Problem 13 - Master Resource Book in JEE Main Physics

Question

Two wires \(A\) and \(B\) have same lengths and made of the same material but \(A\) is thicker than \(B\). Both are subjected to the same extending load. Which will extend more? (a) \(A\) (b) \(B\) (c) Same extension (d) Can't predict

Step-by-Step Solution

Answer

Wire \(B\) will extend more.

Step 1: Identify the Factors

Both wires \(A\) and \(B\) have the same length and are made of the same material. Since the material is the same, both wires will have the same Young's modulus \(E\). Young's modulus is a measure of the elasticity of a material.

Step 2: Understand the Effect of Cross-sectional Area

Wire \(A\) is thicker than wire \(B\), which means the cross-sectional area \(A\) of wire \(A\) is greater than that of wire \(B\). The force \(F\) applied is the same on both wires.

Step 3: Recall the Formula for Extension

The formula for the extension \( \Delta L \) of a wire under a load is given by \( \Delta L = \frac{FL}{AE} \), where \(F\) is the force applied, \(L\) is the original length, \(A\) is the cross-sectional area, and \(E\) is the Young's modulus.

Step 4: Analyze the Relationship

Since \(L\) and \(E\) are the same for both wires, the extension \( \Delta L \) is inversely proportional to the cross-sectional area \(A\). Thus, a wire with a smaller cross-sectional area will extend more under the same load.

Step 5: Determine the Greater Extension

Wire \(B\) has a smaller cross-sectional area than wire \(A\), meaning \(B\) will have a greater extension than wire \(A\).

Key Concepts

Young's Modulus

Young's Modulus is a fundamental concept in the study of mechanical properties of solids. It is a measure of a material's elasticity, expressed as the ratio of stress (force per unit area) over strain (deformation per unit length). In mathematical terms, Young's Modulus, denoted as \(E\), is expressed as:\[ E = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]where:- \(F\) is the force applied,- \(A\) is the cross-sectional area,- \(\Delta L\) is the change in length,- \(L\) is the original length.When two materials exhibit different Young's Moduli, the one with a higher modulus is stiffer and less prone to deformation. In our scenario, since both wires are made of the same material, they have the same Young's Modulus. This implies that their ability to resist deformation under stress is similar. But other factors, such as cross-sectional area, can affect their extension.

Cross-Sectional Area

The cross-sectional area of a wire plays a crucial role in determining how much it will stretch under a given force. The cross-sectional area is essentially the thickness of the wire.In mechanical properties, the cross-sectional area directly influences the extension for a fixed amount of load. According to the extension formula, extension \(\Delta L\) is inversely proportional to the cross-sectional area:\[ \Delta L = \frac{FL}{AE} \]Thus, a smaller area results in a larger extension if all other factors remain constant. For wire \(A\), being thicker means it has a larger cross-sectional area than wire \(B\). Hence, wire \(B\), having a smaller area, will undergo more extension when both wires are subjected to the same force.

Elasticity

Elasticity is an intrinsic property of materials that defines their ability to return to their original form after being deformed. In the context of the problem, elasticity is quantified using Young's Modulus. Materials with high elasticity can undergo deformation under stress but will return to their original configuration upon removal of the force. In simplified terms, elasticity determines the stretch-ability and recoverability of a material. For materials like the wires in the problem, the same Young's Modulus means they share similar elastic properties, indicating their inherent capacity to deal with deformation. However, despite having the same elastic characteristics based on material, the actual physical outcome (how much they will extend) also depends on other factors like the cross-sectional area.

Extension Formula

The extension formula is a key tool for calculating how much a material stretches under stress. For a wire under tension, the formula is:\[ \Delta L = \frac{FL}{AE} \]Let's break down the components:- \(\Delta L\) is the extension, or how much longer the wire gets,- \(F\) is the force applied to the wire,- \(L\) is the original length of the wire,- \(A\) is the cross-sectional area;- \(E\) is Young's Modulus, which describes the wire's elasticity.Understanding this formula is pivotal because it shows that for a constant force and material (same \(E\)), the extension is inversely influenced by the cross-sectional area. This means smaller \(A\) leads to greater \(\Delta L\). That's why wire \(B\), with its smaller cross-sectional area, extends more than wire \(A\).

Material Properties

In mechanical properties of solids, understanding material properties helps predict their behavior under various conditions. These properties include density, melting point, and mechanical properties like elasticity and tensile strength. For the wires in the exercise, the critical shared property is Young's Modulus, indicating their elasticity. Since they're made of the same material, properties like their response to stress are identical. However, differences like cross-sectional area can nuance how they react when the same force is applied. Understanding these material properties can inform practical applications, whether it’s choosing materials for construction or manufacturing processes, to ensure they meet the necessary criteria for strength and flexibility.