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Laws of Motion

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Before the time of Galileo (1564-1642) hardly any attention had been paid to a scientific study of the motions of terrestrial bodies. With regard to celestial bodies, however, the case was different. The regularity of their diurnal revolutions could not escape notice, and a good deal was known 2000 years ago about the motions of the sun and moon and planets among the stars. For the statement of the motions of these bodies uniform motion in a circle was employed as a fundamental type, combinations of motions of this type being constructed to fit the observations. This procedure - which was first employed by the great Greek astronomer Hipparchus (2nd century B.C.), and developed by Ptolemy three centuries later - did not afford any law connecting the motions of different bodies. Copernicus (1473-1543) employed the same system, and greatly simplified the application of it, especially by regarding the earth as rotating and the sun as the centre of the solar system. Kepler (1571-1630) was led by his study of the planetary motions to reject this method of statement as inadequate, and it is in fact incapable of giving a complete representation of the motions in question. In 1609 and 1619 Kepler published his new laws of planetary motion, which were subsequently shown by Newton to agree with the results obtained by experiment for the motion of terrestrial bodies.

The earliest recorded systematic experiments as to the motion of falling bodies were made by Galileo at Pisa in the latter years of the 16th century. Bodies of different substances were employed, and slight differences in their behaviour accounted for by the resistance of the air. The result obtained was that any body allowed to fall from rest would, in a vacuum move re to the earth with constant o G av. , 3' of Gravity. acceleration; that is to say, would move in a straight line, in such a manner that its velocity would increase by equal amounts in any two equal times. This result is very nearly correct, the deviations being so small as to be almost beyond the reach of direct measurement. It has since been discovered, however, that the magnitude of the acceleration in question is not exactly the same at different places on the earth, the range of variation amounting to about 2%. Galileo proceeded to measure the motion of a body on a smooth, fixed, inclined plane, and found that the law of constant acceleration along the line of slope of the plane still held, the acceleration decreasing in magnitude as the angle of inclination was reduced; and he inferred that a body, moving on a smooth horizontal plane, would move with uniform velocity in a straight line if the resistance of the air, and friction due to contact with the plane, could be eliminated. He went on to deal with the case of projectiles, and was led to the conclusion that the motion in this case could be regarded as the result of superposing a horizontal motion with uniform velocity and a vertical motion with constant acceleration, the latter identical with that of a merely falling body; the inference being that the path of a projectile would be a parabola except for deviations attributed to contact with the air, and that in a vacuum this path would be accurately followed. The method of superposition of two motions may be illustrated by such examples as that of a body dropped from the mast of a ship moving at uniform speed. In this case it is found that the body falls relatively to the ship as if the latter were at rest, and alights at the foot of the mast, having consequently pursued a parabolic path relatively to the earth.

The importance of these results, limited though their scope was, can hardly be overrated. They had practically the effect of suggesting an entirely new view of the subject, namely, that a body uninfluenced by other matter might be expected to move, relatively to some base or other, with uniform velocity in a straight line; and that, when it does not move in this way, its acceleration is the feature of its motion which the surrounding conditions determine. The acceleration of a falling body is naturally attributed to the presence of the earth; and, though the body approaches the earth in the course of its fall, it is easily recognized that the conditions under which it moves are only very slightly affected by this approach. Moreover, Galileo recognized, to some extent at any rate, the principle of simple superposition of velocities and accelerations due to different sets of circumstances, when these are combined (see Mechanics). The results thus obtained apply to the motion of a small body, the rotation of which is disregarded. When this case has been sufficiently studied, the motion of any system can be dealt with by regarding it as built up of small portions. Such portion, small enough for the position and motion of each to be sufficiently specified by those of a point, are called "particles." Descartes helped to generalize and establish the notion of the fundamental character of uniform motion in a straight line, but otherwise his speculations did not point in the direc tion of sound progress in dynamics; and the next substantial advance that was made in the principles of the subject was due to Huygens (1629-1695). He attained correct views as to the character of centrifugal force in connexion with Galileo's theory; and, when the fact of the variation of gravity (Galileo's acceleration) in different latitudes first became known from the results of pendulum experiments, he at once perceived the possibility of connecting such a variation with the fact of the earth's diurnal rotation relatively to the stars. He made experiments, simultaneously with Wallis and Wren, on the collision of hard spherical bodies, and his statement of the results (1669) included a clear enunciation of the conservation of linear momentum, as demonstrated for these cases of collision, and apparently correct in certain other cases, mass being estimated by weight. But Huygens's most important contribution to the subject was his investigation, published in 1673, of the motion of a rigid pendulum of any form. This is the earliest example of a theoretical investigation of the rotation of rigid bodies. It involved the adoption of a point of view as to the relation between the motions of bodies of different forms, which practically amounted to a perception of the principle of energy as applied to the case in question.

We owe to Newton (1642-1727) the consolidation of the views which were current in his time into one coherent and universal Galileo- system, sometimes called the Galileo-Newton theory, Newton Theory. but commonly known as the "laws of motion"; T and the demonstration of the fact that the motions of the celestial bodies could be included in this theory by means of the law of universal gravitation. A full account of his results' was first published in the Principia in 1687.

Such statements as that a body moves in a straight line, and that it has a certain velocity, have no meaning unless the base, relative to which the motion is to be reckoned, is defined. Accordingly, in the extension of Galileo's results for the purpose of a universal theory, the establishment of a suitable base of reference is the first step to be taken. Newton assumed the possibility of choosing a base such that, relatively to it, the motion of any particle would have only such divergence from uniform velocity in a straight line as could be expressed by laws of acceleration dependent on its relation to other bodies. He used the term "absolute motion" for motion relative to such a base. Many writers on the subject distinguish such a base as "fixed." The name "Newtonian base" will be used in this article. Assuming such a base to exist, Newton admitted at the outset the difficulty of identifying it, but pointed out that the key to the situation might be found in the identification of forces; that is to say, in the mutual character of laws of acceleration as applied to any given body and any other by whose presence its motion is influenced. In this connexion he took an important step by distinguishing clearly the character of "mass" as a universal property of bodies distinct from weight.

There can be no doubt that the development of correct views as to mass was closely connected with the results of experiments with regard to the collision of hard bodies. Suppose two small smooth spherical bodies which can be regarded as particles to be brought into collision, so that the velocity of each, relative to any base which is unaffected by the collision, is suddenly changed. The additions of velocity which the two bodies receive respectively, relative to such a base, are in opposite directions, and if the bodies are alike their magnitudes are equal. If the bodies though of the same substance are of different sizes, the magnitudes of the additions of velocity are found to be inversely proportional to the volumes of the bodies. But if the bodies are of different substances, say one of iron and the other of gold, the ratio of these magnitudes is found to depend upon something else besides bulk. A given volume of gold is found to count for this purpose for about two and a half times as much as the same volume of iron. This is expressed by saying that the density of gold is about two and a half times that of iron. In fact, experiments upon the changes of velocity of bodies, due to a mutual influence between them, bring to light a property of bodies which may be specified by a quantity proportional to their volumes in the case of bodies which are perceived by other tests to be of one homogeneous substance, but otherwise involving also another factor.

The product of the volume and density of a body measures what is called its "mass." The mass of a body is often loosely defined as the measure of the quantity of matter in it. This definition correctly indicates that the mass of any portion of matter is equal to the sum of the masses of its parts, and that the masses of bodies alike in other respects are equal, but gives no test for comparison of the masses of bodies of different substances; this test is supplied only by a comparison of motions. When, as in the case of contact, a mutual relation is perceived between the motions of two particles, the changes of velocity are in opposite directions, and the ratio of their magnitudes determines the ratio of the masses of the particles; the motion being reckoned relative to any base which is unaffected by the change. It is found that this gives a consistent result; that is to say, if by an experiment with two particles A and B we get the ratio of their masses, and by an experiment with B and a third particle C we get the ratio of the masses of B and C, and thus the ratio of the masses of A and C, we should get the same ratio by a direct experiment with A and C. For the numerical measure of mass that of some standard body is chosen as a unit, and the masses of other bodies are obtained by comparison with this. Masses of terrestrial bodies are generally compared by weighing; this is found by experiment to give a correct result, but it is applicable only in the neighbourhood of the earth. Familiar cases can readily be found of the perception of the mass of bodies, independently of their tendency to fall towards the earth. The mass of any portion of matter is found to be permanent under chemical and other changes, and this fact adds to its importance as a physical quantity. The study of the structure of atoms has suggested a connexion of mass with electrical phenomena which implies its dependence on motion; but this is not inconsistent with the observed fact of its practical constancy, to a high degree of accuracy, for bodies composed of atoms.

The Galileo-Newton theory of motion is that, relative to a suitably chosen base, and with suitable assignments of mass, all accelerations of particles are made up of mutual (so-called) actions between pairs of particles, whereby the two particles forming a pair have accelerations in opposite directions in the line joining them, of magnitudes inversely proportional to their masses. The total acceleration of any particle is that obtained by the superposition of the component accelerations derived from its association with the other particles of the system severally in accordance with this law. The mutual action between two particles is specified by means of a directed quantity to which the term "force" is appropriated. A force is said to act upon each of two particles forming a pair, its magnitude being the product of mass and component acceleration of the particle on which it acts, and its direction that of this component acceleration. Thus each mutual action is associated with a pair of equal forces in opposite directions. Instead of the operation of superposing accelerations, we may compound the several forces acting on a particle by the parallelogram law (see Mechanics) into what may be called the resultant force, the total acceleration of the particle being the same as if this alone acted. The theory depends for its verification and application upon the fact that forces can be identified and classified. They can be recognized by Application their reciprocal character, and it is found to be of the possible to connect them by permanent laws with the Theory. recognizable physical characteristics of the systems in which they occur. A generalization of Galileo's results takes the form that under constant conditions of this kind, force (defined in terms of motion) is constant, and that the superposition of two sets of conditions, if their independence can be secured, results in superposition of the forces associated with them separately. Particular laws of force may be suggested by a study of the simplest cases in which they are manifested, and from them results may be obtained by calculation as to the motions of systems of any given structure. Such results may be tested by direct observation.

It should be noted that, within a limited range of application to terrestrial mechanics, the most convenient way of attacking the question of the relations of forces to the physical statics. conditions of their occurrence may be by balancing their several effects in producing motion; thus avoiding in the first instance both the choice of a base and the consideration of mass. This procedure is useful as a preliminary step in the study of the subject. It does not, however, afford a convenient starting-point for a general theory, because it is apt to involve some confusion of phenomena which, from the point of view of the Galileo-Newton theory, are distinct in character.

Newton's law of gravitation affords the most notable example of the process of verification of a law of force, and incidentally of the Galileo-Newton theory. As a law of acceleration of the planets relatively to the sun, its approximate agreement with Kepler's third law of planetary motion follows readily from a consideration of the character of the acceleration of a point moving uniformly in a circle. Newton tells us that this agreement led him to adopt the law of the inverse square of the distance about 1665-1666, before Huygens's results as to circular motion had been published. At the same time he thought of the possibility of terrestrial gravity extending to the moon, and made a calculation with regard to it. Some years later he succeeded in showing that Kepler's elliptic orbit for planetary motion agreed with the assumed law of attraction; he also completed the co-ordination with terrestrial gravity by his investigation of the attractions of homogeneous spherical bodies. Finally, he made substantial progress with more exact calculations of the motions of the solar system, especially for the case of the moon. The work of translating the law of gravitation into the form of astronomical tables, and the comparison of these with observations, has been in progress ever since. The discovery of Neptune (1846), due to the influence of this planet on the motion of Uranus, may be mentioned as its most dramatic achievement. The verification is sufficiently exact to establish the law of gravitation, as providing a statement of the motions of the bodies composing the solar system which is correct to a high degree of accuracy. In the meantime some confirmation of the law has been obtained from terrestrial experiments, and observations of double stars tend to indicate for it a wider if not universal range. It should be noticed that the verification was begun without any data as to the masses of the celestial bodies, these being selected and adjusted to fit the observations.

The case of electro-magnetic forces between two conductors carrying electric currents affords an example of a statement of motion in terms of force of a highly artificial kind. It can only be contrived by means of complicated mathematical analysis. In this connexion a statement in terms of force is apt to be displaced by more direct and more comprehensive methods, and the attention of physicists is directed to the intervention of the ether. The study of such cases suggests that the statement in terms of force of the relations between the motions of bodies may be only a provisional one, which, though it may summarize the effect of the actual connexions between them sufficiently for some practical purposes, is not to be regarded as representing them completely. There are indications of this having been Newton's own view.

The Newtonian base deserves some further consideration. It is defined by the property that relative to it all accelerations of particles correspond to forces. This test involves only changes of velocity, and so does not distinguish between two bases, each of which moves relatively to the other with uniform velocity without rotation. The establishment of a true Newtonian base presumes knowledge of the motions of all bodies. But practically we are always dealing with limited systems, so any actual determination must always be regarded as to some extent provisional. In the treatment of the relative motions of a limited system, we may use a confessedly provisional base, though it may be necessary to introduce corrections, either exact or approximate, to take account either of the existence of bodies outside the system, or of the rotation of the base employed relative to a more correct one. Such corrections may be made by the device of applying additional unpaired, or what we may call external, forces to particles of the system. These are needed only so far as they introduce differences of accelerations of the several particles. The earth, which is commonly employed as a base for terrestrial motions, is not a very close approximation to being a Newtonian base. Differences of acceleration due to the attractions of the sun and moon are not important for terrestrial systems on a small scale, and can usually be ignored, but their effect (in combination with the rotation of the earth) is very apparent in the case of the ocean tides. A more considerable defect is due to the earth having a diurnal rotation relative to a Newtonian base, and this is never wholly ignored. Take a base attached to the centre of the earth, but without this diurnal rotation. A small body hanging by a string, at rest relatively to the earth, moves relatively to this base uniformly in a circle; that is to say, with constant acceleration directed towards the earth's axis. What is done is to divide the resultant force due to gravitation into two components, one of which corresponds to this acceleration, while the other one is what is called the "weight" of the body. Weight is in fact not purely a combination of forces, in the sense in which that term is defined in connexion with the laws of motion, but corresponds to the Galileo acceleration with which the body would begin to move relatively to the earth if the string were cut. Another way of stating the same thing is to say that we introduce, as a correction for the earth's rotation, a force called "centrifugal force," which combined with gravitation gives the weight of the body. It is not, however, a true force in the sense of corresponding to any mutual relation between two portions of matter. The effect of centrifugal force at the equator is to make the weight of a body there about .35% less than the value it would have if due to gravitation alone. This represents about two-thirds of the total variation of Galileo's acceleration between the equator and the poles, the balance being due to the ellipticity of the figure of the earth. In the case of a body moving relatively tothe earth, the introduction of centrifugal force only partially corrects the effect of the earth's rotation. Newton called attention to the fact that a falling body moves in a curve, diverging slightly from the plumb-line vertical. The divergence in a fall of zoo ft. in the latitude of Greenwich is about T 11n. Foucault's pendulum is another example of motion relative to the earth which exhibits the fact that the earth is not a Newtonian base.

For the study of the relative motions of the solar system, a provisional base established for that system by itself, bodies outside it being disregarded,' is a very good one. No correction for any defect in it has been found necessary; moreover, no rotation of the base relative to the directions of the stars without proper motion has been detected. This is not inconsistent with the law of gravitation, for such estimates as have been made of planetary perturbations due to stars give results which are insignificant in comparison with quantities at present measurable.

For the measurement of motion it must be presumed that we have a method of measuring time. The question of the standard to be employed for the scientific measurement of - time accordingly demands attention. A definition of the measurement dependent on dynamical theory has Time. been a characteristic of the subject as presented by some writers, and may possibly be justifiable; but it is neither necessary nor in accordance with the historical development of science. Galileo measured time for the purpose of his experiments by the flow of water through a small hole under approximately constant conditions, which was of course a very old method. He had, however, some years before, when he was a medical student, noticed the apparent regularity of successive swings of a pendulum, and devised an instrument for measuring, by means of a pendulum, such short periods of time as sufficed for testing the pulse of a patient. The use of the pendulum clock in its present form appears to date from the construction of such a clock by Huygens in 16J7. Newton dealt with the question at the beginning of the Principia, distinguishing what he called "absolute time" from such measures of time as would be afforded by any particular examples of motion; but he did not give any clear definition. The selection of a standard may be regarded as a matter of arbitrary choice; that is to say, it would be possible to use any continuous time-measurer, and to adapt all scientific results to it. It is of the utmost importance, however, to make, if possible, such a choice of a standard as shall render it unnecessary to date all results which have any relation to time. Such a choice is practically made. It can be put into the form of a definition by saying that two periods of time are equal in which two physical operations, of whatever character, take place, which are identical in all respects except as regards lapse of time. The validity of this definition depends on the assumption that operations of different kinds all agree in giving the same measure of time, such allowances as experience dictates being made for changing conditions. This assumption has successfully stood all tests to which it has been subjected. All clocks are constructed on the basis of this method of measurement; that is to say, on the plan of counting the repetitions of some operation, adopted solely on the ground of its being capable of continual repetition with a certain degree of accuracy, and possibly also of automatic compensation for changing conditions. Practically clocks are regulated by reference to the diurnal rotation of the earth relatively to the stars, which affords a measurement on the repetition principle agreeing with other methods, but more accurate than that given by any existing clock. We have, however, good reasons for regarding it as not absolutely perfect, and there are some astronomical data the tendency of which is to confirm this view.

The most important extension of the principles of the subject since Newton's time is to be found in the development of the of theory of energy, the chief value of which lies in the Theory Energy. fact that it has supplied a measurable link connecting E the motions of systems, the structure of which can be directly observed, with physical and chemical phenomena having to do with motions which cannot be similarly traced in detail. The importance of a study of the changes of the vis viva depending on squares of velocities, or what is now called the "kinetic energy" of a system, was recognized in Newton's time, especially by Leibnitz; and it was perceived (at any rate for special cases) that an increase in this quantity in the course of any motion of the system was otherwise expressible by what we now call the "work" done by the forces. The mathematical treatment of the subject from this point of view by Lagrange (1736-1813) and others has afforded the most important forms of statement of the theory of the motion of a system that are available for practical use. But it is to the physicists of the 19th century, and especially to Joule, whose experimental results were published in 1843-1849, that we practically owe the most notable advance that has been made in the development of the subject - namely, the establishment of the principle of the conservation of energy (see Energetics and Energy). The energy of a system is the measure of its capacity for doing work, on the assumption of suitable connexions with other systems. When the motion of a body is checked by a spring, its kinetic energy being destroyed, the spring, if perfectly elastic, is capable of restoring the motion; but if it is checked by friction no such restoration can be immediately effected. It has, however, been shown that, just as the compressed spring has a capacity for doing work by virtue of its configuration, so in the case of the friction there is a physical effect produced - namely, the raising of the temperature of the bodies in contact, which is the mark of a capacity for doing the same amount of work. Electrical and chemical effects afford similar examples. Here we get the link with physics and chemistry alluded to above, which is obtained by the recognition of new forms of energy, interchangeable with what may be called mechanical energy, or that associated with sensible motions and changes of configuration.

Such general statements of the theory of motion as that of Lagrange, while releasing us from the rather narrow and strained view of the subject presented by detailed analysis of motion in terms of force, have also suggested a search for other forms which a statement of elementary principles might equally take as the foundation of a logical scheme. In this connexion the interesting scheme formulated by Hertz (1894) deserves notice. It is important as an addition to the logic of the subject rather than on account of any practical advantages which it affords for purposes of calculation.

Authorities. - Galileo, Dialogues (translations: "The System of the World" and "Mechanics and Local Motion," in T. Salusbury's Mathematical Collections and Translations (1661-1665); Mechanics and Local Motion, by T. Weston (1730); Huygens, Horologium Oscillatorium (1673); Newton, Philosophiae naturalis principia mathematica (1687; translation by A. Motte, 1729); W. W. Rouse Ball, An Essay on Newton's Principia (1893); Whewell, History of the Inductive Sciences (1837); J. Clerk Maxwell, Matter and Motion (1882); H. Streintz, Die physikalischen Grundlagen der Mechanik (1883); E. Mach, Die Mechanik in ihrer Entwickelung historischkritisch dargestellt (1883; 2nd edition (1889 translation) by T. J. McCormack, 1893); K. Pearson, The Grammar of Science (1892); A. E. H. Love, Theoretical Mechanics (1897). H. Hertz, Die Prinzipien der Mechanik (1894, translation by Jones and Walley 1899). (W. H. M.)