PARTICLE EXECUTING SIMPLE HARMONIC MOTION 



255 



Displacement of a Particle Executing Simple Harmonic Motion 



Let Pi in Fig. VII-1 be a particle on a reference circle of radius A 

 moving with uniform speed. The projected position on the vertical axis 

 is wij. If Pi now moves with uniform speed around the circle to succes- 



P36 

 1 



A/ i \ 



/ y 



i \ 

 i 1 



A 



.0 — *— -k— l— i--4p 



\ / 



\ / 



Q 



■-o 



' p l 



m 





Fig. VII-1. 



sive positions P 2 , P3, P4, etc., m\ will assume comparable positions 

 m 2 , w 3 , m 4 , etc., which are the projected positions of P on a diameter of 

 the reference circle. The point m is executing simple harmonic motion. 

 During one complete revolution of P, m will sweep across and back over 

 a diameter taken as the y axis. The frequency with which this sweep 

 occurs is equal to the number of revolutions of P per second. The time 

 in seconds taken to make a complete revolution, or the completed sweep 

 excursion along the diameter, is called the period. The angular displace- 

 ment of P in one revolution is 2w radians. If the angular velocity is co 

 and the period of revolution is T seconds, then 



:tt 



COu 



The sweep motion of m is the simple harmonic counterpart of the 

 circular motion of P, and its displacement at any instant is 



y = A sin 6 



In the reference circle any angular displacement 6 = cot. 



. 2irt 

 y = A sin — - 



Thus 



If this simple harmonic motion executed by m on the y axis were 

 simulated by the point of a pencil held at right angles to the paper, and 

 while doing this the hand is moved to the right with constant speed, you 

 could generate the sine curve of Fig. VII-1 graphically. To produce this 



