Dadourian — Progressive Development of Mechanics. 167 



where F, Ft, and F„. denote, respectively, the magnitudes of 

 the resultant force and of its tangential and normal components. 



(4) Motion of rotation. Here the concept of angular kin- 

 etic reaction is introduced as the counterpart of torques, which 

 come into play whenever a body is given an angular accelera- 

 tion. The quantitative definition of this new type of action is 

 obtained in a way quite analogous to the manner in which the 

 measure of the linear kinetic reaction was obtained. 



Consider a flywheel, which is perfectly balanced and per- 

 fectly free to rotate about a horizontal axis. In order to start 

 the flywheel in motion or to stop it a torque must be applied. 

 Therefore the flywheel experiences an angular kinetic reac- 

 tion when it is given an angular acceleration. If the angular 

 acceleration of the flywheel is observed at different instants 

 and simultaneous observations are made of an arrangement 

 which measures the kinetic reaction it will be found that the 

 following relations hold : 



RJ RJ' RJ" 



= I, (8) 



where <w denotes the angular acceleration, S a the correspond- 

 ing kinetic reaction, and I a constant which is called moment 

 of inertia or angular inertia and which plays a role analogous 

 to that played by the mass of a body in motion of translation. 

 Thus the angular kinetic reaction of a body rotating about a 

 given axis is proportional to the angular acceleration. 



If, on the other hand, equal angular accelerations are given to 

 different bodies or to the same body about different axes and 

 the moments of inertia determined in the foregoing manner 



the following relations will be found 



R a ' __ RJ' __ R a " 



t == i" ~r T7 



(9) 



where co denotes the common angular acceleration. There- 

 fore, the angular kinetic reactions of different bodies, or of the 

 same body relative to different axes, are proportional to the 

 corresponding moments of inertia. It follows therefore that 

 the following relation defines the angular kinetic reaction : 



R a =-Iu, (10) 



where the negative sign is introduced to take into account the 

 fact that R a and a> are oppositely directed. 



The angular kinetic reaction and torques are the only types 

 of angular actions which come into play ; therefore the second 

 section of the action principle states : 



