Spinning Tops Part 3

The precessional motion of the s.h.i.+p, 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 h.o.m.ogeneous, 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-h.o.m.ogeneous, but if its ma.s.s 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...o...b..t, 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 pa.s.s through the centre of the earth's ma.s.s. 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, a.s.sumed roughly to be h.o.m.ogeneous, the resultant attraction is a force through the centre.

[Ill.u.s.tration: FIG. 39.]

I will now consider the attraction on the protuberant equatorial belt indicated by A and B. The sun attracts a pound of matter at B more than it attracts a pound of matter at A, because B is nearer than A, and hence the total resultant force is in the direction M N rather than O O, through the centre of the earth's ma.s.s. But we know that a force in the direction M N is equivalent to a force O O parallel to M N, together with a tilting couple of forces tending to turn the equator edge on to the sun. You will get the true result as to the tilting tendency by imagining the earth to be motionless, and the sun's ma.s.s to be distributed as a circular ring of matter 184 millions of miles in diameter, inclined to the equator at 23.

Under the influence of the attraction of this ring the earth would heave like a great s.h.i.+p on a calm sea, rolling very slowly; in fact, making one complete swing in about three years. But the earth is spinning, and the tilting couple or torque acts upon it just like the forces which are always tending to cause this s.h.i.+p-model to stand upright, and hence it has a precessional motion whose complete period is 26,000 years. When there is no spin in the s.h.i.+p, its complete oscillation takes place in three seconds, and {85} when I spin the gyrostat on board the s.h.i.+p, the complete period of its precession is two minutes. In both cases the effect of the spin is to convert what would be an oscillation into a very much slower precession.

There is, however, a great difference between the earth and the gyrostat.

The forces acting on the top are always the same, but the forces acting on the earth are continually altering. At midwinter and midsummer the tilting forces are greatest, and at the equinoxes in spring and autumn there are no such forces. So that the precessional motion changes its rate every quarter year from a maximum to nothing, or from nothing to a maximum. It is, however, always in the same direction--the direction opposed to the earth's spin. When we speak then of the precessional motion of the earth, we usually think of the mean or average motion, since the motion gets quicker and slower every quarter year.

Further, the moon is like the sun in its action. It tries to tilt the equatorial part of the earth into the plane of the moon's...o...b..t. The plane of the moon's...o...b..t is nearly the same as that of the ecliptic, and hence the average precession of the earth is of much the same kind as if only one of the two, the moon or the sun, alone acted. That is, the general phenomenon of precession of the {86} earth's axis in a conical path in 26,000 years is the effect of the combined tilting actions of the sun and moon.

You will observe here an instance of the sort of untruth which it is almost imperative to tell in explaining natural phenomena. Hitherto I had spoken only of the sun as producing precession of the earth. This was convenient, because the plane of the ecliptic makes always almost exactly 23 with the earth's equator, and although on the whole the moon's action is nearly identical with that of the sun, and about twice as great, yet it varies considerably. The superior tilting action of the moon, just like its tide-producing action, is due to its being so much nearer us than the sun, and exists in spite of the very small ma.s.s of the moon as compared with that of the sun.

As the ecliptic makes an angle of 23 with the earth's equator, and the moon's...o...b..t makes an angle 5 with the ecliptic, we see that the moon's...o...b..t sometimes makes an angle of 29 with the earth's equator, and sometimes only 18, changing from 29 to 18, and back to 29 again in about nineteen years. This causes what is called "Nutation," or the nodding of the earth, for the tilting action due to the sun is greatly helped and greatly modified by it. The result of the variable nature of the moon's action is then that the earth's axis {87} rotates in an elliptic conical path round what might be called its mean position. We have also to remember that twice in every lunar month the moon's tilting action on the earth is greater, and twice it is zero, and that it continually varies in value.

On the whole, then, the moon and sun, and to a small extent the planets, produce the general effect of a precession, which repeats itself in a period of about 25,695 years. It is not perfectly uniform, being performed at a speed which is a maximum in summer and winter; that is, there is a change of speed whose period is half a year; and there is a change of speed whose period is half a lunar month, the precession being quicker to-night than it will be next Sat.u.r.day, when it will increase for about another week, and diminish the next. Besides this, because of 5 of angularity of the orbits, we have something like the nodding of our precessing gyrostat, and the inclination of the earth's axis to the ecliptic is not constant at 23, but is changing, its periodic time being nineteen years. Regarding the earth's centre as fixed at O we see then, as ill.u.s.trated in this model and in Fig. 40, the axis of the earth describes almost a perfect circle on the celestial sphere once in 25,866 years, its speed fluctuating every half year and every half month. But it is not a perfect circle, it is really a wavy {88} line, there being a complete wave every nineteen years, and there are smaller ripples in it, corresponding to the half-yearly and fortnightly periods. But the very cause of the nutation, the nineteen-yearly period of retrogression of the moon's nodes, as it is called, is itself really produced as the precession of a gyrostat is produced, that is, by tilting forces acting on a spinning body.

[Ill.u.s.tration: FIG. 40.]

Imagine the earth to be stationary, and the sun and moon revolving round it. It was Gauss who found that the present action is the same as if the ma.s.ses of the moon and sun were distributed all {89} round their orbits.

For instance, imagine the moon's ma.s.s distributed over her orbit in the form of a rigid ring of 480,000 miles diameter, and imagine less of it to exist where the present speed is greater, so that the ring would be thicker at the moon's apogee, and thinner at the perigee. Such a ring round the earth would be similar to Saturn's rings, which have also a precession of nodes, only Saturn's rings are not rigid, else there would be no equilibrium. Now if we leave out of account the earth and imagine this ring to exist by itself, and that its centre simply had a motion round the sun in a year, since it makes an angle of 5 with the ecliptic it would vibrate into the ecliptic till it made the same angle on the other side and back again. But it revolves once about its centre in twenty-seven solar days, eight hours, and it will no longer swing like a s.h.i.+p in a ground-swell, but will get a motion of precession opposed in direction to its own revolution. As the ring's motion is against the hands of a watch, looking from the north down on the ecliptic, this retrogression of the moon's nodes is in the direction of the hands of a watch. It is exactly the same sort of phenomenon as the precession of the equinoxes, only with a much shorter period of 6798 days instead of 25,866 years.

I told you how, if we knew the moon's ma.s.s or the sun's, we could tell the amount of the forces, or {90} the torque as it is more properly called, with which it tries to tilt the earth. We know the rate at which the earth is spinning, and we have observed the precessional motion. Now when we follow up the method which I have sketched already, we find that the precessional velocity of a spinning body ought to be equal to the torque divided by the spinning velocity and by the moment of inertia[7] of the body about the polar axis. Hence the greater the tilting forces, and the less the spin and the less the moment of inertia, the greater is the precessional speed. Given all of these elements except one, it is easy to calculate that unknown element. Usually what we aim at in such a calculation is the determination of the moon's ma.s.s, as this phenomenon of precession and the action of the tides are the only two natural phenomena which have as yet enabled the moon's ma.s.s to be calculated.

I do not mean to apologize to you for the introduction of such terms as _Moment of Inertia_, nor do I mean to explain them. In this lecture I have avoided, as much as I could, the introduction of mathematical expressions and the use of technical terms. But I want you to {91} understand that I am not afraid to introduce technical terms when giving a popular lecture. If there is any offence in such a practice, it must, in my opinion, be greatly aggravated by the addition of explanations of the precise meanings of such terms. The use of a correct technical term serves several useful purposes.

First, it gives some satisfaction to the lecturer, as it enables him to state, very concisely, something which satisfies his own weak inclination to have his reasoning complete, but which he luckily has not time to trouble his audience with. Second, it corrects the universal belief of all popular audiences that they know everything now that can be said on the subject. Third, it teaches everybody, including the lecturer, that there is nothing lost and often a great deal gained by the adoption of a casual method of skipping when one is working up a new subject.

Some years ago it was argued that if the earth were a sh.e.l.l filled with liquid, if this liquid were quite frictionless, then the moment of inertia of the sh.e.l.l is all that we should have to take into account in considering precession, and that if it were viscous the precession would very soon disappear altogether. To ill.u.s.trate the effect of the moment of inertia, I have hung up here a number of gla.s.ses--one _a_ filled with sand, another _b_ with treacle, a third _c_ with oil, the fourth _d_ with water, {92}

[Ill.u.s.tration: FIG. 41.]

{93} and the fifth _e_ is empty (Fig. 41). You see that if I twist these suspending wires and release them, a vibratory motion is set up, just like that of the balance of a watch. Observe that the gla.s.s with water vibrates quickly, its effective moment of inertia being merely that of the gla.s.s itself, and you see that the time of swing is pretty much the same as that of the empty gla.s.s; that is, the water does not seem to move with the gla.s.s. Observe that the vibration goes on for a fairly long time.

The gla.s.s with sand vibrates slowly; here there is great moment of inertia, as the sand and gla.s.s behave like one rigid body, and again the vibration goes on for a long time.

In the oil and treacle, however, there are longer periods of vibration than in the case of the water or empty gla.s.s, and less than would be the case if the vibrating bodies were all rigid, but the vibrations are stilled more rapidly because of friction.

Boiled (_f_) and unboiled (_g_) eggs suspended from wires in the same way will exhibit the same differences in the behaviour of bodies, one of which is rigid and the other liquid inside; you see how much slower an oscillation the boiled has than the unboiled.

Even on the table here it is easy to show the difference between boiled and unboiled eggs. {94} Roll them both; you see that one of them stops much sooner than the other; it is the unboiled one that stops sooner, because of its internal friction.

I must ask you to observe carefully the following very distinctive test of whether an egg is boiled or not. I roll the egg or spin it, and then place my finger on it just for an instant; long enough to stop the motion of the sh.e.l.l. You see that the boiled egg had quite finished its motion, but the unboiled egg's sh.e.l.l alone was stopped; the liquid inside goes on moving, and now renews the motion of the sh.e.l.l when I take my finger away.

It was argued that if the earth were fluid inside, the effective moment of inertia of the sh.e.l.l being comparatively small, and having, as we see in these examples, nothing whatever to do with the moment of inertia of the liquid, the precessional motion of the earth ought to be enormously quicker than it is. This was used as an argument against the idea of the earth's being fluid inside.

We know that the observed half-yearly and half-monthly changes of the precession of the earth would be much greater than they are if the earth were a rigid sh.e.l.l containing much liquid, and if the sh.e.l.l were not nearly infinitely rigid the phenomena of the tides would not occur, but in regard to the general precession of the earth there is now {95} no doubt that the old line of argument was wrong. Even if the earth were liquid inside, it spins so rapidly that it would behave like a rigid body in regard to such a slow phenomenon as precession of the equinoxes. In fact, in the older line of argument the important fact was lost sight of, that rapid rotation can give to even liquids a quasi-rigidity. Now here (Fig. 42 _a_) is a hollow bra.s.s top filled with water. The frame is light, and the water inside has much more ma.s.s than the outside frame, and if you test this carefully you will find that the top spins in almost exactly the same way as if the water were quite rigid; in fact, as if the whole top were rigid. Here you see it spinning and precessing just like any rigid top. This top, I know, is not filled with water, it is only partially filled; but whether partially or wholly filled it spins very much like a rigid top.

[Ill.u.s.tration: FIG. 42.]

{96}

This is not the case with a long hollow bra.s.s top with water inside. I told you that all bodies have one axis about which they prefer to rotate. The outside metal part of a top behaves in a way that is now well known to you; the friction of its peg on the table compels it to get up on its longer axis. But the fluid inside a top is not constrained to spin on its longer axis of figure, and as it prefers its shorter axis like all these bodies I showed you, it spins in its own way, and by friction and pressure against the case constrains the case to spin about the shorter axis, annulling completely the tendency of the outside part to rise or keep up on its long axis. Hence it is found to be simply impossible to spin a long hollow top when filled with water.

[Ill.u.s.tration: FIG. 43.]

[Ill.u.s.tration: FIG. 44.]

Here, for example, is one (Fig. 42 _b_) that only differs from the last in being longer. It is filled, or partially filled, with water, and you observe that if {97} I slowly get up a great spin when it is mounted in this frame, and I let it out on the table as I did the other one, this one lies down at once and refuses to spin on its peg. This difference of behaviour is most remarkable in the two hollow tops you see before you (Fig. 43). They are both nearly spherical, both filled with water. They look so nearly alike that few persons among the audience are able to detect any difference in their shape. But one of them (_a_) is really slightly oblate like an orange, and the other (_b_) is slightly prolate like a lemon. I will give them both a gradually increasing rotation in this frame {98} (Fig. 44) for a time sufficient to insure the rotation of the water inside. When just about to be set free to move like ordinary tops on the table, water and bra.s.s are moving like the parts of a rigid top. You see that the orange-shaped one continues to spin and precess, and gets itself upright when disturbed, like an ordinary rigid top; indeed I have seldom seen a better behaved top; whereas the lemon-shaped one lies down on its side at once, and quickly ceases to move in any way.

[Ill.u.s.tration: FIG. 45.]

And now you will be able to appreciate a fourth test of a boiled egg, which is much more easily seen by a large audience than the last. Here is the unboiled one (Fig. 45 _b_). I try my best to spin it as it lies on the table, but you see that I cannot give it much spin, and so there is nothing of any importance to look at. But you observe that it is quite easy to spin the boiled {99} egg, and that for reasons now well known to you it behaves like the stones that Thomson spun on the sea-beach; it gets up on its longer axis, a very pretty object for our educated eyes to look at (Fig. 45 _a_). You are all aware, from the behaviour of the lemon-shaped top, that even if, by the use of a whirling table suddenly stopped, or by any other contrivance, I could get up a spin in this unboiled egg, it would never make the slightest effort to rise on its end and spin about its longer axis.

I hope you don't think that I have been speaking too long about astronomical matters, for there is one other important thing connected with astronomy that I must speak of. You see, I have had almost nothing practically to do with astronomy, and hence I have a strong interest in the subject. It is very curious, but quite true, that men practically engaged in any pursuit are almost unable to see the romance of it. This is what the imaginative outsider sees. But the overworked astronomer has a different point of view. As soon as it becomes one's duty to do a thing, and it is part of one's every-day work, the thing loses a great deal of its interest.

We have been told by a great American philosopher that the only coachmen who ever saw the romance of coach-driving are those t.i.tled individuals who pay nowadays so largely for the {100} privilege. In almost any branch of engineering you will find that if any invention is made it is made by an outsider; by some one who comes to the study of the subject with a fresh mind. Who ever heard of an old inhabitant of j.a.pan or Peru writing an interesting book about those countries? At the end of two years' residence he sees only the most familiar things when he takes his walks abroad, and he feels unmitigated contempt for the ingenuous globe-trotter who writes a book about the country after a month's travel over the most beaten tracks in it. Now the experienced astronomer has forgotten the difficulties of his predecessors and the doubts of outsiders. It is a long time since he felt that awe in gazing at a starry sky that we outsiders feel when we learn of the sizes and distances apart of the hosts of heaven. He speaks quite coolly of millions of years, and is nearly as callous when he refers to the ancient history of humanity on our planet as a weather-beaten geologist.

The reason is obvious. Most of you know that the _Nautical Almanac_ is as a literary production one of the most uninteresting works of reference in existence. It is even more disconnected than a dictionary, and I should think that preparing census-tables must be ever so much more romantic as an occupation than preparing the tables of the _Nautical Almanac_. And yet {101} a particular figure, one of millions set down by an overworked calculator, may have all the tragic importance of life or death to the crew and pa.s.sengers of a s.h.i.+p, when it is heading for safety or heading for the rocks under the mandate of that single printed character.

But this may not be a fair sort of criticism. I so seldom deal with astronomical matters, I know so little of the wear and tear and monotony of the every-day life of the astronomer, that I do not even know that the above facts are specially true about astronomers. I only know that they are very likely to be true because they are true of other professional men.

I am happy to say that I come in contact with all sorts and conditions of men, and among others, with some men who deny many of the things taught in our earliest school-books. For example, that the earth is round, or that the earth revolves, or that Frenchmen speak a language different from ours.

Now no man who has been to sea will deny the roundness of the earth, however greatly he may wonder at it; and no man who has been to France will deny that the French language is different from ours; but many men who learnt about the rotation of the earth in their school-days, and have had a plentiful opportunity of observing the heavenly bodies, deny the rotation of the earth. {102} They tell you that the stars and moon are revolving about the earth, for they see them revolving night after night, and the sun revolves about the earth, for they see it do so every day. And really if you think of it, it is not so easy to prove the revolution of the earth. By the help of good telescopes and the electric telegraph or good chronometers, it is easy to show from the want of parallax in stars that they must be very far away; but after all, we only know that either the earth revolves or else the sky revolves.[8] Of course, it seems infinitely more likely that the small earth should revolve than that the whole heavenly host should turn about the earth as a centre, and infinite likelihood is really absolute proof. Yet there is n.o.body who does not welcome an independent kind of proof. The phenomena of the tides, and nearly every new astronomical fact, may be said to be an addition to the proof. Still there is the absence of perfect certainty, and when we are told that these spinning-top phenomena give us a real proof of the rotation of the earth without our leaving the room, we welcome {103} it, even although we may sneer at it as unnecessary after we have obtained it.

[Ill.u.s.tration: FIG. 17.]

You know that a gyrostat suspended with perfect freedom about axes, which all pa.s.s through its centre of gravity, maintains a constant direction in s.p.a.ce however its support may be carried. Its axis is not forced to alter its direction in any way. Now this gyrostat (Fig. 17) has not the perfect absence of friction at its axes of which I speak, and even the slightest friction will produce some constraint which is injurious to the experiment I am about to describe. It must be remembered, that if there were absolutely no constraint, then, even if the {104} gyrostat were _not_ spinning, its axis would keep a constant direction in s.p.a.ce. But the spinning gyrostat shows its superiority in this, that any constraint due to friction is less powerful in altering the axis. The greater the spin, then, the better able are we to disregard effects due to friction. You have seen for yourselves the effect of carrying this gyrostat about in all sorts of ways--first, when it is not spinning and friction causes quite a large departure from constancy of direction of the axis; second, when it is spinning, and you see that although there is now the same friction as before, and I try to disturb the instrument more than before, the axis remains sensibly parallel to itself all the time. Now when this instrument is supported by the table it is really being carried round by the earth in its daily rotation. If the axis kept its direction perfectly, and it were now pointing horizontally due east, six hours after this it will point towards the north, but inclining downwards, six hours afterwards it will point due west horizontally, and after one revolution of the earth it will again point as it does now. Suppose I try the experiment, and I see that it points due east now in this room, and after a time it points due west, and yet I know that the gyrostat is constantly pointing in the same direction in s.p.a.ce all the time, surely it is obvious that the room must {105} be turning round in s.p.a.ce. Suppose it points to the pole star now, in six hours, or twelve, or eighteen, or twenty-four, it will still point to the pole star.

Now it is not easy to obtain so frictionless a gyrostat that it will maintain a good spin for such a length of time as will enable the rotation of the room to be made visible to an audience. But I will describe to you how forty years ago it was proved in a laboratory that the earth turns on its axis. This experiment is usually connected with the name of Foucault, the same philosopher who with Fizeau showed how in a laboratory we can measure the velocity of light, and therefore measure the distance of the sun. It was suggested by Mr. Lang of Edinburgh in 1836, although only carried out in 1852 by Foucault. By these experiments, if you were placed on a body from which you could see no stars or other outside objects, say that you were living in underground regions, you could discover--first, whether there is a motion of rotation, and the amount of it; second, the meridian line or the direction of the true north; third, your lat.i.tude.

Obtain a gyrostat like this (Fig. 46) but much larger, and far more frictionlessly suspended, so that it is free to move vertically or horizontally. For the vertical motion your gymbal pivots ought to be hard steel knife-edges. {106}

[Ill.u.s.tration: FIG. 46.]

As for the horizontal freedom, Foucault used a fine steel wire. Let there be a fine scale engraved crosswise on the outer gymbal ring, and try to discover if it moves horizontally by means of a microscope with cross wires. When this is carefully done we find that there is a motion, {107} but this is not the motion of the gyrostat, it is the motion of the microscope. In fact, the microscope and all other objects in the room are going round the gyrostat frame.

Now let us consider what occurs. The room is rotating about the earth's axis, and we know the rate of rotation; but we only want to know for our present purpose how much of the total rotation is about a vertical line in the room. If the room were at the North Pole, the whole rotation would be about the vertical line. If the room were at the equator, none of its rotation would be about a vertical line. In our lat.i.tude now, the horizontal rate of rotation about a vertical axis is about four-fifths of the whole rate of rotation of the earth on its axis, and this is the amount that would be measured by our microscope. This experiment would give no result at a place on the equator, but in our lat.i.tude you would have a laboratory proof of the rotation of the earth. Foucault made the measurements with great accuracy.

If you now clamp the frame, and allow the spinning axis to have no motion except in a horizontal plane, the motion which the earth tends to give it about a vertical axis cannot now affect the gyrostat, but the earth constrains it to move about an axis due north and south, and consequently the spinning axis tries to put itself parallel {108} to the north and south direction (Fig. 47). Hence with such an instrument it is easy to find the true north. If there were absolutely no friction the instrument would vibrate about the true north position like the compa.s.s needle (Fig. 50), although with an exceedingly slow swing.

[Ill.u.s.tration: FIG. 47.]

It is with a curious mixture of feelings that one first recognizes the fact that all rotating bodies, fly-wheels of steam-engines and the like, are always tending to turn themselves towards the pole star; gently and vainly tugging at their foundations {109} to get round towards the object of their adoration all the time they are in motion.

[Ill.u.s.tration: FIG. 48.]

Now we have found the meridian as in Fig. 47, we can begin a third experiment. Prevent motion horizontally, that is, about a vertical axis, but give the instrument freedom to move vertically in the meridian, like a transit instrument in an observatory {110} about its horizontal axis. Its revolution with the earth will tend to make it change its angular position, and therefore it places itself parallel to the earth's axis; when in this position the daily rotation no longer causes any change in its direction in s.p.a.ce, so it continues to point to the pole star (Fig. 48). It would be an interesting experiment to measure with a delicate chemical balance the force with which the axis raises itself, and in this way _weigh_ the rotational motion of the earth.[9]

Now let us turn the frame of the instrument G B round a right angle, so that the spinning axis can only move in a plane at right angles to the meridian; obviously it is constrained by the vertical component of the earth's rotation, and points vertically downwards.

[Ill.u.s.tration: FIG. 49.]