2. A certain wire of length L, cross-sectional area A is extended by an amount ‘e’by a force F. Given that the young’s modulus of the material of the wire is E,

Show that the;

a. Energy stored in the wire is given by the expression
(5)

b. Energy per unit volume, Ev = x (strain) 2

To derive the expressions for the energy stored in the wire and the energy per unit volume, we need to understand the basic concepts of strain and stress in materials.

1. Strain:
Strain is a measure of the deformation or elongation a material experiences due to applied forces. It is denoted by the symbol ε and is defined as the change in length (∆L) divided by the original length (L) of the material. Mathematically, strain can be expressed as ε = ∆L/L.

2. Stress:
Stress, on the other hand, is a measure of the internal forces that develop within a material when it is subjected to external forces. It is denoted by the symbol σ and is defined as the force applied (F) divided by the cross-sectional area (A) of the material. Mathematically, stress can be expressed as σ = F/A.

Now, let's derive the expressions for the energy stored in the wire and the energy per unit volume:

a. Energy stored in the wire (U):
The energy stored in the wire can be obtained by integrating the work done by the force F to extend the wire by a displacement e. The work done is equal to the force multiplied by the distance moved. In this case, the force is F and the distance moved is e.

Using the formula for work (W = F * d), we can express the energy stored as the work done:
U = F * e

However, we need to express e in terms of L, A, and E. To do this, we use Hooke's Law, which states that the strain (ε) is directly proportional to the stress (σ) in a material. Mathematically, this can be written as:

σ = E * ε

Rearranging this equation, we have:

ε = σ / E

Substituting the values of stress and strain, we get:

ε = F / (A * E)

Now, we can express the change in length (∆L) as:

∆L = L * ε

Substituting the previous equation, we have:

∆L = L * (F / (A * E))

Finally, we can express the extension (e) as:

e = ∆L = L * (F / (A * E))

Substituting this into the expression for energy stored (U), we get:

U = F * e
= F * L * (F / (A * E))
= (F^2 * L) / (A * E)

Therefore, the expression for the energy stored in the wire is U = (F^2 * L) / (A * E).

b. Energy per unit volume (Ev):
The energy per unit volume can be obtained by dividing the energy stored (U) by the volume (V) of the wire. Since the volume is equal to the cross-sectional area (A) multiplied by the length (L), we have:

V = A * L

Therefore, the energy per unit volume (Ev) is given by:

Ev = U / V
= [(F^2 * L) / (A * E)] / (A * L)
= (F^2) / (A * E)

Now, we substitute strain (ε) in terms of stress (σ) and rearrange the equation:

Ev = (σ^2) / E
= (F^2 / (A^2)) / E
= (F^2) / (A * E)

Therefore, the energy per unit volume (Ev) is Ev = (F^2) / (A * E).

And that's how we derive the expressions for the energy stored in the wire and the energy per unit volume!