738 MECHANICS whence, by (5), 02 = __ 2T/D+p This is constant, and therefore the new tangent plane is fixed in space. Let us now find the angular velocity about the fixed line OL of this plane s point of contact Q with the ellipsoid (6). The direction cosines of the instantaneous axis OP are as <a L , 2 , ct> 3 . Those of OQ are as x, y, z. And we have obviously by (4) and (7) = 0. Hence the line OQ lies in the plane containing the instantaneous axis OP and the fixed line OL. The motion of ellipsoid (6) is therefore one of combined sliding and rolling along the new tangent plane. To find the sliding, we must find the angular velocity of Q about the line OL. It is to that about OP, which is ft, in the ratio of the sines of the angles POQ and QOL. But si (x* + if- + s a ) By means of the equations (1) and (7) above we find easily sin 2 POQ ^D> 2 sin 2 QOL" W Hence the angular velocity of Q about OL is Dp, a constant. Xow suppose the plane on which the ellipsoid (6) rolls and slides to become perfectly rough, and to be capable of rotating round OL as an axis, there will no longer be sliding of Q, but the plane will be made to rotate with the constant angular velocity Dp. Thus the time of any portion of the motion of the body will be measured out by the angle of forced rotation of this plane. 256. As a simple example, let us take the case of a quoit, -in which A, the moment of inertia about the axis of figure, is greater than either of the equal quantities B and C, which may be referred to any two perpendicular lilies in the plane of the quoit. The equations become A&i = , so that w 1 is constant, Put for a moment A-B B " n, then we have oi 2 + ?ia>3 = 0, These give by eliminating o> 3 = . Hence w._ co 3 = - n- l d>. 2 = Psin(nt + Q) . The resultant of these is an angular velocity P, about an axis in the plane of the axes of B and C, and making an angle nt + Q with the axis of B. Hence the instantaneous axis describes in the body a right cone whose axis is that of figure ; it moves round it in the same direction as that in which the body is rotating, and with angular velocity n. The fixed cone in space is also, obviously, a right cone and the other rolls on it externally. Cylinder. If instead of a quoit the body be a long stick or cylinder, we have A = B > C, and the equations become Acb 1 + (C-A)a) 2 w 3 = 0, A 2 + ( A - C)w 3 w 1 = , C 3 =0. The last gives w 3 = constant, and, if A-C the first two equations are Thus ii 1 + nu = Q, 1 and cu 2 = - P sin (nt + Q) . This indicates a rotation of the axis of constant angular velocity P in the negative direction. Everything else is as before, but the cone fixed in the body rolls on the inside of that fixed in space. 257. Next let us take the case of a pendulum bob, supported by a flexible but untwistable wire, and containing a gyroscope whose axis is in the direction of the length of the pendulum. Here we may use, for variety, Lagrange s equations. For simplicity we suppose the centres of inertia of the bob and gyroscope to lie in the axis, and the bob to be symmetrical about the direction of the length of the pendulum. Let the moment of inertia of the whole about the axis of symmetry be A when the gyroscope is supposed to be prevented from turning relatively to the bob, and let the other two principal moments about the point of suspension be B. Let that of the gyro scope about its axis be C. Then, if 6 be the inclination to the vertical, <j> the azimuth of the pendulum, and ^ a quantity denot ing the position of the gyroscope with reference to a definite plane in the bob passing through its axis, we easily find -cos0) = V (l -cos0), suppose, where M is the whole mass, and I the distance from the point of suspension to the centre of inertia of the whole. The general treatment of this complex problem cannot be attempted here. We may, however, easily obtain useful and characteristic results in some special simple cases, which will enable us to form a general idea of the nature of the motion. Thus, suppose if possible 6 to be constant. This is the Conical Conical Gyroscopic Pendulum. We easily find the equations gyro- ^ - (1 - cos <?)</> = ft = const. sc r ic (A(l - cos 0) + B cos 0)0- - Cft0 = V . For any assigned values of ft and 0, this shows what will be the corresponding value of 0. But it also shows that if we change simultaneously the signs only of ft and 0, the value of is unaltered. Thus, reversal of the direction of rotation of the gyroscope involves reversal of the direction of motion of the bob, if the time of rota tion is to be unaltered. But to any assigned values of 6 and ft two values of correspond. As cannot, in the case considered, exceed ^ir, the multiplier of 2 is essentially positive. So is V = M</Z. Hence the values of are real ; and one is positive, the other negative. Thus the pendulum, with any rate of rotation of the gyroscope, may be made to move in any horizontal circle ; but the angular velocity will be greater when it is in the same sense as that of the rotation of the gyroscope than when it is in the opposite sense. When is so small that 2 may be neglected, we have B0 2 -Cft0 = V , or 2B0=CftV4BV + C 2 ft 2 . To give a numerical example, let the mass of the gyroscope be
- let B = MZ 2 , C =
l-r, then /-2 If n = 5, c = 10, g = lOl (which are fair approximations to the dimensions of the ordinary form of the instrument), - 4- /TO ^ (T ^~500 V l + (500) 2 Suppose the gyroscope to revolve 100 times per second, then ft = 2007r practically, and The angular velocity, when the gyroscope is not rotating, would be that of the corresponding conical pendulum, = 3162; so that in this case the gyroscopic pendulum would rotate about twice as fast, or only about half as fast, as the ordinary conical pendulum, according as it rotated with or against the gyroscope. If we had taken ft = WTT we should have found = 3-29 or -3 -04, nearly. Thus the slower the gyroscope rotates the slower is the conical pendulum motion in the same direction, and the quicker that in the opposite direction. STATICS OF A CHAIN OR PERFECTLY FLEXIBLE CORD. 258. AXIOM. When a body or system is in equilibrium under the action of any forces, additional constraints ivill not disturb the equilibrium. Compare 193. This principle is of very great use in forming the funda mental equations of fluid equilibrium, and thence those of motion. And we find it of advantage, as will be presently seen, in reducing to elementary geometry the problem of the equilibrium of a chain, or perfectly flexible cord. We may treat this problem, called that of a " catenary," Cate- by any one of the following methods : (1) by invest!- nary. gating, as a question of statics of a particle, the condi tions of equilibrium of a single link ; (2) by imagining
a finite portion of the chain to become rigid in its