20 6 KINETICS OF A RIGID BODY. [369. 



To find v, let G' be the centroid of M + m, so that 

 GG' = mx } /(M+m), G'O' = Mx^(M-\- m). The principal axes of 

 M+m are G'x and the parallels G'y', G'z' to Gy, Gz. The impulse 

 mu at 0' imparts to the mass M+m (see Ex. (4)) a velocity of 

 translation mu/(M-\- m) parallel to G'y', and an angular velocity 

 <o = ;;/# G'O'/(M+ m)q l about 6V, ^ being the radius of inertia 

 otM+m for 6V. The velocity v of (9' is, therefore, 



mu i 

 ~~M+m 



For q we have the relation 



m)f = Mq? + M- GG' 2 + m 



where q^ is the radius of inertia of M for Gz. Substituting this value of 

 q, we find 



v=mu 2 * z> 



and hence 



)qf+ mx? 



It thus appears that F equals mu only in the limiting case when 

 m o, u = oo, whjle lim mu = const. For given values of m and */, F 

 is a maximum for x l = o; i.e. when w strikes the body M at the cen- 

 troid. In this case f= mMu/(Af+m), as it should be, since for 

 direct impact, we have 



F m(u v)=mu m- mu/(M+ m)= mu M/(M+ m). 



(6) A free rigid body turns with angular velocity o> about an instan- 

 taneous axis 1, which is parallel to a centroidal principal axis and meets 

 another centroidal principal axis at a distance GO = x from the cen- 

 troid G ( Fig. 54) . A point P of the body, situated on the principal axis 

 GO at the distance GP=x //-<?#* the centroid, strikes a fixed obstacle; 

 what is the reaction P of the obstacle 9 



The system of impulses to which the angular velocity o> is due reduces 

 to an impulse F= Mu = Mux through G, at right angles to the plane 

 (/, G), and a couple Fx^= Ff /x, where q is the radius of inertia for 

 the centroidal axis 7 parallel to /. The vector of this couple is parallel 

 to / (see Ex. (4) ) . Just after impact, we have, in addition, the im- 



