Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 1511

Standing waves are produced by the superposition of two waves $y_{1}=0.05 \sin (3 \pi t-2 x)\( and \)y_{2}=0.05 \sin (3 \pi t+2 x)$ where \(\mathrm{x}\) and \(\mathrm{y}\) are in meters and \(\mathrm{t}\) is in seconds. The speed (in \(\mathrm{ms}-1\) ) of each wave is \(\ldots \ldots\) (A) \(1.5\) (B) \(3.0\) (C) \(3 \pi / 2\) (D) \(3 \pi\)

Short Answer

Expert verified
The speed of each wave is \(v = \frac{3 \pi}{2}\) m/s.

Step by step solution

01

Identify wavelength and frequency from the equations

We are given the two waves: \(y_{1}=0.05 \sin (3 \pi t-2 x)\) and \(y_{2}=0.05 \sin (3 \pi t+2 x)\). These are sinusoidal waves described by the form \( y = A \sin (\omega t \pm kx) \), where A is the amplitude, ω is the angular frequency, k is the wave number, and ± depends on the direction of each wave. For both waves, we are given \(\omega = 3 \pi\), k = 2. Next, we will find the lambda (wavelength) and frequency (f).
02

Calculate wavelength

The wavelength (lambda) is related to the wave number as follows: \( k = \frac{2 \pi}{\lambda} \) We know k = 2, therefore the wavelength is: \( \lambda = \frac{2 \pi}{k} = \frac{2 \pi}{2} = \pi \) meters.
03

Calculate frequency

Frequency is related to the angular frequency as follows: \( f = \frac{\omega}{2 \pi} \) We know \(\omega = 3 \pi\), and therefore the frequency is: \( f = \frac{3 \pi}{2 \pi} = \frac{3}{2} \) Hz.
04

Determine wave speed

Now we have the wavelength (\(\lambda = \pi\)) and the frequency (\(f = \frac{3}{2}\)). We will use the wave speed equation (\( v = \lambda f\)) to calculate the speed of the wave: \( v = (\pi) \left(\frac{3}{2}\right) = \frac{3 \pi}{2} \) meters per second. The answer is choice (C): \(v = \frac{3 \pi}{2}\) m/s.

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

Two sound waves are represented by $\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\( and \)\mathrm{y}=\mathrm{a} \cos (\omega \mathrm{t}-\mathrm{kx})$. The phase difference between the waves in water is \(\ldots \ldots \ldots\) (A) \((\pi / 2)\) (B) \((\pi / 4)\) (C) \(\pi\) (D) \((3 \pi / 4)\)

When a block of mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude \(\mathrm{A}\) and angular frequency \(\omega\). The total energy of the system comprising of the block and spring is \(\ldots \ldots \ldots\) (A) \((1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}\) (B) \((1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}\) (C) \((1 / 2) \mathrm{ky}^{2}\) (D) \((1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}\)

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\) (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) Periodic time of a simple pendulum is independent of the mass of the bob. Statement \(-2:\) The restoring force does not depend on the mass of the bob. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)

The amplitude for a S.H.M. given by the equation $\mathrm{x}=3 \sin 3 \mathrm{pt}+4 \cos 3 \mathrm{pt}\( is \)\ldots \ldots \ldots \ldots \mathrm{m}$ (A) 5 (B) 7 (C) 4 (D) \(3 .\)

The maximum velocity and maximum acceleration of a particle executing S.H.M. are \(1 \mathrm{~m} / \mathrm{s}\) and \(3.14 \mathrm{~m} / \mathrm{s}^{2}\) respectively. The frequency of oscillation for this particle is...... (A) \(0.5 \mathrm{~s}^{-1}\) (B) \(3.14 \mathrm{~s}^{-1}\) (C) \(0.25 \mathrm{~s}^{-1}\) (D) \(2 \mathrm{~s}^{-1}\)
See all solutions

Recommended explanations on English Textbooks

English Grammar Summary

Read Explanation

Synthesis Essay

Read Explanation

English Language Study

Read Explanation

Summary Text

Read Explanation

Rhetorical Analysis Essay

Read Explanation

Blog

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