Problem 18 - Arihant AIEEE Physics

Question

Calculate the compressional force required to prevent the metallic rod of length \(l \mathrm{~cm}\) and cross-sectional area \(A \mathrm{~cm}^{2}\) when heated through \(t^{\circ} \mathrm{C}\), from expanding along lengthwise. The Young's modulus of elasticity of the metal is \(E\) and mean coefficient of linear expansion is \(\alpha\) per degree celsius: (a) EA\alphat (b) \(\frac{E \text { A\alphat }}{(1+\alpha t)}\) (c) \(\frac{E A \alpha t}{(1-\alpha t)}\) (d) El\alphat

Step-by-Step Solution

Answer

(a) EA\alphat

Step 1: Understanding the problem

We are asked to find the compressional force required to prevent expansion of a metallic rod when heated. The parameters given are: the length of the rod \(l\), cross-sectional area \(A\), temperature change \(t\), Young's modulus \(E\), and the coefficient of linear expansion \(\alpha\).

Step 2: Identifying relevant formulae

The linear expansion of the rod, when not constrained, would be \(\Delta l = \alpha l t\). The force required to prevent this expansion is given by stress times area. Stress is given by Young's modulus times strain (change in length/original length), which gives \(F = E \cdot A \cdot \frac{\Delta l}{l}\).

Step 3: Substituting the expression for expansion

Since \(\Delta l = \alpha l t\), the expression for force becomes \[F = E \cdot A \cdot \frac{\alpha l t}{l} = E \cdot A \cdot \alpha \cdot t\].

Step 4: Verifying against options

Among the given options, (a) \(EA\alphat\) matches the expression derived for the force \[F = E \cdot A \cdot \alpha \cdot t\]. Therefore, option (a) is correct.

Key Concepts

Young's Modulus

Young's Modulus, denoted by the symbol \( E \), is a fundamental property of a material that measures its stiffness. It defines the relationship between stress (force applied) and strain (deformation) in the material. When you apply a force to a metallic rod, it changes its shape slightly. The Young's Modulus tells us how much force is needed to stretch or compress the rod by a certain amount. This is a crucial parameter for materials used in construction and engineering.

In mathematical terms, Young's Modulus is expressed as:

• \( E = \frac{\text{Stress}}{\text{Strain}} \)

Stress is the force applied per unit area, and strain is the relative deformation (change in length divided by the original length). A higher Young's Modulus means the material is stiffer or more rigid and requires more force to deform.

Understanding this concept is important for solving problems involving deformation of objects, such as computing the compressional force needed to prevent the expansion of a rod.

Coefficient of Linear Expansion

The Coefficient of Linear Expansion, represented by \( \alpha \), is another fundamental concept that describes how materials expand or contract with temperature changes. When a material like a metallic rod is heated, its length increases. The coefficient \( \alpha \) indicates how much the length of the rod will change per degree Celsius of temperature change.

The formula to calculate the change in length \( \Delta l \) due to thermal expansion is:

• \( \Delta l = \alpha \cdot l \cdot t \)

where \( l \) is the original length of the rod and \( t \) is the temperature change.

A higher coefficient means the material expands more with temperature. This concept is crucial when designing components that must fit precisely in applications subject to temperature variations, such as engine parts or metal bridges.

Stress and Strain

Stress and strain are core concepts in material science and structural engineering, helping us understand how materials behave under force. Stress is the force applied to a material per unit area, while strain is the material's response to that force in terms of deformation.

Stress is quantified as:

• \( \text{Stress} = \frac{F}{A} \)

where \( F \) is the force applied and \( A \) is the cross-sectional area. Strain is given by:

• \( \text{Strain} = \frac{\Delta l}{l} \)

where \( \Delta l \) is the change in length and \( l \) is the original length.

Understanding how stress and strain relate helps determine how much a component can safely withstand before deforming significantly or failing, which directly links to Young's Modulus in evaluating material rigidity.

Metallic Rod

A metallic rod, often used in examples involving thermal and mechanical properties, serves as a perfect case study to understand the principles of thermal expansion and elasticity. In our problem, the metallic rod undergoes heating, causing it to try to expand in length.

The physical properties of the rod, such as its length \( l \), cross-sectional area \( A \), and material-specific properties like Young’s Modulus \( E \) and Coefficient of Linear Expansion \( \alpha \), play vital roles in determining the compressive force required to prevent its expansion.

When analyzing a metallic rod under thermal stress, it is important to consider:

• Its material composition, dictating \( E \) and \( \alpha \).
• The applied temperature changes and resulting physical changes.
• The potential constraints or forces needed to manage those changes as in this problem scenario.

Understanding how these variables interact is crucial for successful application in real-world engineering contexts.