MOTION OF AN OSCILLATING ROD. 161 



The equations of motion are, for the rod, 



and for the ring 



m-jjr=mff—2> r V + t % —t 1 , 





Substituting from (13, 14, 15), we have 

 d z X X' 



m ~^ r=m ^"" 2 17( X ~f~ L ) + 2 7 (Xl ~ X ^ ' (l6) 



and for the ring 



t dt z =m l g-2j(X l -X). . . (17) 





Let x denote the displacement of the axis of the rod 

 from its position of rest, and x x the displacement of the 

 centre of the ring from its position of rest ; then 



X = x +L X + -, 

 2 



X 1 =a? I + L a . 



Substituting in equations (16) and (17), and cancelling 

 those terms which are in equilibrium by the statical equa- 

 tions, we obtain 



d z x X ! X X , _. 



m j^ =— 2j-# — 2jX + 2jX If . . . 418) 



d z x T XX , % 



m r^ = 2-x-2jx i; (19) 



by addition we have 



d z x d z x x X' , v 



