Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 18: Problem 5

Assuming that the wave speed on a stretched string depends on the tension \(F\) and linear mass density \(\mu\) as \(v \propto F^{a} / \mu^{b}\), use dimensional analysis to show that \(a=\frac{1}{2}\) and \(b=\frac{1}{2}\).

Short Answer

Expert verified
The wave speed on a stretched string depends on the tension \(F\) and linear mass density \(\mu\) as \(v \propto F^{1/2} / \mu^{1/2}\).

Step by step solution

01

Identify the dimensions of the key quantities

The key quantities in the expression \(v \propto F^{a} / \mu^{b}\) are \(v\) (speed), \(F\) (force), and \(\mu\) (linear mass density). Speed has dimensions of distance over time (L/T), force is mass times acceleration i.e. (M L/T²) and linear density is mass per length i.e. (M/L).
02

Substitute the dimensions into the given expression

Replace each quantity in the expression \(v \propto F^{a} / \mu^{b}\) with its dimension. This gives dimension of \(v\) is \(L/T\), dimension of \(F\) is \((M L/T²)^{a}\) and dimension of \(\mu\) is \((M/L)^{b}\).
03

Equate dimensions

According to the principle of dimensional analysis, the dimensions on both sides of an equation must be the same. This gives us \((L/T) = (M^{a} L^{a} / T^{2a}) / (M^{b} / L^{b})\).
04

Simplify and equate powers

Simplify the above equation to \((L/T) = (M^{a - b}L^{a + b} / T^{2a})\). For the unit of each dimension (Length: \(L\), Mass: \(M\) and Time: \(T\)) on the left to equal the unit of each dimension on the right, the powers of \(L\), \(M\), and \(T\) on the right must equal the powers of \(L\), \(M\), and \(T\) on the left. This gives us three equations: \(1 = a + b\), \(0 = a - b\) and \(-1 = -2a\).
05

Solve the equations

These are simultaneous equations. Solving these gives \(a = \frac{1}{2}\) and \(b = \frac{1}{2}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 string fixed at both ends is \(8.36 \mathrm{~m}\) long and has a mass of \(122 \mathrm{~g} .\) It is subjected to a tension of \(96.7 \mathrm{~N}\) and set vibrating. (a) What is the speed of the waves in the string? (b) What is the wavelength of the longest possible standing wave? (c) Give the frequency of that wave.

The equation of a transverse wave traveling along a very long string is given by $$ y=(6.0 \mathrm{~cm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{m}) x+(4.0 \pi \mathrm{rad} / \mathrm{s}) t] $$ Calculate \((a)\) the amplitude, \((b)\) the wavelength, \((c)\) the frequency, \((d)\) the speed, \((e)\) the direction of propagation of the wave, and \((f)\) the maximum transverse speed of a particle in the string.

Four sinusoidal waves travel in the positive \(x\) direction along the same string. Their frequencies are in the ratio \(1: 2: 3: 4\) and their amplitudes are in the ratio \(1: \frac{1}{2}: \frac{1}{3}: \frac{1}{4}\), respectively. When \(t=0\), at \(x=0\), the first and third waves are \(180^{\circ}\) out of phase with the second and fourth. Plot the resultant waveform when \(t=0\) and discuss its behavior as \(t\) increases.

Vibrations from a \(622-\mathrm{Hz}\) tuning fork set up standing waves in a string clamped at both ends. The wave speed for the string is \(388 \mathrm{~m} / \mathrm{s}\). The standing wave has four loops and an amplitude of \(1.90 \mathrm{~mm} .\) (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

The equation of a transverse wave traveling in a string is given by $$ y=(0.15 \mathrm{~m}) \sin [(0.79 \mathrm{rad} / \mathrm{m}) x-(13 \mathrm{rad} / \mathrm{s}) t] $$ (a) What is the displacement at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s}\) ? (b) Write down the equation of a wave that, when added to the given one, would produce standing waves on the string. (c) What is the displacement of the resultant standing wave at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Dynamics

Read Explanation

Particle Model of Matter

Read Explanation

Atoms and Radioactivity

Read Explanation

Oscillations

Read Explanation

Electrostatics

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.

Sign-up for free