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Chapter 16: Problem 7

A sinusoidal wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in \(30.0 \mathrm{s}\) Also, a given maximum travels \(425 \mathrm{cm}\) along the rope in \(10.0 \mathrm{s} .\) What is the wavelength?

Short Answer

Expert verified
The wavelength of the sinusoidal wave is \(0.32\) meters or \(32\) centimeters.

Step by step solution

01

Finding the Frequency

The frequency \(f\) of the wave is the number of vibrations divided by the time. In this case, the oscillator completes 40 vibrations in 30 seconds. Therefore, the frequency is given by \(f=\frac{40 \text{ vibrations}}{30 s}=1.33 Hz\).
02

Calculating Speed of the Wave

The speed \(v\) of the wave is calculated by dividing the distance it travels by the time taken. The given maximum of the wave travels 425 cm (or 4.25 meters) in 10 seconds, leading to a wave speed of \(v=\frac{425\text{ cm}}{10 s}=42.5 cm/s\) or \(v=\frac{4.25\text{ m}}{10 s}=0.425 m/s\).
03

Finding the Wavelength

The wavelength \(位\) of a wave is the speed of the wave divided by the frequency of the wave. With the calculated wave speed and frequency, the wavelength is given by \(位=\frac{v}{f}=\frac{0.425 m/s}{1.33 Hz}=0.32 m\) or \(32 cm\).

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Key Concepts

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

Understanding Frequency
Frequency is an important concept when dealing with waves. It tells us how many times something happens in a specific time period. For waves, it specifically tells us how many waves pass a point in one second.
Imagine an oscillator鈥攁 device that creates motion back and forth. When it vibrates, it creates waves. If it completes 40 vibrations in 30 seconds, we can calculate the frequency, denoted by the symbol \(f\), using:
  • Formula: \(f = \frac{\text{Number of Vibrations}}{\text{Time}}\)
  • Example Calculation: \(f = \frac{40}{30} = 1.33 \text{ Hz}\) (Hertz)
Frequency is measured in Hertz, where one Hertz (Hz) equals one cycle per second.
Understanding frequency helps us know how often a wave repeats itself over time.
Wave Speed Simplified
Wave speed is about how fast a wave travels from one point to another. It鈥檚 determined by how far the wave moves over a set period of time. Think of it like calculating the speed of a car, except instead of a car, we have a wave moving over a distance.
To find the wave speed \(v\), we use the formula:
  • Formula: \(v = \frac{\text{Distance}}{\text{Time}}\)
  • Example Calculation: Wave traveled 4.25 meters in 10 seconds, so \(v = \frac{4.25 \text{ m}}{10 \text{ s}} = 0.425 \text{ m/s}\)
Wave speed, therefore, reflects how quickly the wave's pattern progresses through space.
The Nature of Sinusoidal Waves
A sinusoidal wave is a smooth, periodic oscillation that looks like a sine wave if you plot it on a graph. These waves are common in nature and technology because they describe repetitive phenomena purely.
They can be produced by devices like oscillators. Here鈥檚 how they manifest:
  • They have peaks (crests) and valleys (troughs).
  • The distance between consecutive peaks or troughs is the wavelength.
  • The maximum height of the peaks or depth of the troughs represents amplitude.
Sinusoidal waves are the backbone of many physical phenomena like sound and light waves due to their repetitive and predictable patterns.
Exploring Oscillators
An oscillator is a system or device that moves or vibrates back and forth. It's like a pendulum on a clock or even a child swinging on a playground. These movements create waves in the medium they are attached to or affect.
Oscillators are the engine behind wave creation:
  • A steady, repetitive motion is generated.
  • This motion transmits energy through the medium.
  • In the case of a rope or string, it creates rolling waves along its length.
Understanding oscillators is key in physics and engineering, allowing us to create and control waves for various applications, such as in radio waves or electronic circuits.

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Most popular questions from this chapter

Consider the sinusoidal wave of Example \(16.2,\) with the wave function $$y=(15.0 \mathrm{cm}) \cos (0.157 x-50.3 t)$$ At a certain instant, let point \(A\) be at the origin and point \(B\) be the first point along the \(x\) axis where the wave is \(60.0^{\circ}\) out of phase with point \(A .\) What is the coordinate of point \(B ?\)

Sinusoidal waves \(5.00 \mathrm{cm}\) in amplitude are to be transmitted along a string that has a linear mass density of \(4.00 \times 10^{-2} \mathrm{kg} / \mathrm{m} .\) If the source can deliver a maximum power of \(300 \mathrm{W}\) and the string is under a tension of \(100 \mathrm{N}\) what is the highest frequency at which the source can operate?

For a certain transverse wave, the distance between two successive crests is \(1.20 \mathrm{m},\) and eight crests pass a given point along the direction of travel every 12.0 s. Calculate the wave speed.

A string on a musical instrument is held under tension \(T\) and extends from the point \(x=0\) to the point \(x=L .\) The string is overwound with wire in such a way that its mass per unit length \(\mu(x)\) increases uniformly from \(\mu_{0}\) at \(x=0\) to \(\mu_{L}\) at \(x=L .\) (a) Find an expression for \(\mu(x)\) as a function of \(x\) over the range \(0 \leq x \leq L\). (b) Show that the time interval required for a transverse pulse to travel the length of the string is given by $$\Delta t=\frac{2 L\left(\mu_{L}+\mu_{0}+\sqrt{\mu_{L} \mu_{0}}\right)}{3 \sqrt{T}(\sqrt{\mu_{L}}+\sqrt{\mu_{0}})}$$

At \(t=0,\) a transverse pulse in a wire is described by the function $$y=\frac{6}{x^{2}+3}$$ where \(x\) and \(y\) are in meters. Write the function \(y(x, t)\) that describes this pulse if it is traveling in the positive \(x\) direction with a speed of \(4.50 \mathrm{m} / \mathrm{s}\).
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