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Chapter 18: Problem 32

A \(1.48\) -m-long wire has a mass of \(8.62 \mathrm{~g}\) and is held under a tension of \(122 \mathrm{~N}\). The wire is held rigidly at both ends and set into vibration. Calculate \((a)\) the speed of waves on the wire, (b) the wavelengths of the waves that produce one- and twoloop standing waves on the wire, and ( \(c\) ) the frequencies of the waves in \((b)\).

Short Answer

Expert verified
The speed of waves on the wire is approximately 486.5 m/s. The wavelengths that produce one- and two-loop standing waves on the wire are approximately 2.96m and 1.48m respectively. The frequencies of the waves that correspond to these wavelengths are approximately 164.2 Hz and 328.4 Hz respectively.

Step by step solution

01

Calculate the speed of waves on the wire

First, calculate the linear density \(\mu\) of the wire by dividing its mass \(m\) by its length \(L\), both in SI units. The mass should be converted from grams to kilograms. Then, substitute the values of \(T\) and \(\mu\) into the equation \(v = \sqrt{\frac{T}{\mu}}\) to find the speed \(v\).
02

Calculate the wavelengths that produce one- and two-loop standing waves

The wavelength of a standing wave can be calculated with \(\lambda = \frac{2L}{n}\), where \(L\) is the length and \(n\) is the number of loops. For one-loop and two-loop standing waves, \(n=1\) and \(n=2\) respectively.
03

Calculate the frequencies of the waves

The frequency of a wave can be calculated with \(f = \frac{v}{\lambda}\), where \(v\) is the speed and \(\lambda\) is the wavelength. Substitute the values obtained for \(v\) and \(\lambda\) into the equation to find \(f\). Do it twice, once for \(n=1\) and once for \(n=2\)
04

Review your answer

Verify that your answers are consistent with the given conditions and they make sense in the context of the problem. Return to any previous steps if necessary.

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Most popular questions from this chapter

A \(15.0-\mathrm{cm}\) violin string, fixed at both ends, is vibrating in its \(n=1\) mode. The speed of waves in this wire is \(250 \mathrm{~m} / \mathrm{s}\), and the speed of sound in air is \(348 \mathrm{~m} / \mathrm{s}\). What are \((a)\) the frequency and \((b)\) the wavelength of the emitted sound wave?

Calculate the speed of a transverse wave in a string of length 2.15 m and mass \(62.5 \mathrm{~g}\) under a tension of \(487 \mathrm{~N}\).

The equation of a transverse wave traveling along a very long string is given by $$ y=(6.0 \mathrm{~cm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{m}) x+(4.0 \pi \mathrm{rad} / \mathrm{s}) t] $$ Calculate \((a)\) the amplitude, \((b)\) the wavelength, \((c)\) the frequency, \((d)\) the speed, \((e)\) the direction of propagation of the wave, and \((f)\) the maximum transverse speed of a particle in the string.

The equation of a transverse wave traveling along a string is given by $$ y=(2.30 \mathrm{~mm}) \sin [(1822 \mathrm{rad} / \mathrm{m}) x-(588 \mathrm{rad} / \mathrm{s}) t] $$ Find \((a)\) the amplitude, \((b)\) the frequency, \((c)\) the velocity, \((d)\) the wavelength of the wave, and ( \(e\) ) the maximum transverse speed of a particle in the string.

A string fixed at both ends is \(8.36 \mathrm{~m}\) long and has a mass of \(122 \mathrm{~g} .\) It is subjected to a tension of \(96.7 \mathrm{~N}\) and set vibrating. (a) What is the speed of the waves in the string? (b) What is the wavelength of the longest possible standing wave? (c) Give the frequency of that wave.
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