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Chapter 11: Problem 21

A wave on a string has equation $$ y(x, t)=(4.0 \mathrm{mm}) \sin (\omega t-k x) $$ where \(\omega=6.0 \times 10^{2} \mathrm{rad} / \mathrm{s}\) and $k=6.0 \mathrm{rad} / \mathrm{m} .$ (a) What is the amplitude of the wave? (b) What is the wavelength? (c) What is the period? (d) What is the wave speed? (e) In which direction does the wave travel?

Short Answer

Expert verified
Answer: The properties of the wave are as follows: - Amplitude: \(A = 4.0\, mm\) - Wavelength: \(\lambda = \frac{2\pi}{6.0\, \mathrm{rad/m}}\) - Period: \(T =\frac{2\pi}{6.0 \times 10^2\, \mathrm{rad/s}}\) - Wave speed: \(v=\frac{\lambda}{T}\) - Direction: Positive x-direction

Step by step solution

01

Identify the amplitude of the wave

In the wave equation provided, the amplitude can be identified as the coefficient of the sine function. In this case, it is given as \(4.0\, mm\). Thus, the amplitude of the wave is \(A = 4.0\, mm\).
02

Determine the wavelength of the wave

To determine the wavelength, we need to make use of the \(k\) value given in the equation. The relationship between the wavenumber, \(k\), and the wavelength, \(\lambda\), is given by the formula \(k = \frac{2\pi}{\lambda}\). In the equation, k is given as \(k=6.0\, \mathrm{rad/m}\), and we can solve for \(\lambda\): $$\lambda =\frac{2\pi}{k}$$ $$\lambda =\frac{2\pi}{6.0\, \mathrm{rad/m}}$$ Calculating this value, we can find the wavelength of the wave.
03

Find the period of the wave

To find the period, we need to make use of the provided angular frequency, \(\omega\). The period, \(T\), is related to the angular frequency through the formula \(\omega = \frac{2\pi}{T}\). The equation provides the angular frequency as \(\omega = 6.0 \times 10^2\, \mathrm{rad/s}\). We can solve for the period T: $$T =\frac{2\pi}{\omega}$$ $$T =\frac{2\pi}{6.0 \times 10^2\, \mathrm{rad/s}}$$ Calculating this value, we can find the period of the wave.
04

Calculate the wave speed

To find the wave speed, we can use the relationship between wavelength, period, and wave speed. The wave speed \(v\) is given by the formula \(v=\frac{\lambda}{T}\). Using the wavelength and period derived in Steps 2 and 3, we can calculate the wave speed.
05

Determine the direction of the wave

To determine the direction of the wave, we need to analyze the signs in the wave equation. The equation is given as \(y(x, t)=(4.0\, \mathrm{mm}) \sin (\omega t-k x)\). The wave propagates in the positive x-direction if the term inside the sine function is of the form \((\omega t - kx)\); conversely, if the term is of the form \((\omega t + kx)\), the wave would propagate in the negative x-direction. The given equation has the form \((\omega t - kx)\), so the wave is moving in the positive x-direction.

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Most popular questions from this chapter

The formula for the speed of transverse waves on a spring is the same as for a string. (a) A spring is stretched to a length much greater than its relaxed length. Explain why the tension in the spring is approximately proportional to the length. (b) A wave takes 4.00 s to travel from one end of such a spring to the other. Then the length is increased \(10.0 \% .\) Now how long does a wave take to travel the length of the spring? [Hint: Is the mass per unit length constant?]
A traveling sine wave is the result of the superposition of two other sine waves with equal amplitudes, wavelengths, and frequencies. The two component waves each have amplitude \(5.00 \mathrm{cm} .\) If the superposition wave has amplitude \(6.69 \mathrm{cm},\) what is the phase difference \(\phi\) between the component waves? [Hint: Let \(y_{1}=A \sin (\omega t+k x)\) and $y_{2}=A \sin (\omega t+k x-\phi) .$ Make use of the trigonometric identity (Appendix A.7) for \(\sin \alpha+\sin \beta\) when finding \(y=y_{1}+y_{2}\) and identify the new amplitude in terms of the original amplitude. \(]\)
What is the speed of the wave represented by \(y(x, t)=\) $A \sin (k x-\omega t),\( where \)k=6.0 \mathrm{rad} / \mathrm{cm}\( and \)\omega=5.0 \mathrm{rad} / \mathrm{s} ?$
An underground explosion sends out both transverse (S waves) and longitudinal (P waves) mechanical wave pulses (seismic waves) through the crust of the Earth. Suppose the speed of transverse waves is \(8.0 \mathrm{km} / \mathrm{s}\) and that of longitudinal waves is \(10.0 \mathrm{km} / \mathrm{s} .\) On one occasion, both waves follow the same path from a source to a detector (a seismograph); the longitudinal pulse arrives 2.0 s before the transverse pulse. What is the distance between the source and the detector?
Tension is maintained in a string by attaching one end to a wall and by hanging a \(2.20-\mathrm{kg}\) object from the other end of the string after it passes over a pulley that is \(2.00 \mathrm{m}\) from the wall. The string has a mass per unit length of \(3.55 \mathrm{mg} / \mathrm{m} .\) What is the fundamental frequency of this string?
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