88 



KINETICS OF A PARTICLE. 



[168. 



cal circle, a constraining force F' (Fig. 23) must be intro- 



duced such that the resultant 

 R of F and F' shall produce 

 the acceleration required for 

 motion in the circle. Thus, for 

 instance, for uniform motion 

 in a circle the resulting ac- 

 celeration must be directed 

 towards the centre and must 

 be =a> 2 #, if a is the radius and 

 a) the constant angular veloc- 

 ity. We have, therefore, in 

 this case R mo^a along the 

 radius, F= mg vertically down- 



wards ; and hence, denoting by 9 the angle made by the radius 



CP with the vertical (Fig. 23), 



F' 2 = F 2 + R 2 + 2 FR cos 6 



a cos 6). 



The constraint F', which is thus seen to vary with the angle 

 6, can be resolved into a tangential component Ft and a normal 

 component F n '. As in our problem the velocity is to remain of 

 constant magnitude, the tangential constraint must just counter- 

 balance the tangential component F t = mgsin of gravity. The 

 normal constraint FJ not only counterbalances the normal com- 

 ponent F n =mgcos of gravity, but also furnishes the centrip- 

 etal force R = mco 2 a required for motion in the circle ; i.e. 



0). 



168. In the general case of a particle of mass m acted upon 

 by any given forces and constrained to any fixed curve, it is 

 convenient to resolve both the resultant Fof. the given forces 

 and the constraint F 1 along the tangent and the normal plane. 



