A wire elongates by ℓm m when a load w is hung from it. If the wire goes over a pulley and two w

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A wire elongates by ℓm m when a load w is hung from it. If...

A wire elongates by $\ell m m$ when a load $\mathrm{w}$ is hung from it. If the wire goes over a pulley and two weights $w$ each are hung at the two ends, the elongation of the wire will be (in mm)
(A) $\ell$
(B) $2 \ell$
(C) Zero
(D) $\frac{\ell}{2}$

Key Concepts

Tension in a Uniform Wire

Tension refers to the force transmitted through a wire when it is subject to pulling forces from either end. In a uniform wire under equilibrium, the tension remains constant throughout its length. Whether the wire is loaded from one end or pulled from both ends in a balanced manner, the tension in each segment is determined by the magnitude of the forces applied, which in turn governs the extension of the wire according to its elastic properties.

Springs in Series

When a uniform elastic wire is effectively divided into sections, such as when forces are applied at different points, the wire can be thought of as a set of springs connected in series. The overall extension of the system is the sum of the extensions of the individual sections. Even though each section may have a different effective stiffness, the same tension acts uniformly in each segment, which allows the analysis of complex loading configurations using series spring models.

Hooke's Law

Hooke's Law states that, within the elastic limit of a material, the extension produced is directly proportional to the load applied. This principle underlies the linear relationship between force and displacement in elastic objects, and it serves as the basis for analyzing how forces cause deformations in materials such as wires or springs.

Elasticity and Young's Modulus

The concept of elasticity describes a material's ability to return to its original shape after the removal of a load. Young's modulus is a material-specific constant that quantifies the relationship between tensile stress and strain. It is used to determine the extent to which a wire will elongate under a given load, linking the microscopic properties of the material to its macroscopic behavior.

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