730 



PLANE MOTION. 



93 



;o that T=2Tra/v\ then, since P' describes the circle of radius 

 in the same time T, we have for the velocity of P' the expres- 

 >ion 2TTV/T, or substituting for T its value, v*/a, as above. 



172. . Simple harmonic motion is a rectilinear motion in which 

 he distance x of the moving point P x (Fig. 44) from a fixed 

 >rigin O in the line of motion is a simple harmonic function of 

 he time, i.e. a function of the form 



vhere a, o>, e are constants. 



If the positions P of a point moving uniformly in a circle be 

 )rojected at every instant on any diameter A A' of the circle, it 

 s easy to see that the motion 

 )f the projection P x along the 

 liameter is simply harmonic. For 

 lenoting the constant angular 

 elocity of P by &>, the angle 

 4OP will be = ft>/ if the time be 

 :ounted from the point A. Hence 

 he distance OP x =x of the point 

 P x from the centre O, or the dis- 

 )lacement of P x at the time t y is 



;r=tfCOSG)/, Fig. 44. 



vhere a is the radius of the circle. This radius a=OA is 

 :alled the amplitude of the simple harmonic motion. 



173. While P moves uniformly in the circle, its projection 

 evidently performs oscillations from A through O to A' and 

 3ack through O to A. 



The time T of completing one whole oscillation forward and 

 Backward is called the period of the simple harmonic motion ; 

 t is obviously equal to the period of the motion of P in the 

 :ircle ; i.e. 



T= 2 -^- (14) 



