CHAPTER 22 



The Effects of Constancy 



22/1. A variable behaves as a step-function over some given 

 period of observation if it changes value at only a finite number of 

 discrete instants, at which it changes value instantaneously and 

 by a finite jump. 



22/2. An example of a step-function in a system will be given 

 to establish the main properties. 



Suppose a mass m hangs downwards suspended on a massless 

 strand of elastic. If the elastic is stretched too far it will break 

 and the mass will fall. Let the elastic pull with a force of k 

 dynes for each centimeter increase from its unstretched length, 

 and, for simplicity, assume that it exerts an opposite force when 

 compressed. Let x, the position of the mass, be measured verti- 

 cally downwards, taking as zero the position of the elastic when 

 there is no mass. 



If the mass is started from a position vertically above or below 

 the point of rest, the movement will be given by the equation 



j t { m T) =gm - kx • ■ • (1) 



where g is the acceleration due to gravity. This equation is not 

 in canonical form, but may be made so by writing x = x v 

 dx/dt = x 2 , when it becomes 

 dx ± 



dx» k 



dt 6 m 1 



(2) 



If the elastic breaks, k becomes 0, and the equations become 



dx x _ 



j- 



dXo 



Assume that the elastic breaks if it is pulled longer than X. 



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