In a wire stretched by hanging a weight from its end, the elastic potential energy per unit volume in terms of the longitudinal strain s and modulus of elasticity Y is ?

To understand the elastic potential energy per unit volume in a wire subjected to longitudinal strain, we can derive the formula step by step. The relationship between strain, stress, and elastic potential energy is fundamental in materials science and physics.

Key Concepts in Elasticity

Before diving into the formula, let's clarify some essential concepts:

• Longitudinal Strain (s): This is defined as the change in length per unit original length of the material. Mathematically, it is expressed as s = ΔL / L₀, where ΔL is the change in length and L₀ is the original length.
• Modulus of Elasticity (Y): This is a measure of a material's stiffness, defined as the ratio of stress (force per unit area) to strain. In simple terms, it describes how much a material will deform under a given load.

Understanding Stress and Strain

Stress (σ) in a material can be expressed as:

σ = F / A

where F is the force applied and A is the cross-sectional area. According to Hooke’s Law, for elastic materials, we can relate stress and strain through the modulus of elasticity:

σ = Y * s

Deriving Elastic Potential Energy per Unit Volume

The elastic potential energy (U) stored in a material can be expressed in terms of stress and strain. The general formula for the energy per unit volume in a linearly elastic material is:

U = (1/2) * σ * s

Now, substituting the expression for stress (σ) from Hooke’s Law:

U = (1/2) * (Y * s) * s

This simplifies to:

U = (1/2) * Y * s²

Final Expression

Thus, the elastic potential energy per unit volume in terms of longitudinal strain and modulus of elasticity is:

U = (1/2) * Y * s²

Example Application

Imagine you have a steel wire with a modulus of elasticity of 200 GPa, and you stretch it with a longitudinal strain of 0.01 (1%). To find the elastic potential energy per unit volume, you would substitute these values into the derived formula:

U = (1/2) * 200 GPa * (0.01)²

This calculation would give you the energy stored in the wire due to the applied strain, illustrating how materials behave under tensile stress.

By understanding these principles and equations, you can analyze various applications in engineering and materials science, from designing structures to evaluating the performance of different materials under load.