736 MECHANICS Then we have the six equations (in which a = 0) 2(TO*/)=2(Y)4-H + H , - yx) = 2(a;Y - t/X) , 2/ft(2/;- s/0 = 2(yZ - sY) - cH - c H , S,tn(H - xz) = 2(2X - a-Z) + cS + c H . The fourth equation, as before, determines 6, and we have then four equations to determine , E , H, H . The remaining equation de termines only the sum Z + Z . In fact by more or less perfect fitting we can throw more or less of the force parallel to the axis on one or other of the bearings. There is really no indeterminateness in nature, but we cannot get the information required to evaluate separately Z and Z . 250. When impulsive forces are applied to the body, exactly the same methods may be employed, with the ex ception that u u must be written for x, &c., arid a> - w for 0. The quantities X, Y, Z, H, H, Z, E , H , Z now denote impulses and not forces. As a single example of the use of these formula;, take the case of a body rotating, under the action of no force, about an axis through its centre of ine.rtia. Here ~S,(mx) = 0, &c., and zis con stant. The first two of the six equations last written show that the pairs of forces H, E and H, H form couples. The fourth equation gives 6 = tat ; and with this the remaining two become 2(m3!/).w 2 = -cH-c H = -(c-c )H, S(mzx) . o> 2 = -cs.- c E = -(c-cf)s.. The multipliers of o> 2 are each zero if the axis of rotation be a principal axis, and thus, in this case, there is no stress perpendicular to the axis. When the axis is not a principal axis the left hand terms are generally finite but they vary as the body turns. It is easy to see, however, that together these terms constitute a constant couple always in a plane passing through the axis, rotating with the body and dependent directly on the square of the angular velocity. Thus, to analyse the factor ~2,(mzy), we note that z is constant, and where r, a were the polar coordinates of the mass m at time t = 0. Hence ~2.(mzy] = sin ut2(mzx L ) + cos ut^mzy-^ , where a; 1 = rcosa and y 1 ^rsina were the coordinates of m at t = 0. Similarly we have ~S,(mzx) = coswt^mzxj) - sin cot^mzijj) . These expressions prove the preceding statements. One 251. As a final instance of impulse in this branch of the point subject, suppose that a rigid plate, moving anyhow in its suddenly Qwn p} an6j j^g one O f j^s p om t s suddenly fixed, what will be the subsequent motion? Let the position in space at which the point is to be fixed be chosen as origin, and let the axis of x be chosen so as to pass through the centre of inertia at the moment of fixture. Then, if n, v be the velocities of the centre of inertia, o> the angular velocity about it, a its distance from the point to be fixed, the conditions of the impact are u = u - E/M , v = v- H/M , ca = u+ Ha/ M.k , three equations with five unknown quantities. But the conditions that the point in question is reduced to rest are evidently u = , v - u a = . These furnish the requisite additional data, and the solution is complete. If we eliminate H between the two equa tions which contain it, we have & 2 o> + av = k-w + av , whence by the relation between v and a/ we have These equal quantities, each multiplied by M, represent respectively the moment of momentum about the point before and after its fixture. 252. Thus, as a little consideration will show, we might have solved the problem at once, so far as the impulsive change of motion is concerned, by noticing that as the im pulse is applied at the origin, the moment of momentum about that point will not be altered by it. In fact many problems, which present serious complexity when treated by the direct methods, are solved with comparative ease by such general considerations as the conservation of moment of momentum, or the conservation of energy. The first principle holds good when there is no resultant couple, or impulsive couple, round the origin ; the second when no work on the whole is done by or against the forces or impulses. 253. We have given instances of pure sliding, and of Corn- pure rolling, in one plane, and will now give a single 1)in ^ instance of combined rolling and sliding. A common but 10 ; 11 111(1 instructive case of the problem we propose to consider is m(r> that of a hoop thrown forwards and at the same time made to rotate, so that after a time it stops, and finally rolls backwards to the hand. Other cases are furnished by a "following stroke "or a "screw-back" with a billiard ball. Let the axis of x be parallel to the motion of translation of a sphere or cylinder moving on a horizontal plane. Then we have, if F be the friction, a the radius of the hoop or ball, the rate of change of momentum = F, and that of moment of momentum about the centre = Fa. So long as sliding continues, F is constant, and equal to the product /u,M^ of the normal pressure and the coefficient of kinetic friction. Hence at time t, if ?/ and o> be the initial velocities of translation and of rotation, These equations cease to be true when the sliding ceases, i.e., when we have pure rolling, of which the geometrical condition is This gives + au - fig(a 2 /k- + 1 }t = , At time ^ and ever after we have k 2 (u + aaif,) _ aii n - k 2 ca + a 2 Hence, if the body be projected in the positive direction, its ultimate motion will be in the negative direction if au - & 2 <o be negative, i.e., if the initial angular velocity be positive, and greater than aujk 2 ; which is -f nja in the case of a sphere. Thus, at starting, the linear velocity of the point of contact with the plane must bear to that of translation of the ball a ratio of over 7:2 if it is to stop and return. In the case of a hoop this ratio must be at least 2 : 1. 254. We pass now to the case of a rigid body one point Rigid. only of which is fixed. As we have already seen ( 234) ^^ this has only to be compounded with the motion of the point whole mass, supposed concentrated at the point, in order fixed. to give the most general motion of which a rigid body is capable. The geometrical processes which have been applied to this problem, though in many respects of great power and elegance, cannot be introduced here. We will therefore give the more important results in a brief ana lytical form, and then geometrically exhibit their application. Recurring to the general equations 2m(o$ - yx) = 2(xY - i/X) = N", &c. , we may transform the left hand members as follows. Let u x , oiy, o>z be the angular velocities of the body about the fixed axes of x, y, z respectively. Then ( 77) we have X = ZUy - 7/COz , &C.
Fro n these we have three equations of the type