Write down the expression for the elastic potential energy of a stretched wire .

Explanation

Consider a wire whose un-stretch length is L and area of cross section is A. Let a force produce an extenion l.
The work done by the force F is equal to the energy gained by the wire.
The work done in stretching the wire by dl, dW = F dl
The total work done in stretchng the wire from 0 to l is
`W = int_(0)^(1)` ....(1)
From Young's modulus of elasticity,
`Y = (F)/(A) xx (L)/(l) rArr (YAl)/(L)` ..(2)
Substituting equation (2) in equation (1) we get
`W = int_(0)^(1)(YAl)/(L)dl = (YAl^(2))/(L.2) = (1)/(2).Fl`
`W = int_(0)^(1) (YAl')/(L) dl' = (YA)/(L)((l'^(2))/(2))|_(0)^(l) = (YA)/(2)(l^(2))/(2) = (1)/(2)((YAl)/(L))t = (1)/(2)Fl`
`W = (1)/(2)Fl` = Elastic potential energy.