/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 A heavy rope \(6.00 \mathrm{~m}\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 20

A heavy rope \(6.00 \mathrm{~m}\) long and weighing \(29.4 \mathrm{~N}\) is attached at one end to a ceiling and hangs vertically. A \(0.500 \mathrm{~kg}\) mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the (a) bottom of the rope, (b) middle of the rope, and (c) top of the rope? (d) Is the tension in the middle of the rope the average of the tensions at the top and bottom of the rope? Is the wave speed at the middle of the rope the average of the wave speeds at the top and bottom? Explain.

Short Answer

Expert verified
The wave speed varies along the rope鈥檚 length due to the variation in tension, with higher speed at the top due to higher tension and lower speed at the bottom because of lower tension. The tension or wave speed at the middle isn't precisely the average of its corresponding values at the top and bottom due to the non-linear relationship with the square root in the wave speed equation.

Step by step solution

01

Calculate Linear Mass Density

Calculate the linear mass density (\(\mu\)) of the rope using its direct weight and its length \(\mu = \frac{W}{g \cdot L}\) where \(W = 29.4 \, \mathrm{N}\), \(L = 6 \, \mathrm{m}\), and \(g = 9.8 \, \mathrm{m/s^2}\).
02

Calculate Tension at Different Locations

Find out the tension at the bottom, middle, and top ends of the rope. At the bottom (\(T_b\)), the tension comes from the weight of the suspended mass (\(m = 0.500 \, \mathrm{kg}\)) i.e., \(T_b = m \cdot g\). For the middle (\(T_m\)), the tension is contributed by the weight of the rope and mass below it i.e., \(T_m = (m + \frac{W}{g} \cdot \frac{L}{2}) \cdot g\). But at the top (\(T_t\)), the tension is due to the total weight of the rope and the mass i.e., \(T_t = (m + \frac{W}{g}) \cdot g\).
03

Calculate Wave Speed at Different Locations

Calculate the wave speed (\(v\)) for bottom, middle, and top regions using the equation \(v = \sqrt{\frac{T}{\mu}}\).
04

Compare Tensions and Wave Speeds

Establish relationships between tensions and wave speeds at different locations. For tensions, \(T_t > T_m > T_b\). Now, if the wave speeds obey the same relation at these locations, it means that the wave speed is directly proportional to the square root of tension. At the same time, determine whether the tension or speed at the middle point is the average of the values at the top and bottom.

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 91影视!

Key Concepts

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

Transverse Waves
Transverse waves are waves that move perpendicular to the direction of the wave itself. A simple example of a transverse wave is a wave moving along a rope. Here, the fluctuations occur up and down, while the wave travels horizontally through the medium.
Transverse waves are vital in understanding wave dynamics, especially on ropes and strings where the medium provides the directional path.
  • In the classroom, this type of motion is illustrated using a shaking rope, where the peaks and troughs are visually evident.
  • The wave speed depends on both the tension applied to the medium and the medium's mass characteristics.
Understanding these waves is crucial for examining how tension and mass density affect their properties.
Tension in Ropes
When considering waves in ropes, understanding tension is essential. Tension is the force exerted by the rope when it is pulled tight by forces acting from opposite ends.
In the scenario provided with a rope hanging vertically, the tension varies along its length due to gravitational forces.
  • At the bottom of the rope, tension is primarily due to the weight of any additional mass, here calculated as the product of mass and gravity: \( T = m \cdot g \).
  • Moving upwards, the tension increases because it includes the weight of the rope itself below that point.
  • At the top, the tension is at its maximum, balancing the entire weight of the rope plus any suspended mass.
Understanding how tension varies along a rope helps in calculating how waves propagate through the rope.
Linear Mass Density
Linear mass density (\( \mu \)) is a measure of a rope鈥檚 mass per unit length. This is a significant factor when considering wave speed, as it influences how quickly a wave can move through the object.
It is calculated using the formula: \( \mu = \frac{W}{g \cdot L} \), where \( W \) is the weight, \( g \) is the acceleration due to gravity, and \( L \) is the length.
  • Higher linear mass densities imply heavier ropes, making it challenging for the wave to propagate quickly.
  • Lower densities, like lightweight strings, allow faster wave movement.
  • Changes in density can alter how waves behave at different sections of a rope, affecting overall communication between the rope's ends.
Examining linear mass density is crucial when solving problems involving the speed of waves in ropes.
Harmonic Motion
Harmonic motion is a form of periodic motion where an object moves back and forth through an equilibrium position in a regular pattern.
In the context of waves in ropes, harmonic motion is essential for understanding how waves oscillate. The response of the medium follows the principles of harmonic motion, typically resulting in sinusoidal waves.
  • Each point on the rope undergoes repetitive, periodic motion, creating a continuous wave.
  • Wave speed is a key aspect of harmonic motion, influenced by both the tension and linear mass density of the rope.
  • The regularity and predictability of harmonic motion help in accurately modeling and solving physical problems involving waves.
Mastery of harmonic motion is vital for comprehending wave phenomena and solving related exercises effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For a violin, estimate the length of the portions of the strings that are free to vibrate. (a) The frequency of the note played by the open E5 string vibrating in its fundamental standing wave is 659 Hz. Use your estimate of the length to calculate the wave speed for the transverse waves on the string. (b) The vibrating string produces sound waves in air with the same frequency as that of the string. Use \(344 \mathrm{~m} / \mathrm{s}\) for the speed of sound in air and calculate the wavelength of the E5 note in air. Which is larger: the wavelength on the string or the wavelength in air? (c) Repeat parts (a) and (b) for a bass viol, which is typically played by a person standing up. Start your calculation by estimating the length of the bass viol string that is free to vibrate. The G2 string produces a note with frequency \(98 \mathrm{~Hz}\) when vibrating in its fundamental standing wave.

Ultrasound Imaging. Sound having frequencies above the range of human hearing (about \(20,000 \mathrm{~Hz}\) ) is called ultrasound. Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of \(1500 \mathrm{~m} / \mathrm{s}\) For a good, detailed image, the wavelength should be no more than 1.0 \(\mathrm{mm} .\) What frequency sound is required for a good scan?

(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?

You are exploring a newly discovered planet. The radius of the planet is \(7.20 \times 10^{7} \mathrm{~m}\). You suspend a lead weight from the lower end of a light string that is \(4.00 \mathrm{~m}\) long and has mass \(0.0280 \mathrm{~kg}\). You measure that it takes \(0.0685 \mathrm{~s}\) 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 \(0.0390 \mathrm{~s}\) 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?

Adjacent antinodes of a standing wave on a string are \(15.0 \mathrm{~cm}\) apart. A particle at an antinode oscillates in simple harmonic motion with amplitude \(0.850 \mathrm{~cm}\) and period \(0.0750 \mathrm{~s}\). The string lies along the \(+x\) -axis and is fixed at \(x=0 .\) (a) How far apart are the adjacent nodes? (b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern? (c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Astrophysics

Read Explanation

Fields in Physics

Read Explanation

Nuclear Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Wave Optics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.