Two wires are made of the same material and have the same...

Two wires are made of the same material and have the same volume. However, wire 1 has crosssectional area $3 A$. If length of wire 1 increased by $\Delta x$ on applying force $F$, how much force is needed to stretch wire 2 by the same amount ? (a) $4 F$ (b) $6 F$ (c) $9 \mathrm{~F}$ (d) $F$

Key Concepts

Hooke's Law

Hooke's Law is the principle that, within the elastic limit of a material, the extension produced in the material is directly proportional to the applied force. In the context of stretching wires or rods, the change in length (extension) is given by ?x = (F·L)/(A·Y), where F is the force, L is the original length, A is the cross?sectional area, and Y is the Young’s modulus of the material.

Young’s Modulus

Young’s modulus is a measure of the stiffness of a material. It is defined as the ratio of tensile stress to tensile strain and appears in the relation ?x = (F·L)/(A·Y). A higher Young’s modulus indicates that the material is stiffer and will exhibit less extension for a given force, making it a key concept for problems involving elastic deformation.

Stress and Strain in Materials

Stress is defined as the force per unit area (? = F/A), while strain is the fractional change in length (? = ?x/L). The linear relationship between stress and strain in the elastic region of a material’s behavior is central to understanding how materials deform under load and is directly connected to Young’s modulus by the equation ? = Y·?.

Effect of Geometry on Elastic Deformation

The geometry of an object, specifically its length and cross?sectional area, plays a crucial role in its elastic deformation. According to ?x = (F·L)/(A·Y), for a given force and material, a longer object will extend more, and an object with a larger cross?sectional area will extend less. This concept is essential for comparing how objects of different dimensions respond to the same applied forces.

Volume Constraint in Comparing Rods

When two rods or wires are constrained to have the same volume, changes in one geometric parameter (like cross?sectional area) affect another (like length). For example, if one wire has a larger cross?sectional area than the other at constant volume, it must be proportionally shorter. This relationship is important when comparing the elastic properties of wires with different geometries, as it affects the magnitude of extension under a given force.

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