this body (Fig. 27) has them; but I have here constrained it to rotate
about the axis D D, and you all observe the effect of the unbalanced
centrifugal forces, which is nearly great enough to tear the framework in
pieces. The higher the speed the more important this want of balance is. If
the speed is doubled, the centrifugal forces become four times as great;
and modern mechanical engineers with their quick speed engines, some of
which revolve, like the fan-engines of torpedo-boats, at 1700 revolutions
per minute, require to pay great attention to this subject, which the older
engineers never troubled their {58} heads about. You must remember that
even when want of balance does not actually fracture the framework of an
engine, it will shake everything, so that nuts and keys and other
fastenings are pretty sure to get loose.
I have seen, on a badly-balanced machine, a securely-fastened pair of nuts,
one supposed to be locking the other, quietly revolving on their bolt at
the same time, and gently lifting themselves at a regular but fairly rapid
rate, until they both tumbled from the end of the bolt into my hand. If my
hand had not been there, the bolts would have tumbled into a receptacle in
which they would have produced interesting but most destructive phenomena.
You would have somebody else lecturing to you to-night if that event had
come off.
Suppose, then, that our earth were spinning about any other axis than its
present axis, the axis of figure. If spun about any diameter of the equator
for example, centrifugal forces would just keep things in a state of
unstable equilibrium, and no great change might be produced until some
accidental cause effected a slight alteration in the spinning axis, and
after that the earth would wobble very greatly. How long and how violently
it would wobble, would depend on a number of circumstances about which I
will not now venture to guess. If you {59} tell me that on the whole, in
spite of the violence of the wobbling, it would not get shaken into a new
form altogether, then I know that in consequence of tidal and other
friction it would eventually come to a quiet state of spinning about its
present axis.
You see, then, that although every body has three axes about which it will
rotate in a balanced fashion without any tendency to wobble, this balance
of the centrifugal forces is really an unstable balance in two out of the
three cases, and there is only one axis about which a perfectly stable
balanced kind of rotation will take place, and a spinning body generally
comes to rotate about this axis in the long run if left to itself, and if
there is friction to still the wobbling.
To illustrate this, I have here a method of spinning bodies which enables
them to choose as their spinning axis that one principal axis about which
their rotation is most stable. The various bodies can be hung at the end of
this string, and I cause the pulley from which the string hangs to rotate.
Observe that at first the disc (Fig. 28 _a_) rotates soberly about the axis
A A, but you note the small beginning of the wobble; now it gets quite
violent, and now the disc is stably and smoothly rotating about the axis B
B, which is the most important of its principal axes. {60}
[Illustration: FIG. 28.]
Again, this cone (Fig. 28 _b_) rotates smoothly at first about the axis A
A, but the wobble begins and gets very great, and eventually the cone
rotates smoothly about the axis B B, which is the most important of its
principal axes. Here again is a rod hung from one end (Fig. 28 _d_).
See also this anchor ring. But you may be more interested in this limp ring
of chain (Fig. 28 _c_). See how at first it hangs from the cord vertically,
and how the wobbles and vibrations end in its becoming a perfectly circular
ring lying all in a horizontal plane. This experiment illustrates also the
quasi-rigidity given to a flexible body by rapid motion.
To return to this balanced gyrostat of ours (Fig. 13). It is not
precessing, so you know that the weight W just balances the gyrostat F. Now
if I leave the instrument to itself after I give a downward impulse to F,
not exerting merely a steady pressure, you will notice that F swings to the
right for the reason already given; but it swings too fast and too far,
just like any other swinging body, and it is easy from what I have already
said, to see that this wobbling motion (Fig. 29) should be the result, and
that it should continue until friction stills it, and F takes its permanent
new position only after some time elapses.
You see that I can impose this wobble or nodding {62} motion upon the
gyrostat whether it has a motion of precession or not. It is now nodding as
it processes round and round--that is, it is rising and falling as it
precesses.
[Illustration: FIG. 29.]
Perhaps I had better put the matter a little more clearly. You see the same
phenomenon in this top. If the top is precessing too fast for the force of
gravity the top rises, and the precession diminishes in consequence; the
precession being now too slow to balance gravity, the top falls a little
and the {63} precession increases again, and this sort of vibration about a
mean position goes on just as the vibration of a pendulum goes on till
friction destroys it, and the top precesses more regularly in the mean
position. This nodding is more evident in the nearly horizontal balanced
gyrostat than in a top, because in a top the turning effect of gravity is
less in the higher positions.
When scientific men try to popularize their discoveries, for the sake of
making some fact very plain they will often tell slight untruths, making
statements which become rather misleading when their students reach the
higher levels. Thus astronomers tell the public that the earth goes round
the sun in an elliptic path, whereas the attractions of the planets cause
the path to be only approximately elliptic; and electricians tell the
public that electric energy is conveyed through wires, whereas it is really
conveyed by all other space than that occupied by the wires. In this
lecture I have to some small extent taken advantage of you in this way; for
example, at first you will remember, I neglected the nodding or wobbling
produced when an impulse is given to a top or gyrostat, and, all through, I
neglect the fact that the instantaneous axis of rotation is only nearly
coincident with the axis of figure of a precessing gyrostat or top. And
indeed you may generally {64} take it that if all one's statements were
absolutely accurate, it would be necessary to use hundreds of technical
terms and involved sentences with explanatory, police-like parentheses; and
to listen to many such statements would be absolutely impossible, even for
a scientific man. You would hardly expect, however, that so great a
scientific man as the late Professor Rankine, when he was seized with the
poetic fervour, would err even more than the popular lecturer in making his
accuracy of statement subservient to the exigencies of the rhyme as well as
to the necessity for simplicity of statement. He in his poem, _The
Mathematician in Love_, has the following lines--
"The lady loved dancing;--he therefore applied
To the polka and waltz, an equation;
But when to rotate on his axis he tried,
His centre of gravity swayed to one side,
And he fell by the earth's gravitation."
Now I have no doubt that this is as good "dropping into poetry" as can be
expected in a scientific man, and ----'s science is as good as can be
expected in a man who calls himself a poet; but in both cases we have
illustrations of the incompatibility of science and rhyming.
[Illustration: FIG. 17.]
The motion of this gyrostat can be made even more complicated than it was
when we had {65} nutation and precession, but there is really nothing in it
which is not readily explainable by the simple principles I have put before
you. Look, for example, at this well-balanced gyrostat (Fig. 17). When I
strike this inner gymbal ring in any way you see that it wriggles quickly
just as if it were a lump of jelly, its rapid vibrations dying away just
like the rapid vibrations of any yielding elastic body. This strange
elasticity is of very great interest when we consider it in relation to the
molecular properties of matter. Here again (Fig. 30) we have an example
which is even more interesting. I have supported the cased {66} gyrostat of
Figs. 5 and 6 upon a pair of stilts, and you will observe that it is moving
about a perfectly stable position with a very curious staggering kind of
vibratory motion; but there is nothing in these motions, however curious,
that you cannot easily explain if you have followed me so far.
[Illustration: FIG. 30.]
Some of you who are more observant than the others, will have remarked that
all these precessing gyrostats gradually fall lower and lower, just as they
would do, only more quickly, if they were not spinning. And if you cast
your eye upon the third statement of our wall sheet (p. 49) you will
readily understand why it is so.
"Delay the precession and the body falls, as gravity would make it do if it
were not spinning." {67} Well, the precession of every one of these is
resisted by friction, and so they fall lower and lower.
I wonder if any of you have followed me so well as to know already why a
spinning top rises. Perhaps you have not yet had time to think it out, but
I have accentuated several times the particular fact which explains this
phenomenon. Friction makes the gyrostats fall, what is it that causes a top
to rise? Rapid rising to the upright position is the invariable sign of
rapid rotation in a top, and I recollect that when quite vertical we used
to say, "She sleeps!" Such was the endearing way in which the youthful
experimenter thought of the beautiful object of his tender regard.
All so well known as this rising tendency of a top has been ever since tops
were first spun, I question if any person in this hall knows the
explanation, and I question its being known to more than a few persons
anywhere. Any great mathematician will tell you that the explanation is
surely to be found published in _Routh_, or that at all events he knows men
at Cambridge who surely know it, and he thinks that he himself must have
known it, although he has now forgotten those elaborate mathematical
demonstrations which he once exercised his mind upon. I believe that all
such statements are made in error, but I cannot {68} be sure.[6] A partial
theory of the phenomenon was given by Mr. Archibald Smith in the _Cambridge
Mathematical Journal_ many years ago, but the problem was solved by Sir
William Thomson and Professor Blackburn when they stayed together one year
at the seaside, reading for the great Cambridge mathematical examination.
It must have alarmed a person interested in Thomson's success to notice
that the seaside holiday was really spent by him and his friend in spinning
all sorts of rounded stones which they picked up on the beach.
And I will now show you the curious phenomenon that puzzled him that year.
This ellipsoid (Fig. 31) will represent a waterworn stone. It is lying in
its most stable state on the table, and I give it a spin. You see that for
a second or two it was inclined to go on spinning about the axis A A, but
it began to wobble violently, and after a while, when these wobbles
stilled, you saw that it was spinning nicely with its axis B B vertical;
but then a new series of wobblings began and became more violent, and when
they ceased you saw that the object had at length reached a settled state
of {69} spinning, standing upright upon its longest axis. This is an
extraordinary phenomenon to any person who knows about the great
inclination of this body to spin in the very way in which I first started
it spinning. You will find that nearly any rounded stone when spun will get
up in this way upon its longest axis, if the spin is only vigorous enough,
and in the very same way this spinning top tends to get more and more
upright.
[Illustration: FIG. 31.]
I believe that there are very few mathematical explanations of phenomena
which may not be given in quite ordinary language to people who have an
ordinary amount of experience. In most cases the symbolical algebraic
explanation must be given first by somebody, and then comes the time for
its translation into ordinary language. This is the foundation of the new
thing called Technical Education, which assumes that a {70} workman may be
taught the principles underlying the operations which go on in his trade,
if we base our explanations on the experience which the man has acquired
already, without tiring him with a four years' course of study in
elementary things such as is most suitable for inexperienced children and
youths at public schools and the universities.
[Illustration: FIG. 32.]
[Illustration: FIG. 33.]
With your present experience the explanation of the rising of the top
becomes ridiculously simple. If you look at statement _two_ on this wall
sheet (p. 48) and reflect a little, some of you will be able, without any
elaborate mathematics, to give the simple reason for this that Thomson gave
me sixteen years ago. "Hurry on the precession, and the body rises in
opposition to gravity." Well, as I am not touching the top, and as the body
does rise, we look at once for something that is hurrying on the
precession, and we naturally look to the way in which its peg is rubbing on
the table, for, with the exception of the atmosphere this top is touching
nothing else than the table. Observe carefully how any of these objects
precesses. Fig. 32 shows the way in which a top spins. Looked at from
above, if the top is spinning in the direction of the hands of a watch, we
know from the fourth statement of our wall sheet, or by mere observation,
that it also precesses in the direction of the hands {71} of a watch; that
is, its precession is such as to make the peg roll at B into the paper. For
you will observe that the peg is rolling round a circular path on the
table, G being nearly motionless, and the axis A G A describing nearly a
cone in space whose vertex is G, above the table. Fig. 33 {72} shows the
peg enlarged, and it is evident that the point B touching the table is
really like the bottom of a wheel B B', and as this wheel is rotating, the
rotation causes it to roll _into_ the paper, away from us. But observe that
its mere precession is making it roll _into_ the paper, and that the spin
if great enough wants to roll the top faster than the precession lets it
roll, so that it hurries on the precession, and therefore the top rises.
That is the simple explanation; the spin, so long as it is {73} great
enough, is always hurrying on the precession, and if you will cast your
recollection back to the days of your youth, when a top was supported on
your hand as this is now on mine (Fig. 34), and the spin had grown to be
quite small, and was unable to keep the top upright, you will remember that
you dexterously helped the precession by giving your hand a circling motion
so as to get from your top the advantages as to uprightness of a slightly
longer spin.
[Illustration: FIG. 34.]
I must ask you now by observation, and the application of exactly the same
argument, to explain the struggle for uprightness on its longer axis of any
rounded stone when it spins on a table. I may tell you that some of these
large rounded-looking objects which I now spin before you in illustration,
are made hollow, and they are either of wood or zinc, because I have not
the skill necessary to spin large solid objects, and yet I wanted to have
objects which you would be able to see. This small one (Fig. 31) is the
largest solid one to which my fingers are able to give sufficient spin.
Here is a very interesting object (Fig. 35), spherical {74} in shape, but
its centre of gravity is not exactly at its centre of figure, so when I lay
it on the table it always gets to its position of stable equilibrium, the
white spot touching the table as at A. Some of you know that if this sphere
is thrown into the air it seems to have very curious motions, because one
is so apt to forget that it is the motion of its centre of gravity which
follows a simple path, and the boundary is eccentric to the centre of
gravity. Its motions when set to roll upon a carpet are also extremely
curious.
[Illustration: FIG. 35.]
Now for the very reasons that I have already given, when this sphere is
made to spin on the table, it always endeavours to get its white spot
uppermost, as in C, Fig. 35; to get into the position in which when not
spinning it would be unstable.
[Illustration: FIG. 36.]
The precession of a top or gyrostat leads us at once to think of the
precession of the great spinning body on which we live. You know that the
earth {75} spins on its axis a little more than once every twenty-four
hours, as this orange is revolving, and that it goes round the sun once in
a year, as this orange is now going round a model sun, or as is shown in
the diagram (Fig. 36). Its spinning axis points in the direction shown,
very nearly to the star which is called the pole star, almost infinitely
far away. In the figure and model I have greatly exaggerated the elliptic
nature of the earth's path, as is quite usual, although it may be a little
misleading, because the earth's path is much more nearly circular than many
people imagine. As a matter of fact the earth is about three million miles
nearer the sun in winter than it is in summer. This seems at first
paradoxical, but we get to understand it when we reflect that, because of
the slope of the earth's axis to the ecliptic, we people who live in the
northern hemisphere have the sun less vertically above us, and have a
shorter day in the winter, and hence each square foot of our part of the
earth's surface receives much less heat every day, and so we feel colder.
Now in about 13,000 years the earth will have precessed just half a
revolution (_see_ Fig. 38); the axis will then be sloped towards the sun
when it is nearest, instead of away from it as it is now; consequently we
shall be much warmer in summer and colder in winter than we are now. Indeed
we shall then be much worse off than the southern {77} hemisphere people
are now, for they have plenty of oceanic water to temper their climate. It
is easy to see the nature of the change from figures 36, 37, and 38, or
from the model as I carry the orange and its symbolic knitting-needle round
the model sun. Let us imagine an observer placed above this model, far
above the north pole of the earth. He sees the earth rotating against the
direction of the hands of a watch, and he finds that it precesses with the
hands of a watch, so that spin and precession are in opposite directions.
Indeed it is because of this that we have the word "precession," which we
now apply to the motion of a top, although the precession of a top is in
the same direction as that of the spin.
[Illustration: FIG. 37.]
[Illustration: FIG. 38.]
The practical astronomer, in explaining the _luni-solar precession of the
equinoxes_ to you, will not probably refer to tops or gyrostats. He will
tell you that the _longitude_ and _right ascension_ of a star seem to
alter; in fact that the point on the ecliptic from which he makes his
measurements, namely, the spring equinox, is slowly travelling round the
ecliptic in a direction opposite to that of the earth in its orbit, or to
the apparent path of the sun. The spring equinox is to him for heavenly
measurements what the longitude of Greenwich is to the navigator. He will
tell you that aberration of light, and parallax of the stars, {80} but more
than both, this precession of the equinoxes, are the three most important
things which prevent us from seeing in an observatory by transit
observations of the stars, that the earth is revolving with perfect
uniformity. But his way of describing the precession must not disguise for
you the physical fact that his phenomenon and ours are identical, and that
to us who are acquainted with spinning tops, the slow conical motion of a
spinning axis is more readily understood than those details of his
measurements in which an astronomer's mind is bound up, and which so often
condemn a man of great intellectual power to the life of drudgery which we
generally associate with the idea of the pound-a-week cheap clerk.
[Illustration: FIG. 22.]
The precession of the earth is then of the same nature as that of a
gyrostat suspended above its centre of gravity, of a body which would be
stable and not top-heavy if it were not spinning. In fact the precession of
the earth is of the same nature as that of this large gyrostat (Fig. 22),
which is suspended in gymbals, so that it has a vibration like a pendulum
when not spinning. I will now spin it, so that looked at from above it goes
against the hands of a watch, and you observe that it precesses with the
hands of a watch. Here again is a hemispherical wooden ship, in which there
is a gyrostat with its axis vertical. It is in stable {81} equilibrium.
When the gyrostat is not spinning, the ship vibrates slowly when put out of
equilibrium; when the gyrostat is spinning the ship gets a motion of
precession which is opposite in direction to that of the spinning.
Astronomers, beginning with Hipparchus, have made observations of the
earth's motion for us, and we have observed the motions of gyrostats, and
we naturally seek for an explanation of the precessional motion of the
earth. The equator of the earth makes an angle of 23-1/2deg with the
ecliptic, which is the plane of the earth's orbit. Or the spinning axis of
the earth is always at angle of 23-1/2deg with a perpendicular to the
ecliptic, and makes a complete revolution in 26,000 years. The surface of
the water on which this wooden ship is floating represents the ecliptic.
The axis {82} of spinning of the gyrostat is about 23-1/2deg to the
vertical; the precession is in two minutes instead of 26,000 years; and
only that this ship does not revolve in a great circular path, we should
have in its precession a pretty exact illustration of the earth's
precession.
The precessional motion of the ship, or of the gyrostat (Fig. 22), is
explainable, and in the same way the earth's precession is at once
explained if we find that there are forces from external bodies tending to
put its spinning axis at right angles to the ecliptic. The earth is a
nearly spherical body. If it were exactly spherical and homogeneous, the
resultant force of attraction upon it, of a distant body, would be in a
line through its centre. And again, if it were spherical and
non-homogeneous, but if its mass were arranged in uniformly dense,
spherical layers, like the coats of an onion. But the earth is not
spherical, and to find what is the nature of the attraction of a distant
body, it has been necessary to make pendulum observations all over the
earth. You know that if a pendulum does not alter in length as we take it
about to various places, its time of vibration at each place enables the
force of gravity at each place to be determined; and Mr. Green proved that
if we know the force of gravity at all places on the surface of the earth,
although we may know nothing about the {83} state of the inside of the
earth, we can calculate with absolute accuracy the force exerted by the
earth on matter placed anywhere outside the earth; for instance, at any
part of the moon's orbit, or at the sun. And hence we know the equal and
opposite force with which such matter will act on the earth. Now pendulum
observations have been made at a great many places on the earth, and we
know, although of course not with absolute accuracy, the attraction on the
earth, of matter outside the earth. For instance, we know that the
resultant attraction of the sun on the earth is a force which does not pass
through the centre of the earth's mass. You may comprehend the result
better if I refer to this diagram of the earth at midwinter (Fig. 39), and
use a popular method of description. A and B may roughly be called the
protuberant parts of the earth--that protuberant belt of matter which makes
the {84} earth orange-shaped instead of spherical. On the spherical portion
inside, assumed roughly to be homogeneous, the resultant attraction is a
force through the centre.
[Illustration: FIG. 39.]