A flutist assembles her flute in a room where the speed of sound is 342 m/s. When she plays the note A, it is in perfect tune with a 440 Hz tuning fork. After a few minutes, the air inside her flute has warmed to where the speed of sound is 346 m/s. How far does she need to extend the 'tuning joint' of her flute to be in tune with the tuning fork?

Verified step by step guidance

1. Determine the wavelength of the sound wave when the flute is in tune with the tuning fork at the initial speed of sound (342 m/s). Use the formula for wavelength: $\lambda =\frac{v}{f}$, where $v$ is the speed of sound and $f$ is the frequency.
2. Calculate the new wavelength of the sound wave when the speed of sound increases to 346 m/s, using the same formula: $\mathrm{\lambda \text{'}}=\frac{\mathrm{v\text{'}}}{f}$, where $\mathrm{v\text{'}}$ is the new speed of sound.
3. Recognize that the length of the flute corresponds to half the wavelength of the sound wave for the fundamental frequency. Therefore, the length of the flute when in tune is $L=\frac{\lambda }{2}$ initially and $\mathrm{L\text{'}}=\frac{\mathrm{\lambda \text{'}}}{2}$ after the air warms.
4. Find the difference in the flute's length required to stay in tune by calculating $\mathrm{\Delta L}=\mathrm{L\text{'}}-L$. Substitute the expressions for $L$ and $\mathrm{L\text{'}}$ in terms of the wavelengths.
5. Simplify the expression for $\mathrm{\Delta L}$ to find the required extension of the tuning joint. This will give the distance the flutist needs to extend the flute to match the tuning fork's frequency.

Key Concepts

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium, such as air. It varies with temperature, humidity, and pressure. In this scenario, the speed of sound increases from 342 m/s to 346 m/s as the air inside the flute warms, affecting the pitch of the sound produced.

Frequency and Wavelength

Frequency is the number of oscillations or cycles of a wave per second, measured in Hertz (Hz). The wavelength is the distance between successive crests of a wave. The relationship between speed, frequency, and wavelength is given by the equation: speed = frequency × wavelength, which is crucial for understanding how changes in speed affect the sound produced by the flute.

Tuning and Length Adjustment

In wind instruments like flutes, the pitch can be adjusted by changing the effective length of the air column inside the instrument. Extending the tuning joint increases the length, lowering the pitch. To remain in tune with a specific frequency, the flutist must calculate the necessary extension based on the change in speed of sound and the desired frequency.