/*! 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 47 A standing wave has wave numbe... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 47

A standing wave has wave number $2.0 \times 10^{2} \mathrm{rad} / \mathrm{m} .$ What is the distance between two adjacent nodes?

Short Answer

Expert verified
Answer: The distance between two adjacent nodes in the standing wave is 0.0157 meters.

Step by step solution

01

Determine the wavelength of the wave

We are given the wave number k = \(2.0 \times 10^{2} \mathrm{rad} / \mathrm{m}\). To find the wavelength, we can use the expression \(k = \frac{2\pi}{位}\). Rearrange the equation to solve for 位: \(位 = \frac{2\pi}{k}\) Now, plug in the given value of k: \(位 = \frac{2\pi}{2.0 \times 10^{2} \mathrm{rad} / \mathrm{m}}\)
02

Evaluate the expression for the wavelength

Calculate the value for the wavelength: \(位 = \frac{2\pi}{2.0 \times 10^{2} \mathrm{rad} / \mathrm{m}} \approx \frac{6.28}{200} \approx 0.0314 \mathrm{m}\)
03

Find the distance between adjacent nodes

The distance between adjacent nodes in a standing wave is half the wavelength. Therefore, the distance between two adjacent nodes can be found by dividing the wavelength by 2: Distance between adjacent nodes = \(\frac{位}{2}\) Now plug in the value we found for the wavelength: Distance between adjacent nodes = \(\frac{0.0314 \mathrm{m}}{2}\)
04

Calculate the distance between adjacent nodes

Evaluate the expression for the distance between adjacent nodes: Distance between adjacent nodes = \(\frac{0.0314 \mathrm{m}}{2} = 0.0157 \mathrm{m}\) The distance between two adjacent nodes in the standing wave is 0.0157 meters.

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Most popular questions from this chapter

Show that the amplitudes of the graphs you made in Problem 74 satisfy the equation \(A^{\prime}=2 A \cos (\omega t),\) where \(A^{\prime}\) is the amplitude of the wave you plotted and \(A\) is \(5.0 \mathrm{cm},\) the amplitude of the waves that were added together.
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(a) Use a graphing calculator or computer graphing program to plot \(y\) versus \(x\) for the function $$ y(x, t)=(5.0 \mathrm{cm})[\sin (k x-\omega t)+\sin (k x+\omega t)] $$ for the times \(t=0,1.0 \mathrm{s},\) and \(2.0 \mathrm{s}\). Use the values \(k=\pi /(5.0 \mathrm{cm})\) and \(\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s} .\) (b) Is this a traveling wave? If not, what kind of wave is it?
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