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^ji (t - t )}, 

 and this can be expressed in the form* 



A cos (t VfO + B sin (t V/A). 



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

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

 so must be given by an equation of the form 



x A cos (t /V/M) + B sin 



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



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

 we have 



x = - A \/JL sin t VA + B*j, cos 



Now put t = and we find 



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

 that x = # and a? = # when t = 0, is 



x = # cos ^/-t + -7 sn 



V/4 



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

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

 2?r 

 >' 



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

 motion with period 2?r/\//A, in this form a is called the amplitude 

 of the motion, it is the greatest value of x, and e determines the 

 phase of the motion. 



43. Composition of simple harmonic motions. We 



consider the case where the moving point has a simple harmonic 



2_ 

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



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. 



I, 4 



