/*! 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 The tension in a wire clamped at... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 20

The tension in a wire clamped at both ends is doubled without appreciably changing the wire's length between the clamps. What is the ratio of the new to the old wave speed for transverse waves traveling along this wire?

Short Answer

Expert verified
The ratio of the new to the old wave speed is \( \sqrt{2} \).

Step by step solution

01

Understand the Relationship Between Tension and Wave Speed

The speed of a wave on a string or a wire, denoted as \( v \), is determined by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the wire and \( \mu \) is the linear density. Since the wire's length and mass do not change, \( \mu \) remains constant.
02

Express Wave Speeds With Old and New Tensions

Let \( T_1 \) be the old tension and \( T_2 = 2T_1 \) be the new tension. The old wave speed \( v_1 = \sqrt{\frac{T_1}{\mu}} \) and the new wave speed \( v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{2T_1}{\mu}} \).
03

Calculate the Ratio of New to Old Wave Speed

Simplify \( v_2 \) using the relationship between the new and old tensions: \( v_2 = \sqrt{2} \times \sqrt{\frac{T_1}{\mu}} = \sqrt{2} \times v_1 \).
04

Express the Final Ratio

The ratio of the new wave speed \( v_2 \) to the old wave speed \( v_1 \) is \( \frac{v_2}{v_1} = \sqrt{2} \).

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.

Tension
Tension is a force that stretches a material, often experienced in strings and wires. When you pull on a wire, you're applying tension. The amount of tension influences how tightly the wire is held together. In the context of waves, tension plays a crucial role in determining wave speed. The formula for wave speed on a string or wire is given by \( v = \sqrt{\frac{T}{\mu}} \). In this equation, \( T \) represents the tension. - As tension increases, the wave speed increases. - Conversely, less tension results in slower waves.Let's consider a scenario where the tension in a wire is doubled. This alteration has significant effects on how the wave travels through it. By understanding the relationship, we can make predictions about changes in wave behavior.
Transverse Waves
Transverse waves are a type of wave where the disturbance moves perpendicular to the direction of the wave's travel. Imagine a ripple moving through a pond; the water moves up and down, while the wave travels outward. Here鈥檚 what you need to know about transverse waves in wires: - They require tension in the medium to propagate effectively. - The disturbance occurs at right angles to the direction of wave movement. Consider the wire under tension: as the tension is increased, the transverse wave can travel faster due to a tighter, more strongly held medium. This is because higher tension reduces the time it takes for the wave to move across each part of the wire.
Linear Density
Linear density, denoted as \( \mu \), represents the mass of the wire per unit length, calculated as \( \mu = \frac{m}{L} \) where \( m \) is the mass and \( L \) is the length of the wire. It is a crucial factor in determining how waves travel through a medium.Key points about linear density:- It stays constant if the wire's length and mass remain unchanged.- Changes in linear density affect wave speed inversely, meaning as \( \mu \) increases, wave speed decreases, and vice versa.Understanding linear density helps predict wave behavior. In instances where other factors like tension change but linear density remains the same, calculations can focus directly on those changing parameters. This simplifies solving equations relating to wave speed without needing to adjust for \( \mu \).

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 traveling wave on a string is described by $$ y=2.0 \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right] $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) For \(t=0,\) plot \(y\) as a function of \(x\) for \(0 \leq x \leq 160 \mathrm{~cm}\). (b) Repeat (a) for \(t=0.05 \mathrm{~s}\) and \(t=0.10 \mathrm{~s}\). From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.

These two waves travel along the same string: \(y_{1}(x, t)=(4.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t)\) \(y_{2}(x, t)=(5.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad})\) What are (a) the amplitude and (b) the phase angle (relative to wave 1 ) of the resultant wave? (c) If a third wave of amplitude \(5.00 \mathrm{~mm}\) is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

A wave has an angular frequency of \(110 \mathrm{rad} / \mathrm{s}\) and a wavelength of \(1.80 \mathrm{~m}\). Calculate (a) the angular wave number and (b) the speed of the wave.

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of \(347 \mathrm{~m}\), a linear density of \(3.35 \mathrm{~kg} / \mathrm{m},\) and a tension of \(65.2 \mathrm{MN},\) what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes?

String \(A\) is stretched between two clamps separated by distance \(L\). String \(B\), with the same linear density and under the same tension as string \(A,\) is stretched between two clamps separated by distance \(4 L\). Consider the first eight harmonics of string \(B\). For which of these eight harmonics of \(B\) (if any) does the frequency match the frequency of (a) \(A\) 's first harmonic, (b) \(A\) 's second harmonic, and (c) \(A\) 's third harmonic?
See all solutions

Recommended explanations on Physics Textbooks

Scientific Method Physics

Read Explanation

Wave Optics

Read Explanation

Particle Model of Matter

Read Explanation

Atoms and Radioactivity

Read Explanation

Electrostatics

Read Explanation

Electric Charge Field and Potential

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.