/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 A steel cable has a cross-sectio... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 18

A steel cable has a cross-sectional area \(2.83 \times 10^{-3} \mathrm{m}^{2}\) and is kept under a tension of \(1.00 \times 10^{4} \mathrm{N}\). The density of steel is \(7860 \mathrm{kg} / \mathrm{m}^{3}\). Note that this value is not the linear density of the cable. At what speed does a transverse wave move along the cable?

Short Answer

Expert verified
The speed of the transverse wave is approximately 21.21 m/s.

Step by step solution

01

Identify the formula for wave speed

The speed of a transverse wave on a stretched string is given by the formula:\[ v = \sqrt{\frac{T}{\mu}} \]where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear density of the string.
02

Calculate the linear density

The linear density \( \mu \) of the cable can be determined from the cross-sectional area \( A \) and the density \( \rho \) of the steel:\[ \mu = \rho \times A \]Substitute the given values:\[ \mu = 7860 \mathrm{kg/m}^3 \times 2.83 \times 10^{-3} \mathrm{m}^2 \]\[ \mu = 22.2318 \mathrm{kg/m} \]
03

Substitute the values into the wave speed formula

Using the formula for wave speed:\[ v = \sqrt{\frac{T}{\mu}} \]Substitute \( T = 1.00 \times 10^4 \mathrm{N} \) and \( \mu = 22.2318 \mathrm{kg/m} \):\[ v = \sqrt{\frac{1.00 \times 10^4 \mathrm{N}}{22.2318 \mathrm{kg/m}}} \]
04

Solve for wave speed

Calculate the value inside the square root:\[ \frac{1.00 \times 10^4}{22.2318} \approx 449.98 \]Now take the square root of 449.98:\[ v \approx \sqrt{449.98} \approx 21.21 \mathrm{m/s} \]

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Key Concepts

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

Transverse Wave
A transverse wave is a wave where the particle displacement is perpendicular to the direction of wave propagation. Imagine shaking one end of a string up and down, the wave travels horizontally along the string, transporting energy without transporting matter. This is how transverse waves work.
  • Energy Transmission: The main purpose of a transverse wave is to transfer energy from one point to another.
  • Displacement: Occurs perpendicular to the motion, such as ripples on the water, vibrating strings, and light waves.
  • Characteristics: Each point on the wave moves up and down as the wave travels horizontally.
Transverse waves differ from longitudinal waves, which move particles in the same direction as the wave. Understanding this helps in visualizing how waves, such as those on a string, appear and function.
Linear Density
Linear density, symbolized by \( \mu \), is a measure of mass per unit length of a string or cable. It indicates how much material is packed along a given length, essential for calculating wave speed.
  • Calculation: Determined by multiplying material density (\( \rho \)) by the cross-sectional area (\( A \)). For the steel cable exercise, \( \mu = 7860 \text{ kg/m}^3 \times 2.83 \times 10^{-3} \text{ m}^2 = 22.2318 \text{ kg/m} \).
  • Significance: High linear density implies a heavier cable, affecting how fast a wave can travel through it.
  • Units: Typically expressed in kilograms per meter (kg/m).
Knowing the linear density helps in determining the speed of waves on strings, crucial in musical instruments and engineering applications.
Tension in String
Tension in a string or cable plays a pivotal role in wave dynamics, particularly for transverse waves. It is essentially the force exerted along the length of the string.
  • Effect on Wave Speed: Higher tension generally leads to an increase in wave speed. It means more energy is being transmitted along the string.
  • Direct Relationship: The wave speed \( v \) is calculated using the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension.
  • Units and Measurement: Expressed in newtons (N), it accounts for the force trying to pull the string apart.
Balancing tension correctly can manipulate wave speeds, which is useful in designing cables and musical strings to achieve desired performance characteristics.

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Most popular questions from this chapter

A wave traveling in the \(+x\) direction has an amplitude of \(0.35 \mathrm{m}\), a speed of \(5.2 \mathrm{m} / \mathrm{s},\) and a frequency of \(14 \mathrm{Hz}\). Write the equation of the wave in the form given by either Equation 16.3 or 16.4

A monatomic ideal gas \((\gamma=1.67)\) is contained within a box whose volume is \(2.5 \mathrm{m}^{3} .\) The pressure of the gas is \(3.5 \times 10^{5} \mathrm{Pa} .\) The total mass of the gas is \(2.3 \mathrm{kg}\). Find the speed of sound in the gas.

Light is an electromagnetic wave and travels at a speed of \(3.00 \times\) \(10^{8} \mathrm{m} / \mathrm{s} .\) The human eye is most sensitive to yellow- green light, which has a wavelength of \(5.45 \times 10^{-7} \mathrm{m} .\) What is the frequency of this light?

Hearing damage may occur when a person is exposed to intensity level at the center of the field is \(60.0 \mathrm{dB}\). When all the people shout togcther, the intensity level increases to 109 dB. Assuming that each person generates the same sound intensity at the center of the field, how many people are at the game?

An observer stands \(25 \mathrm{m}\) behind a marksman practicing at a rifle range. The marksman fires the rifle horizontally, the speed of the bullets is \(840 \mathrm{m} / \mathrm{s},\) and the air temperature is \(20^{\circ} \mathrm{C} .\) How far does each bullet travel before the observer hears the report of the rifle? Assume that the bullets encounter no obstacles during this interval, and ignore both air resistance and the vertical component of the bullets" motion.
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