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Chapter 16: Problem 43

What are (a) the lowest frequency, (b) the second lowest frequency, and (c) the third lowest frequency for standing waves on a wire that is \(10.0 \mathrm{~m}\) long. has a mass of \(100 \mathrm{~g}\), and is stretched under a tension of \(250 \mathrm{~N} ?\)

Short Answer

Expert verified
The lowest frequency is 7.91 Hz, the second lowest is 15.82 Hz, and the third lowest is 23.73 Hz.

Step by step solution

01

Understanding the Problem

We need to find the frequencies of the standing waves on a wire. The wire's length is \(10.0\, \text{m}\), its mass is \(100 \text{ g} = 0.1\, \text{kg}\), and it's under a tension of \(250 \text{ N}\).
02

Find Linear Mass Density

Linear mass density \(\mu\) is the mass per unit length of the wire. It is given by \( \mu = \frac{m}{L} \). Substituting the given values, \( \mu = \frac{0.1}{10} = 0.01\, \text{kg/m} \).
03

Calculate Wave Speed

The speed of waves on a stretched wire is given by \( v = \sqrt{\frac{T}{\mu}} \). Substitute the tension \(T = 250\, \text{N}\) and \(\mu = 0.01\, \text{kg/m}\), we get: \[ v = \sqrt{\frac{250}{0.01}} = \sqrt{25000} = 158.11\, \text{m/s} \]
04

Calculate Fundamental Frequency (Lowest Frequency)

The fundamental frequency (first harmonic) \( f_1 \) is given by: \( f_1 = \frac{v}{2L} \). Substituting the values, we find: \[ f_1 = \frac{158.11}{2 \times 10} = 7.91\, \text{Hz} \]
05

Calculate Second Harmonic Frequency (Second Lowest Frequency)

The second harmonic frequency \( f_2 \) is twice the fundamental frequency: \[ f_2 = 2 \times f_1 = 2 \times 7.91 = 15.82\, \text{Hz} \]
06

Calculate Third Harmonic Frequency (Third Lowest Frequency)

The third harmonic frequency \( f_3 \) is three times the fundamental frequency: \[ f_3 = 3 \times f_1 = 3 \times 7.91 = 23.73\, \text{Hz} \]

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Key Concepts

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

Wave Speed
Wave speed is the speed at which waves travel along a medium, in this case, a wire. It is a crucial concept in understanding standing waves.
The wave speed on a wire is determined by two main factors: tension and the linear mass density of the wire. The formula used to calculate wave speed is
  • \[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the wire, and \( \mu \) is the linear mass density.
In our example, the tension is given as 250 N, and the wave speed was calculated to be 158.11 m/s using this formula.
Understanding wave speed is essential because it directly influences the frequency of the standing waves forming on the wire, which we'll explore more in the following sections.
Harmonic Frequencies
Harmonic frequencies occur when waves on a medium, like a wire, form resonating standing wave patterns. These frequencies are integral multiples of the fundamental frequency.
The fundamental frequency is also known as the first harmonic. For a wire of length \( L \), the formula for the fundamental frequency \( f_1 \) is
  • \[ f_1 = \frac{v}{2L} \]
where \( v \) is the wave speed. This leads to the calculation of higher harmonic frequencies:
  • The second harmonic, \( f_2 \), is twice the fundamental: \( f_2 = 2 \times f_1 \)
  • The third harmonic, \( f_3 \), is three times the fundamental: \( f_3 = 3 \times f_1 \)
By substituting the known wave speed and wire length into this formula, we calculated frequencies of 7.91 Hz, 15.82 Hz, and 23.73 Hz for the first three harmonics.
These harmonic frequencies are important when tuning musical instruments or engineering specific resonance conditions.
Linear Mass Density
Linear mass density, symbolized as \( \mu \), is the mass of a wire per unit of length. It is calculated using the equation:
  • \[ \mu = \frac{m}{L} \]
where \( m \) is the mass, and \( L \) is the length of the wire.
In our problem, the mass of the wire is 0.1 kg, and its length is 10 m, resulting in a linear mass density of 0.01 kg/m.
Linear mass density is a key factor in determining wave speed, which in turn affects the harmonic frequencies.
A higher linear mass density indicates a heavier wire for the same length and leads to a slower wave speed.
Understanding this relationship helps in designing systems where precise control of wave characteristics is necessary, like in manufacturing musical strings or various engineering applications.

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Most popular questions from this chapter

The equation of a transverse wave traveling along a very long string is \(y=6.0 \sin (0.020 \pi x+4.0 \pi t),\) where \(x\) and \(y\) are expressed in centimeters and \(t\) is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave, and (f) the maximum transverse speed of a particle in the string. (g) What is the transverse displacement at \(x=3.5 \mathrm{~cm}\) when \(t=0.26 \mathrm{~s} ?\)

Four waves are to be sent along the same string, in the same direction: \(y_{1}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t)\) \(y_{2}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.7 \pi)\) \(y_{3}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+\pi)\) \(y_{4}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+1.7 \pi) .\) What is the amplitude of the resultant wave?

A wave has an angular frequency of \(110 \mathrm{rad} / \mathrm{s}\) and a wavelength of \(1.80 \mathrm{~m}\). Calculate (a) the angular wave number and (b) the speed of the wave.

A \(120 \mathrm{~cm}\) length of string is stretched between fixed supports. What are the (a) longest, (b) second longest, and (c) third longest wavelength for waves traveling on the string if standing waves are to be set up? (d) Sketch those standing waves.

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of \(347 \mathrm{~m}\), a linear density of \(3.35 \mathrm{~kg} / \mathrm{m},\) and a tension of \(65.2 \mathrm{MN},\) what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes?
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