Question

show work done per unit volume in stretching a wire is 1/2 ×stress×strain

Answer 1

hope this will help u .........

Answer 2

To prove:

The work done per unit volume in stretching a wire is 1/2 X stress X strain

Proof:

Let a wire of original length L and the area of cross-section A and Young's modulus Y be extended using a force F such that the new length becomes x.

According to the formula for Young's Modulus,

Y = FL / Ax

or F = YAx / L    - (1)

Let dx be a further small extension in the length of the wire.

Then, the work done dW required in the extension gets stored in the wire as potential energy.

We know that dW = F dx

or dW = (YAx / L) dx

Since Y, A and L are constant for a given wire.

Integrating both sides,

$\int\ {dW}=\frac{YA}{L} \int\ {x} \, dx$

Putting limits on both sides,

$\int\limits^W_0 {dW} = \frac{YA}{L} \int\limits^x_0 {x} \, dx$

or W = YA / L (x²/2)

From equation 1,

W = Fx/2

Multiplying and dividing by A.L

W = $\frac{1}{2} X \frac{F}{A} X \frac{x}{L} X AL$

Substituting A.L = Volume V, F/A = Stress and x/L = strain we get:

W = 1/2 X Stress X Strain X Volume

oe W/V = 1/2 X Stress X Strain

or Work Done per unit volume = 1/2 X Stress X Strain

∴Hence, proved