GENERAL PRINCIPLES 5 



show that, finally, the value <o/ is expressed by : <o/ = ; so 

 that the law of a harmonic motion becomes : 

 s = a sin ^~- 



The instantaneous velocity attains its greatest value at the 

 moment of departure and diminishes to the end of the swing ; 

 from M 1 to M it accelerates gradually and decreases progressively 

 from M to M 2 . It follows that the acceleration always tends to 

 bring the moving body back to its initial position M at the centre 

 of the oscillation. 



Using the expression -=r, for the instantaneous velocity v we 

 At 



have : 



V = - COS 27TT 



its value at departure is the maximum, because <t = O, 

 cos 2n = 1 ; therefore v = -~- 



The expression gives for the acceleration, the formula 

 y~ 



J. JL 



Familiar examples of these periodic movements are afforded 

 by the moving elements of the plane, the saw, the piston, etc. 

 Thus with the sawyer and the filer, the speed of the tool falls 

 to zero at the end of each stroke. Whatever may .be the 

 movement, it obeys a more or less complex law connecting 

 space and time, the only elements considered in kinetics. In 

 short, every movement can be expressed by an equation. 



3. Representation and Registration of Movement. Given the 



path of a moving point, we know that it can be either rectilineal 

 or curvilineal (R or C). In the former case, the straight line 

 represents a path of movement, XX', the motive power being 

 able to cause movement either in the direction XX' or in the 

 opposite direction X'X. A rectilineal trajectory has therefore a 

 path and two opposite directions of movement (fig. 5). 



FIG. 5. 

 X' 



If the speed is MM' in a second it can be represented by the 

 straight line MM', provided that the point has moved from M 

 to M' with a uniform speed. 



