A wire fixed at the upper end stretches by length l by applying a force F. The work done in stretching is

Explanation

Suppoese a wire of length L, area of cross-section A and radius r is suspended from a rigid support. Let the free end of wire be loaded by a weight F=mg and l be the elongation of wire by appllying load.
So, `L+l` be the length of wire.
Young's modulus `=("Longitudinal stress")/("Longitudinal strain")`
`Y=(F//A)/(l//L)=(FL)/(Al)`
`F=(YAl)/(L)" "..........(i)`
Let, f be the restroing force and x be the conrresponding extension. Then,
`f=(YAx)/(L)" ".......(ii)`
Let, dW be the work done for further small extension dx
dW=fdx
`dW=((YAx)/(L))dx`
[using equation (ii)]
Total amount of work done in stretching the wire from x=0 to x=l can be calculated by integrating dW.
`:.W=int_(0)^(l)dW`
`W=int_(0)^(l)(YAx)/(L)dx=(YA)/(L)int_(0)^(l)xdx`
`W=(YA)/(L)[(x^(2))/(2)]_(0)^(l)`
`W=(YA)/(L)[(l^(2))/(2)-(0^(2))/(2)]`
`W=(YAl^(2))/(2L)`
`W=(1)/(2)((YAl^(2))/(L))(l)`
`W=(1)/(2)F.l" "("":'(YAl)/(L)=F)`
which represents the work done by stretching force on a wire.