691 om- The consideration of simultaneous rotations is very important. osition Suppose a plane figure to rotate in its own plane, with angular f velocity u>, about the origin. Then it is obvious that TO>, in a stations direction perpendicular to r, is the velocity of a point whose dis- bout timce from the origin is r. The components are, therefore, arallel x=-yu,y = X(a. If the rotation be about the point a, b, these become
- = -(y-b)u, y = (x-a)(a.
Hence, when there is any number of simultaneous rotations about parallel axes, we have If we write and we have x = - (y- /3}Q, y=(x-a.}&. These are the component velocities which the point x, y would have if there were only a single rotation, with angular velocity 12, about an axis passing through the point o, $. When 2(co) = fl=0, we see that sc = 2(&w), ?/=-2(aw), so that all points of the figure have equal velocities. This is the case of pure translation. Here o and & are (in general) each infinite; i.e., we have as resultant a vanishing angular velocity about an infinitely distant axis. lolling 74. As any displacement of a plane figure in its own f curve plane is equivalent to a rotation, we may represent a series
- une of displacements by a series of rotations. Also if we
know the positions, in the figure itself, of the points which are successively the axes, and likewise the position which each of them occupies in space at the instant when the rotation about it takes place, we can construct the whole motion. Let them be O, A, B, C, &c., and 0, a, b, c, &c., respectively (fig. 27). Then the figure turns about O till A coincides with a. Next it turns about A (or a) till B coincides with b, and so on. Hence the motion will be represented by the rolling of the poly gon OABC, fixed in the moving figure, Q on the polygon Oabc fixed in the plane of the motion. In the limit, when the axis con tinuously shifts its position in the figure while the rotation goes on round it, the polygons become plane curves. Thus we have the fundamental proposition that any motion of a plane figure in its own plane can be represented by the rolling of a curve attached to it, on a curve fixed in space. Both curves are situated at an infinite distance when the motion is one of pure translation. Displace- nent of entre. Kinematics of a Rigid Figure. 75. When a spherical cap, or skin, moves on the surface O f a sphere of equal radius with which it is everywhere in contact J we ma y ma ke the construction of 71 with great circles bisecting the arcs AA and BB . Two great circles (unless they coincide) always intersect at the extremities of one definite diameter. The case of coincidence is met exactly as it was in 7 1 . Hence every motion of a spherical skin on a sphere is equivalent to a rotation about a definite axis through the centre of the sphere. Thus any number of successive or simultaneous rotations about axes passing through one point can be compounded into a single rotation about an axis passing through that point. And the con struction of 74 can be carried out with spherical polygons or curves, so that we see that any motion of a rigid figure, one point of which is fixed, can be represented by the rolling of a pyramid or cone, fixed in the figure, upon another fixed in space. 76. The law of composition of simultaneous angular velocities about axes which pass through one point is pre- Fig. 28. cisely the same as that for simultaneous linear velocities Com- of a moving point. The following simple geometrical pro- position cess establishes the proposition for two intersecting axes ; of , and it is easy to see that it can be extended to any number about U of such. Let OA and OB (fig. 28) represent, the two axes, axes and let the lengths of these lines (both drawn in the positive direction for the rotation about them) repre sent the angular velocities corre spending. Then a point P, in the angle between the positive ends of the axes, is raised above the plane of the paper by rotation about OA, but depressed below it by the rota tion about OB. The amounts of the elevation and depression are pro portional to the distance from either axis, and to the angular velocity about it, conjointly. Hence they will annihilate one another if, perpendiculars PM, PN being drawn to the axes, we have OA.PM = OB. PN. This is equivalent to saying that the areas of the triangles OAP, OBP, are equal, which necessitates that P should lie on the diagonal of the parallelogram of which OA, OB are conterminous sides. Let OC be the diagonal of this parallelogram. From what has been said above it is evi dent that the displacement of any point in the plane is necessarily proportional to the algebraic sum of the moments of OA and OB about it, and therefore ( 46) to the moment of OC. Hence all points in the line OC remain at rest, and the figure turns about that line with an angular velocity represented by its length. This analogy to moments shows the reason for the remarkable proposition that angular velocities, about axes which intersect, are to be compounded according to the same law as linear velocities. 77. Any proposition regarding simultaneous linear Analogy velocities or accelerations has thus its counterpart in angular between velocities and accelerations. Thus, as we have seen ( 36) liu ^ ar that under acceleration in one plane, always perpendicular to !))L u i ar the direction of motion, a point moves with uniform velocity, veloci- so, if a figure be rotating about one axis, and have angular ties. acceleration about a second axis always perpendicular to the first, the direction of the axis about which it rotates is changed, but not the angular velocity. It is to be noted that in such a case the direction of the axis changes not only in space, but also in the rotating figure itself. This, however, is merely the result of 36 in a slightly altered form. If o> 2 be the angular velocity of a figure about the line which, for Com- the moment, coincides with the axis of z, the consequent displace- position ments during time St of a point x, y, z are ( 73) O f a ngu- $x= - yw z tit , y = X(i>z5t . lar velo- Of course similar results hold for the angular velocities about lines clties for the moment coinciding with the axes of x and of y. The joint * effect therefore is found by adding the various separate values ! n obtained by permuting the letters x, y, z in cyclical order. Thus 1U| The right hand members of these equations vanish if (i). These correspond to the two equations of the instantaneous axis, and reproduce, in an analytical form, the result of 76. The angular velocity about this axis is (2 = V| + ]! + !. For it is clear that the direction cosines of the displacement of x, y, z are proportional to zaiy yeo z , x<a z zw x , y&x x^y,
showing that it takes place in a line perpendicular to the plane