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Chapter 18: Problem 2

By rocking a boat, a child produces surface water waves on a previously quiet lake. It is observed that the boat performs 12 oscillations in \(30 \mathrm{~s}\) and also that a given wave crest reaches shore \(15 \mathrm{~m}\) away in \(5.0 \mathrm{~s}\). Find \((a)\) the frequency, \((b)\) the speed, and (c) the wavelength of the waves.

Short Answer

Expert verified
The frequency of the waves is 0.4 Hz, the speed is 3 m/s, and the wavelength is 7.5 meters.

Step by step solution

01

Calculate the Frequency

Frequency of any periodic motion is given by the total number of oscillations divided by the total time taken. From the given data, the boat makes 12 oscillations in 30 seconds. So, the frequency (\(f\)) is \(f = \frac{Oscillations}{Time} = \frac{12}{30} Hz\)
02

Calculate the Speed of the waves

The speed of the waves can be calculated by dividing the total distance the wave travels by the time it takes to travel that distance. Given that a wave crest reaches shore 15m away in 5 seconds, we can calculate the speed (\(v\)) as follows: \(v = \frac{Distance}{Time} = \frac{15m}{5s} m/s\)
03

Calculate the Wavelength

Lastly, we can use the wave equation \(v = fλ\) (where \(v\) is the speed, \(f\) is the frequency and \(λ\) is the wavelength) to solve for the wavelength (\(λ\)). By rearranging the equation, we get \(λ = \frac{v}{f}\). Substituting our previously obtained values for \(v\) and \(f\), we get \(λ = \frac{3 m/s}{0.4 Hz}\)

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Most popular questions from this chapter

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.

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