91Ó°ÊÓ

A simple harmonic oscillator at the point \(x=0\) generates a wave on a rope. The oscillator operates at a frequency of 40.0 \(\mathrm{Hz}\) and with an amplitude of 3.00 \(\mathrm{cm} .\) The rope has a linear mass density of 50.0 \(\mathrm{g} / \mathrm{m}\) and is stretched with a tension of 5.00 \(\mathrm{N}\) .(a) Determine the speed of the wave. (b) Find the wavelength. (c) Write the wave function \(y(x, t)\) for the wave, Assume that the oscillator has its maximum upward displacement at time \(t=0\) . (d) Find the maximum transverse acceleration of points on the rope. (e) In the discussion of transverse waves in this chapter, the force of gravity was ignored. Is that a reasonable approximation for this wave? Explain.

Step by step solution

01

Calculate the Speed of the Wave

The speed of a wave on a rope is determined by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the rope (5.00 N) and \( \mu \) is the linear mass density of the rope (0.050 kg/m). Convert the mass density from g/m to kg/m first if needed. Plug in the values:\[ v = \sqrt{\frac{5.00}{0.050}} = \sqrt{100} = 10.0 \, \text{m/s} \].Thus, the speed of the wave is 10.0 m/s.

02

Calculate the Wavelength

The wavelength \( \lambda \) of the wave is found using the wave speed and frequency. Use the formula \( \lambda = \frac{v}{f} \), where \( v = 10.0 \, \text{m/s} \) (from Step 1) and \( f = 40.0 \, \text{Hz} \). Substitute these values:\[ \lambda = \frac{10.0}{40.0} = 0.25 \, \text{m} \].The wavelength of the wave is 0.25 m.

03

Write the Wave Function

To write the wave function \( y(x,t) \), use the standard form \( y(x,t) = A \sin(kx - \omega t + \phi) \). The amplitude \( A \) is given as 0.03 m. The wave number \( k = \frac{2\pi}{\lambda} \) and the angular frequency \( \omega = 2\pi f \). Since the maximum displacement is at \( t=0 \), \( \phi \) is zero. Calculate:\[ k = \frac{2\pi}{0.25} = 8\pi \, \text{m}^{-1} \],\[ \omega = 2\pi \times 40 = 80\pi \, \text{rad/s} \].The wave function is \( y(x, t) = 0.03 \sin(8\pi x - 80\pi t) \).

04

Find Maximum Transverse Acceleration

The maximum transverse acceleration \( a_{max} \) is given by \( a_{max} = A\omega^2 \). Using \( A = 0.03 \, \text{m} \) and \( \omega = 80\pi \, \text{rad/s} \):\[ a_{max} = 0.03 \times (80\pi)^2 = 0.03 \times 6400\pi^2 \approx 1897.96 \, \text{m/s}^2 \].The maximum transverse acceleration is approximately 1897.96 m/s².

05

Evaluate the Effect of Gravity on the Wave

Gravity's influence is generally negligible for transverse waves on a rope, especially when the tension is significantly higher compared to gravitational force. The force of gravity can affect the rope's overall tension slightly, but as long as the tension in the rope is maintained (as given by 5.00 N), ignoring gravity is a reasonable approximation. This can be significant only if the wave's vertical span reaches significant proportions of the rope's length or when the tension is low.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Harmonic Oscillators

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. In our exercise, this concept is represented by the oscillator that generates the wave on the rope. Harmonic oscillators are foundational in understanding wave mechanics since they help to produce periodic motion, leading to wave generation.

Key characteristics:

• Frequency (40 Hz in this context) which describes how often the motion occurs each second.
• Amplitude (3.00 cm here), which is the maximum extent of the oscillation from the equilibrium position.

The oscillator's consistent periodic motion is crucial for producing a wave with repetitive and predictable behavior. This repetitiveness and predictability are what make harmonic oscillations essential to understanding wave mechanics, especially in simple systems like ropes or strings.

Wave Speed

Wave speed is a measure of how fast the wave travels through a medium. It's influenced by factors like tension in the medium and its linear mass density. In our context, the wave's speed can be calculated using the formula \[v = \sqrt{\frac{T}{\mu}},\] where \( T \) represents the tension applied to the rope and \( \mu \) is the linear mass density.

In practice:

• The given tension is 5.00 N, and after converting mass density to kilograms per meter, we have \( \mu = 0.050 \) kg/m.
• Substituting these values gives a wave speed of 10.0 m/s.

This speed determines how quickly wave crests move along the rope, providing insights into energy propagation and wave dynamics within the medium. Understanding wave speed is essential as it dictates not just travel time, but also the energy distribution along the medium.

Wave Function

The wave function describes the displacement of points in the medium through which the wave travels, as functions of both time and position. It provides an equation to predict where each point on the medium will be at any time.

In our problem:

• The wave function is given by \( y(x,t) = A \sin(kx - \omega t + \phi) \).
• The amplitude \( A \) is 0.03 m, which refers to the highest point of the wave.
• The wave number \( k = \frac{2\pi}{\lambda} = 8\pi \) m\(^{-1}\) and angular frequency \( \omega = 80\pi \) rad/s are calculated considering wavelength and frequency respectively.
• Since the maximum displacement occurs at \( t=0 \), the phase shift \( \phi \) is zero.

The resulting wave function: \( y(x, t) = 0.03 \sin(8\pi x - 80\pi t) \).

This function is fundamental for predicting the wave's behavior across different points and times, central to both theoretical and practical wave analysis.

Transverse Waves

Transverse waves are waves in which the oscillation or vibration occurs perpendicular to the direction of the wave's advance. In the example with the rope, each point on the rope moves up and down while the wave travels horizontally.

Features include:

• The maximum transverse acceleration, essential for understanding how quickly points move along their path. This is calculated with \( a_{max} = A\omega^2 \) yielding around 1897.96 m/s² in this problem.
• Ignoring gravity's role is typically a good assumption unless the rope's tension is low or the wave's vertical motion is substantial compared to the rope’s length.

Recognizing these characteristics helps in comprehending how energy and motion manifest in waves, enabling clear visualization and application of how waves interact in various media.