4 8 



KINETICS OF A PARTICLE. 



[88. 



88. The name principle of areas is due to the kinematical 

 meaning of the left-hand member (comp. Part I., Arts. 227-232). 

 As xdyydx represents twice the infinitesimal sector described 

 by the radius vector of the point (x, y) during the element of 

 time, the quantity xdy/dtydx/dt is twice the sectorial velocity 

 about the origin. Introducing polar co-ordinates by putting 

 jr=r cos 6, y=rs'm 0, we have xdyydx^r^dQ, and denoting by 

 6 1 the sector described in the time /, 



dt dt y dt~ dt 



The kinematical meaning of equation (15), after dividing it 

 by m, can therefore be stated as follows : the time-rate of change 

 of twice the sectorial velocity about any point is equal to the 

 moment of the acceleration about the same point. 



89. The dynamical meaning of equation (15) appears by 

 considering that mdx]dt, mdy/dt are the components of the 



momentum mv of the moving 

 particle (Fig. 15). The prod- 

 uct mvp of the momentum and 

 its perpendicular distance from 

 the origin is called the moment 

 of momentum, or the angular 

 momentum, of the particle about 

 the origin. 



It appears from Fig. 1 5 that 

 we have 



r v dx 



Fig. 15. 



\ 



The angular momentum is evidently nothing but twice the 

 sectorial velocity multiplied by the mass, just as momentum is 

 linear velocity times mass. 



The dynamical meaning of equation (15) can therefore be 

 expressed as follows : the time-rate of change of angular momen- 



