732 MECHANICS pass through it. Here again there is indeterminateness, because there are two places at which friction comes in, and we do not know at which it is most freely exerted. But if the whole be on the point of slipping, we have as before the additional data These lead to the equation (1- ju 2 )cosa + 2/, If we introduce an angle v, such that = seca solid. this equation becomes cos 2 a cos (a - v} = - = - cos 2 - , a( 1 + fj. 2 ) a 2 the right-hand member of which must necessarily be less than 1. This determines the lowest position of the lower end consistent with equilibrium, and the mere change of sign of /A, and therefore of v, alters it into the equation for the highest. The signs of the friction terms are changed when the direction of slipping is supposed to be reversed. KINETICS OF A. BIGID SOLID. Rotation 234. The motion of a rigid body is, as we have seen, of^rigid completely determined when we know the motion of one of its points and the relative motion of the body about that point. The point usually chosen is the centre of inertia of the body, and the investigation of its motion comes under the kinetics of a particle, which we have already sufficiently discussed. For we are permitted to suppose the whole mass to be concentrated at that point, and to be acted on by all the separate forces, each unaltered in direction and magnitude. Hence we may now confine ourselves to the study of the motion about the centre of inertia which, for the moment, we may look on as fixed. To illustrate, in a very simple manner, the new concep tions which are required for the study of this question, let us take a uniform circular ring of matter, of radius E, revolving with angular velocity w about an axis through its centre, and perpendicular to its plane. Its moment of momentum is obviously M . Rco . R or MR 2 . co. Its kinetic energy is pl(llco) 2 or PIR 2 . co 2 . If it be acted on by a couple C, in its plane, C is the rate of increase of the moment of momentum, or MR 2 . co = C. The work done by the couple in time 8t is and the increase of kinetic energy is MR 3 . o)co5< . By equating these we have (after dividing both sides by <D) the same equation as we obtained from the rate of increase of moment of momentum, It will be observed that these equations are of exactly the same form as those for the motion of a particle parallel to one of the coordinate axes, only that <o takes the place of a velocity (such as x) while the expression ME 2 takes the place of M, and- the right-hand side is the moment of a force, not a force simply. 235. Hence, generally, we define as follows : DBF. The " moment of inertia " of a body about any axis is the sum of the products of the mass of each particle of the body into the square of its (least) distance from the axis. The following theorem enables us at once to find the Moment of iner tia. moment of inertia about any line, as axis, from that about a parallel axis through the centre of inertia. Let the line be chosen as the axis of z, then the moment of inertia about it is ~2,m(x" + y 2 ) . But, if x, y be the coordinates of the centre of inertia, |, ?j the coordinates of m with reference to that centre, we have and the above expression for the moment of inertia becomes 2z(x 2 + f + 2xiJ + 2J/7J + 1 2 + r; 2 ) . By the property of the centre of inertia, 109, Hence the above expression consists of two parts: 2/(| 2 + 77 -) the moment of inertia about a parallel axis through the centre of inertia, and 5(m) . (x 2 + 7/ 2 ) the moment of inertia of the whole mass supposed concentrated at its centre of inertia. 236. Hence we need study only the moments of inertia about axes passing through the centre of inertia. But we will commence with an origin assumed at hazard. If the direction cosines of an axis through the origin be X, p, v, the square of the distance of the mass m at x, y, z from it is x 2 + y 2 + z 2 - (x + py 4- c) 2 . Hence the moment of inertia is ?/ 2 + z 2 - (x + py + i>z) 2 ) = 2wi((2/ 2 + z 2 ) A 2 + (z 2 + x which may be written | == AA 2 + 2G 3 If we measure off, on the axis, a quantity p whose square is the reciprocal of f, and call its terminal coordi nates , 77, , this equation becomes by multiplying both sides by p 2 As the moment of inertia is essentially a positive quantity, this equation represents an ellipsoid. It must of course have three principal axes ; and, when these are taken as the coordinates axes, the terms in 17, i;, and in the above expression must disappear. 237. Hence at every point of every rigid body there Prim are three " principal axes " of inertia, at right angles to one axes another. One of them is the axis of absolute maximum in moment, another that of absolute minimum. Our equation now becomes, when referred to these axes, or, dividing by p 2 , Thus the moment of inertia about any axis is found from those about the principal axes at that point by multiplying each by the square of the corresponding direction cosine, and adding the results. For the quantity A was written originally as 2m(?/ 2 + z 2 ), i.e., it is the moment of inertia about the axis of or. "We see also that, at every point of a body, there are three rectangular axes such that the expressions ^(rnxij), 2(myz), ^(mzx) vanish when these are taken as coordinate axes. To find how these axes are distributed in a body, let us suppose Distri it referred to the principal axes through its centre of inertia, and tion o let M&j, M&:!, M&* be the moments of inertia about them. The princi quantities k lt k. 2 , k 3 are called the principal "radii of gyration." axes. Then, by the results above, the moment of inertia about a line Radiu A, p, v through the point a, /3, 7 is of gyi 1 - M { a 2 + & + 7 2 - (Aa For a principal axis this is to be a maximum or minimum, with
the sole condition