Problem 914 - JEE Physics

Question

On stretching a wire what is the elastic energy stored per unit volume? (A) \([\mathrm{F} \ell / 2 \mathrm{AL}]\) (B) [FA/2L] (C) \([\mathrm{FL} / 2 \mathrm{~A}]\) (D) \([\mathrm{FL} / 2]\)

Step-by-Step Solution

Answer

The correct expression for the elastic energy stored per unit volume is \(u = (Y * ∆l^2) / (2 * l^2)\), which is not among the given options.

Step 1: Write down the formula for elastic energy stored

The elastic potential energy (U) stored in a stretched wire is given by the formula: U = (1/2) * F * ∆l where F is the applied force and ∆l is the change in length of the wire.

Step 2: Write down the formula for Hooke's Law

Hooke's Law relates the force (F) applied on the wire to the extension (∆l) and the spring constant (k): F = k * ∆l

Step 3: Relate the spring constant with the Young's modulus and the dimensions of the wire

The spring constant (k) is related to the Young's modulus (Y), the length (l), and the area (A) of the wire as follows: k = (Y * A) / l

Step 4: Substitute the spring constant and force in the formula for elastic energy

Substituting the expressions for spring constant (k) and force (F) in the equation for elastic potential energy (U), we get: U = (1/2) * (Y * A * ∆l^2) / l

Step 5: Calculate the elastic energy stored per unit volume

Divide the elastic potential energy (U) by the volume (V) of the wire to find the energy stored per unit volume (u), where V = A * l: u = U / V = ((1/2) * (Y * A * ∆l^2) / l) / (A * l) Simplifying the expression, we get: u = (Y * ∆l^2) / (2 * l^2) Now let's compare this expression with the given options. (A) [\(\mathrm{F} \ell / 2 \mathrm{AL}\)] - This option is incorrect as it does not include the Young's modulus (Y) and has the incorrect relationship. (B) [FA/2L] - This option is incorrect as it does not involve the Young's modulus (Y) and the extension (∆l). (C) [\(\mathrm{FL} / 2 \mathrm{~A}\)] - This option is incorrect as it does not involve the Young's modulus (Y) and the extension (∆l). (D) [\(\mathrm{FL} / 2\)] - This option is incorrect as it does not have the correct relationship for the energy stored per unit volume. None of the given options match our derived expression for the elastic energy stored per unit volume. Therefore, the exercise does not provide the correct option. The correct expression should be: \(u = (Y * ∆l^2) / (2 * l^2)\).

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle that describes how elastic materials behave when subjected to an external force. It states that the force needed to extend or compress a spring by a distance \( \Delta l \) is proportional to that distance. In mathematical terms, the law is expressed as:
\[ F = k \cdot \Delta l \]
Where:

• \( F \) is the force applied,
• \( k \) is the spring constant or stiffness of the spring,
• \( \Delta l \) is the change in length of the material.

Hooke's Law is only applicable within the elastic limit of materials. This means materials will return to their original shape when the force is removed. However, if the force exceeds this limit, permanent deformation occurs. Understanding this concept is crucial for calculations involving elastic systems.

Young's Modulus

Young's Modulus is a measure of the stiffness of an elastic material. It is a property that describes the elasticity of solid materials and is defined as the ratio of stress (force per unit area) to strain (proportional deformation in length) in the linear region of the stress-strain curve. The formula for Young's Modulus (\( Y \)) is given by:
\[ Y = \frac{F / A}{\Delta l / l} \]
Where:

• \( F \) is the force applied,
• \( A \) is the cross-sectional area,
• \( \Delta l \) is the change in length,
• \( l \) is the original length.

Young's Modulus helps us understand how a material will deform under different loads. A high Young's Modulus value indicates a material that is rigid and less prone to deformation. It's an important factor in engineering applications where material flexibility and durability are concerns.

Spring Constant

The spring constant, denoted as \( k \), quantifies the stiffness of a spring. It is a proportionality constant in Hooke's Law that relates the applied force to the displacement in a spring. The spring constant is determined by the material's properties and geometry.
For a wire, the spring constant can be derived using Young's Modulus (\( Y \)), the cross-sectional area (\( A \)), and the original length (\( l \)):
\[ k = \frac{Y \cdot A}{l} \]
This relationship helps convert material properties into practical applications for simple mechanical systems. A larger spring constant means a stiffer spring that requires more force to compress or stretch.
Understanding spring constants is essential for designing systems that involve any type of spring or elastic component, ensuring they perform to expected standards.

Elastic Potential Energy

Elastic Potential Energy is the energy stored in an elastic material due to its deformation. In the context of stretching or compressing a spring, this energy is given by the formula:
\[ U = \frac{1}{2} \cdot F \cdot \Delta l \]
Where:

• \( U \) is the elastic potential energy,
• \( F \) is the force applied,
• \( \Delta l \) is the extension or compression length.

By substituting Hooke's Law (\( F = k \cdot \Delta l \)) into this formula, we get a relation involving the spring constant:
\[ U = \frac{1}{2} \cdot k \cdot (\Delta l)^2 \]
Elastic potential energy plays a vital role in mechanisms that involve springs, from simple toys to complex machinery. It is a measure of the energy available to perform work when the force is released. In engineering, it's pivotal for tasks such as energy recovery and shock absorption.