If a metal wire of length L, having area of cross-section A...

If a metal wire of length $L$, having area of cross-section $A$ and Young's modulus $Y$, behaves as a spring of spring constant $k$. The value of $k$ is:
(a) $\frac{Y A}{L}$
(b) $\frac{Y A}{2 L}$
(c) $\frac{2 Y A}{L}$
(d) $\frac{Y L}{A}$

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Key Concepts

Geometric Properties and Elastic Behavior

The dimensions of a material, specifically cross-sectional area and length, play a significant role in its elastic behavior. The effective spring constant for a wire in tension can be directly related to the area (which influences the force-bearing capacity) and inversely related to the length (which affects the amount of extension for a given force).

Spring Constant

The spring constant, often denoted as k, is a measure of a spring's stiffness, defined as the force required to produce a unit displacement. For a wire under tension, it is derived from the material's Young’s modulus and its geometric dimensions.

Young's Modulus

This is a material property that quantifies the stiffness of a material by relating the stress (force per unit area) to the strain (proportional deformation) in the elastic region. It provides a measure of how much a material will deform under a given load.

Hooke's Law

Hooke's Law states that the extension of an elastic object is directly proportional to the force applied to it, as long as the deformation is within the elastic limit. This concept is critical in defining the effective spring constant for a material.

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