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Holding Up Under Stress. A string or rope will braak apart if it is placed under too much tensile stress \([\mathrm{Eq} \text { . }(11.8)]\) . Thicker ropes can withstand more tension without breaking because the thicker the rope, the greater the cross-sectional area and the smaller the stress. One type of steel has density 7800 \(\mathrm{kg} / \mathrm{m}^{3}\) and will break if the tensile stress exceeds \(7.0 \times 10^{8} \mathrm{N} / \mathrm{m}^{2} .\) You want to make a guitar string from 4.0 \(\mathrm{g}\) of this type of steel. In use, the guitar string must be able to withstand a tension of 900 \(\mathrm{N}\) without breaking. Your job is the following: (a) Determine the maximum length and minimum radius the string can have. (b) Determine the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate.

Step by step solution

01

Determine the Volume of the String

Start by converting the mass of the steel from grams to kilograms: 4.0 g = 0.004 kg. Given the density \( \rho = 7800 \, \text{kg/m}^3 \), use the formula for density, \( \rho = \frac{m}{V} \), to find the volume \( V \) of the string:\[ V = \frac{m}{\rho} = \frac{0.004}{7800} \approx 5.13 \times 10^{-7} \, \text{m}^3 \].

02

Determine the Cross-Sectional Area

The tensile stress \( \sigma \) is given by \( \sigma = \frac{F}{A} \), where \( F \) is the force and \( A \) is the cross-sectional area.Rearrange this formula to find \( A \):\[ A = \frac{F}{\sigma} = \frac{900}{7.0 \times 10^{8}} \approx 1.29 \times 10^{-6} \, \text{m}^2 \].

03

Calculate the Radius of the String

The cross-sectional area of a circular string is given by \( A = \pi r^2 \). Solve for the radius \( r \):\[ r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{1.29 \times 10^{-6}}{\pi}} \approx 6.41 \times 10^{-4} \, \text{m} \] or \( 0.641 \, \text{mm} \).

04

Calculate the Maximum Length of the String

Use the volume \( V = A \times L \) to find the maximum length \( L \):\[ L = \frac{V}{A} = \frac{5.13 \times 10^{-7}}{1.29 \times 10^{-6}} \approx 0.398 \, \, \text{m} \].

05

Calculate the Highest Fundamental Frequency

The fundamental frequency is given by \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension and \( \mu \) is the linear mass density. Calculate \( \mu \):\[ \mu = \frac{m}{L} = \frac{0.004}{0.398} \approx 0.0101 \, \text{kg/m} \].Substitute the values into the formula:\[ f = \frac{1}{2 \times 0.398} \sqrt{\frac{900}{0.0101}} \approx 475.5 \, \text{Hz} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density

Density is a core property of materials that describes how much mass is present within a certain volume. It is calculated using the formula \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. This concept is crucial when dealing with objects like a guitar string, where you need to know the available mass and its distribution over a specific volume.

• If a string is crafted from a type of steel with a density of 7800 kg/m³, it helps determine how much volume this mass will occupy.
• Using the mass of the steel (converted from grams to kilograms), you can find the volume by rearranging the density formula: \( V = \frac{m}{\rho} \).

Understanding density is essential in this case to figure out the dimensions of the string, ensuring it can hold the necessary tension without breaking.

Tension

Tension refers to the force applied along the length of an object, like a string, which pulls it taut. It's a critical factor for ensuring that materials remain intact under stress.

• In this scenario, tension plays a pivotal role in determining the strength and durability of the guitar string.
• The formula for tensile stress \( \sigma = \frac{F}{A} \) highlights how tension \( F \) relates to the cross-sectional area \( A \) and stress.

The tension must not exceed a certain threshold for the string to avoid snapping, and this helps in calculating the minimum cross-sectional area necessary to sustain a specific tension level.

Fundamental Frequency

The fundamental frequency refers to the lowest frequency at which a system, such as a guitar string, vibrates. This frequency is integral to the sound produced by the string.

• The formula \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \) calculates this frequency, where \( L \) is the string length, \( T \) is the tension, and \( \mu \) is the linear mass density.
• The string length and tension directly affect the frequency, with higher tension and shorter lengths typically producing higher frequencies.

This fundamental understanding helps in designing string instruments where the desired pitch or note is essential for musical harmony.

Cross-Sectional Area

The cross-sectional area of a string determines how well it can withstand tension before snapping. It is the area of the string slice you'd see if you cut the string perpendicular to its length. In the case of a circular cross-section, it is calculated using \( A = \pi r^2 \), where \( r \) is the radius.

• A larger cross-sectional area means that the string can withstand more force without experiencing too much stress.
• Knowing the tension and maximum stress a material can endure allows you to determine the necessary cross-sectional area to avoid breakage.

Calculating this ensures that the string is both sturdy and functional, meeting the physical demands of its intended use while maintaining performance.