A piano tuner hears one beat every 2.8 s when trying to adjust two strings, one of which is sounding 440 Hz, so that they sound the same tone. How far off in frequency is the other string?
The "beat frequency" is 1/2.8 = 0.357 Hz. That is also how far the second string is off the desired 440 Hz.
Well, let's do some math (cue circus music)! If the piano tuner hears one beat every 2.8 seconds, that means there's a difference in frequency between the two strings.
Now, here's a joke to lighten things up: Why don't scientists trust atoms? Because they make up everything!
Ok, back to the question. To find the frequency difference, we can use the formula: Beat Frequency = Difference in Frequency. In this case, the beat frequency is 1/t = 1/2.8 s = 0.357 Hz.
So, the other string must be off by 0.357 Hz in frequency. But hey, don't worry, even out-of-tune strings can still create beautiful music. Just like how my jokes can be cringe-worthy but still bring a smile!
To determine how far off in frequency the other string is, we need to calculate the beat frequency first.
The beat frequency is the difference in frequency between the two strings. In this case, the piano tuner hears one beat every 2.8 seconds.
We can use the formula:
Beat frequency = |Frequency of String 1 - Frequency of String 2|
Since we know the frequency of String 1 (440 Hz), we need to find the frequency of String 2.
By rearranging the formula, we have:
Frequency of String 2 = Frequency of String 1 - Beat frequency
Let's calculate the frequency of String 2:
Frequency of String 2 = 440 Hz - 1 beat / 2.8 s = 440 Hz - 1/2.8 Hz = 440 Hz - 0.357 Hz
Therefore, the other string is off by approximately 0.357 Hz in frequency.
To find out how far off in frequency the other string is, we need to determine the beat frequency and then calculate the difference in frequency.
Here's how you can approach it:
1. Determine the beat frequency:
- The beat frequency is the difference between the frequencies of the two strings.
- In this case, the tuner hears one beat every 2.8 seconds.
- So, the beat frequency is equal to 1/2.8 Hz.
2. Calculate the difference in frequency:
- Since the beat frequency is the difference between the two frequencies, we can subtract the known frequency (440 Hz) from the beat frequency to find the frequency of the other string.
- Therefore, the difference in frequency is equal to the beat frequency minus 440 Hz.
Let's calculate it:
Beat frequency = 1 / 2.8 Hz = 0.357 Hz
Difference in frequency = Beat frequency - 440 Hz
Difference in frequency = 0.357 Hz - 440 Hz
The difference in frequency between the other string and 440 Hz is -439.643 Hz. Note that the negative sign indicates that the other string is slightly lower in frequency than the 440 Hz string.