166 Dadourian — Progressive Development of Mechanics. 



which corresponds to a common transverse acceleration f. 

 Thus the transverse kinetic reactions of different particles 

 having equal transverse accelerations are proportional to their 

 transverse masses. 



It follows, therefore, that the quantitative definition of the 

 transverse kinetic reaction is given by the relation 



R n = - m n f„ , (6) 



where the subscripts are introduced in order to emphasize the 

 fact that the magnitudes involved in (6) relate to the normal, 

 or transverse, direction. 



When the same system of units are used to measure 

 i?r, R n , f and f m it will be found that m- and m n are 

 equal. Therefore, when a particle has a transverse as well as a 

 longitudinal acceleration, the total linear kinetic reaction is the 

 sum of R and R re . Thus, 



R, = R r + R,, 



= - m(f T + t) 



/. v* 



= — m I v T 



\ ' P 



= — m\ , (V) 



where v denotes velocity, Vr the tangential component of 

 the acceleration, and v the total acceleration of the particle; 

 while p denotes the radius of curvature of the path, measured 

 from the center of curvature. 



The linear kinetic reaction and forces are the only types of 

 linear action which a body experiences. Therefore, the first 

 section of the action principle leads to the relations : 



2F; + R; = , 



or F = — Rz 



= mv , (HI) 



where F denotes the resultant force. 



The last relation, which is called force equation, is then 

 resolved into two component equations which correspond to 

 the directions parallel to the tangent and to the normal to the 

 path of the particle. Thus 



v 



and F = m\/ v * + -^ , 



P 



