Question

Asbestos is a fibrous silicate mineral with remarkably high tensile strength. But it is no longer used because airborne asbestos particles can cause lung cancer. Grunerite, a type of asbestos, has a tensile strength of \(3.5 \times 10^{2} \mathrm{~kg} / \mathrm{mm}^{2}\) (thus, a strand of grunerite with a \(1-\mathrm{mm}^{2}\) cross-sectional area can hold up to \(3.5 \times 10^{2} \mathrm{~kg}\) ). The tensile strengths of aluminum and Steel No. 5137 are \(2.5 \times 10^{4} \mathrm{lb} / \mathrm{in}^{2}\) and \(5.0 \times 10^{4} \mathrm{lb} / \mathrm{in}^{2},\) respectively. Calculate the cross-sectional areas (in \(\mathrm{mm}^{2}\) ) of wires of aluminum and of Steel No. 5137 that have the same tensile strength as a fiber of grunerite with a crosssectional area of \(1.0 \mu \mathrm{m}^{2}\).

Short answer

The cross-sectional area for aluminum is 306.3 mm\textsuperscript{2} and for Steel No. 5137 is 153.52 mm\textsuperscript{2}.

Step-by-step solution

Convert Grunerite's tensile strength to lbs/in\textsuperscript{2}

First, convert Grunerite's tensile strength from kg/mm\textsuperscript{2} to lbs/in\textsuperscript{2}. Use the conversion factors: 1 kg = 2.20462 lbs and 1 mm\textsuperscript{2} = 0.00155 in\textsuperscript{2}. Therefore, \[3.5 \times 10^2~\mathrm{kg/mm^2} = 3.5 \times 10^2 \times 2.20462 \times 0.00155~\mathrm{lb/in^2} = 1.19 \times 10^4~\mathrm{lb/in^2} \]

Calculate the tensile strength of a grunerite fiber with a cross-sectional area of 1.0 \( \text{mm}^2 \times \text{mm} \text{}^{2} \tilde{}\)

Given the cross-sectional area of 1 mm\textsuperscript{2} for a grunerite fiber, the tensile strength is already provided: \[\boxed{3.5 \times 10^2~\mathrm{kg/mm^2}}~ = 350 kg/mm^2 \]

Convert the tensile strength to lbs

Given tensile strength in lbs/in\textsuperscript{2} for Aluminum and Steel No. 5137, we know the tensile strength of grunerite in lbs/in\textsuperscript{2} from step 1: \[1.19 \times 10^4~\mathrm{lb/in^2}=1.19 \times{10^4} \]

Calculate the cross-sectional area for aluminum

Using the tensile strength of aluminum (2.5 \times 10^{4} \text{lbs/in}^{2}), set up the equation to find the cross-sectional area: \[\frac{{1.19 \times 10^4}}{{2.5 \times 10^4}} \approx 0.476~ \text{in}^{2} \] Convert this value to mm\textsuperscript{2} using the conversion factor 1 \text{in}^{2}= 645.16 \text{mm}^{2}: \[0.476~ \text{in}^{2} \times 645.16 = 306.3~ \text{mm}^{2} \]

Calculate the cross-sectional area for Steel No. 5137

Using the tensile strength of Steel No. 5137 (5.0 \times 10^{4} \text{lbs/in}^{2}), set up the equation to find the cross-sectional area: \[\frac{{1.19 \times 10^4}}{{5.0 \times 10^4}} \approx 0.238~ \text{in}^{2} \] Convert this value to mm\textsuperscript{2} using the conversion factor 1 \text{in}^{2}= 645.16 \text{mm}^{2}: \[0.238~ \textbf{in}^{2} \times 645.16 = 153.52~ \text{mm}^{2} \]

Key concepts

asbestos mineral

Asbestos is a group of minerals composed of fibrous silicate crystals, renowned for their durability and heat resistance. Historically, asbestos was widely used in construction, insulation, and other industries due to these properties.
However, inhaling airborne asbestos fibers is extremely hazardous and can lead to serious lung diseases, including asbestosis, mesothelioma, and lung cancer.
Grunerite, known as amosite, is one type of asbestos with a tensile strength of 350 kg/mm². This means a strand of grunerite with a 1 mm² cross-sectional area can hold up to 350 kg before breaking. Due to its health dangers, asbestos use has been heavily regulated and mostly banned worldwide.

• Key takeaway 1: Asbestos is highly durable.
• Key takeaway 2: It poses severe health risks when airborne.
• Key takeaway 3: Regulations limit its use considerably.

unit conversion

Unit conversion is an essential skill in fields like physics and engineering, enabling the comparison of different measurements. In the given problem, we need to convert tensile strength from kg/mm² to lbs/in². The key conversions used are:
1 kg = 2.20462 lbs
1 mm² = 0.00155 in²
Using these conversions, we find that the tensile strength of grunerite, 350 kg/mm², converts to approximately 11,900 lbs/in².
This conversion is crucial for comparing grunerite's tensile strength with materials like aluminum and steel, which are measured in lbs/in². Understanding and performing these conversions correctly ensures accurate and meaningful comparisons.

• Step-by-step guide: First, multiply the force by the kg-to-lb conversion factor.
• Then, multiply by the mm²-to-in² conversion factor.

• Key takeaway: Proper unit conversion allows for accurate comparisons across different systems.

cross-sectional area

The concept of cross-sectional area is central to understanding tensile strength calculations. The cross-sectional area of a material, usually measured in square millimeters (mm²) or square inches (in²), is the area of a slice perpendicular to the length of the material.

• For calculation purposes, we use this area to determine how a material will perform under stress. For example, in the problem, we find the cross-sectional areas necessary for aluminum and steel to match the tensile strength of a grunerite fiber with a given area.
• For aluminum (with a tensile strength of 25,000 lbs/in²), we calculate a required area of approximately 306.3 mm².
• For Steel No. 5137 (with a tensile strength of 50,000 lbs/in²), we calculate a required area of about 153.52 mm².

This ensures that both materials will have the same maximum load-bearing capacity as the grunerite strand. Properly calculating and understanding cross-sectional areas is vital for designing and assessing materials in engineering and construction.

• Key takeaway: Cross-sectional area helps determine a material's tensile strength and load-bearing capabilities.