E L A 



ELASTIC bodies, equilibrium of.(Whewell.) 



This subject may be comprised under three heads. (1.) Elasticity of 

 Extension and Compression, as in the case of a string stretched by a 

 force. (2.) Elasticity of Flexure, as when wires and laminae of different 

 metals and other substances exert a force to unbend themselves when 

 forcibly bent. (3.) The Elasticity of Torsion, as when twisted threads 

 of metal exert a force to untwist themselves. Our view of these several 

 subjects must necessarily be very limited and imperfect. 



1. Elasticity of Extension. 



1. When an elastic string of given length is stretched by a given force, 

 to find its length. 



The increase of length is proportional to the tension. Let i be the 

 measure of the extensibility of the string, whose length at first is a j t 

 the force or weight with which the string is stretched, which of course 

 measures the tension ; then the increase of length = a it, and the length 

 I when stretched will .'. be 



a -}- a it, or a (1 + i f) 



We may determine i, if we know the original length of the string, and 

 its length for any given value of t. It may be convenient to know it in 

 terms of the force which will draw out the string to double its length. 

 Let E be this force j hence 



a (1 -f- i E) = 2 a, and t = -g. 

 Hence the length of the string under a tension t becomes 



=(>+*) 



E may be expressed by a length of the given string, whose weight 

 would draw the string a to double its length. E is then called the mo- 

 dulus of elasticity. 



2. A uniform elastic string hangs vertically, stretched by its own 

 weight .* to find its length. 

 The same notation being retained, 



Cor. 1. I 



Cor. 2. Since I a f 1 + ~-\ , it appears that the weight of the 



string stretches it half as much, as if it were ail collected at the lowe*t 

 point 

 98 



