/*! 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 12 The mass of a string is 5.0 103 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 12

The mass of a string is 5.0 103 kg, and it is stretched so that the tension in it is 180 N. A transverse wave traveling on this string has a frequency of 260 Hz and a wavelength of 0.60 m. What is the length of the string?

Short Answer

Expert verified
The length of the string is approximately 676.72 meters.

Step by step solution

01

Understand the Problem

We are given the mass of the string (5.0 kg), tension in the string (180 N), frequency of the wave (260 Hz), and wavelength of the wave (0.60 m). We need to find the length of the string.
02

Find the Wave Speed

Use the wave speed formula, where wave speed \( v = \text{frequency} \times \text{wavelength} \). Thus, \( v = 260 \text{ Hz} \times 0.60 \text{ m} = 156 \text{ m/s} \).
03

Relate Wave Speed to Mass and Tension

The wave speed on a string is also given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension and \( \mu \) is the linear mass density of the string. Substitute \( v = 156 \text{ m/s} \) and \( T = 180 \text{ N} \) to find \( \mu \).
04

Solve for Linear Mass Density \( \mu \)

Rearrange the equation to find \( \mu \): \( \mu = \frac{T}{v^2} = \frac{180}{156^2} \approx 0.00739 \text{ kg/m} \).
05

Calculate the Length of the String

We use \( \mu = \frac{m}{L} \), where \( m = 5.0 \text{ kg} \) is the mass and \( L \) is the length of the string. Rearrange to find \( L: L = \frac{m}{\mu} = \frac{5.0}{0.00739} \approx 676.72 \text{ m} \).

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.

Wave Speed
Wave speed refers to how fast a wave travels along a medium, such as a string. It plays a crucial role in understanding wave behavior. In the context of a vibrating string, wave speed can be calculated using the formula:
\[ v = \text{frequency} \times \text{wavelength}, \]
where the frequency of the wave is how many oscillations occur per second (in Hz) and the wavelength is the distance between successive crests of the wave.
  • In our example, the frequency is 260 Hz, and the wavelength is 0.60 m.
  • Therefore, the wave speed: \( v = 260 \times 0.60 = 156 \text{ m/s} \).
Understanding the wave speed is essential because it helps link the physical properties of the string with how the wave propagates through it.
Linear Mass Density
Linear mass density is a measure of a string's mass per unit length. It significantly influences the wave speed on the string. A higher mass density results in slower wave propagation, while a lower density allows waves to travel faster.
The formula for linear mass density \( \mu \) is given as:
\[ \mu = \frac{m}{L}, \]
where \( m \) is the total mass of the string, and \( L \) is its length.
  • In this problem, the mass is 5.0 kg, and with a calculated \( \mu \) of 0.00739 kg/m, we can derive the string length.
  • Knowing \( \mu \) allows you to better understand how mass and tension interact to affect wave speed.
Tension in a String
Tension in a string directly impacts how quickly waves can travel through it. The force that stretches the string, known as tension \( T \), determines the string's stiffness, affecting the wave speed.
The relationship between wave speed, tension, and mass density is expressed as:
\[ v = \sqrt{\frac{T}{\mu}}. \]
  • For instance, with a tension of 180 N and a wave speed of 156 m/s, this formula helps calculate the linear mass density.
  • Greater tension typically increases wave speed because the wave force spreads more efficiently along a tighter, stiffer string.
Understanding tension is key to predicting wave behavior, making it an essential concept in wave dynamics.
Transverse Waves
Transverse waves are a type of wave where particle displacement is perpendicular to the direction of wave propagation. When a transverse wave travels along a string, it causes the string to move up and down, while the wave energy moves horizontally.
  • These waves are characterized by their wave speed, frequency, and wavelength.
  • Understanding transverse waves in the context of a string helps in visualizing the wave behavior and calculating essential properties like speed and tension.
Transverse waves are common in daily life, such as waves on a string instrument or on the surface of water, and provide a practical example of wave mechanics.

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

(a) A uniform rope of mass m and length L is hanging straight down from the ceiling. A small-amplitude transverse wave is sent up the rope from the bottom end. Derive an expression that gives the speed v of the wave on the rope in terms of the distance y above the bottom end of the rope and the magnitude g of the acceleration due to gravity. (b) Use the expression that you have derived to calculate the speeds at distances of 0.50 m and 2.0 m above the bottom end of the rope.

The middle C string on a piano is under a tension of 944 N. The period and wavelength of a wave on this string are 3.82 ms and 1.26 m, respectively. Find the linear density of the string.

At a distance of 3.8 \(\mathrm{m}\) from a siren, the sound intensity is \(3.6 \times 10^{-2} \mathrm{W} / \mathrm{m}^{2}\) . Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.

Multiple-Concept Example 11 provides a model for solving this type of problem. A wireless transmitting microphone is mounted on a small platform that can roll down an incline, directly away from a loudspeaker that is mounted at the top of the incline. The loudspeaker broadcasts a tone that has a fixed frequency of \(1.000 \times 10^{4} \mathrm{Hz}\) , and the speed of sound is 343 \(\mathrm{m} / \mathrm{s}\) . At a time of 1.5 s following the release of the platform, the microphone detects a frequency of 9939 \(\mathrm{Hz}\) . At a time of 3.5 s following the release of the platform, the microphone detects a frequency of 9857 Hz. What is the acceleration (assumed constant) of the platform?

The sound intensity level at a rock concert is 115 dB, while that at a jazz fest is 95 dB. Determine the ratio of the sound intensity at the rock concert to the sound intensity at the jazz fest.
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Famous Physicists

Read Explanation

Torque and Rotational Motion

Read Explanation

Electrostatics

Read Explanation

Kinematics Physics

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.