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Chapter 12: Problem 5

A stecl wire 4.00 \(\mathrm{m}\) long has a mass of 0.0600 \(\mathrm{kg}\) and is stretched with a tension of 1000 \(\mathrm{N}\) . What is the speed of prop- agation of a transverse wave on the wire?

Short Answer

Expert verified
The speed of the wave is approximately 258.2 m/s.

Step by step solution

01

Calculate Linear Density

To find the speed of the wave, we first calculate the linear density (mass per unit length) of the wire, represented by \( \mu \). This is calculated using the formula \( \mu = \frac{m}{L} \) where \( m = 0.0600 \, \text{kg} \) is the mass of the wire and \( L = 4.00 \, \text{m} \) is the wire's length. Plugging in the values, we get \( \mu = \frac{0.0600}{4.00} = 0.0150 \, \text{kg/m} \).
02

Use the Wave Speed Formula

The speed of a transverse wave on a wire is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T = 1000 \, \text{N} \) is the tension and \( \mu = 0.0150 \, \text{kg/m} \) is the linear density. Substitute these values into the equation to get \( v = \sqrt{\frac{1000}{0.0150}} \).
03

Calculate the Square Root

Calculate the division first: \( \frac{1000}{0.0150} = 66666.67 \). Then, find the square root: \( v = \sqrt{66666.67} \approx 258.2 \, \text{m/s} \).

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Key Concepts

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

Linear Density
In the context of waves on a wire, understanding linear density is essential. Linear density, represented as \( \mu \), is the mass of the wire per unit length. It provides a means to relate mass and length in wave calculations.
To calculate linear density, use the formula \( \mu = \frac{m}{L} \). Here, \( m \) is the mass of the wire and \( L \) is its length.
In the exercise example, the mass \( m = 0.0600 \, \text{kg} \) and the length \( L = 4.00 \, \text{m} \). Thus, the linear density is \( \mu = \frac{0.0600}{4.00} = 0.0150 \, \text{kg/m} \). This value is crucial for further steps in calculating wave speed.
Wave Speed Formula
To find the speed of a transverse wave on a wire, we use the wave speed formula \( v = \sqrt{\frac{T}{\mu}} \). This formula highlights the relationship between tension, linear density, and wave speed.
  • \( v \) is the wave speed
  • \( T \) is the tension in the wire
  • \( \mu \) is the linear density, calculated earlier
For the given problem, the tension \( T = 1000 \, \text{N} \) and the linear density \( \mu = 0.0150 \, \text{kg/m} \).
By substituting these values, the equation becomes \( v = \sqrt{\frac{1000}{0.0150}} \). The solution to this equation provides the wave speed.
Tension in a Wire
Tension is a key factor in wave propagation on a wire. In this exercise, the tension is given as \( 1000 \, \text{N} \).
Tension affects how quickly a wave travels along the wire. Higher tension results in faster wave speed, while lower tension slows it down.
It is crucial for determining wave speed through the wave speed formula \( v = \sqrt{\frac{T}{\mu}} \). Thus, the accuracy of tension measurements directly impacts the calculation of wave speed.
Mass Per Unit Length
Mass per unit length is synonymous with linear density, \( \mu \). It helps in understanding how much mass is distributed along each meter of the wire. The calculation itself involves dividing the wire's mass by its length.
Knowing the mass per unit length is important in physics, especially in wave mechanics. It informs the behavior of waves on strings, wires, and even in air columns.
In scenarios like the exercise example, calculating mass per unit length provides the necessary data for further determining how fast a wave can travel across a medium such as a tensioned wire.

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Most popular questions from this chapter

A very noisy chain saw operated by a tree surgeon emits a total acoustic power of 20.0 \(\mathrm{W}\) uniformly in all directions. At what distance from the source is the sound level equal to (a) 100 \(\mathrm{dB}\) . (b) 60 \(\mathrm{dB} ?\)

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\(\bullet\) One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at 120 \(\mathrm{Hz}\) . The other end passes over a pulley and supports a 1.50 \(\mathrm{kg}\) mass. The linear mass density of the rope is 0.0550 \(\mathrm{kg} / \mathrm{m}\) . (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 \(\mathrm{kg}\) ?

\(\cdot\) Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 \(\mathrm{cm}\) long (a) if the pipe is open at both ends; (b) if the pipe is closed at one end. (c) For each of the preceding cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 \(\mathrm{Hz}\) to \(20,000 \mathrm{Hz}\) ?

\(\bullet\) Guitar string. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) frequency 245 Hz \()\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 \(\mathrm{m} / \mathrm{s}\) , find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?
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