A wire of length L has a linear mass density μand area of cross-section A and the Young's modulu

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A wire of length L has a linear mass density μand area of...

A wire of length $L$ has a linear mass density $\mu$ and area of cross-section $A$ and the Young's modulus $Y$ is suspended vertically from a rigid support. The extension produced in the wire due to its own weight is (a) $\frac{\mu g L^{2}}{Y A}$ (b) $\frac{\mu g L^{2}}{2 Y A}$ (c) $\frac{2 \mu g L^{2}}{Y A}$ (d) $\frac{2 \mu g L^{2}}{3 Y A}$

A wire of length $L$ has a linear mass density $\mu$ and area of cross-section $A$ and the Young's modulus $Y$ is suspended vertically from a rigid support. The extension produced in the wire due to its own weight is
(a) $\frac{\mu g L^{2}}{Y A}$
(b) $\frac{\mu g L^{2}}{2 Y A}$
(c) $\frac{2 \mu g L^{2}}{Y A}$
(d) $\frac{2 \mu g L^{2}}{3 Y A}$

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Key Concepts

Integration in Continuum Mechanics

Integration is used to sum the contributions of each differential element to the total extension. By integrating across the entire length of the wire, one can account for the continuously varying tension, leading to the overall expression for the extension due to the wire's own weight.

Variable Tension and Differential Element Analysis

Since the tension in the wire due to its own weight varies along its length, considering a small differential element of the wire is crucial. Analyzing an infinitesimally small segment allows us to set up a relationship between the local force and the local extension, which takes into account the variable weight acting on different segments.

Young's Modulus

Young's modulus is a measure of the stiffness of a material. It relates the stress (force per unit area) applied to a material to the resulting strain (relative deformation). In problems involving stretching of materials, it is used to determine how much a material will extend under a given load.

Linear Mass Density

Linear mass density describes the mass per unit length of a material. In the context of self-weight extension problems, it helps to quantify how the weight (force due to gravity) is distributed along the length of the wire, affecting its total elongation.

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