MOTION ABOUT A CENTER OF FORCE 271 



This is the equation of an ellipse. 



Thus the most general motion possible for the particle consists 

 in describing the same ellipse over and over again. The period is 

 2 TT/P, this being the time required for cos pt and sin pt both to 

 repeat their values. 



215. The axes of x, y are still undetermined ; let us imagine 

 them to be the principal axes of the ellipse. 



Then if we suppose the time measured from one of the instants 

 at which the particle is at one of the extremities of the major 

 axis, we shall have equations of the form 



x = A cos pt^ 

 y = B sin pt. 



Thus pt is the eccentric angle of the ellipse described by the 

 particle, so that the eccentric angle increases with uniform angular 



nj 



velocity p or A The motion repeats itself as soon as p increases 



f/A Ira 



by 2 TT. Thus the frequency is p or -vl > while the period is 2 TT A| 



216. This motion is realized experimentally in the motion of 

 a pendulum which is not constrained to move in one vertical 

 plane, but of which the deviations from the vertical 



/*t 



remain small. 



Let the pendulum be of length a, and let its bob be 

 displaced from its equilibrium position to some near 

 point P, such that the angle PCO may be treated as 

 small. Calling this angle 6, the weight of the bob may 

 be resolved into mg cos along CP, which is exactly neu- 

 tralized by the tension of the string, and a force mg sin -Q 

 along PO. If 6 is small, ski 6 may be put. equal to 0, and FIG. 134 



OP 



this in turn to Thus the bob may be supposed to experience 



Ck 



a force- OP along OP. The motion is therefore of the kind 

 a/ 



which has been described, the value of //, being > and the value 



I d 



of p therefore being A) Thus we see that a hanging weight 



