KINETICS OF A RIGID BODY. [362. 



resultant momentum has the components o, Max, o ; it is 

 therefore perpendicular to the plane through axis and centroid. 

 Hence the resultant impulse R at O must be equal in magnitude 

 and direction to the momentum Mv = Mex of the centroid, due 

 to the rotation CD about the instantaneous axis 1. 



The resultant angular momentum is found just as in Art. 317 ; 



it is = o>V' 2 + Z? 2 -f- J %2 and has the components Eco along Ox, 

 Deo along Oy, and Ceo along the instantaneous axis Oz. 



It follows that a pure rotation of angular velocity co about 

 an axis 1 can be imparted to a free rigid body by the combined 

 action of an impulse R and an impulsive couple H. The im- 

 pulse R = M(ox is perpendicular to the plane (/, G), and passes 

 through the foot O of the perpendicular let fall from the 

 centroid G on the axis; it vanishes only when x= OG = o; i.e. 

 when the instantaneous axis passes through the centroid. The 

 remarks of Art. 318 apply to the couple without change. 



362. As mentioned above, it is often more convenient to 

 take the centroid G as origin for the reduction of the impulses. 

 To reduce the system of impulses R, H, determined in the j 

 preceding article, to G as origin and to parallel axes (Fig. 46), 



it is only necessary to apply R \ 

 and R at G\ we then have 

 the resultant impulse R = Mu>x 

 at G, and the couple formed by 

 / R at O, and -R at G. The 



- * - 1 moment of this couple is Rx 

 = Mo)x z ; its vector is parallel 



,^ - -> - ^ - -, to the instantaneous axis /, and 



* / ' can therefore be added alge- 



*/ . braically to H n , while H x and H y \ 



remain unchanged. Thus the j 



components of the resulting couple for the reduction to the j 

 centroid are H x =-E<* y H y =-Da>, H z =(C-M^ 2 )oo=Coo f 

 where C f = CMx 2 is the moment of inertia about the cen- \ 



