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A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Short Answer

Expert verified
The fundamental has \( \lambda = 3.00 \) m and \( f \approx 20.67 \) Hz; the second overtone has \( \lambda = 1.00 \) m and \( f = 62.0 \) Hz; the fourth harmonic has \( \lambda = 0.75 \) m and \( f \approx 82.67 \) Hz.

Step by step solution

01

Understanding Wave Properties on a String

For a string fixed at both ends, the wavelengths of the standing waves are given by \( \lambda_n = \frac{2L}{n} \), where \( L = 1.50 \) m is the length of the string, and \( n \) is the harmonic number. The speed \( v \) of waves on the string is 62.0 m/s.
02

Calculating Wavelength for the Fundamental Frequency

The fundamental frequency, also known as the first harmonic, corresponds to \( n = 1 \). Use the formula \( \lambda_1 = \frac{2 \times 1.50}{1} = 3.00 \text{ m} \).
03

Finding Fundamental Frequency from Wave Speed and Wavelength

The frequency \( f_n \) of a wave is given by \( f_n = \frac{v}{\lambda_n} \). For the fundamental frequency: \( f_1 = \frac{62.0}{3.00} \approx 20.67 \text{ Hz} \).
04

Calculating Wavelength for the Second Overtone

The second overtone is the third harmonic, which corresponds to \( n = 3 \). Use the formula \( \lambda_3 = \frac{2 \times 1.50}{3} = 1.00 \text{ m} \).
05

Finding Frequency for the Second Overtone

For the second overtone: \( f_3 = \frac{62.0}{1.00} = 62.0 \text{ Hz} \).
06

Calculating Wavelength for the Fourth Harmonic

The fourth harmonic corresponds to \( n = 4 \). Use the formula \( \lambda_4 = \frac{2 \times 1.50}{4} = 0.75 \text{ m} \).
07

Finding Frequency for the Fourth Harmonic

For the fourth harmonic: \( f_4 = \frac{62.0}{0.75} \approx 82.67 \text{ Hz} \).

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Key Concepts

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

Fundamental Frequency
In the context of waves on a string, the fundamental frequency is the simplest form of vibration, where the string has the least amount of nodes and antinodes required to sustain a wave. This is also referred to as the first harmonic. The fundamental frequency is crucial because it sets the baseline for understanding other harmonics or overtones in a wave system. It is represented when the string completes exactly one half of a wavelength within the span of the string length.
To calculate the fundamental frequency (\(f_1\)), you use the equation \(f_1 = \frac{v}{\lambda_1}\). Here, \(v\) is the wave speed, and \(\lambda_1\) is the wavelength of the fundamental frequency. This calculation helps us understand how fast the wave oscillates at its simplest harmonic state.
Harmonics
Harmonics refer to the multiple frequencies at which a system, such as a string, naturally oscillates. Beyond the fundamental frequency, there exist other harmonics that make the string vibrate in more complex patterns. These are the 2nd, 3rd, etc., harmonics, also called overtones.
The general expression for the wavelength of harmonics in a string fixed at both ends is given by \(\lambda_n = \frac{2L}{n}\), where \(L\) is the length of the string, and \(n\) is the harmonic number. Harmonics are integral because they embody the different modes of vibration a string can possess. Each mode comes with its own unique wavelength and frequency. For example, the third harmonic has three half-wavelength segments in the length of the string, leading to a different wavelength and frequency than the fundamental.
Wave Speed
Wave speed is an essential property that defines how quickly a wave travels through a medium. In this problem, we consider transverse waves on a string. The wave speed (\(v\)) depends on both the tension of the string and its linear density, although these specifics are not covered in our direct calculation.
To find the wave speed, the equation \(v = f \times \lambda\) is utilized, where \(f\) is the frequency and \(\lambda\) is the wavelength of the wave. This relation highlights the interdependence of wave properties and allows us to determine any one property if the others are known. In our problem, the speed is given by 62.0 m/s, helping us calculate frequencies and wavelengths of different harmonics.
Wave Properties
Wave properties encompass various aspects such as wavelength, frequency, amplitude, and speed. Each of these contributes to the behavior and characteristics of the wave.
  • Wavelength (\(\lambda\)): This is the distance over which the wave's shape repeats. Shorter wavelengths mean more waves are packed into a given space.
  • Frequency (\(f\)): This defines how many wave cycles pass a point in a given time period, usually a second. It is measured in hertz (Hz).
  • Amplitude: Though not directly covered in the problem, amplitude represents the wave's height and is related to the wave's energy.
  • Speed (\(v\)): This is the rate at which a wave propagates through a medium. Relating to all other properties through the formula \(v = f \times \lambda\), it underscores the interconnectedness of the elements.
Understanding these properties allows us to dissect and predict wave behavior in different mediums and situations.

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Most popular questions from this chapter

(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed \(v\), frequency \(f\), amplitude \(A\), and wavelength \(\lambda\). Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) \(x = \lambda/2\), (ii) \(x = \lambda\)/4, and (iii) \(x = \lambda\)8, from the left-hand end of the string. (b) At each of the points in part (a), what is the amplitude of the motion? (c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

A large rock that weighs 164.0 N is suspended from the lower end of a thin wire that is 3.00 m long. The density of the rock is 3200 kg/m\(^3\). The mass of the wire is small enough that its effect on the tension in the wire can be ignored. The upper end of the wire is held fixed. When the rock is in air, the fundamental frequency for transverse standing waves on the wire is 42.0 Hz. When the rock is totally submerged in a liquid, with the top of the rock just below the surface, the fundamental frequency for the wire is 28.0 Hz. What is the density of the liquid?

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 Hz. The other end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/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 kg?

You are exploring a newly discovered planet. The radius of the planet is \(7.20 \times 10^7\) m. You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You measure that it takes 0.0685 s for a transverse pulse to travel from the lower end to the upper end of the string. On the earth, for the same string and lead weight, it takes 0.0390 s for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 70.0 Hz, amplitude 5.00 mm, and wavelength 0.600 m. (a) How long does it take the wave to travel a distance of 8.00 m along the length of the string? (b) How long does it take a point on the string to travel a distance of 8.00 m, once the wave train has reached the point and set it into motion? (c) In parts (a) and (b), how does the time change if the amplitude is doubled?

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