Choose OX, OY so that C-C'=0, and put A'- A=a, B'—B=ß [Compare below, § 124 (2) and (1).] And for σ, the angular velocity of spinning, the obvious proposition stated in the preceding large print gives if 1 and γ 1 γ be the curvatures of the projections on the tangent plane of the fixed and moveable traces. [Compare below, § 124 From (1) and (2) it follows that When one of the surfaces is a plane, and the trace on the other is a line of curvature (§ 130), the rolling is direct. When the trace on each body is a line of curvature, the rolling is direct. Generally, the rolling is direct when the twists of infinitely narrow bands (§ 120) of the two surfaces, along the traces, are equal and in the same direction. 112. Imagine the traces constructed of rigid matter, and all the rest of each body removed. We may repeat the motion with these curves alone. The difference of the circumstances now supposed will only be experienced if we vary the direction of the instantaneous axis. In the former case, we can only do this by introducing more or less of spinning, and if we do so we alter the trace on each body. In the latter, we have always the same moveable curve rolling on the same fixed curve; and therefore a determinate line perpendicular to their common tangent for one component of the rotation; but along with this we may give arbitrarily any velocity of twisting round the common tangent. The consideration of this case is very in rolling. Curve rolling on curve. Angular velocity of structive. It may be roughly imitated in practice by two stiff wires bent into the forms of the given curves, and prevented from crossing each other by a short piece of elastic tube clasping them together. First, let them be both plane curves, and kept in one plane. We have then rolling, as of one cylinder on another. Let p be the radius of curvature of the rolling, p of the fixed, cylinder; the angular velocity of the former, V the linear velocity of the point of contact. We have O' Also, before O'Q' and OQ can coincide in direction, the former must evidently turn through an angle 0+0'. Therefore wt=0+0'; and by eliminating ◊ and 6', and dividing by t, we get the above result. It is to be understood, that as the radii of curvature have been considered positive here when both surfaces are convex, the negative sign must be introduced for either radius when the corresponding curve is concave. Hence the angular velocity of the rolling curve is in this rolling in a case equal to the product of the linear velocity of the point of contact by the sum or difference of the curvatures, according as the curves are both convex, or one concave and the other plane. Plane curves not in same plane. convex. 113. When the curves are both plane, but in different planes, the plane in which the rolling takes place divides the angle between the plane of one of the curves, and that of the other produced through the common tangent line, into parts whose sines are inversely as the curvatures in them respectively; and the angular velocity is equal to the linear velocity Plane curves not plane. of the point of contact multiplied by the difference of the projections of the two curvatures on this plane. The projections of in same the circles of the two curvatures on the plane of the common tangent and of the instantaneous axis coincide. For, let PQ, Pp be equal arcs of the two curves as before, and let PR be taken in the common tangent (i.e., the intersection of the planes of the curves) equal to each. Then QR, PR are ultimately perpendicular to PR. Also, 4 QRp=a, the angle between the planes of the curves. Also the instantaneous axis is evidently perpendicular, and there- A good example of this is the case of a coin spinning on a table (mixed rolling and spinning motion), as its plane becomes gradually horizontal. In this case the curvatures become more and more nearly equal, and the angle between the planes of the curves smaller and smaller. Thus the resultant angular velocity becomes exceedingly small, and the motion of the point of contact very great compared with it. ing on 114. The preceding results are, of course, applicable to tor- Curve rolltuous as well as to plane curves; it is merely requisite to sub- curve: two stitute the osculating plane of the former for the plane of the freedom. latter. degrees of 115. We come next to the case of a curve rolling, with or Curve rollwithout spinning, on a surface. ing on surface: three degrees of It may, of course, roll on any curve traced on the surface, freedom. When this curve is given, the moving curve may, while rolling along it, revolve arbitrarily round the tangent. But the com ing on sur degrees of freedom. Curve roll- ponent instantaneous axis perpendicular to the common tanface: three gent, that is, the axis of the direct rolling of one curve on the other, is determinate, § 113. If this axis does not lie in the surface, there is spinning. Hence, when the trace on the surface is given, there are two independent variables in the motion; the space traversed by the point of contact, and the inclination of the moving curve's osculating plane to the tangent plane of the fixed surface. Trace pre scribed and permitted. 116. If the trace is given, and it be prescribed as a condino spinning tion that there shall be no spinning, the angular position of the rolling curve round the tangent at the point of contact is determinate. For in this case the instantaneous axis must be in the tangent plane to the surface. Hence, if we resolve the rotation into components round the tangent line, and round an axis perpendicular to it, the latter must be in the tangent plane. Thus the rolling, as of curve on curve, must be in a normal plane to the surface; and therefore (§§ 114, 113) the rolling curve must Two degrees be always so situated relatively to its trace on the surface that the projections of the two curves on the tangent plane may be of coincident curvature. of freedom. The curve, as it rolls on, must continually revolve about the tangent line to it at the point of contact with the surface, so as in every position to fulfil this condition. We Let a denote the inclination of the plane of curvature of the trace, to the normal to the surface at any point, a' the same for the plane of the rolling curve; reckon a as obtuse, and a' acute, opposite sides of the tangent plane. 1 1 their curvatures. ρ when the two curves lie on Then Angular velocity of direct rolling. which fixes a' or the position of the rolling curve when the point of contact is given. Let be the angular velocity of rolling about an axis perpendicular to the tangent, a that of twisting about the tangent, and let 1 V be the linear velocity of the point of contact. Then, since cos a' P and 1 P cos a (each positive when the curves lie on opposite sides Angular ve of the tangent plane) are the projections of the two curvatures on a plane through the normal to the surface containing their common tangent, we have, by § 112, locity of direct rolling. a' being determined by the preceding equation. Let and r denote the tortuosities of the trace, and of the rolling curve, respectively. Then, first, if the curves were both plane, we see that one rolling on the other about an axis always perpendicular to their common tangent could never change the inclination of their planes. Hence, secondly, if they are both tortuous, such rolling will alter the inclination of their osculating planes by an indefinitely small amount (TT) ds during rolling which shifts Angular ve the point of contact over an arc ds. Now a is a known function tangent. of s if the trace is given, and therefore so also is a'. But aa-a' is the inclination of the osculating planes, hence locity round surface. 117. Next, for one surface rolling and spinning on another. Surface on First, if the trace on each is given, we have the case of § 113 or § 115, one curve rolling on another, with this farther condition, that the former must revolve round the tangent to the two curves so as to keep the tangent planes of the two surfaces coincident. prescribed: of freedom. It is well to observe that when the points in contact, and the Both traces two traces, are given, the position of the moveable surface is one degree quite determinate, being found thus:-Place it in contact with the fixed surface, the given points together, and spin it about the common normal till the tangent lines to the traces coincide. Hence when both the traces are given the condition of no spinning cannot be imposed. During the rolling there must in general be spinning, such as to keep the tangents to the two traces coincident. The rolling along the trace is due to rotation round the line perpendicular to it in the tangent plane. The whole rolling is the resultant of this rotation and a rotation about the tangent line required to keep the two tangent planes coincident. |