Question

A wire \(4.75 \times 10^{3} \mathrm{cm}\) long when loaded is seen to stretch \(9.55 \times 10^{-2} \mathrm{cm} .\) Find the strain in the wire, using the formula strain \(=\) elongation \(\div\) length.

Short answer

The strain in the wire is \(2.01 \times 10^{-5}\).

Step-by-step solution

Identify Elongation and Original Length

First, identify the elongation (the amount the wire stretched) and the original length of the wire before it was loaded. Elongation is given as \(9.55 \times 10^{-2} \mathrm{cm}\) and the original length is \(4.75 \times 10^{3} \mathrm{cm}\).

Apply the Strain Formula

Use the formula for strain which is the ratio of elongation to the original length. In mathematical terms, strain \(= \frac{\text{Elongation}}{\text{Original Length}}\).

Calculate the Strain

Substitute the values of elongation and original length into the strain formula: \[\text{Strain} = \frac{9.55 \times 10^{-2} \mathrm{cm}}{4.75 \times 10^{3} \mathrm{cm}}\].

Simplify the Calculation

Perform the division to find the strain. \[\text{Strain} = \frac{9.55 \times 10^{-2}}{4.75 \times 10^{3}}\] which simplifies to \(2.01 \times 10^{-5}\), by dividing the numbers and subtracting the exponents of 10.

Key concepts

Material Deformation

Material deformation refers to the change in shape or size of a material when it undergoes a force or load. When materials like metals, plastics, or wood are subjected to external forces, they might stretch, compress, or twist. In technical terms, deformation is classified into two types: elastic deformation, which is reversible, and plastic deformation, which is permanent.

In our exercise, the wire's elongation—its extension under load—is a measurable deformation. By calculating the degree of deformation compared to the original length, we quantify how much the material has stretched, which is essential in understanding materials' behavior under stress and designing structures and components in engineering.

Elasticity in Materials

Elasticity in materials describes their ability to return to their original shape after the deforming force is removed. This property is pivotal in materials that undergo deformation as part of their normal operation, like springs or bridges. The elasticity is governed by a material's modulus of elasticity or Young's modulus, which is a measure of stiffness and is constant for a given material.

In the context of our wire stretching example, the strain calculation tells us the extent to which the wire exhibits elastic behavior. If the strain remains within the material's elastic limit, the wire would return to its original length once the loading force is removed.

Physics of Solids

The physics of solids encompasses the study of how solid materials behave under various conditions, including stress, temperature, and pressure changes. It is important in predicting the performance of materials in different scenarios, like construction, manufacturing, and even nanotechnology.

In the provided exercise, the concept of strain is a fundamental part of the physics of solids. It gives us insight into the performance of the wire when subjected to a load. This understanding is critical because it helps us determine whether a material will function as required in a solid structure or fail under stress.

Technical Mathematics

Technical mathematics involves the application of mathematical principles to solve problems related to engineering and applied sciences. It includes a wide range of topics from basic arithmetic to complex calculus, often taught in a way that emphasizes their practical application.

In solving the strain problem from the exercise, we use technical mathematics to manipulate the given values through mathematical operations. The correct use of exponents and division is crucial for reaching the correct answer. Adequately simplifying the result ensures that the strain is presented in a meaningful and understandable way. This precision is vital in technical fields where even the smallest error can have significant consequences.