Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 242 Hz, while the other produces 288 Hz. When a tuning fork is sounded together with the 242-Hz tone, a beat frequency of 26.0 Hz is produced. When the same tuning fork is sounded together with the 288-Hz tone, a beat frequency of 20.0 Hz is produced. What is the frequency of the tuning fork?
Beat frequency
f =| f1-f2|
242<f<288
f-242=26
288-f=20
f=268 Hz
Well, it sounds like those two flutes are really marching to the beat of their own off-key drum! Now, let's solve this musical mystery, shall we?
The beat frequency is the difference between the frequencies of the two sounds. So, using some basic subtraction skills, we can find that the difference between the 242 Hz tone and the tuning fork frequency is 26 Hz, while the difference between the 288 Hz tone and the tuning fork frequency is 20 Hz.
Now, since we have two pieces of information, we can put on our thinking caps and do some math magic! By subtracting 242 Hz from 288 Hz, we get a difference of 46 Hz. Similarly, subtracting 26 Hz from 20 Hz, we find a difference of -6 Hz.
Okay, let's get back to the beat frequency with the positive value of 26 Hz. This means that the tuning fork must produce a frequency that is higher than the 242 Hz tone. So, if we add 26 Hz to 242 Hz, we get a frequency of 268 Hz.
Now, let's put on our detective hats again and use this newfound frequency of 268 Hz to solve the next equation. The beat frequency with the 288 Hz tone is 20 Hz, which means the tuning fork frequency must be less than 288 Hz. If we subtract 20 Hz from 288 Hz, we find a frequency of 268 Hz.
Ah! What a coincidence! The frequency of the tuning fork that causes both beat frequencies is 268 Hz!
So, my friend, the frequency of the tuning fork is a harmonious 268 Hz. Now, let's hope those flutes catch up to the beat!
To find the frequency of the tuning fork, we need to determine the difference in frequency between the tuning fork and each flute.
Let's denote the frequency of the tuning fork as "f", and the difference in frequency between the tuning fork and each flute as "Δf1" and "Δf2" respectively.
From the given information, we know:
- Frequency of the first flute (f1) = 242 Hz
- Frequency of the second flute (f2) = 288 Hz
- Beat frequency when the tuning fork is sounded with the first flute (f-beat1) = 26 Hz
- Beat frequency when the tuning fork is sounded with the second flute (f-beat2) = 20 Hz
We can calculate the difference in frequency between the tuning fork and each flute using the following equations:
Δf1 = f - f1
Δf2 = f - f2
Now, let's calculate the values of Δf1 and Δf2:
Δf1 = f - f1 = f - 242 Hz
Δf2 = f - f2 = f - 288 Hz
Since the beat frequency is the absolute difference between the two frequencies played together, we can also use the following equations to find the difference in frequency:
Δf1 = f - f2 - f-beat1
Δf2 = f - f1 - f-beat2
Using the given values:
Δf1 = f - 288 Hz - 26 Hz
Δf2 = f - 242 Hz - 20 Hz
Now we have two equations for Δf1 and Δf2:
Δf1 = f - 314 Hz
Δf2 = f - 222 Hz
These two equations are equal to our initial equations, so we can equate them:
f - 314 Hz = f - 242 Hz
f - 222 Hz = f - 288 Hz
Simplifying these equations:
-314 Hz = -242 Hz
-222 Hz = -288 Hz
We can see that these equations are inconsistent, which means there is no frequency of the tuning fork that satisfies both conditions.
Therefore, based on the given information, it is not possible to determine the frequency of the tuning fork.
To find the frequency of the tuning fork, we need to understand how beat frequencies work. Beat frequency is the difference in frequencies between two slightly out-of-tune tones.
In this case, we have two situations:
1) When the tuning fork is sounded with the 242 Hz tone, a beat frequency of 26 Hz is produced.
2) When the tuning fork is sounded with the 288 Hz tone, a beat frequency of 20 Hz is produced.
Let's calculate the beat frequencies in these two scenarios:
1) Beat frequency = frequency of the tuning fork - frequency of the 242 Hz tone
26 Hz = tuning fork frequency - 242 Hz
tuning fork frequency = 26 Hz + 242 Hz = 268 Hz
2) Beat frequency = frequency of the tuning fork - frequency of the 288 Hz tone
20 Hz = tuning fork frequency - 288 Hz
tuning fork frequency = 20 Hz + 288 Hz = 308 Hz
So, we have found two possible frequencies for the tuning fork - 268 Hz and 308 Hz. However, since the tuning fork produces a single frequency, we know that the correct frequency must be the average of these two values.
Average frequency = (268 Hz + 308 Hz) / 2 = 536 Hz / 2 = 268 Hz
Therefore, the frequency of the tuning fork is 268 Hz.