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Chapter 16: Problem 50

A thin aluminum rod of length \(L=2.00 \mathrm{~m}\) is clamped at its center. The speed of sound in aluminum is \(5000 . \mathrm{m} / \mathrm{s}\). Find the lowest resonance frequency for vibrations in this rod.

Short Answer

Expert verified
Answer: The lowest resonance frequency for the given aluminum rod is 1250 Hz.

Step by step solution

01

Identify given values

We are given the length of the aluminum rod, \(L = 2.00 \mathrm{~m}\), and the speed of sound in aluminum, \(v = 5000 \mathrm{~m/s}\).
02

Apply the formula for the fundamental frequency

We will use the formula for the fundamental frequency of a freely vibrating rod clamped at its center: \(f_1 = \frac{v}{2L}\). Substitute the given values of \(v = 5000 \mathrm{~m/s}\) and \(L = 2.00 \mathrm{~m}\) into the formula: \(f_1 = \frac{5000 \mathrm{~m/s}}{2(2.00 \mathrm{~m})}\)
03

Calculate the fundamental frequency

Perform the calculation in the formula: \(f_1 = \frac{5000 \mathrm{~m/s}}{4.00 \mathrm{~m}} = 1250 \mathrm{~Hz}\) Therefore, the lowest resonance frequency for vibrations in the rod is \(1250 \mathrm{~Hz}\).

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Most popular questions from this chapter

You are playing a note that has a fundamental frequency of \(400 .\) Hz on a guitar string of length \(50.0 \mathrm{~cm}\). At the same time, your friend plays a fundamental note on an open organ pipe, and 4 beats per seconds are heard. The mass per unit length of the string is \(2.00 \mathrm{~g} / \mathrm{m}\). Assume the velocity of sound is \(343 \mathrm{~m} / \mathrm{s}\). a) What are the possible frequencies of the open organ pipe? b) When the guitar string is tightened, the beat frequency decreases. Find the original tension in the string. c) What is the length of the organ pipe?

Two sources, \(A\) and \(B\), emit a sound of a certain wavelength. The sound emitted from both sources is detected at a point away from the sources. The sound from source \(\mathrm{A}\) is a distance \(d\) from the observation point, whereas the sound from source \(\mathrm{B}\) has to travel a distance of \(3 \lambda .\) What is the largest value of the wavelength, in terms of \(d\), for the maximum sound intensity to be detected at the observation point? If \(d=10.0 \mathrm{~m}\) and the speed of sound is \(340 \mathrm{~m} / \mathrm{s}\), what is the frequency of the emitted sound?

A source traveling to the right at a speed of \(10.00 \mathrm{~m} / \mathrm{s}\) emits a sound wave at a frequency of \(100.0 \mathrm{~Hz}\). The sound wave bounces off of a reflector, which is traveling to the left at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\). What is the frequency of the reflected sound wave detected by a listener back at the source?

Two people are talking at a distance of \(3.0 \mathrm{~m}\) from where you are, and you measure the sound intensity as \(1.1 \cdot 10^{-7} \mathrm{~W} / \mathrm{m}^{2}\). Another student is \(4.0 \mathrm{~m}\) away from the talkers. What sound intensity does the other student measure?

In a suspense-thriller movie, two submarines, \(X\) and Y, approach each other, traveling at \(10.0 \mathrm{~m} / \mathrm{s}\) and \(15.0 \mathrm{~m} / \mathrm{s}\), respectively. Submarine X "pings" submarine Y by sending a sonar wave of frequency \(2000.0 \mathrm{~Hz}\). Assume that the sound travels at \(1500.0 \mathrm{~m} / \mathrm{s}\) in the water. a) Determine the frequency of the sonar wave detected by submarine Y. b) What is the frequency detected by submarine \(X\) for the sonar wave reflected off submarine Y? c) Suppose the submarines barely miss each other and begin to move away from each other. What frequency does submarine Y detect from the pings sent by X? How much is the Doppler shift?
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