One end of a uniform wire of length L and of weight W is...

One end of a uniform wire of length $L$ and of weight $W$ is attached rigidly to a point in the roof, and a weight $W_{1}$ is suspended from its lower end. If $S$ is the area of cross-section of the wire, the stress in the wire at a height ( $3 L / 4$ ) from its lower end is (A) $W_{1} / S$ (B) $\left[W_{1}+(W / 4)\right] / S$ (C) $\left[W_{1}+(3 W / 4)\right] / S$ (D) $\left[W_{1}+W\right] / S$

One end of a uniform wire of length $L$ and of weight $W$ is attached rigidly to a point in the roof, and a weight $W_{1}$ is suspended from its lower end. If $S$ is the area of cross-section of the wire, the stress in the wire at a height ( $3 L / 4$ ) from its lower end is
(A) $W_{1} / S$
(B) $\left[W_{1}+(W / 4)\right] / S$
(C) $\left[W_{1}+(3 W / 4)\right] / S$
(D) $\left[W_{1}+W\right] / S$

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Key Concepts

Stress

Stress is defined as the force exerted per unit area. In mechanics, it quantifies how internal forces within a material react when an external load is applied, and it is essential for analyzing whether a material will withstand applied forces without yielding or breaking.

Tension Variation in a Hanging Wire

In a hanging wire or cable, the tension is not uniform along its length; rather, it varies with height. This variation occurs because each point along the wire must support the weight of everything below it, including any additional loads and the weight of the wire itself.

Uniform Weight Distribution

A uniform wire has a constant weight per unit length. This characteristic means that the weight of any segment of the wire is proportional to its length, a concept which allows one to determine the weight of a specific fraction of the wire when calculating the total load supported at a given point.

Cross-sectional Area

The cross-sectional area of a wire is critical in determining the stress, as stress is calculated by dividing the force (or tension) acting on the wire by this area. A larger cross-sectional area results in lower stress for the same force, influencing the material's performance and safety under load.

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