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Chapter 19: Problem 34

If a violin string is tuned to a certain note, by what factor must the tension in the string be increased if it is to emit a note of double the original frequency (that is, a note one octave higher in pitch)?

Short Answer

Expert verified
The tension in the violin string must be increased by a factor of 4 to emit a note of double the original frequency.

Step by step solution

01

Understand the problem

In essence, the exercise is asking to find out how much the tension in a violin string must be increased if the emitted note has to be doubled. By understanding that doubling the frequency increases the pitch by an octave, the problem nicely lies within the realm of understanding waves and vibrations.
02

Apply the formula for frequency

The formula for the frequency (\(f\)) of a vibrating string under tension is given by: \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \(f\) is the frequency, \(L\) is the length of the string, \(T\) is the tension, and \(\mu\) is the linear mass density. Since we're dealing with the same string in both cases, the length and density of the string remain constants. Therefore, if the frequency is doubled, the relationship becomes: \( 2f = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} \), where \(T'\) is the new tension.
03

Rearrange and compute

By rearranging the equations, we can find an expression in terms of the new tension \(T'\). Squaring both equations and solving for \(T'\) yields: \(T' = T \times (\frac{2f}{f})^2 \). Since \(\frac{2f}{f} = 2\), we have: \(T' = 4T\).
04

Interpret the result

This result implies that the tension in the string must be increased by a factor of 4 for the frequency to be doubled (or for the note to be one octave higher). This makes intuitive sense because for the frequency to be double, a greater tension is needed to propagate the waves faster.

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