464 KENNELLY AND AFFEL. 



Appendix I. 



Elementary Theory of Simple Vibration. 



We may suppose a material particle of mass m grams at the point P,. 

 Figure 24, to rotate in a circular orbit, and in the positive or counter- 

 clockwise direction, about the center O, with uniform angular velocity 

 CO radians per second, as indicated by the arrow. If the radius of the 

 orbit is x cms., we may assume that there is a constant force sx dynes 

 in the direction PO, along the radius vector, so that this centripetal 

 force is proportional to the displacement x from the center O. The 

 vector displacement, measured positively outwards from O at time t 

 seconds, is 



X = .To e^"' cm Z (12) 



the epoch being selected such that .Tq = OP, when i = 0. 



The instantaneous velocity of the particle will be directed along 

 the tangent PP^ at P, and its vector value will be : 



X = jco.ro e^"' = jco.r cm/sec Z (13) 



The instantaneous acceleration of the particle will also be directed 

 along the tangent QQ^ at Q, a point 90° advanced in phase beyond P, 

 and 



.V = {juf .To e^"' = - co^.r cm/sec2 Z (14) 



The forces acting on the particle at any instant, such as that indicated 



s 

 in Figure 25, will then be (1) the elastically restoring force —sx=jx 



CO 



dynes in the direction PO, or opposite to the direction of x. (2) the 

 force opposing the velocity, or —rx dynes acting in the direction 

 PT or OC, assuming that this force acts in simple proportion to the 

 instantaneous velocity, and (3) the force opposing acceleration, or 

 inertia force, acting in the direction Q^Q, or OB = — mx = —jmcox 

 dynes. The vector sum of these forces, in conjunction with a rotatory 

 impressed force F dynes along OD, which sustains the motion, must be 

 zero. 



The reactive forces (1) and (3) must be in mutual opposition at any 



instant, because (1) is j x, and the other is — jnmx. Their relative 



CO 



magnitudes, however, depend upon the value of the angular velocity 



