/*! 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 46 How many nodes are there when a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 46

How many nodes are there when a rope fixed at both ends vibrates in its fourth harmonic?

Short Answer

Expert verified
There are 5 nodes in the fourth harmonic.

Step by step solution

01

Understand Harmonics

Harmonics are the different frequencies at which a string or rope can vibrate. The first harmonic (fundamental frequency) has one antinode and two nodes (one at each fixed end). Each subsequent harmonic increases the number of antinodes by one.
02

Identify Number of Harmonics

For the fourth harmonic, we need to know the relationship between the harmonic number and the number of nodes.
03

Use the Formula

The number of nodes () in the nth harmonic is given by the formula = n + 1. For the fourth harmonic (n = 4), substitute the value into the formula: = 4 + 1.
04

Calculate

Calculate the number of nodes: = 4 + 1 = 5.

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.

Harmonic Frequencies
When a rope or string fixed at both ends vibrates, it can do so at several specific frequencies. These frequencies are known as harmonics. The fundamental frequency (or the first harmonic) is the lowest frequency at which the string vibrates. Higher harmonics are integer multiples of this fundamental frequency. For example:
  • The second harmonic is twice the frequency of the first.
  • The third harmonic is three times the frequency of the first, and so on.
These harmonic frequencies play a crucial role in phenomena like musical instruments, where the different harmonics give each instrument its unique sound. Understanding harmonic frequencies helps us predict and analyze the vibrational behavior of strings and other systems.
Nodes in Vibrating Strings
In the context of vibrating strings, nodes are points along the string that remain stationary and do not move. These points occur at fixed intervals along the string and are essential for forming standing waves. Each harmonic has a specific pattern of nodes and antinodes (points of maximum vibration).
  • For the first harmonic, there are two nodes: one at each end of the string.
  • As the harmonic number increases, so does the number of nodes.
The fourth harmonic, for instance, has five nodes (n + 1, where n is the harmonic number). Nodes are where the wave pattern crosses the mean position and are critical for determining the string's vibrational modes.
Fundamental Frequency
The fundamental frequency is the lowest possible frequency at which a string fixed at both ends can vibrate. This fundamental frequency, also known as the first harmonic, forms the basis for all other harmonics. When considering the fundamental frequency:
  • It has two nodes, one at each end of the string.
  • It has one antinode in the middle, where the vibration amplitude is highest.
The formula for the fundamental frequency depends on the string's length, tension, and mass per unit length. Harmonics beyond the fundamental frequency are whole-number multiples of this base frequency, creating more complex patterns of nodes and antinodes.

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 mass is oscillating up and down on a vertical spring. If the mass is increased, will the period of oscillation increase, decrease, or stay the same? Will the frequency increase, decrease, or stay the same? Explain.

What is the fundamental frequency on a 6 -m rope that is tied at both ends if the speed of the waves is \(18 \mathrm{m} / \mathrm{s} ?\)

How many antinodes are there when a rope fixed at both ends vibrates in its third harmonic?

A mass is oscillating up and down on a vertical spring. When the mass is below the equilibrium point and moving downward, what is the direction of its acceleration? Is the mass speeding up or slowing down? Explain.

If the breakers at a beach are separated by \(5 \mathrm{m}\) and hit shore with a frequency of \(0.3 \mathrm{Hz}\), at what speed are they traveling?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Kinematics Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Electrostatics

Read Explanation

Classical Mechanics

Read Explanation

Scientific Method 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.