Convergency of Fourier's series. the series for (§) is more convergent than a harmonic series for its co-efficients. If Displace ment of a for its co-efficients. 78. We now pass to the consideration of the displacement rigid body. of a rigid body or group of points whose relative positions are unalterable. The simplest case we can consider is that of the motion of a plane figure in its own plane, and this, as far as kinematics is concerned, is entirely summed up in the result of the next section. Displacements of a 79. If a plane figure be displaced in any way in its own plane figure plane, there is always (with an exception treated in § 81) one in its plane. point of it common to any two positions; that is, it may be moved from any one position to any other by rotation in its own plane about one point held fixed. To prove this, let A, B be any two points of the plane figure in its first position, A', B' the positions of the same two after B' a displacement. The lines AA', BB' will = not be parallel, except in one case to be presently considered. Hence the line equidistant from A and A' will meet that equidistant from B and B' in some point O. Join OA, OB, OA', OB'. Then, evidently, because OA' OA, OB′ = OB, B and A'B' AB, the triangles OA'B' and OAB are equal and similar. Hence O is similarly situated with regard to A'B' and AB, and is therefore one and the same point of the plane figure in its two positions. If, for the sake of illustration, we actually trace the triangle OAB upon the plane, it becomes OA'B' in the second position of the figure. 80. If from the equal angles A'OB', AOB of these similar ments of a in its plane. triangles we take the common part A'OB, we have the remain- Displaceing angles AOA', BOB' equal, and each of them is clearly plane figure equal to the angle through which the figure must have turned round the point 0 to bring it from the first to the second position. The preceding simple construction therefore enables us not only to demonstrate the general proposition, § 79, but also to determine from the two positions of one line AB, A'B' of the figure the common centre and the amount of the angle of rotation. 81. The lines equidistant from A and A', and from B and B', are parallel if AB is parallel to A'B'; and therefore the construction fails, the point O being A infinitely distant, and the theorem B becomes nugatory. In this case the 82. It is not necessary to suppose the figure to be a mere flat disc or plane-for the preceding statements apply to any one of a set of parallel planes in a rigid body, moving in any way subject to the condition that the points of any one plane in it remain always in a fixed plane in space. 83. There is yet a case in which the construction in § 79 is nugatory-that is when AA' is parallel to BB', but AB intersects A'B. In this case, however, it is easy to see at once that this point of intersection is the point O required, although the former method would not have enabled us to find it. A' A B B' of displace 84. Very many interesting applications of this principle Examples may be made, of which, however, few belong strictly to our ment in one subject, and we shall therefore give only an example or two. Thus we know that if a line of given length AB move with plane. Examples Composi tion of rota its extremities always in two fixed lines OA, OB, any point in B P Q A it as P describes an ellipse. It is required to find the direction of motion of P at any instant, i.e., to draw a tangent to the ellipse. BA will pass to its next position by rotating about the point Q; found by the method of § 79 by drawing perpendiculars to OA and OB at A and B. Hence P for the instant revolves about Q, and thus its direction of motion, or the tangent to the ellipse, is perpendicular to QP. Also AB in its motion always touches a curve (called in geometry its envelop); and the same principle enables us to find the point of the envelop which lies in AB, for the motion of that point must evidently be ultimately (that is for a very small displacement) along AB, and the only point which so moves is the intersection of AB, with the perpendicular to it from Q. Thus our construction would enable us to trace the envelop by points. A B 85. Again, suppose AB to be the beam of a stationary engine having a reciprocating motion about A, and by a link BD turning a crank CD about C. Determine the relation between the angular velocities of AB and CD in any position. Evidently the instantaneous direction of motion of B is D transverse to AB, and of D transverse to CD-hence if AB, CD produced meet in O, the motion of BD is for an instant as if it turned about O. From this it may easily be seen that AB OD OB CD if the angular velocity of AB be w, that of CD is A similar process is of course applicable to any combination of machinery, and we shall find it very convenient when we come to consider various dynamical problems connected with virtual velocities. 86. Since in general any movement of a plane figure in its tions about plane may be considered as a rotation about one point, it is evident that two such rotations may in general be compounded parallel axes. Composition of rotations about into one; and therefore, of course, the same may be done with any number of rotations. Thus let A and B be the points of the figure about which in succession the rotations are to take parallel axes place. By rotation about A, B is brought say to B', and by a rotation about B', A is brought to A'. The construction of § 79 gives us at once the point 0 and the amount of rotation about it which singly gives the same effect as those about A and B in succession. But there is one case of exception, viz., when the rotations about A and B are of equal amount and in opposite directions. In this case A'B' is evidently parallel to AB, and therefore the compound result is a translation only. A B' B That is, if a body revolve in succession through equal angles, but in opposite directions about two parallel axes, it finally takes a position to which it could have been brought by a simple translation perpendicular to the lines of the body in its initial or final position, which were successively made axes of rotation; and inclined to their plane at an angle equal to half the supplement of the common angle of rotation. of rotations tions in one 87. Hence to compound into an equivalent rotation a rota- Composition tion and a translation, the latter being effected parallel to the and transla plane of the former, we may decompose the translation into plane. two rotations of equal amount and opposite direction, compound one of them with the given rotation by § 86, and then compound the other with the resultant rotation by the same process. Or we may adopt the following B' C' A B far simpler method. Let OA be p Composition of rotations and transla tions in one plane. Omission of the second orders of tities. In general if the origin be taken as the point about which rotation takes place in the plane of xy, and if it be through an angle 0, a point whose co-ordinates were originally x, y, will have them changed to 88. In considering the composition of angular velocities and higher about different axes, and other similar cases, we may deal small quan- with infinitely small displacements only; and it results at once from the principles of the differential calculus, that if these displacements be of the first order of small quantities, any point whose displacement is of the second order of small quantities is to be considered as rigorously at rest. Hence, for instance, if a body revolve through an angle of the first order of small quantities about an axis belonging to the body which during the revolution is displaced through an angle or space, also of the first order, the displacement of any point of the body is rigorously what it would have been had the axis been fixed during the rotation about it, and its own displacement made either before or after this rotation. Hence in any case of motion of a rigid system the angular velocities about a system of axes moving with the system are the same at any instant as those about a system fixed in space, provided only that the latter coincide at the instant in question with the moveable ones. Superposition of sinall motions. 89. From similar considerations follows also the general principle of Superposition of small motions. It asserts that if several causes act simultaneously on the same particle or rigid body, and if the effect produced by each is of the first order of small quantities, the joint effect will be obtained if we consider the causes to act successively, each taking the point or system in the position in which the preceding one left it. It is evident at once that this is an immediate deduction from the fact that the second order of small quantities may be with rigorous accuracy neglected. This principle is of very great use, as we shall find in the sequel; its applications are of constant occurrence. |