A 4.0-m-long steel wire has a cross-sectional area of 0.050...

A 4.0-m-long steel wire has a cross-sectional area of 0.050 cm$^2$. Its proportional limit has a value of 0.0016 times its Young's modulus (see Table 11.1). Its breaking stress has a value of 0.0065 times its Young's modulus. The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit? (b) How much will the wire stretch under this load? (c) What is the maximum weight that the wire can support?

Key Concepts

Breaking Stress

Breaking stress, also known as the ultimate tensile strength, is the maximum stress that a material can withstand before it fails or fractures. It is a critical parameter in design and safety, ensuring that loads remain within the material's capacities to avoid catastrophic failure.

Proportional Limit

The proportional limit is the highest stress at which stress and strain remain directly proportional to each other. Beyond this limit, the material behavior may become non-linear and plastic deformation can occur, which is undesirable in applications requiring elastic recovery.

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 principle allows one to calculate the extension or compression of a material under a given load, provided the material has not exceeded its proportional limit.

Strain

Strain is a measure of deformation representing the displacement between particles in the material body relative to a reference length. It quantifies how much the material elongates or compresses when subjected to stress.

Stress

Stress is defined as the force applied per unit area of a material and is a measure of the internal forces generated within the material when an external load is applied. In engineering, understanding stress is critical for determining how a material will behave under various loading conditions.

Young's Modulus

Young's modulus is the ratio of stress to strain in the elastic region of a material's stress-strain curve. It is a fundamental property that indicates the stiffness of a material, and it plays a crucial role in predicting how much a material will deform under a given load.

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