Y - OSCILLATIONS AND CYCLIC MOTIONS. 



axes pivoted in bearings fastened to bodies themselves in motion. 

 Such motions will be treated in 94. 



A very simple case of the above process is encountered in 

 treating the motion of a particle m sliding on a horizontal rod, 

 revolving about a vertical axis, at a distance r from the axis. Let 

 the angle made by the rod with a fixed horizontal line be cp, then 

 the velocity perpendicular to the rod is rep'. The velocity along the 

 rod being r f , the kinetic energy of the body m is 



146) T=~ 



Since (p does not appear in T, (p is a cyclic coordinate. If there is 

 no force tending to change the angle y> we have 



dT 



147) P(p = = mr*<p' = c, 



from which we obtain 



148 ) f' = ^ 



to eliminate cp'. Thus we get 



149) 

 Supposing that there is no potential energy we have 



150) 



illustrating the general property of d> mentioned, ~mr' 2 being the 



quadratic function of the remaining velocity r' and ^ g being 



the quadratic function of the constant c, which contains as a coeffi- 

 cient a function of the coordinate r. We may now, ignoring the 

 coordinate <p, use the differential equation for r, 



d 



d (V$\ 

 dt(dr') 



-5 > or 

 dr 



1R1\ r $ c 



m W = ^ = ^r*' 



We accordingly see that the system acts as if, there being no rotation, 

 it possessed an amount of potential energy C&, producing the force 



s>2 



^3 directed from the center. This example accordingly illustrates 



the effect of ignored cyclic motions in producing an apparent potential 

 energy, but it does not illustrate the effect of linear terms in <p, for 

 they disappear in this example, which is chosen on account of its 

 very simplicity. The example hardly seems to illustrate the case of 

 concealed motions, for the fact of there being a rotation cp r could 

 with difficulty be concealed. Nevertheless this is exactly what 



