A Review problem. A wire of length L, Young's modulus Y, and...

A Review problem. A wire of length $L,$ Young's modulus $Y,$ and cross-sectional area $A$ is stretched elastically by an amount $\Delta L$ . By Hooke's law, the restoring force is $-k \Delta L .$ (a) Show that $k=Y A / L .$ (b) Show that the work done in stretching the wire by an amount $\Delta L$ is $$ W=\frac{1}{2} Y A \frac{(\Delta L)^{2}}{L} $$

Key Concepts

Work Done in Elastic Deformation

Work done in the context of elastic deformation is the energy required to stretch or compress an elastic material. This work is computed by integrating the force over the displacement, which, for an elastic material obeying Hooke's law, leads to a quadratic dependence on the amount of stretch or compression.

Elastic Potential Energy

Elastic potential energy is the energy stored in an elastic material when it is deformed under an applied force. This energy can be recovered when the material returns to its original shape. It is typically calculated by integrating the force over the displacement, leading to an expression that is proportional to the square of the displacement.

Hooke's Law

Hooke's law states that, for elastic deformations, the force exerted by an elastic object such as a spring is directly proportional to the displacement from its equilibrium position. This principle is central in relating the applied force to the resulting elongation or compression in elastic materials.

Young's Modulus

Young's modulus is a measure of a material's stiffness, defined as the ratio of tensile (or compressive) stress to tensile (or compressive) strain in the elastic region of the material. It quantifies how much a material will deform under a given load and is intrinsic to the material's properties.

Stress and Strain

Stress is defined as the force per unit area applied to a material, while strain is the relative deformation caused by this stress, measured as the change in length over the original length. The linear relationship between stress and strain in the elastic region, as expressed by Young's modulus, is fundamental to understanding how materials respond to applied forces.

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