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Dynamics 54

system, forces, force, effective, particle and external

DYNAMICS.

54. Natural bodies, with which the general theory has to deal, are continuous or apparently continuous distributions of matter, either solid, fluid or gaseous. One way in which they may be treated is to conceive them as an aggregate of particles—large but finite in number, and separated by small but finite intervals— which act on one another with forces of direct attraction or repul sion. This is commonly known as Boscovich's hypothesis (R. G. Boscovich, a Treatise on Natural Philosophy, Venice, 1 758) it enables us to formulate after the manner of §§ 22, 3o, the princi ples of linear and angular momentum.

Principle of d'Alembert.-55.

Another method of treatment is to assume a principle first stated by d'Alembert (J. le R. d'Alembert, Traite de dynamique, 1743). According to Newton's second law (§ 15) the possession of an acceleration F, in any direction, by a particle of mass M implies that a force of magni tude .MF acts upon it in that direction. This force, which we may call the effective force, is the resultant of all the forces which act on the particle. When the particle forms part of a material "system," the latter forces may be divided into two classes: (I) the external forces acting from outside the system, and (2) the internal forces due to the reaction of other particles in the system. D'Alembert's principle assumes that the internal forces constitute by themselves a system in equilibrium, and hence, that the effective forces constitute a system which as a whole is statically equivalent to the system of external forces.

Accordingly we have, for any system, three equations of the type /(mi)=E(X), (6i) in which denotes a summation embracing all the particles of the system. In these equations m denotes the mass of a typical particle, and x, y, z its coordinates referred to any rectangular system of axes, so that my, mE are the components of the effective force on m; X, F, Z are the components of the external force on this particle. Equation (6r) is obtained by resolving parallel to Ox, and (62) by taking moments about Oz.

Writing (6i) and (62) in the equivalent forms we see that they express the principles of linear and angular mo mentum, which are thus shown to be derivable from either of the two fundamental assumptions just stated. It will be observed that neither principle is restricted to rigid bodies : the importance of assuming rigidity lies in the fact that it renders the six equa tions of types (63) and (64) suf ficient in number to determine the motion of a body.

56. An obviously equivalent statement of d'Alembert's prin ciple is that the system of exter nal forces is in equilibrium with the system of effective forces re versed. This concept enables us to treat problems in dynamics by the methods of statics. For example, in the problem of the conical pendulum (fig. 21) we have a mass M attached to a fixed point 0 by an inextensible string, and describing a circular path about a vertical axis through 0. The circular motion can be shown to involve an acceleration of the mass along the radius MN, and if this acceleration has a con stant value f we may take account of the motion by assuming a reversed effective force, of magnitude M.f, to act as shown. Then we have, in effect, a problem in statics, since the reversed effective force must be in equilibrium with the external forces on M, viz., its weight M.g acting vertically and the force T imposed upon it by the tension of the string. (R. V. S.) dealing with the elementary notions of mechanics, as outlined in this article, are very numerous: the reader may be referred to A. E. H. Love's Theoretical Mechanics, or to H. Lamb's Statics, Dynamics and Higher Mechanics. The last of these treatises deals with higher developments of the subject as outlined in the article DYNAMICS; E. T. Whittaker's Treatise on Analytical Dynamics may also be consulted. Philosophical aspects are treated in the first part of J. Ward's Naturalism and Agnosticism. For a short account of Newton's researches cf. S. Brodetsky, Sir Isaac Newton.