367-] 



INITIAL MOTION OF FREE BODY. 



201 



ducing pure rotation about the instantaneous axis / provided 

 it satisfies the condition 



just mentioned. This can j 



be done, if R and H are T 



known for the centroid, by 

 transferring R to parallel po- 

 sitions so as to reduce the 

 components H n and H x to 

 zero. Thus, to destroy 

 H z =Cco we have only to 

 transfer R from G along 

 the axis of x to a point O 1 

 (Fig. 47) at a distance 



Fig. 47. 



'=.^ from G such that xx l = q^ J where ~x=OG, and ^ is the 

 radius of inertia for the centroidal instantaneous axis 7. For 

 then the couple resulting from the transfer has a vector along 

 the axis of z equal to Rx^ Ma)xx l = M^ H g . 



Next to destroy H x Ea> = a&mzx, we transfer the point 

 of application of R parallel to the instantaneous axis / to a point 

 O lt at a distance O 1 O l =z l from O', such that Rz l =H xt 

 whence z = ^mzx/Mx. 



If the point O 1 be taken as origin of reduction, the system 

 of impulses reduces to R at O v and the couple // = //.= )&, 

 whose vector is parallel to R. Thus the central axis, which has 

 of course the direction of R, and is therefore perpendicular to 

 the plane (/, G), meets this plane at a point O x whose co-ordi- 

 nates are x^ q^/'x^ z l = E/mx. It is easy to see that these 

 results agree with the developments of Arts. 309, 310, the 

 centre of percussion, if it exists, being situated on the central 

 axis. 



367. Twist or Screw Motion. In the most general case the 

 motion of a rigid body consists of an angular velocity o> about 

 the instantaneous axis /and a simultaneous velocity of transla- 

 tion a along this axis (Art. 357). 



