How many beats will be heard if two identical flutes each try to play middle c (262 Hz), but one is at 5.0 Celsius and the other is at 25 Celsius?
To calculate the number of beats that will be heard when two identical flutes try to play middle C at different temperatures, we need to understand the concept of beat frequencies.
When two sound waves with slightly different frequencies overlap, they interfere with each other, resulting in a periodic increase and decrease in amplitude known as beats. The frequency of the beat is equal to the difference between the frequencies of the two sound waves.
In this case, we have two identical flutes playing middle C (262 Hz) but at different temperatures, which affect the speed of sound waves. The speed of sound increases with an increase in temperature.
To calculate the beat frequency, we need to find the frequency difference between the two flutes. Since the temperature difference is given (25 Celsius - 5 Celsius = 20 Celsius), we can use the formula:
Δf = α * f0 * ΔT
Where:
- Δf is the frequency difference
- α is the temperature coefficient of air (roughly 0.6 Hz/Celsius for middle frequencies)
- f0 is the original frequency (262 Hz)
- ΔT is the temperature difference (20 Celsius)
Plugging in the values, we get:
Δf = 0.6 Hz/Celsius * 262 Hz * 20 Celsius
Δf = 0.6 Hz/Celsius * 5240 Hz
Δf = 3144 Hz
Therefore, the frequency difference between the two flutes is 3144 Hz.
To convert this frequency difference to the number of beats heard per second, we divide it by the speed of sound. The speed of sound at 5 Celsius and 25 Celsius is slightly different, but for simplicity, we can assume it as the average speed of sound (343 m/s).
Number of beats = Δf / speed of sound
Number of beats = 3144 Hz / 343 m/s
Number of beats ≈ 9.16 beats per second
In conclusion, if two identical flutes attempt to play middle C (262 Hz), with one flute at 5 Celsius and the other at 25 Celsius, approximately 9.16 beats per second will be heard.