Lord Kelvin Part 12

GYROSTATIC ACTION

Thomson in his lectures and otherwise gave a great deal of attention to the motion of gyrostats, and to the effect of the inclusion of gyrostats in a system on its properties. Reference has been made to the treatment of "gyrostatic domination" in "Thomson and Tait." A gyrostat consists of a disk or wheel with a ma.s.sive rim, which revolves within a case or framework, by which the whole arrangement can be moved about, or supported, without interfering with the wheel. The ordinary toy consisting of wheel with a ma.s.sive rim, and a light frame, is an example. But much larger and more carefully made instruments, in which the wheel is entirely enclosed, give the most interesting experiments.

The body seems to have its properties entirely altered by the rotation of the wheel, and of course the case prevents any outward change from being visible.

[Ill.u.s.tration: FIG. 15.]

Figure 15 shows one form of gyrostat mounted on a horizontal frame, held in the hands of an experimenter. The axis of the fly-wheel is vertical within the tubular part of the case; the fly-wheel is within the part on which is engraved an arrow-head to show the direction of rotation. Round the case in the plane of the wheel is a projecting rim sharpened to an edge, on which the gyrostat can be supported in other experiments. To the rim are screwed two projecting pivots, which can turn in bearings on the two sides of the frame as shown. The centre of ma.s.s of the wheel is on the level of these pivots, so that the instrument will remain with either end of the axis up.

If the fly-wheel be not in rotation, the experimenter can carry the arrangement about, and the fly-wheel and case move with it as if the gyrostat were merely an ordinary rigid body. But now remove the gyrostat from the frame, and set the wheel in rotation. This is done by an endless cord wrapped round a small pulley fast on the axle (to which access is obtained by a hole just opposite in the case) and pa.s.sed also round a larger pulley on the shaft of a motor. When the motor is started the cord must be tightened only very gently at first, so that it slips on the pulley, otherwise the motor would be r.e.t.a.r.ded, and possibly burned by the current. The fly-wheel gradually gets up speed, and then the cord can be brought quite tight so that no slipping occurs. When the speed is great enough the cord is cut with a stroke from a sharp knife and runs out.

The gyrostat is now replaced on its pivots in the frame, with its axis vertical, and moved about as it was before. If the experimenter, holding the frame as shown, turns round in the direction of the arrow, which is that of rotation, nothing happens. If, however, he turns round the other way, the gyrostat immediately turns on its pivots so as to point the other end of the axis up. If the experimenter continues his turning motion, the gyrostat is now quiescent: for it is being carried round now in the direction of rotation. Thus, with no gravitational stability at all (since the centre is on a level with the pivots) the gyrostat is in stable equilibrium when carried round in the direction of rotation, but is in unstable equilibrium when carried round the opposite way.

Thus, if the observer knew nothing of the rotation of the fly-wheel, and could see and feel only the outside of the case, the behaviour of the instrument might well appear very astonishing.

This is a case of what Thomson and Tait call "gyrostatic domination,"

which is treated very fully in their Sections 345 (vi) to 345 (xxviii) of Part I. It may be remarked here that this case of motion may be easily treated mathematically in an exceedingly elementary manner, and the instability of the one case, and the stability of the other, made clear to the beginner who has only a notion of the composition of angular momenta about different axes.

A year or two ago it was suggested by Professor Pickering, of Harvard, that the fact that the outermost satellite of Saturn revolves in the direction opposite to the planet's rotation, may be due to the fact that originally Saturn rotated in the direction of the motion of this moon, but inasmuch as his motion round the sun was opposite in direction to his rotation, he was turned, so to speak, upside down, like the gyrostat! The other satellites, it is suggested, were thrown off later, as their revolution is direct. Professor Pickering refers to an experiment (similar to that described above) which he gives as new.

Thomson had shown this experiment for many years, as an example of the general discussion in "Thomson and Tait," and its theory had already been explicitly published.[21]

Many other experiments with gyrostats used to be shown by Thomson to visitors. Many of these are indicated in "Thomson and Tait." The earth's precessional motion is a gyrostatic effect due to the differential attraction of the sun, which tends to bring the plane of the equator into coincidence with the ecliptic, and so alters the direction of the axis of rotation. Old students will remember the balanced globe--with inclined material axis rolling round a horizontal ring--by which the kinematics of the motion could be studied, and the displacement of the equinoxes on the ecliptic traced.

[Ill.u.s.tration: FIG. 16.]

Another example of the gyrostatic domination discussed in "Thomson and Tait" is given in the very remarkable address ent.i.tled "A Kinetic Theory of Matter," which Sir William Thomson delivered to Section A of the British a.s.sociation at Montreal, in 1884. Figure 16 shows an ordinary double "coach spring," the upper and lower members of which carry two hooked rods as shown. If the upper hook is attached to a fixed support, and a weight is hung on the lower, the spring will be drawn out, and the arrangement will be in equilibrium under a certain elongation. If the weight be pulled down further and then left to itself, it will vibrate up and down in a period depending upon the equilibrium elongation produced by the weight. The same thing will happen if a spiral spring be subst.i.tuted for the coach spring. A spherical case, through which the hooked rods pa.s.s freely, hides the internal parts from view.

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

Figure 17 shows two hooked rods, as in the former case, attached by swivels to two opposite corners of a frame formed of four rods jointed together at their ends. Each of these is divided in the middle for the insertion of a gyrostat, the axis of which is pivoted on the adjacent ends of the two halves of the rod. A spherical case, indicated by the circle, again hides the internal arrangement from inspection, but permits the hooked rods to move freely up and down. The swivels allow the frame, gyrostats and all, to be turned about the line of the hooks.

If now the gyrostats be not in rotation, the frame will be perfectly limp, and will not in the least resist pull applied by a weight. But if the gyrostats be rotated in the directions shown by the circles, with arrow-heads drawn round the rods, there will be angular momentum of the whole system about the line joining the hooks, and if a weight or a force be applied to pull out the frame along that line, the pull will be resisted just as it was in the other case by the spring. Moreover, equilibrium will be obtained with an elongation proportional to the weight hung on, and small oscillations will be performed just as if there were a spring in the interior instead of the gyrostats.

According as the frame is pulled out, or shortened, the angular momentum of the gyrostats about the line joining the hooks is increased or diminished, and the frame, carrying the gyrostats with it, turns about the swivels in one direction or the other, at the rate necessary to maintain the angular momentum at a constant value. But this will not be perceived from without.

The rotation of the fly-wheels thus gives to the otherwise limp frame the elasticity which the spring possesses; without dissection of the model the difference cannot be perceived. This ill.u.s.trates Thomson's idea that the elasticity of matter may be due to motion of molecules or groups of molecules of the body, imbedded in a connecting framework, deformed by applied forces as in this model, and producing displacements which are resisted in consequence of the motion.

And here may be mentioned also Thomson's explanation of the phenomenon, discovered by Faraday, of the rotation of the plane of a beam of polarised light which is pa.s.sed along the lines of force of a magnetic field. This rotation is distinct altogether from that which is produced when polarised light is pa.s.sed along a tube filled with a solution of sugar or tartaric acid. If the ray be reflected after pa.s.sage, and made to retraverse the medium, the rotation is annulled in the latter case, it is doubled in the former. This led Thomson to the view that in sugar, tartaric acid, quartz, etc., the turning is due to the structure of the substance, and in the magnetic field to rotation already existing in the medium. He used to say that a very large number of minute spiral cavities all in the same direction, and all right-handed or all left-handed, in the sugar or quartz, would give the effect; on the other hand, the magnetic phenomenon could only be produced by some arrangement a.n.a.logous to a very large number of tops, or gyrostats, imbedded in the medium with their axes all in one direction (or preponderatingly so) and all turning the same way. The rotation of these tops or gyrostats Thomson supposed to be caused by the magnetic field, and to be essentially that which const.i.tutes the magnetisation of the medium.

Let the frame of the gyrostatic spring-balance described above, turn round the line joining the hooks so as to exactly compensate, by turning in the opposite direction, the angular momentum about that line given by the fly-wheels; then the arrangement will have no angular momentum on the whole; and a large number of such balances, all very minute and hooked together, will form a substance without angular momentum in any part. But now by the equivalent of a magnetic force along the lines of the hooks, let a different angular turning of the frames be produced; the medium will possess a specific angular momentum in every part. If a wave of transverse vibrations which are parallel to one direction (that is, if the wave be plane-polarised) enter the medium in the direction of the axes of the frames, the direction of vibration will be turned as the wave proceeds, that is, the plane of polarisation will be turned round.

More recent research has shown an effect of a magnetic field on the spectrum of light produced in the field, and viewed with a spectroscope in a direction at right angles to the field--the Zeeman effect, as it is called--and the explanation of this effect by equations of moving electric charges, which are essentially gyrostatic equations, is suggestive of an a.n.a.logy or correspondence between the systems of moving electrons which const.i.tute these charges, and some such gyrostatic molecules as Thomson imagined. It has been pointed out that the Zeeman effect, in its simple forms at least, can be exactly imitated by the motion of an ordinary pendulum having a gyrostat in its bob, with its axis directed along the suspension rod.[22]

ELECTROSTATICS AND MAGNETISM

In the ten years from 1863 to 1873 Thomson was extremely busy with literary work. In 1872, five years after the publication of the treatise on _Natural Philosophy_, and just before the appearance of the Elements, Messrs. Macmillan & Co. published for him a collection of memoirs ent.i.tled _Reprint of Papers on Electrostatics and Magnetism_. The volume contains 596 pages, and the subjects dealt with range from the "Uniform Motion of Heat and its Connection with the Mathematical Theory of Electricity" (the paper already described in Chapter II above) and the discussion of Electrometers and Electrostatic Measuring Instruments, to a complete mathematical theory of magnetism. The subject of electrostatics led naturally to the consideration of electrical measuring instruments as they existed forty years ago (about 1867), and their replacement by others, the indications of which from day to day should be directly comparable, and capable of being interpreted in absolute units. Down to that time people had been obliged to content themselves with gold-leaf electroscopes, and indeed it was impossible for accurate measuring instruments to be invented until a system of absolute units had been completely worked out. The task of fixing upon definitions of units and of realising them in suitable standards had been begun by the British a.s.sociation, and it was as part of the Report of that Committee to the Dundee Meeting in 1867 that Thomson's paper on Electrometers first appeared.

It was there pointed out that an electrometer is essentially an instrument for measuring differences of electric potential between conductors, by means of effects of electrostatic force. Such a difference is what a gold-leaf electroscope indicates for its gold leaves and the walls surrounding the air-s.p.a.ce in which they are suspended. As electroscopes used to be constructed, these walls were made of gla.s.s imperfectly covered, if at all, by conducting material, and the electroscope was quite indefinite and uncertain in its action.

The instrument was also, as made, quite insensitive. Recently, however, it has been rehabilitated in reputation, and brought into use as a very sensitive indicator of effects of radio-activity.

Thomson described in this paper six species of electrometers of his own devising. The best known of these are his quadrant electrometer and his attracted-disk electrometers. The former is to be found in some form or other in every laboratory nowadays, and need not be described in detail.

The action is of two conductors--the two pairs of opposite quadrants of a shallow, horizontal, cylindrical box, made by dividing the box into four by two slits at right angles--upon an electrified slip of aluminium suspended by a two-thread suspension within the box, with its length along one of the slits. The two pairs of opposite quadrants are at the potential difference to be measured, and the slip of aluminium, or "needle," has each end urged round from a quadrant at higher potential towards one at a lower, and these actions conspire to turn the slip against its tendency to return to the position in which the two threads are in one plane. Thus the deflection (measured by the displacement of a reflected ray of light used as index) gives an indication of the amount of the potential difference.

The electrification of the "needle" was kept up by enclosing the quadrantal box within an electrified Leyden jar, to the interior coating of which contact is made by a platinum wire, depending from the needle to sulphuric acid contained in the jar. The whole apparatus was enclosed in a conducting case connected to earth. This made its action perfectly definite. Variations of this electrification of the jar were shown by an attached attracted-disk electrometer, the principle of which we shall merely indicate.

The quadrant electrometer has now been vastly increased in sensibility by the use of a single quartz fibre as suspension. By the invention of this fibre, which is exceedingly strong and is, moreover, so definite in its elastic properties that it comes back at once exactly to its former zero state after twist, Mr. C. V. Boys has increased the delicacy of all kinds of suspended indicators many fold. But it ought to be remembered that a Dolezalek electrometer, with some hundred or more times the sensibility of the bifilar instrument, was only made possible by its predecessor.

Attracted-disk-electrometers simply measure, either by weighing or by the deflection of a spring, the attractive force between two parallel disks at different potentials. From the determination of this force, and the measurement of the distance between the disks (or better, of an alteration of the distance) a difference of potentials can be determined, and a unit for it obtained, which is in direct and known relation to ordinary dynamical units. Thomson's "Absolute Electrometer"

was designed specially for accurate determinations of this kind. Another form, called the Long Range Electrometer, was devised for the measurement of the potentials of the charged conductors in electric machines and Leyden jars.

Accurate determinations of the sparking resistance between parallel plates charged to different potentials in air were made by means of attracted-disk-electrometers in the course of some important experiments described in the _Electrostatics and Magnetism_. These results have been much referred to in later researches.

A small attracted-disk-electrometer was used as indicated above to keep a watch on the electrification of the Leyden jar of the quadrant instrument, and a small induction machine was added, by turning which the operator could make good any loss of charge of the jar.

This electrical machine was an example of an apparatus on precisely the same principle as the Voss or Wimshurst machines of the present day. In it by a set of moving carriers, influenced by conductors, the charges of the latter were increased according to a compound interest principle only interfered with by leakage to the air or by the supports. Several forms of this machine, on the same principle, were constructed by Thomson, and described in 1868; but he afterwards found that he had been antic.i.p.ated by C. F. Varley in 1860. Still later it was discovered that a similar instrument had been made a century before by Nicholson, and called by him the "Revolving Doubler."

The experiments which Thomson made on atmospheric electricity at the old College tower, and by means of portable electrometers in Arran and elsewhere, can only be mentioned. They led no doubt to some improvements on electrometers which he made, the method of bringing the nozzle of a water-dropper, or a point on a portable electrometer to the potential of the air, by the inductive action on a stream of water-drops in the one case, or the particles of smoke from a burning match in the other. He invented a self-acting machine, worked by a stream of water-drops, for acc.u.mulating electric charges, on the principle of the revolving doubler. It was this apparently that led to the machines with revolving carriers, to which reference has been made above.

The mathematical theory of magnetism which Thomson gave in 1849, in the _Phil. Trans. R.S._, was, when completed by various later papers, a systematic discussion of the whole subject, including electromagnetism and diamagnetism. To a large extent the ground covered by the 1849 paper had been traversed before by Poisson, and partially by Murphy and Green; but Thomson stated that one chief object of his memoir was to formally construct the theory without reference to the two magnetic fluids, by means of which the facts of experiment and conclusions of theory had so far been expressed. He found it, however, convenient to introduce the idea of positive and negative magnetic matter (attracting and repelling as do charges of positive and negative electricity), which are to be regarded as always present in equal amounts, not only in a magnet as a whole, but in every portion of a magnet; and at first sight this might appear like a return to the magnetic fluids. But it amounts on the whole rather to a conception of a magnet as a conglomeration of doublets of magnetic matter (that is, very close, equal and inseparable charges of the two kinds of matter), the arrangement of which can be changed by the action of magnetic force. This idea is set forth now in all the books on magnetism and electricity. There can be no doubt that the systematic presentment of the subject by Thomson, and the theorems and ideas of magnetic force and magnetic permeability by which he rendered the clear, and therefore mathematical, notions of Faraday explicitly quant.i.tative, had much influence in furthering the progress of electrical science, and so leading on the one hand to the electromagnetic theories of Maxwell, and on the other to modern research on the magnetic properties of iron, and to the correct ideas which now prevail as to construction of dynamo-electric machines and motors.

CHAPTER XII

THE AGE OF THE EARTH

From his student days throughout his life, Lord Kelvin took a keen interest in geological questions. He was always an active member of the Geological Society of Glasgow, and was its president for twenty-one years (1872-1893). The distribution of heat in the substance of the earth was the subject of his inaugural dissertation as Professor of Natural Philosophy; and previously, as a student, he had written an essay on "The Figure of the Earth," for which he had been awarded a University Gold Medal. He never ceased to ponder over the problems of terrestrial physics, and he wrote much on the subject. His papers are to be found as Appendices to Thomson and Tait's _Natural Philosophy_, and in vol. ii of his _Popular Lectures and Addresses_, which is devoted to geology and general physics.

His conclusions regarding the age of the earth have been referred to in the last chapter. The first allusion to the subject was contained (see p. 65 above) in his inaugural dissertation "_De Caloris distributione in Terrae Corpus_"; but he returned to it again in a communication made to the Royal Society of Edinburgh in December, 1865, and ent.i.tled "The Doctrine of Uniformity in Geology briefly refuted." On February 27, 1868, he delivered to the Geological Society of Glasgow an address ent.i.tled "On Geological Time," in which the necessity for limiting geological and other changes to an almost infinitesimal fraction of the vast periods at that time demanded was insisted on, and which gave rise to much discussion.

The address began with a protest against the old uniformitarian view of geological changes as expressed by Playfair in his _Ill.u.s.trations of the Huttonian Theory_. The first objection taken to the idea that "in the continuation of the different species of animals and vegetables that inhabit the earth, we discern neither a beginning nor an end; in the planetary motions where geometry has carried the eye so far, both into the future and the past, we discover no mark either of the commencement or the termination of the present order" is, that the stability of the motions of the heavenly bodies, to which reference is made in this statement, is founded upon what is essentially an approximate calculation, which leaves out, by intention, the consideration of frictional resistance.

He points out, for example, that the friction which accompanies the relative motion of the waters of the earth and the land is attended by the production of heat, and that, by the doctrine of the conservation of energy, heat cannot be produced without a disappearance of an equivalent quant.i.ty of energy, either of motion or of position. The chief source of this energy is the earth's rotation. Since the earth turns under the moon and the tidal spheroid--that is, the earth's shape as distorted by the heaping up of the waters in the tides--remains on the whole stationary with respect to the moon, the solid matter of the earth turns under the distribution of the water, held more or less fixed by the moon, as does a fly-wheel under a stationary friction band round its rim. Then just as the band held fixed r.e.t.a.r.ds the fly-wheel, so the earth must be r.e.t.a.r.ded in its rotation by this water-brake. In the earth's rotation there is a store of kinetic energy which, roughly estimated, would not be exhausted in less than ten million million years, although drawn upon continuously by friction, or other actions, at the rate of one million horse-power; so that, no immediate catastrophe, such as that we should be involved in by the stoppage or considerable r.e.t.a.r.dation of the spinning motion of the earth, is possible. But it was pointed out by Thomson that the best results of astronomical observation show that the earth would in one hundred years fall behind a perfect time-keeper, with which its rotation kept pace at the beginning of the time, by about twenty seconds. The tendency is to make the earth turn slower, and the moon to increase its distance and move more slowly in its...o...b..t, but with a resultant effect towards coincidence of the period of the earth's rotation with that of revolution of the moon round the earth. After this coincidence has been attained, however, the solar tides will tend to make the moon fall in towards the earth.

If then the earth be rotating more and more slowly, as time goes on, at present, it must have been rotating more rapidly in past time. A thousand million years ago, at the present rate of r.e.t.a.r.dation, the earth must have been rotating one seventh part of its speed faster than it is rotating at present, and this would give for centrifugal force at the surface one thousand million years ago, greater than the centrifugal force at present, in the ratio of 64 to 49. Apparently therefore the earth must have solidified at a much later date than that epoch, a date when it was rotating much more nearly with the angular speed which it has now; otherwise the figure of the earth would have deviated much more from the spherical form than it actually does. On the other hand, one hundred million years ago centrifugal force would be only three per cent. greater than it is at present, and consolidation of the earth at that less remote period would give a shape to the earth not very different from that which it now possesses. The argument therefore from tidal r.e.t.a.r.dation would cut down the time available for geological and biological changes to something not much more than one hundred million years, perhaps to less.

A second argument for limitation of the time available for such processes is derived from the sun's heat. The sun cannot be regarded as a miraculous body producing its light and heat from nothing. Changes of the const.i.tution of the sun must be continually proceeding, to account for its enormous radiation of energy into s.p.a.ce, a radiation of which only an infinitesimal part is received by the bodies of the solar system, and a still more minute portion by the earth. The effects of the sun's light and heat on the earth show how enormous must be the quant.i.ty of energy lost from the sun in a year. How is this loss of energy to be accounted for? What is the physical change which gives rise to it? In 1854 Thomson put forward the theory that the sun's heat is kept up by the falling in of meteors on the sun's surface, but he afterwards saw reason to abandon that view. Helmholtz had advocated the theory that the sun was a body heated by the coming together of the matter composing it by its mutual attraction, a process which, although the sun is now a continuous ma.s.s, is to be regarded as still going on. It is easy to calculate the exhaustion of potential energy caused by the coming together of the matter of the sun from universal dispersion through infinite s.p.a.ce to a sphere of uniform density of the present size of the sun. The result is about as much energy as would be generated by burning seven million million million million million tons of coal. The amount radiated in each hour is about as much as would be generated by burning something like nine tons of coal every hour on every square yard of the sun's surface. It is certain that the sun must be still contracting, and if it contracts sufficiently to just make good this expenditure by the further exhaustion of potential energy involved in the closer aggregation of the matter, it must diminish in radius in each year by as much as 130 feet.

The amount of energy generated by the falling together of the matter of the sun from universal diffusion to the dimensions which the sun has at present, is only about 13,000,000 times the amount now radiated per annum. In Thomson's paper Pouillet's estimate of the energy radiated per second is used, and this number is raised to 20,000,000. Taking the latter estimate, the whole potential energy exhausted by the condensation of the sun's ma.s.s to uniform density would suffice for only 20,000,000 years' supply. But the sun is undoubtedly of much greater density in the central parts than near the surface, and so the energy exhausted must be much greater than that stated above. This will raise the number of years provided for. On the other hand, a considerable amount of energy would be dissipated during the process of condensation, and this would reduce the period of radiation estimated.

Thomson suggests that 50,000,000, or 100,000,000, years is a possible estimate.

It is not unlikely that the rate of radiation in past time, when the sun had not nearly condensed to its present size, was so much less than it is at present that the period suggested above may have to be considerably augmented. Another source of radiation, which seems to be regarded by some authorities as a probable, if not a certain, one, has been suggested in recent years--the presence of radio-active substances in the sun. So far as we know, Lord Kelvin did not admit that this source of radiation was worthy of consideration; but of course, granted its existence to an extent comparable with the energy derivable from condensation of the sun's ma.s.s, the "age of the sun's heat" would have to be very greatly extended. These are matters, however, on which further light may be thrown as research in radio-activity progresses.

Lord Kelvin was engaged when seized with his last illness in discussing the changes of energy in a gaseous, or partially gaseous, globe, slowly cooling and shrinking in doing so; and a posthumous paper on the subject will shortly be published which may possibly contain further information on this question of solar physics.

But Thomson put forward a third argument in the paper on Geological Time, which has always been regarded as the most important. It is derived from the fact, established by abundant observations, that the temperature in the earth's crust increases from the surface inwards; and that therefore the earth must be continually losing heat by conduction from within. If the earth be supposed to have been of uniform temperature at some period of past time and in a molten state, and certain a.s.sumptions as to the conductive power and melting point of its material be made, the time of cooling until the gradient of temperature at the surface acquired its present value can be calculated. This was done by Thomson in a paper published in the _Transactions, R.S.E._, in 1862. We propose to give here a short sketch of his argument, which has excited much interest, and been the cause of some controversy.