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Chapter 17: Problem 35

Resonant Frequencies A string that is stretched between fixed supports separated by \(75.0 \mathrm{~cm}\) has resonant frequencies of 420 and \(315 \mathrm{~Hz}\), with no intermediate resonant frequencies. What are (a) the lowest resonant frequency and (b) the wave speed?

Short Answer

Expert verified
The lowest resonant frequency is 105 Hz. The wave speed is 157.5 m/s.

Step by step solution

01

- Understand the problem

A string is stretched between fixed supports with a separation of 75.0 cm. It has resonant frequencies of 420 Hz and 315 Hz with no intermediate resonant frequencies.
02

- Identify harmonics

Given that there are no intermediate resonant frequencies, 420 Hz and 315 Hz correspond to consecutive harmonics. Let鈥檚 denote 420 Hz as the n-th harmonic and 315 Hz as the (n-1)-th harmonic.
03

- Set up the harmonic relationship

The resonant frequency for the n-th harmonic is given by: \( f_n = \frac{n}{2L}v \) and for the (n-1)-th harmonic: \( f_{n-1} = \frac{n-1}{2L}v \) where \( f_n = 420 \mathrm{~Hz} \), \( f_{n-1} = 315 \mathrm{~Hz} \), and \( L = 75.0 \mathrm{~cm} = 0.75 \mathrm{~m} \).
04

- Create frequency ratio

The ratio of these consecutive frequencies is: \( \frac{f_n}{f_{n-1}} = \frac{420}{315} = \frac{n}{n-1} \)
05

- Solve for n

Solving the ratio equation: \( \frac{420}{315} = \frac{n}{n-1} \) gives: \( 420(n-1) = 315n \) \( 420n - 420 = 315n \) \( 105n = 420 \) \( n = 4 \).
06

- Find the lowest resonant frequency

The lowest resonant frequency corresponds to the first harmonic (fundamental frequency). Thus, if 420 Hz is the 4th harmonic, the fundamental frequency \( f_1 = \frac{420}{4} \), so: \( f_1 = 105 \mathrm{~Hz} \).
07

- Calculate the wave speed

The wave speed ( v ) is given by the formula: \( v = f \times \text{wavelength} \). The wavelength for the 4th harmonic is: \( \text{wavelength} = \frac{2L}{n} \). Therefore: \( \text{wavelength}_4 = \frac{2 \times 0.75}{4} = 0.375 \mathrm{~m} \). The wave speed: \( v = 420 \times 0.375 = 157.5 \mathrm{~m/s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonics
In vibrating systems like strings, harmonics are specific frequencies at which the system naturally resonates. These frequencies are integral multiples of the fundamental frequency. When a string vibrates, it can do so in patterns that include full, half, or even smaller fractions of wavelengths fitting between its fixed ends. Each pattern is known as a harmonic, with the first harmonic corresponding to the case where the string fits exactly half a wavelength. For a string with fixed ends, the nth harmonic has nodes (points of no displacement) at both ends and an antinode (point of maximum displacement) where the string reaches its peak amplitude in the middle. The nth harmonic frequency (or nth resonant frequency) is expressed as: ewline ewline ewline ewline n

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