KINEMATICS. 



[178. 



it appears that the motion of P y is simply harmonic of the same 

 period and amplitude with the motion of P x , but differing by 

 -7T/2 in phase. 



178. Simple harmonic motions occur very frequently in 

 applied mechanics and mathematical physics. A particular 

 case has been treated in Arts. 121-124. As another example 

 we may mention the apparent motion of a satellite about its 

 primary as seen from any point in the plane of the motion, pro- 

 vided the satellite be regarded as moving uniformly in a circle 

 relatively to its primary. Thus the moons of Jupiter, as seen 

 from the earth, have approximately a simple harmonic motion. 



179. A mechanism for producing simple harmonic motion 

 can readily be constructed as follows. The end A (Fig. 45) 



of a crank rotating uniformly 

 about the axis O, carries a pin 

 running in the slot AB of a T 

 bar ABD whose axis (produced) j 

 passes through the centre O of! 

 the crank circle. The T bar is 

 constrained by guides to move 

 back and forth along the line OD ; its motion is evidently simply! 

 harmonic, the uniform circular motion of the crank being trans-l 

 formed into rectilinear motion. Compare Art. 151. 



180. Exercises. 



(i) Show that the maximum acceleration of the simple harmonic 

 motion is numerically equal to the acceleration in the corresponding 

 uniform circular motion. 



. (2) Find the time of one oscillation from equation (15) without 

 reference to the circular motion. 



(3) In the mechanism shown in Fig. 45, if the length of the crank 

 2 feet and the number of revolutions 15 per minute, find the velocity^ 

 .and acceleration of the end D of the T bar : (a) when at elongation ; 

 .() when at quarter stroke ; (<r) when at the middle of the stroke. 



