Question

If \(\mathrm{x}\), longitudinal strain is produced in a wire of young's modulus \(\mathrm{y}\) then energy stored in the material of the wire per unit volume is...... (A) \(\mathrm{yx}^{2}\) (B) \(2 \mathrm{yx}^{2}\) (C) \((1 / 2) \mathrm{y}^{2} \mathrm{x}\) (D) \((1 / 2) \mathrm{yx}^{2}\)

Short answer

The energy stored in the material of the wire per unit volume is \(\frac{1}{2} Yx^2\). Therefore, the correct answer is (D) \(\frac{1}{2} \mathrm{yx}^{2}\).

Step-by-step solution

Recall the formula for elastic potential energy

The elastic potential energy, U, stored in a wire when it is subjected to longitudinal strain is given by the formula: \(U = \frac{1}{2} \cdot F \cdot \Delta L\) Where F is the force exerted on the wire and ΔL is the change in length of the wire.

Express force in terms of stress and area

Stress is the force applied per unit area. We can express the force acting on the wire (F) as the product of stress and the wire's cross-sectional area (A): \(F = \sigma \cdot A\)

Relate stress with Young's modulus and strain

Young's modulus (Y) is defined as the ratio of stress to strain: \(\sigma = Y \cdot x\) Where x is the longitudinal strain.

Substitute stress in the Force equation

Substitute the expression for stress in terms of Young's modulus and strain into the force equation: \(F = Y \cdot x \cdot A\)

Substitute force and change in length in the elastic potential energy formula

Substitute the expression for force and change in length (ΔL) in terms of strain (x) and original length (L) into the elastic potential energy formula: \(U = \frac{1}{2} \cdot (Y \cdot x \cdot A) \cdot (x \cdot L)\)

Simplify the elastic potential energy expression

Simplify the expression for elastic potential energy: \(U = \frac{1}{2} \cdot Y \cdot x^{2} \cdot A \cdot L\)

Divide the elastic potential energy by volume

Divide the elastic potential energy by the volume of the wire (V = A ⋅ L) to find the energy stored per unit volume: \(\frac{U}{V} = \frac{1}{2} \cdot Y \cdot x^{2}\) So, the energy stored in the material of the wire per unit volume is (1/2)yx². Therefore, the correct answer is (D) \((1 / 2) \mathrm{yx}^{2}\).

Key concepts

Young's Modulus

Young's Modulus is a fundamental concept in the study of materials under stress. It is denoted by the symbol \( Y \) and defines the stiffness of a material. In simple terms, it quantifies how much a material will deform under a given stress. When you apply a force to an object, it experiences stress and consequently deforms. Young's Modulus is the ratio of this stress to the strain produced. This is expressed through the formula:\[ Y = \frac{\text{Stress}}{\text{Strain}} \]Here, stress is defined as the force per unit area, and strain is the relative deformation, which is the change in length divided by the original length of a material. Materials like metals have high Young's Modulus because they are relatively stiff and don't deform easily under stress. Knowing Young's Modulus allows engineers to predict how much force a material can endure before deforming.

Longitudinal Strain

Longitudinal Strain occurs when a wire or any object changes its length under the action of an applied force. It is dimensionless since it is a ratio of lengths. Longitudinal strain is calculated by dividing the change in length \( \Delta L \) by the original length \( L \):\[ \text{Longitudinal Strain} = \frac{\Delta L}{L} \]It reflects how much the material stretches or compresses along the line of force. This concept helps in understanding how flexible or ductile a material is. Strain values are typically small numbers since the deformations are subtle compared to the original dimensions, especially in metals and other stiff materials.

Stress and Strain

Stress and strain are pivotal in understanding how materials react under external forces. Stress is the measure of force per unit area applied to a material - calculated as \( \sigma = \frac{F}{A} \), where \( F \) is the applied force and \( A \) is the cross-sectional area.Strain measures the deformation produced by this stress and can be described as \( \epsilon = \frac{\Delta L}{L} \), which is the alteration in length per unit original length. In the context of elastic materials, where they regain their original shape after the force is removed, stress is proportional to strain. This linear relationship is described by Hooke's Law, a concept that helps predict the behavior of materials under different loads.

Energy Stored in Wire

The energy stored in a wire, especially when stretched or compressed, is known as elastic potential energy. When you apply a force to a wire, causing it to stretch, work is done, and this work gets stored as potential energy in the wire. The formula for the energy stored (per unit volume) due to elongation is driven by the properties of the material, expressed as \( \frac{1}{2} Y x^2 \).

• \( Y \) is Young's Modulus, indicating the material's stiffness.
• \( x \) is the longitudinal strain, the change in length relative to the original length.

This equation shows that how much energy is stored depends on the stiffness of the wire and the extent of deformation. Higher stiffness and more deformation lead to more stored energy. This is vital for understanding the limits of mechanical stresses in applications like cable manufacturing or in structural engineering.