Elastic Potential Energy in a Stretched Wire
When a wire experiences tensile stress, work must be done against the inter-atomic forces to elongate the wire. The work done on the wire is stored as elastic potential energy, which can be released when the wire returns to its original length after the force is removed. For a wire of original length L and cross-sectional area A subjected to a deforming force F, if the length is elongated by a small amount l, we can establish the relationship:
Mathematical Relationships
Using Young's modulus (Y) which relates stress (σ) and strain (ε), we can express the work done (W) in increasing the length of the wire:
- The force can be described as:
$$F = YA \frac{l}{L}$$
- The incremental work done, $dW$ is related to the force and an infinitesimal change in length, leading to:
$$dW = F \cdot dl = YA \frac{l}{L} dl$$
- The total work done in stretching from length L to L + l can be integrated:
$$W = \int_0^{l} \frac{1}{2} YA \frac{l}{L} dl = \frac{1}{2} Y \cdot \text{strain}^2 \cdot \text{volume of the wire}$$
- Hence, the elastic potential energy per unit volume is:
$$u = \frac{1}{2} \sigma \epsilon$$
This relationship illustrates that the stored elastic potential energy is not just a function of the material properties but also of the geometry and the deformation applied.