1902 Encyclopedia > Dynamics

Dynamics




DYNAMICS properly means that science which treats of the action of force. Defining force as that which affects the motion of matter, it appears that the study of dynamics will lead to the consideration of the motion of material systems, and the laws in accordance with which this motion is changed by the mutual actions of the bodies forming such systems. But there is a sense in which we may con-template the geometrical results of the motion of bodies without studying the forces under which, or the time dur-ing which, it takes place; and hence there are many pro-blems which at first sight we might be disposed to include under the head of dynamics, but which also belong to the domain of pure mathematics, and may therefore more pro-perly be considered as a branch of geometry. On the other hand, there is a branch of dynamics which treats of pure motion without taking any account of its subject or the means by which it is produced or changed. In this branch, to which the term kinematics, though first employed by Ampere in a wider sense, may with propriety be confined, it may seem that no consideration of matter or of force is involved ; but, unlike the class just alluded to, the pro-blems which come under this head involve explicitly the element of time, and it is only after studying the laws of dynamics that we are able to furnish a theoretical measure of time satisfying the demands of the human mind. Thus any subject in which the measurement of time is involved enters on this account into the domain of dynamics.

Measurement of Time.—For ordinary purposes the rota-tion of the earth furnishes a sufficiently exact means of measuring time, and the observation of the transit of a known star is the best method we possess of determining the error of a clock ; but that the fundamental conception of the measurement of intervals of time is based upon other foundation than the diurnal rotation of our planet at once appears from the fact that we see no inconsistency in asking whether the length of the day is the same now as it was 2000 years ago. If our primary conceptions of the measure-ment of time were derived from the earth's rotation, the absolute constancy of the length of the day would be amatter of definition. But it is not to the motion of the earth or of any other single body that we are indebted for our highest conception of the measurement of time—it is rather to the dynamical principle expressed in the first law of motion ; and hence it is that the theoretical measurement of time, and of other physical quantities which explicitly involve time, must find a place under the head of dynamics. Kinematics may therefore properly be treated as a branch of dynamics, and for its discussion, as well as for the enunication and explanation of the laws of motion, the reader is referred to the article on MECHANICS.

Perhaps there is nothing which appears to present a subject for study simpler than that afforded by the properties of space, and hence it is that geometry attained so high a reputation and made such rapid advances among the ancients. It was easy to construct material standards of length and by their means to measure approximately the linear dimensions of limited portions of space, the human mind being only too ready to believe in the constancy of the dimensions of the standards constructed ; and thus the properties of space presented a subject which, at the very outset, afforded a facility for investigation which was waut-iug in the study of other physical quantities. The great simplification introduced by this belief in the permanence of the dimensions of material standards will be apparent if we consider the position in which we should be placed by the adoption of a different hypothesis. Once admit the supposition that the properties of a figure, as regards dimensions or form, depend explicitly on its position in space, or upon time, either by a process of growth in them-selves or because space is changing its character, and the whole subject of geometry will require reconsideration.

Displacement.-—A number of points or figures may be connected in accordance with such geometrical conditions that if one or more be displaced in a given manner the displacements of all the others may be determined. The determination of the displacement of each in terms of the given displacements is a problem in pure mathematics, and the branch of geometry which treats of such questions may be called the science of displacement. If we suppose the figures here contemplated to be material bodies, and the geometrical conditions to be determined by means of material constraints such as links, guides, teeth, and the like, the science of displacement thus applied becomes that of mechanism, and it is only necesary here to call attention to the following statements. First, in the study of dis-placements, or of pure mechanism, no account is taken of any but the geometrical properties of the bodies displaced, while the forces engaged in producing the displacement are entirely neglected : the consideration of the mechanical properties of the materials of which the parts of a machine are constructed, the forces acting between those parts, and the best means of " fitting " them, belongs to applied mechanics and machine construction. Secondly, the ele-ment of time is altogether left out of consideration; for, although it may be argued that the displacement of each part of the system takes place in the same interval of time, and that the geometrical conditions enable us to compare the displacements experienced by different parts during the same time, and thus lead us to a comparison of velocities, yet it must be observed that this is only a comparison amounting simply to a relation between corresponding displacements, and does not involve time explicitly, since the whole displacement may take place in a time as long or as short as we please, for we do not consider it. More-over, the actual motion of any part may be made uniform or varying in any arbitrary manner without any account being taken of it. In fact it is simply two or more con-figurations of the material system which are compared together, and, though for the sake of distinction we call one the initial and another the final configuration, we might as well distinguish them in any other manner and without stating which follows the other. Indeed we contemplate them as co-existent during the act of comparison. Hence we may complete the science of displacement or pure mechanism without ever considering force, or being able to measure time or even to define equal intervals.





Kinematics.—If to our conceptions of space and of dis-placement we couple that of time as a measurable quantity, we are led to compare the rates of non-simultaneous as well as of simultaneous displacements, and are consequently obliged to measure the rate at which displacement occurs by the change of position experienced in a definite interval of time by the body, figure, or point we are regarding. Rate of change of position measured thus we call velocity. The next step in the same direction is the consideration of the rate at which velocity changes, or acceleration, and thus the association of our conception of space with that of time as a measurable quantity opens up to us that branch of dynamics which we call kinematics.

Matter.—Having considered displacement in connection with the time during which it occurs, the next step leads us to take account of the thing displaced, and here we are obliged to contemplate matter directly. Matter, like time and space, we do not attempt to define, but treat it as a primary conception, its more obvious properties making themselves known to all through daily experience.

Force.—The change of the motion of material bodies brings us at once, through the introduction furnished by the first law of motion, to the conception of force, which may be defined in terms of three primary quantities, viz., space, time, and matter. The second law of motion expresses the manner in which matter is affected by force, and teaches us how to measure force by the observation of its effects.

The science of dynamics in its restricted sense is that which treats of the consequences arising from the relations of matter to force, and before we can proceed in this science beyond the first step we must become acquainted with the second law of motion, while kinematics requires for its complete development only the first law of motion, its range being thereby sharply defined and separated from that of the rest of dynamics. The laws of motion, like other natural laws, must be understood to express merely the properties of natural bodies as we find them, and within the degree of accuracy to which our experiments can be relied on. We might, of course, have started with any hypotheses we liked respecting the relations of force to matter, and upon these hypotheses and our conceptions of time and space have constructed a purely theoretical system of dynamics which would have been perfectly self-con-sistent ; but our conclusions might, or might not, have agreed with observations of natural phenomena. If we found an agreement between the results of the application of our theory to special problems and the solutions of the corresponding problems as worked out objectively in nature, we should have reason to believe that our hypotheses agreed with the facts, or, in other words, that they were true, and we should then raise them to the dignity of natural laws. It is on evidence of this kind that our acceptation of all natural laws is based. If our conclusions were inconsistent with natural phenomena our system of dynamics would be an abstract, instead of a natural, science—if, indeed, it might be called a science at all—and would be valuable merely as an intellectual exercise. In the case of such an abstract science we are not even bound to adopt the axioms respecting the properties of space which are usually accepted, but may confer upon our " space" any number of dimensions and any properties we please.

Stress.—Though the conception of a single force is con-venient, it nevertheless results from a mere process of mental abstraction. We never meet with a single isolated force in nature, but each is accompanied by an equal and opposite force acting in the same straight line, and when we speak of one without the other we do so merely for the sake of brevity. The third law of motion implies this statement, though it has also a wider signification. The action and reac-tion which are thus always inseparably linked together may be conveniently called a stress, of wdiich the two forces are opposite aspects. Thus it appears that there is nothing in nature corresponding to what we are accustomed to call a single force; stresses, indeed, abound, and may be produced whenever we please, but we are completely ignorant of their existence except when they change the relative velocities of different portions of matter. Then, and then only, do they appeal to our senses.

Statics.—The investigation of the conditions under which a system of stresses produces no displacement of the bodies between which they act constitutes the science of statics, and will be discussed under the head of MECHANICS.

Measurement of Force.—Since force can be defined in terms of space, time, and matter, it follow that the measurement of a force ought to involve measurements of these three quantities and of them only. Now it is plain that any force whatever may be chosen as the unit in terms of which other forces should be expressed, provided it is capable of being reproduced at all times and in all places with precision. We all now believe that the quantity of matter in a body is unchanged by changing its position or by the simple lapse of time, and we also believe that the region of space which we inhabit is sufficiently homoloidal to allow us to compare distances in different directions, at different places, and at different times. Moreover, the first law of motion, as has been stated above, provides, when proper precautions are taken, a method of measuring time which satisfies the requirements of the mind, while the rotation of the earth affords a practical measure of time sufficiently exact for the most refined experiments we can execute. Therefore a unit of force which depends only on the units of length, mass, and time will be the same at all places, and, so far as our experience allows us to judge, at all times. Such a unit is termed an absolute unit. Not only force but every other quantity dealt with in dynamical science, as well as every physical quantity whose relations to space, mass, and time are known, can be measured in terms of a unit of its own kind which depends only on the fundamental units of length, mass, and time, and is then said to be expressed in absolute measure. The three primary units must be chosen in an arbitrary manner, and their permanence must be considered a matter of definition; but when these have been once fixed, all the absolute units derived from them are perfectly determinate and invariable. If a foot, a pound, and a second be chosen as units, the corresponding absolute unit of force is called a poundal; while if the primary units be a centimetre, a gramme, and a second, the unit of force is called a dyne.

For the definitions of the derived dynamical units and the investigation of their dependence on the fundamental units, the reader may refer to the article on MECHANICS.





From what has been said it will appear that the whole difficulty in fixing upon a system of dynamical units lies in the determination of the fundamental units of length, mass, and time in such a manner that their constancy can be relied upon. The unit of mass offers very little difficulty in this respect. Long experience has taught us which are the most permanent of the varieties of matter we have at com-mand. We have good reason to believe that a piece of platinum or an alloy of platinum and iridium may be exposed to pure air at ordinary temperatures for an in-definite time without any increase or diminution of its mass whatever. Such a piece of metal may therefore with propriety be chosen as a national standard of mass, the absolute constancy of the quantity of matter constituting it being accepted on definition, as we are unable to test it by any standard in which we have more confidence than we have in itself. The British and French national standards of mass are of platinum, but the new standards recently constructed in Paris consist of an alloy of platinum and iridium.

The determination of a unit of length is not so simple as that of the unit of mass. In this case, as in the preceding, we avail ourselves of the properties of a material standard, but we know that however indestructible the standard itself may be its dimensions depend upon the pressure to which it is exposed, its temperature, and in some cases upon other accidents, such as the magnetic force in the neighbourhood, &x. Hence the only course open to us is to determine as far as possible all the causes of variation in the length of our standard, and carefully to define its condition with respect to these variables, so that it shall be a standard only under the circumstances thus defined. Having thus defined the condition of the material standard with respect to all the variables upon which we know its length to depend, we must consider the absolute constancy of its length at all times and places to be a matter of definition until we have discovered other causes which affect it. It has been pro-posed that the wave length in vacuo of a particular kind of light, as for instance that corresponding to one of the sodium lines, should be taken as the unit of length, and its period as the unit of time. These units are probably more constant than those afforded by any material standards or vibrating springs which we can construct; but a belief in their absolute constancy implies complete confidence in the constancy of the properties of the interstellar medium, and of the sodium molecule.

The determination of a satisfactory means of measuring time seems to offer greater difficulties than the measure-ment of mass or of space, though the difficulties are of the same character as those we have just considered. The great difficulty consists in defining what is meant by the equality of two intervals of time which do not commence simultaneously. Eemembering that it is upon the proper-ties of matter alone that we can rely for assistance, we might construct a spring and define as equal lapses of time those intervals during which this spring executes the same number of vibrations, the temperature, &c, being kept constant. But if we were to construct a number of such springs, though a perfect agreement might obtain between them at first, we should find after a considerable period that the measurements of time derived from different springs did not agree, while our knowledge is insufficient to enable us to apply to each the corrections necessary to lead us to a consistent result. Now there may be no reason why we should prefer one spring above all the others, and thus it appears that a definition of equal intervals of time based upon the behaviour of any single spring is too arbitrary to be satisfactory. If, however, we found a large number of springs, constructed of different materials and differently affected by temperature and other known causes of variation, continue to give perfectly consistent results, the theory of probability would lead us to place a high value upon the measure of time thus afforded. Now, we have stated that our highest conception of the measurement of time is derived from the dynamical principle expressed in the first law of motion, but when we come to apply this it is impossible to determine a priori whether in the case of two given bodies there is no stress acting between them or between one of them and some third object. Consequently, the only course open to us is to examine the motion of a large number of material systems, making such corrections for the action of stresses which we know to be in operation as our theoretical dynamics will enable us to determine; and, if after this we find that several independent systems afford the same measurement of time, while those systems which lead to a different result disagree also among themselves, we must accept the measurement of time afforded by the first set as the true measure, and attribute the discrepancies manifested by the other systems to some unknown stresses, which it should be our subsequent business to discover.

Work.—The contemplation of a stress, together with a re-lative displacement of theportions of matter between which it acts, introduces us to the conception of work. If we con-sider a stress, together with the distance through which the solicited bodies are capable of moving relative to one another in obedience to the stress, the object of our contem-plation is the work which may be done under the given conditions of the system, and this we call energy. The sub-ject of which natural philosophy treats is the transformation of energy, which in all its phases takes place in accordance with two great principles, known respectively as the principles of the conservation and the dissipation of energy. Of these two principles the former rests upon a much higher scientific basis than the latter. In order to lose our faith in the principle of the conservation of energy we must give up our belief in the fundamental principles of dynamics expressed in the laws of motion ; but as regards the dissi-pation of energy we can say little more than that all the operations of nature with which we are acquainted take place in accordance with this principle. Clerk Maxwell has, however, shown that it is possible to subvert the prin-ciple of the dissipation of energy by the simple exercise of a sufficiently high order of intelligence. For the statement and discussion of these two principles see ENERGY.

It is the work of the natural philosopher to explain the operations of nature in accordance with the principles of dynamics, and we consider that we understand any pheno-
menon when we have shown it to consist of a motion of matter and determined the character of this motion. Thus it is that dynamics forms the foundation of every branch of natural philosophy, and a thorough appreciation of the principles of conservation and dissipation of energy is the only safe guide in physical investigations. (w. G.)



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