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Determination of Work done in Stretching Wire

Consider a wire as the system. If the length of the wire in which there is a tension F is changed from L to L+ dL, the infinitesimal amount of work that is done equal to 

dW = -F dL 




The minus sign used because a positive value of dL means an expression of  the wire, for which work must be done on the wire, i.e, negative  work. For a finite change a length, 

W = - Integral (2 to 1) FdL.

If  we limit the problem to within the elastic limit, where E is the modulus of elasticity, s is the stress, e is the strain, and A is cross - sectional area, then 

F = sA = EeA , since s/e = E 

de = dL / L 

dW = - FdL = -Ee AL de

W = - AeL Integral (2 to 1) e de =  - AEL ( e2^2 - e1^2 ) 

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