132 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



where the two factors are the tension in the stretched state and the increase 



of length. 



Referring to the definition in Article 138 we see that the potential energy 

 of the stretched string is Xe 2 s , when the extension is uniform, and is 

 i\j"f 2 cfo in any case, the standard state being that in which the string has its 

 natural length. This potential energy is equal to the work that would be 

 done by the internal forces between the parts of the string if the string were 

 allowed to contract freely to its natural length. 



148. Spring. In the same way, for a spring whose mass is neglected, 

 whether the spring is extended or contracted, the potential energy when the 



length is I is X ' ~ - , where 1 Q is the natural length, and X the modulus of 



^o 

 elasticity. 



149. Localisation of Potential Energy. The potential energy of a 

 gravitating system and the potential energy of a stretched string are two 

 examples of the potential energy that arises from internal forces between the 

 parts of a system. 



But the two cases present a marked difference. In the case of the string 

 we are able definitely to assign a certain amount of the potential energy to 

 every element of the string, that amount depending only on the state of the 

 element. We can do this in such a way that every change in the energy so 

 assigned to the element corresponds to a definite change in the state of that 

 element. We may therefore say that the energy is located in the element. 

 The amount so located in an infinitesimal element is %\6 2 ds Q . We can think 

 of it as possessed by the element, just as kinetic energy is possessed by a 

 particle. 



In the case of the gravitating system we are not able to assign a certain 

 amount of the potential energy to any part of the system, in such a way that 

 changes of the energy so assigned correspond to changes in the state of that 

 part, independently of changes in the position of the part relative to other 

 parts. We cannot, in any way that shall be completely satisfactory, locate 

 some portion of the energy in one part of the system, another portion in 

 another part of the system, and so on. For instance in the case of the heavy 

 body near the Earth's surface we cannot locate the energy in the body, nor in 

 the Earth, nor in any definite proportion some of it in the body and some in 

 the Earth. We have to think of it as possessed by the system, not by the 

 bodies composing the system. 



It is however frequently convenient, when discussing the motion of a body 

 in a conservative field, to attach, in imagination, the potential energy of the 

 system to the body, and when we do this we shall speak of the energy as 

 " potential energy of the body in the field." For example, the potential of a 

 gravitating system at a point, with its sign changed, is the potential energy of 

 a unit mass at the point in the field of the gravitating system. In the case of 

 a heavy body near the Earth's surface the expression Mgh of Article 139 is 

 the potential energy of the body in the field of the Earth's gravitation, when 

 its centre of inertia is at a height h above a particular horizontal plane. 



