Spinning Tops Part 2

[Ill.u.s.tration: 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, a.s.suming 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 h.o.m.ogeneous 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 h.o.m.ogeneous 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 n.o.body 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 j.a.pan, 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.

[Ill.u.s.tration: FIG. 26.]

If a rotating body is in balance, not only does the axis of rotation pa.s.s through the centre of gravity (or rather centre of ma.s.s) of the body, but {57} the axis of rotation must be one of the three princ.i.p.al axes through the centre of ma.s.s of the body. Here, for example, is an ellipsoid of wood; A A, B B, and C C (Fig. 27) are its three princ.i.p.al 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 h.o.m.ogeneous sphere, every diameter of which is a princ.i.p.al axis.

[Ill.u.s.tration: FIG. 27.]

Every body has three such princ.i.p.al axes through its centre of ma.s.s, and 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 circ.u.mstances 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 fas.h.i.+on 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 ill.u.s.trate this, I have here a method of spinning bodies which enables them to choose as their spinning axis that one princ.i.p.al 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 princ.i.p.al axes. {60}

[Ill.u.s.tration: 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 princ.i.p.al 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 ill.u.s.trates 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.

[Ill.u.s.tration: 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 s.p.a.ce 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 ill.u.s.trations of the incompatibility of science and rhyming.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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 a.s.sumes 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.

[Ill.u.s.tration: FIG. 32.]

[Ill.u.s.tration: 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 s.p.a.ce 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.

[Ill.u.s.tration: 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 ill.u.s.tration, 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: FIG. 37.]

[Ill.u.s.tration: 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...o...b..t, 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 a.s.sociate with the idea of the pound-a-week cheap clerk.

[Ill.u.s.tration: 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 s.h.i.+p, in which there is a gyrostat with its axis vertical. It is in stable {81} equilibrium.

When the gyrostat is not spinning, the s.h.i.+p vibrates slowly when put out of equilibrium; when the gyrostat is spinning the s.h.i.+p 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 with the ecliptic, which is the plane of the earth's...o...b..t. Or the spinning axis of the earth is always at angle of 23 with a perpendicular to the ecliptic, and makes a complete revolution in 26,000 years. The surface of the water on which this wooden s.h.i.+p is floating represents the ecliptic. The axis {82} of spinning of the gyrostat is about 23 to the vertical; the precession is in two minutes instead of 26,000 years; and only that this s.h.i.+p does not revolve in a great circular path, we should have in its precession a pretty exact ill.u.s.tration of the earth's precession.