Question

derive an expression for the energy stored in a stretched wire

Answer 1

Let a wire of length l, cross sectional area A and Young's modulus of elasticity of wire is Y.
if F force applied on wire, wire stretched x .

now from formula, Y = Fl/Ax
F= YAx/l

now, workdone by force on wire = F.dx
= YAx/l dx
= (YA/l) xdx
= (YA/l) × x²/2
= (YAx/l) × x/2

put F = YAx/l

so, workdone on wire = 1/2 × F × x

we know, energy stored in the stretched wire = workdone by force on it.
= 1/2 × F × x

this is the expression of energy stored in wire , where F is force applied on wire and x is extension of wire.