Question

The elastic energy density of a stretched wire is given by:

• A. Stress \(\times\) strain
• B. stress/ strain
• C. 1/2 \(\times\) stress \(\times\) strain
• D. strain/stress

Hint:

The formula for elastic energy density is analogous to other energy formulas like kinetic energy (\(\frac{1}{2}mv^2\)) or capacitor energy (\(\frac{1}{2}CV^2\)). The factor of 1/2 arises because the force (or stress) is not constant but increases linearly from zero as the material is deformed.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Elastic energy density is the elastic potential energy stored per unit volume in a deformed material. It is equal to the work done per unit volume to stretch the material.
Step 2: Key Formula or Approach:
The work done (\(dW\)) in stretching a wire is given by \(dW = F \, dx\), where F is the applied force and dx is the elongation.
The energy density (\(u\)) is the work done per unit volume (\(V\)). For an elastic deformation, this can be calculated as the area under the stress-strain curve.
Stress (\(\sigma\)) is \(F/A\) and strain (\(\epsilon\)) is \(x/L\).
For a material obeying Hooke's Law, the stress is proportional to the strain (\(\sigma = E\epsilon\)), so the stress-strain graph is a straight line through the origin.
Step 3: Detailed Explanation:
The energy density \(u\) is the area of the triangle under the stress-strain curve:
\[ u = \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In the stress-strain graph, the base is the strain (\(\epsilon\)) and the height is the corresponding stress (\(\sigma\)).
Therefore, the energy density is:
\[ u = \frac{1}{2} \times \sigma \times \epsilon \]
\[ u = \frac{1}{2} \times \text{stress} \times \text{strain} \]
Step 4: Final Answer:
The elastic energy density of a stretched wire is given by \( \frac{1}{2} \times \text{stress} \times \text{strain} \).