47, 48] SIMPLE HARMONIC MOTION. 49 



It follows, by changing the epoch from which time is measured, 

 that the complete solution must be of the form 



a cos y/j, (t - )}, 



and this can be expressed in the form* 



A cos (t *Jfj) + B sin (t V/-0- 



Let the moving point have at time t = a position denoted 

 by # and a velocity denoted by # ; we know, that at any time t, 

 x must be given by an equation of the form 



x = A cos (t VA&) + B sin (t V/*)- 

 To determine the constant A put t 0, we have a? = A. 



To determine the constant B, differentiate with respect to t, 

 we have 



x = A tJn sin (t V* + B *j, cos 



Now put = and we find 



XQ 



Hence the solution of the equation x = /*#, with the conditions 

 that x = XQ and x = X Q when t = 0, is 



x = # cos i A + -^- sin 



It is to be observed that the whole motion is periodic, that 

 is repeats itself after equal intervals of time ; the period is 



The equation x = a cos (t \l p e) represents simple harmonic 

 motion with period ^TT/^/JL, in this form a is called the amplitude 

 of the motion, it is the greatest value of a?, and e determines the 

 phase of the motion. 



48. Composition of simple harmonic motions. We 



consider the case where the moving point has a simple harmonic 



motion of period -y- parallel to each of the axes x and y, the 

 V/4 



acceleration in each case being directed towards the origin. 



* The student who is acquainted with the methods of solving linear differential 

 equations will recognize that this is the known form of the complete primitive of 

 the differential equation. 



T, 4 



