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Chapter 16: Problem 3

A string fixed at both ends is \(7.50 \mathrm{~m}\) long and has a mass of \(0.120 \mathrm{~kg}\). It is subjected to a tension of \(96.0 \mathrm{~N}\) and set oscillating. (a) What is the speed of the waves on the string? (b) What is the longest possible wavelength for a standing wave? (c) Give the frequency of that wave.

Short Answer

Expert verified
(a) Wave speed = 40 m/s, (b) Longest wavelength = 15 m, (c) Frequency = 2.67 Hz.

Step by step solution

01

Identify Given Data

The length of the string \(L = 7.50 \text{ m}\), the mass of the string \(m = 0.120 \text{ kg}\), the tension \(T = 96.0 \text{ N}\). We will use these to find (a) wave speed, (b) longest wavelength, and (c) frequency.
02

Calculate Linear Density

The linear density \( \mu \) of the string is given by \( \mu = \frac{m}{L} = \frac{0.120\, \text{kg}}{7.50\, \text{m}} \). Calculate this to use in the wave speed formula.
03

Calculate Wave Speed

The speed of the wave \( v \) on a string is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension and \( \mu \) is the linear density calculated previously. Substitute the values to find \( v \).
04

Determine Longest Wavelength

In a fixed-fixed string, the longest possible wavelength for a standing wave is twice the length of the string. Thus, \( \lambda_{\max} = 2L = 2 \times 7.50 \text{ m} \).
05

Calculate Frequency of the Wave

Use the wave speed \( v \) and the longest wavelength \( \lambda_{\max} \) to find the frequency \( f \) using the formula \( f = \frac{v}{\lambda_{\max}} \). Insert values for \( v \) and \( \lambda_{\max} \) to calculate \( f \).

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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 a crucial concept in wave mechanics and refers to how fast a wave travels through a medium, like a string. In the case of a string fixed at both ends, the wave speed depends on two main factors: the tension in the string and the linear density of the string.

To find the wave speed (\( v \)), we use the formula:\[ v = \sqrt{\frac{T}{\mu}} \]where:
  • \( T \) is the tension in the string (measured in newtons, N), and
  • \( \mu \) is the linear density of the string (mass per unit length, measured in kilograms per meter, kg/m).

Linear density is calculated by dividing the mass of the string by its length. This speed is fundamentally how quickly each part of the string "communicates" the wave motion along its length. Understanding wave speed is essential for addressing problems involving oscillating strings.
Standing Wave
A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions along a medium, like a string. The result is a pattern that appears to be "standing still," characterized by nodes and antinodes.

Nodes are points where there is no movement, while antinodes are points with maximum movement. In a string fixed at both ends, standing waves are integral as they define the possible vibrational modes. The longest standing wave occurs when the string vibrates in its fundamental frequency, where there is only one node at each end. The pattern crucially depends on the string's fixed conditions and length.
Wavelength
Wavelength is the distance between two consecutive points in phase in a wave, such as between two crests or troughs. It's a vital property of waves that dictates many wave behaviors, including standing waves.

For a string fixed at both ends, the longest wavelength corresponds to the fundamental frequency. This occurs when the string vibrates in one complete cycle, with nodes only at the ends.

The formula for the maximum wavelength (\( \lambda_{\text{max}} \)) of a standing wave on a string is:\[ \lambda_{\text{max}} = 2L \]where:
  • \( L \) is the length of the string.

Knowing the wavelength helps in determining wave speed and frequency, connecting it to broader wave mechanics concepts.
Frequency
Frequency refers to how many complete waves, or cycles, pass a given point per second. It is measured in hertz (Hz) and is a crucial concept when dealing with oscillating strings.

For a string, the frequency of a standing wave is related both to its wave speed and its wavelength. The formula to compute the frequency (\( f \)) is:\[ f = \frac{v}{\lambda} \]where:
  • \( v \) is the wave speed,
  • \( \lambda \) is the wavelength of the wave.

Frequency determines the pitch of sound if the string produces sound waves. It's important to understand how frequency varies with changes in wave speed and wavelength.
Tension in String
Tension is the force transmitted through a string when it is pulled tight by forces acting from opposite ends. It's a fundamental factor affecting both wave speed and the formation of standing waves on the string.

Higher tension results in increased wave speed, assuming linear density remains constant. This is because greater tension means the waves can travel faster along a tighter string.

Mathematically, tension (\( T \)) directly influences wave speed as shown in the formula:\[ v = \sqrt{\frac{T}{\mu}} \]where:
  • \( \mu \) is the linear density.

Understanding tension's role is crucial in practical applications, from musical instruments to engineering systems where wave dynamics are important.

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Most popular questions from this chapter

A string oscillates according to the equation $$ y^{\prime}=(0.80 \mathrm{~cm}) \sin \left[\left(\frac{\pi}{3} \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(40 \pi \mathrm{s}^{-1}\right) t\right] $$ What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose superposition gives this oscillation? (c) What is the distance between nodes? (d) What is the transverse speed of a particle of the string at the position \(x=2.1 \mathrm{~cm}\) when \(t=0.50 \mathrm{~s} ?\)

A wave has an angular frequency of \(110 \mathrm{rad} / \mathrm{s}\) and a wavelength of \(1.50 \mathrm{~m}\). Calculate (a) the angular wave number and (b) the speed of the wave.

Two waves are generated on a string of length \(4.0 \mathrm{~m}\) to produce a three-loop standing wave with an amplitude of \(1.0 \mathrm{~cm}\). The wave speed is \(100 \mathrm{~m} / \mathrm{s}\). Let the equation for one of the waves be of the form \(y(x, t)=y_{m} \sin (k x+\omega t)\). In the equation for the other wave, what are (a) \(y_{m}\), (b) \(k\), (c) \(\omega\), and (d) the sign in front of \(\omega\) ?

The equation of a transverse wave traveling along a very long string is \(y=3.0 \sin (0.020 \pi x-4.0 \pi t)\), 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: $$ \begin{aligned} &y_{1}(x, t)=(5.00 \mathrm{~mm}) \sin (4 \pi x-400 \pi t) \\ &y_{2}(x, t)=(5.00 \mathrm{~mm}) \sin (4 \pi x-400 \pi t+0.8 \pi) \\ &y_{3}(x, t)=(5.00 \mathrm{~mm}) \sin (4 \pi x-400 \pi t+\pi) \\ &y_{4}(x, t)=(5.00 \mathrm{~mm}) \sin (4 \pi x-400 \pi t+1.8 \pi) \end{aligned} $$ What is the amplitude of the resultant wave?
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