SIMPLE HARMONIC MOTION 355 



radius making an angle nt with OQ (Fig. 117), then Qj the pro- 

 jection of Q on the horizontal diameter, is such that 



f i 



a 2 + sn e 



Q! is evidently the initial point. 



Pj, the projection of P on the horizontal diameter, is such that 



BSPi V a *+ (~) sin (n/ + e), which satisfies the condition 

 v/t/ 



V/j, \ 2 ffiy> 



a? -f ( -2J sin (nZ + e) , and consequently -JT? = n 2 a;. 



Here again the motion of the body is the horizontal projection 

 of the motion of a particle describing a circular path of radius 



jP\2 



) with uniform angular velocity n radians per second. 



Periodic time = 

 n 



ft 



Frequency = - 



Amplitude = Va 2 + ( 



The angle is spoken of as the Epoch of the simple harmonic 

 motion. 



179. The resultant of two simple harmonic motions of the same 

 period and in the same straight line is a simple harmonic motion. 



Let #! = Aj sin (nt + ej 



and # a = A 2 sin (nt + e z ) 



be the two simple harmonic motions. 



Then x + ;r 2 A. l sin (nt + e x ) + A 2 sin (nt f e 2 ) is the resultant. 



Let OR be drawn making an angle nt with the vertical line 

 OY (Fig. 118). Let OPj^ and OP 2 make angles l and e 2 respec- 

 tively with OR ; also OP a = A x and OP 2 = A 2 . Complete the 

 parallelogram, of which OP X and OP 2 are adjacent sides, OP 

 being the diagonal of this parallelogram and Q the horizontal 

 projection of P. 



Then OQ = OQ 2 + Q 2 Q 



= A 2 sin (nt + e 2 ) 4- A x sin (nt + ej 



