230 MOTION OF SYSTEMS OF PAETICLES 



186. THEOREM. The kinetic energy of any system of moving par- 

 ticles is equal to the kinetic energy of motion relative to the center 

 of gravity of the particles, plus the kinetic energy of a single par- 

 ticle of mass equal to the aggregate mass of the system, moving 

 with the center of gravity. 



Let the particles be m l at x ly y l} z lt etc., and let the coordinates 

 be measured with the center of gravity taken as origin. Let the 

 velocities be denoted by u lf v lt w lt etc., and let these also be meas- 

 ured relative to a frame moving with the center of gravity, so that 



dx. 

 u, = ^, etc. 



Let the velocity of the center of gravity referred to any frame 

 of reference, moving or fixed (provided only that the directions of 

 the axes do not turn), have components u, v, w. Then the velocity 

 of the particle m^ is compounded of the velocity of the particle, rel- 

 ative to the center of gravity of the system, of components u lt v lt 

 w v together with the velocity of the center of gravity, of com- 

 ponents u, v, w. Thus the whole velocity of the particle m l has 



components 



u + u lt v -h v lt w + w r 



The kinetic energy of the first particle is accordingly 



1 m l [(u + ^) 2 + (v + vtf + (w + wtf\, 

 so that the kinetic energy of the system is 



[(u + u) 2 + (v + v) 2 + (w + wf}> 

 or, on expanding squares, 



Since, when the center of gravity is taken as origin, the coor- 

 dinates of the particles are x l) y l) z l) etc., we have, by equations (8), 



