208 KINETICS OF A RIGID BODY. [371. 



371. Analytically, the continuous motion of a free rigid body 

 is given by the six equations of motion, (4) or (5), Art. 223, and 

 (6) or (7), Art 224. As pointed out in Art. 233, the motion of 

 the centroid and the motion of the body about the centroid can 

 be considered separately. The former is given by the equations 

 (8), Art. 226, viz. : 



Mx = R xy MJ = R y , Mz=R (i) 



where M is the mass of the body ; x, j>, z are the components 

 of the accelerations of the centroid along any three fixed rec- 

 tangular axes ; and R m R y , R z are the components along the 

 same axes of the resultant R of all the external forces acting 

 on the body. 



The motion of the body about the centroid is the same as if 

 the centroid were fixed (Art. 229). It is therefore best studied 

 by taking the centroid G as origin ; all the developments of 

 Arts. 324-356 will then apply without change, except that the 

 centroid G must be substituted for the fixed point O. The 

 general equations (3), Art. 326, or Euler's equations (4), Art. 

 328, can be used to determine the motion about the centroid. 



The integration of Euler's equations gives the angular 

 velocities a> lt <0 2 , < 3 about the three centroidal principal axes of 

 the body. The position of the body, i.e. the relation of this 

 system of principal axes to a system of axes through the cen- 

 troid, parallel to a fixed system, can be determined by means of 

 Euler's angles 6, <, ty (see Arts. 333-335), or by means of the 

 9 cosines a v a 2 , a B , b lt b^ 3 , c lt c^ C B (Arts. 336, 337). 



372. Kinetic Energy. As the instantaneous motion consists 

 of an angular velocity o> about the instantaneous axis / and a 

 velocity of translation u along this axis, the velocity v of any 

 point of the body, at the distance r from /, is ?; = V 2 +a>V 2 . 

 Hence the kinetic energy (comp. Art. 235) has the expression 



o> 



) = 



