38 KINETICS OF A PARTICLE. [67. 



III. Free Curvilinear Motion. 



I. GENERAL PRINCIPLES. 



67. Let j be the acceleration of a particle of mass m at the 

 time /; Fthe resultant of all the forces acting on the particle ; 

 then its equation of motion is (Art. 35) 



mj=F. 



In curvilinear motion (Fig. 12) the direction of j and F differs 

 from the direction of the velocity v ; and the angle i/r between/ 

 and v varies in general in the course 

 of time. As shown in kinematics (Part 

 L, Art. 159), the acceleration can be 

 resolved into a tangential component 

 j t =zdv/dt=*d?s/dP and a normal com- 

 ponent j n = iP/p t where p is the radius 

 of curvature of the path. Hence, if the 

 resultant force F which has the direc- 

 tion of j be resolved into a tangential 



force F t = Fcos-*lr, and a normal force F n = Fsm^ t the above 

 equation of motion will be replaced by the following two equa- 

 tions : 



m%=F m v - = F n . (i) 



dt p 



In the particular case when the normal component F n is con- 

 stantly directed towards a fixed point it is called centripetal 

 force. 



68. The formulae (i) show how the force .F affects the veloc- 

 ity of the particle and the curvature of the path. The change 

 of the magnitude of the velocity is due to the tangential force 

 F t alone. If this component be zero, i.e. if the resultant force 

 F be constantly normal to the path, the velocity v will remain 

 of constant magnitude. The curvature of the path, i/p, and 



