690 MECHANICS Hence, in the resolved part of the motion, the logarithm of the amplitude is diminished, every half vibration, by This process of solution is only applicable to resistance of harmonic vibrations when n is greater than k. When n is not greater than k the auxiliary curve can no longer be a logarithmic spiral, for the moving particle never describes more than a finite angle about the pole ; and then the geo metrical method ceases to be simpler than the analytical one. Relative 69. What we have said about composition of motions motion. j s merely a particular case of the general question of relative motion, which in its main principles is exceedingly simple. It is entirely comprehended in the following pro positions, which may be regarded as almost self-evident. Given the motion of A with regard to a point 0, and that of B with regard to A, to find that of B with regard to 0. By compounding the vectors of relative position OA, AB, we have at once the required vector OB. Thus it is obvious that we have only to add the separate com ponents of the velocity of A with regard to O, and those of B with regard to A, to obtain those of B with regard to 0. And, of course, the same rule applies to the accelerations. If x, y, z be the coordinates of A (referred to 0) at time t ; x , I/, d those of B referred to parallel axes from A; f, ij, those of B referred to ; we have at once Kevolv- ing axes. They give, by differentiation with regard to t, =x + x , &c., %=x + x, &c., which constitute the analytical proof of the statement above. 70. Hence we have the solution of the further question : Given the motions of A and B with regard to 0, to find the relative motion of B with regard to A. In this case, of course, before compounding, the vector of A must have its sign changed. Another very important case is that in which the motion is referred to axes which are themselves moving. So long as their directions remain unchanged, this reduces itself to the former investigation as a mere question of changed origin ; so that we need consider only the effect of the change of direction of the axes. And this is at once de- ducible from the results of last section. For we have only to consider, instead of the moving point, its projections on the moving axes, and find their velocities and accelerations relative to fixed axes. Thus, if the rectangular axes of x and y be fixed, and those of and TJ be rotating in the same plane, we have a datum of the form, 6 = angle 0x =jf) , giving the position of the moving axes in terms of the time. Let P be the moving point, and PM perpendicular to Of (fig. 22). Then, as the polar coordinates of M are , 0, we have, for its velocity, along Of , along MP . But these must be combined with the velocity of P relative to M, which con sists of i] along MP and ^6 parallel to fO. Thus the velocities parallel to fixed lines corresponding to the instantaneous positions of Of and Orj are, respectively, Fig. 22. - yd and 17 + f $ . In the same way it is easy to see, by 47, that the corresponding components of the acceleration are M Kinematics of a Rigid Plane Figure, displaced in its own Plane. 71. When a rigid plane figure is displaced anyhow in its own plane, the displacement may always be regarded a result of a definite rotation about a definite axis per- )endicular to the plane. The proof of this follows at once from the fact that, under the assigned conditions, the figure has only three degrees of freedom ; and consequently its position is determinate whenever the positions of any two of its points are given. Also, a single rotation can, in general, be found which will transfer these two points from one pair of assigned positions to another. Let A, B, A ,B (fig. 23) be successive positions of two Doints of the figure. Bisect AA by the line Oa perpendi- Motior a plane figure ii its plar.> : Fig. 23. Fig. 24. cular to it, and let Ob do the same for BB . Let these perpendiculars meet in O. Then it is clear that the two triangles OAB, OA B are similar and equal. Hence AB may be regarded as having passed to the position A B by rotation about an axis through perpendicular to the plane of the paper, The angle of rotation is AOA or BOB . The construction fails when Oa and Ob coincide, but in this case it is evident that the required point O is the point of intersection of BA and B A (fig. 24). It also fails when the bisecting perpendiculars are parallel (fig. 25). But then AA and BB are equal and parallel, and the dis placement is a pure transla tion, the same for every point of the plane figure, which may be regarded as an infinitely small rotation about an infinitely distant axis. A. Fig. 25. 72. Since any displacement in one plane corresponds Com- in general to a rotation, any two or more rotations about positio parallel axes can always be compounded into a single one. , , . Of two equal and opposite rotations the resultant is simple a1l o ut translation. This is evident from fig. 26. In both cases parall* A and B are the initial posi- axes. tions, A and B the final A , . B positions of the two axes. In the first we begin with the rotation about A, in the second with that about B. 73. When these equal rotations are simultaneous instead of successive, the figure becomes a rectangle; i.e., the translation is per- A A Fig. 26. B pendicular to the line joining the axes. For in this case we may suppose the two rotations to be each broken up into successive equal but infinitesimal instalments. And the principles of infinitesimals show that two such instal ments, either about the same or about different axes, pro duce the same ultimate effects whether they be applied simul taneously or successively. The general principle of which Super this is a particular case is called the "principle of super- positi< position of small motions" It is merely an application of of s the fact that infinitesimals of the second order may be
neglected in comparison with those of the first order.