Question

A block of material of length \(0.9 \mathrm{~m}\) is subject to a force which causes the material to extend to a length of \(0.93 \mathrm{~m}\). Calculate the strain in the material and express it as both a proper fraction and a decimal fraction stating your answer to four decimal places.

Short answer

Answer: The strain in the material is 1/30 as a proper fraction, and 0.0333 as a decimal fraction rounded to four decimal places.

Step-by-step solution

Identify given information and required output

The original length of the material, \(L_1\), is given as 0.9 meters, and the length after the force is applied, \(L_2\), is given as 0.93 meters. Our task is to find the strain.

Calculate the change in length

To find the change in length, subtract the original length from the final length: ΔL = L_2 - L_1 = 0.93 - 0.9 = 0.03.

Calculate the strain as a proper fraction

Strain is defined as the change in length divided by the original length: Strain = \(\frac{ΔL}{L_1}\) = \(\frac{0.03}{0.9}\). To simplify this fraction, divide both the numerator and denominator by 0.03: Strain = \(\frac{0.03}{0.9}\) = \(\frac{1}{30}\).

Convert the strain to a decimal fraction

To find the decimal equivalent of the strain, divide the numerator by the denominator: Strain = \(\frac{1}{30}\) = 1 ÷ 30 = 0.0333\bar{3}. Round the decimal fraction to four decimal places: Strain = 0.0333. In conclusion, the strain in the material is \(\frac{1}{30}\) as a proper fraction, and 0.0333 as a decimal fraction rounded to four decimal places.

Key concepts

Engineering Mechanics

Understanding the impacts of forces on materials is a fundamental aspect of engineering mechanics, an essential field in civil, mechanical, and aerospace engineering. It involves analyzing the stability, strength, and motion of structures under various loads. When a force is applied to a material, it may undergo deformation—this is where the concept of strain becomes important.

Strain measures how much a material deforms under a load, relative to its original size. It is a dimensionless quantity, meaning it has no units, as it represents the ratio of the length change to the original length. In the given exercise, the length of a material changes when a force is applied, therefore constituting a classic problem within the study of engineering mechanics.

Material Deformation

Material deformation is the change in shape or size of an object due to an applied force or a change in temperature. There are two types of deformation: elastic and plastic. Elastic deformation is temporary and reversible once the load is removed, while plastic deformation is permanent.

In the exercise provided, the extension of the material's length from 0.9 m to 0.93 m is an instance of deformation. Assuming the material has not exceeded its elastic limit, this deformation is likely elastic. Engineers need to calculate the extent of this deformation, termed as 'strain', to ensure that materials will perform adequately under expected loads and to avoid structural failure.

Strain as a Decimal Fraction

In engineering, we often express strain as a decimal fraction to make it more understandable and usable, especially in calculations and analytical models. A decimal number is simply a way of representing fractions in a base-10 number system that is easy for us to interpret. The step-by-step solution provided goes through the process of converting strain from a change in length to a dimensionless ratio, and finally to a decimal fraction to precision which, in this case, is four decimal places.

Calculating strain in this manner allows engineers and designers to quickly assess the degree of deformation, compare materials or design scenarios, and apply this data to formulas and simulations used in predicting material behaviors under various loads.

Proper Fraction to Decimal Conversion

Conversion from a proper fraction to a decimal is a valuable mathematical skill not only in engineering but in many scientific calculations. A proper fraction is where the numerator is less than the denominator. To convert a proper fraction to decimal, divide the numerator by the denominator.

In the textbook solution, the strain is presented first as a proper fraction, \(\frac{1}{30}\), which is an exact representation. By dividing 1 by 30, we convert this into a decimal that can be rounded to any required degree of accuracy. For this exercise, it's rounded to four decimal places, providing the decimal strain value of 0.0333. This process highlights the importance of understanding and being able to toggle between different numerical formats for accurate and efficient problem-solving.