MOTION OF AN OSCILLATING ROD. 165 



Hence the equation of motion of the rod takes the form 

 V ( / z /A/ \\\ sin pt 



-( m r- 2 {L + -i))-r\- 



Since q is greater than p, and mq z — 2 1 — +- J is posi- 

 tive, and mp z — 2 I f- + y) is negative, it follows that both 



R and S are positive quantities. The motion of the rod 

 is therefore made up of two harmonic motions of unequal 

 periods and unequal amplitudes. 



To determine the motion of the ring differentiate the 

 last equation twice, and substitute in (18); we then 

 obtain 



Yl 



2\m(q 2 



sinpt sin qt 



p 



By some substitutions this equation may be represented 

 in the following simpler form : — 



2VI /sinpt sin qt\ 

 I — mj(q z —p z )\ p q J 



If we differentiate this equation we find that the ring 

 .11 be a 

 filled :— 



will be at rest whenever the following condition is ful- 



cos qt — cospt=o. 

 This condition is satisfied when 



2717T 



q+p 



