Question

Show that strain energy per unit volume of a strained wire is 1/2 (stress) (strain).

Answer 1

we know, Young's modulus is the ratio of stress to strain. e.g., Y = stress/strain.
if we assume a wire of length L, area of cross section A is fixed at one end and stretched by suspended a load M from the other end due to this extension of wire is x.

so, Young's modulus of wire , Y = FL/Ax
or, F = YAx/L
we know, workdone , W = F.dx
so, W = (YAx/L)dx
or, W = 1/2 × YAx²/L
W = 1/2 × {YAx/L} × x
W = 1/2 × F × x

hence, Work done = 1/2 × load × extension

now, potential energy = workdone by external applied force
= 1/2 × load × extension.
so, strain energy = 1/2 × load × extension
now, strain energy per unit volume = 1/2 × load × extension/volume
= 1/2 × load × extension/(cross section area × length)
= 1/2 × (load/cross section area) × (extension/length)
= 1/2 × stress × strain

hence, energy per unit volume = 1/2 × stress × strain

Answer 2

HEY DEAR ... ✌️

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=) When a wire is stretched , work is done against the inter - atomic forces . This work is stored in the wire in the form of elastic potential energy . Suppose , the length of wire is L and area of cross-section is A . Suppose , on applying a force F along the length of the wire , the length increases by x . Then ,

The young's modulus Y of the material of the wire is
Y = Stress / Strain = F L / A x

Thus , the force necessary to increase the length of wire by x is given by
F = YA / L x .

Therefore , in increasing the original length L to L + l , that is , from x = 0 , to X = l , the work done is

( Derivation is attached )


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HOPE , IT HELPS ... ✌️