Again, from the geometry, we see that y cos x = a + (x − x′) sin a: Also, since no sliding takes place between the cylinder and the plane, it is clear that dt2 {m'a3 cos3 a+ (m + m') (a2 sina + k)}=-m'a'g sin x cos a: d'x the value of is therefore constant during the whole motion; dt2 de' d' de may now be readily obtained by the values of 2 dt2 the aid of the equations (5), (6), (7), and will be constant dx dy during the whole motion. Knowing the values of dt' dt2, we may immediately obtain the values of x, y, 0, x', in terms of t, if the initial values of x, x', dx dx' The values of R and F may also be readily obtained from the equations (1), (2), (3), (4). (4) A bullet is fired with a given velocity into a body in a direction passing through the centre of gravity of the body; the body is initially at rest and is capable of free motion, not being under the action of any forces: to determine the velocities of the bullet and of the body when the bullet has traversed any space within the body, the resistance of the body to the motion of the bullet being supposed to be a constant force. Let k denote the constant retarding force, m the mass of the bullet, μ of the body, ẞ the initial velocity of the bullet; then, if u and v denote the velocities of the bullet and of the body when the bullet has traversed a space x within the body, Camus; 2k ημ 2k ημ (m +μ) (m + μ) x} *. Mém. de l'Acad. des Sciences de Paris, 1738, p. 147. (5) A rough cylinder, the centre of gravity of which is not in its axis, is placed, in a position nearly coinciding with one of stable equilibrium, on a board resting on a smooth horizontal plane to find the length of the simple pendulum which vibrates isochronously with the oscillations of the system. If c be the shortest distance of the centre of gravity of the board from the surface of the cylinder, k the radius of gyration of the cylinder about a line, parallel to its axis, through its centre of gravity, and m, m', the masses of the cylinder and board, then the length of the pendulum is given by the equation m'c2 lh = k2 + m+ m2 CHAPTER X. DYNAMICAL PRINCIPLES. SECT. 1. Vis Viva. THE term Vis Viva was first introduced into the language of Mechanics by Leibnitz, in a memoir published in the Acta Eruditorum for the year 1695, entitled Specimen dynamicum pro admirandis naturæ legibus circa corporum vires et mutuas actiones detegendis et ad suas causas revocandis: it was intended by its author to signify the force of a body in actual motion, called otherwise its Vis Motrix or Moving Force, as distinguished from the statical pressure of a body, which has merely a tendency to motion, against a fixed obstacle; the statical force of a body he designated by the appellation of Vis Mortua. Leibnitz contended, in opposition to the received doctrine of the Cartesians, that the proper measure of the Vis Viva or Moving Force of a body, is the product of its mass into the square of its velocity, the measure adopted by the disciples of Descartes having been the same as that of the Quantity of Motion, namely, the product of the mass and the first power of the velocity. This contrariety of opinion in respect to the estimation of Moving Force, gave rise to one of the most memorable controversies in the annals of philosophy; almost all the mathematicians of Europe ultimately arranging themselves as partizans, either of the Cartesian or of the Leibnitzian doctrine. Among the adherents of Leibnitz may be mentioned John and Daniel Bernoulli, Poleni, Wolff, 'sGravesande, Camus, Muschenbroek, Papin, Hermann, Bulfinger, Koenig, and eventually Madame du Châtelet; while in the opposite ranks may be named Maclaurin, Clarke, Stirling, Desaguliers, Catalan, Robins, Mairan, and Voltaire. The Vis Motrix, or, as Leibnitz expressed it, the Vis Viva of a moving body was regarded as a power inherent in the body, by which it is able to encounter a certain amount of resistance before losing the whole of its velocity the question reduced itself, therefore, to the determination of an appropriate measure of this amount of resistance, to which the Moving Force was supposed to be proportional. Leibnitz regarded the product of the mass of the body and the space through which it must move, under the action of a given retarding force, to lose the whole of its velocity, as the correct measure of the whole resistance expended in the destruction of its motion, and therefore as a proper representative of the Vis Motrix or Vis Viva of the body. Now, by the theory of uniform acceleration, mv2 = 2mfs, m being the mass of the body, and s the space which it must describe, under the action of a constant retarding force f, to lose the whole of its velocity v hence it is evident that, according to the doctrine of Leibnitz, mu2 will represent the body's Vis Viva. On the other hand, the Cartesians estimated the whole resistance necessary for the destruction of the body's velocity by the product of the mass of the body and the whole time of the action of the given retarding force; and therefore, by the formula mu mft, it would follow that my is the proper measure of the Vis Motrix, or, in the language of Leibnitz, of the Vis Viva of the body. The memorable controversy of the Vis Viva, after raging for the space of about thirty years, was finally set to rest by the luminous observations of D'Alembert in the preface to his Dynamique, who declared the whole dispute to be a mere question of terms, and as having no possible connection with the fundamental principles of Mechanics. Since the publication of D'Alembert's work, the term Vis Viva has been used to signify merely the algebraical product of the mass of a moving body and the square of its velocity, while the words Moving Force have been universally employed, agreeably to the definition given by Newton in the Principia, in the signification of the product of the mass of a body and the accelerating force to which it is conceived to be subject, no physical theory whatever in regard to the absolute nature of force being supposed to be involved in these definitions. For = additional information respecting the controversy of the Vis Viva, the reader is referred to Montucla's Histoire des Mathématiques, Tom. III.; Hutton's Mathematical Dictionary under the word 'Force'; and Whewell's History of the Inductive Sciences. The Principle of the Conservation of Vis Viva is comprehended in the following proposition: If a system of particles, any number of which are rigidly connected together, move from one position to another, either with or without constraint, under the action of finite accelerating forces, external or internal; the change of the vis viva of the whole system will be independent of the actions of the particles arising from their mutual connections, and will be equal to the sum of the changes which would be experienced by the vires viva of the particles, were each to move unconnectedly from its original to its new position through a thin smooth fixed tube, under the action of the very accelerating forces to which it is subject in the actual state of the motion. This Principle immediately furnishes us with a first integral of the differential equations of motion, which is frequently of great use; especially if the co-ordinates of the position of the moving system involve only one independent variable, as in the problem of the Centre of Oscillation, when the Principle is sufficient for the complete determination of the motion. The Principle employed by Huyghens' as the basis of his investigations on the problem of the Centre of Oscillation, constitutes under an indirect form a particular instance of the Principle of the Conservation of Vis Viva. John Bernoulli', however, was the first who enunciated the theory of the Conservation of Vis Viva, a name which he gave to the Principle, as a general law of nature, from which he deduced that of 1 Si pendulum e pluribus ponderibus compositum, atque e quiete dimissum, partem quamcunque oscillationis integræ confecerit, atque inde porro intelligantur pondera ejus singula, relicto communi vinculo, celeritates acquisitas sursum convertere, ac quousque possunt ascendere; hoc facto, centrum gravitatis ex omnibus compositæ, ad eandem altitudinem reversum erit, quam ante inceptam oscillationem obtinebat. Horolog. Oscillator. p. 126. 2 Opera, passim. |