of nearly the same general design,
one balanced properly and the other not, consume very different amounts of
coal in making the same journey at the same speed.
[Illustration: FIG. 26.]
If a rotating body is in balance, not only does the axis of rotation pass
through the centre of gravity (or rather centre of mass) of the body, but
{57} the axis of rotation must be one of the three principal axes through
the centre of mass of the body. Here, for example, is an ellipsoid of wood;
A A, B B, and C C (Fig. 27) are its three principal axes, and it would be
in balance if it rotated about any one of these three axes, and it would
not be in balance if it rotated about any other axis, unless, indeed, it
were like a homogeneous sphere, every diameter of which is a principal
axis.
[Illustration: FIG. 27.]