A rod of length 1.05 m having negligible mass is supported...

A rod of length $1.05 \mathrm{~m}$ having negligible mass is supported at its ends by two wires of steel (wire A) and aluminium (wire B) of equal lengths as shown in Fig. 9.15. The cross-sectional areas of wires $\mathrm{A}$ and $\mathrm{B}$ are $1.0 \mathrm{~mm}^{2}$ and $2.0 \mathrm{~mm}^{2}$. respectively. At what point along the rod should a mass $m$ be suspended in order to oroduce (a) equal stresses and (b) equal strains in both steel and aluminium wires.

Key Concepts

Static Equilibrium

Static equilibrium involves the condition where the sum of all forces and moments (torques) acting on a system is zero. In the context of the rod supported by two wires, applying static equilibrium principles allows for the determination of the effective loads on each wire and the conditions under which equal stresses or strains can be obtained. This involves balancing moments about a point and distributing the applied load appropriately along the rod.

Hooke's Law

Hooke's law states that, within the elastic limit of a material, the strain is directly proportional to the applied stress. This linear relationship is expressed by the formula stress equals Young's modulus times strain. The concept is crucial for understanding how materials deform similarly or differently when the same load is applied, and hence it informs the analysis leading to conditions for equal stresses or equal strains.

Young's Modulus

Young's modulus, also known as the modulus of elasticity, is a material property that measures stiffness, defined as the ratio of stress to strain in the elastic region. Since different materials have different Young's moduli, equal stress does not necessarily lead to equal strain. In this problem, the disparity between the moduli of steel and aluminium must be considered when determining the point of mass suspension to satisfy specific conditions.

Strain

Strain is the measure of deformation of a material expressed as the ratio of the change in length to the original length. Equal strain conditions imply that both wires experience the same relative extension. In problems involving different materials, achieving equal strains requires accounting for the material properties such as the modulus of elasticity, since the same stress can cause different amounts of strain depending on the material.

Stress

Stress is defined as the force per unit area acting on a material and is a measure of the internal forces that develop within a body to resist deformation. In the context of the problem, ensuring equal stresses in different wires means that the forces in each wire divided by their respective cross-sectional areas are made identical, which requires considering both the distribution of the applied load and the geometric differences between the wires.

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