32 KINETICS OF A PARTICLE. [61. 



as no permanent deformation is produced) , directly proportional to the 

 extension or change of length produced. 



Thus, let an elastic string, whose natural length is /, assume the 

 length s when its tension is T\ then Hooke's law can be expressed in 

 the form 



where k is a constant. To determine this constant for a given string, 

 we may observe the length / x assumed by the string under a known 

 tension, say the tension TI = mg, produced by suspending a given mass 

 m from the string (the weight of the string itself being neglected). We 

 then have 



71 =(/i-/). 

 Hence, dividing, 



T _ s - I 

 mg 4 - / 



or, denoting by e the extension l l due to the weight mg, 



T="*f(s-l). (16) 



61. By means of this relation we can determine the motion of a 

 particle of mass m attached to a fixed point O by means of an elastic 

 string, if the string be stretched and then let go. We shall assume the 

 particle and string to lie on a smooth horizontal table, so as to eliminate 

 the effect of the weight of the particle. 



The equation of motion is 



whence, putting for shortness ^Jg/e = K, 



s = 1+ C\ cos K/ + C 2 sin */, 



If the initial length of the string at the time /= o be s , the constant 

 .are readily determined, and we find 



s = /-f (.r /) cos K/, 



v = K (s /) sin*/. (18) 



