3I7-] BODY WITH FIXED POINT. 1? i 



a couple H on a rigid body at rest, with a fixed point, is to 

 impart to the body a rotation CD whose magnitude and axis can 

 be found by determining co x) co y) co z from (2). 



Any system of impulses acting on the body can be reduced, for 

 the fixed point (9 as origin, to a resultant R and a couple H ; the 

 effect of the couple has just been stated ; that of R consists merely 

 in producing a pressure on the fixed point. To find this pressure, 

 determine e>= Vo) x 2 -f u-\-w? from (2); the velocity of the cen- 

 troid can then be found and its components substituted in (i), 

 Art. 314. 



317. The axis / of the rotation produced by a given couple 

 H is not, in general, perpendicular to the plane of the couple. 

 Imagine the angular velocity co to be represented by its rotor, 

 i.e. by a length co laid off from O on the axis /, and the couple H 

 by its vector, i.e. by a length //'laid off from O on the perpen- 

 dicular to the plane of the couple. The relation between the 

 rotor co and the vector H producing it will best appear if we take 

 the axis of rotation / as axis of z. We then have x coy, 

 y = cox, z=o, and the momenta mcoy, mcox, o of the particles 

 reduce to a resultant and couple at O as follows. The resultant 

 momentum has the components : 



county = My co, co^mx = Mxco, O ; 



it is equal to Mco V^ 2 4- y 2 = Mar, where r is the distance of the 

 centroid from the axis /, and is perpendicular to the plane 

 through axis and centroid. The couple has the components 



oftmzx= Eco, u&myz = Deo, co^m (x* +y 2 ) = Ceo. 

 The equations (19) and (20) of Art. 238 reduce therefore to 



Mxco =R y +A y , o =R z +A g , (3) 

 -Eco =H X , -Dco = H y , Cco = H x . (4) 



These equations can also be derived directly from the equations 

 (i) and (2) above, since in the present case we have x= coy, 



