2 9 7-] 



BODY WITH FIXED AXIS. 



159 



The kinetic energy is given by (2). The work of a force F in 

 a plane perpendicular to the axis, at the distance/ from the axis, 

 is F-pdO for an infinitesimal rotation of angle dO ; hence, the sum 

 of the elementary works of all the forces = 



296. While thus the motion of a rigid body about a fixed axis 

 is given by a single equation, the other equations of motion of a 

 rigid body are required to determine the reactions of the fixed axis 

 (comp. Part II., Art. 227). 



The axis will be fixed if any two of its points A, B are 

 fixed. The reaction of the fixed point A can be resolved into 

 three components A x , A y , A g , that of B 

 into B x , B y , B x . By introducing these re- 

 actions the body becomes free ; and the 

 system composed of these reactions, of the 

 external forces, and of the reversed effec- 

 tive forces must be in equilibrium. We 

 take again the axis of rotation as axis of z 

 (Fig. 34) so that the ^-co-ordinates of the 

 particles are constant, and hence z=o, 

 i? = o; and we put OA=a, OB=b. Then 

 the six equations of motion are (see Art. 

 223 (4) and Art. 224 (6)) : 



Fig. 34. 





^m (xy yx) = *Z(x YyX). 



297. It remains to introduce into these equations the values 

 for x, y. As the motion is a pure rotation, we have (see Part I., 

 Art. 245) x= <oy, y = a*x\ hence, x= u>y aPx, y=zwxaPy. 

 Summing over the whole body, we find 



^mx = &) %my <tF%mx = Mwy May^x, 

 '= MwxMw*y, 



