What Was the Original Frequency of the Piano's Out of Tune A Note?

[ortegavs]

Homework Statement

Ok so a piano tuner plays an out of tune A note on his piano and then strikes his 440 hz tuning fork. He notices a beat of 2 Hz. When he loosens the piano string and plays the note again, the beat goes to 4 hz. What was the frequency of the note before he loosened the string? a.438 b. 440 c. 442. d. 444

Homework Equations

f1-f2=beat f
v=square root of T/m/L

The Attempt at a Solution

Ok so I know that its either 442 or 438Hz since the only way I can get 2 beats is 440-438 or 442-440. I also know that the string is loosen then T must go down and since its proportion to v, then v must go down also. Since v=fλ and since λ stays constant since the sting length does not change then frequency must go down. That means that the final frequency must be lower than 440 because if it was higher it would have to go up to 444. If its 440-436= 4hz and this means that the starting note had to be below 440. Is this the right way to think about? And I don't understand the relationship between change in v with change in beat frequency? For example in V goes up should beat f go up and vice versa.

 

[rl.bhat]

It can be either 442 or 438.

In case of initial 442 Hz, initially the beat frequency is 2 Hz. When he loosen the string, the frequency may become 436 making the beat frequency 4 Hz.
Same thing happens in the case of initial 438 Hz and final 436 Hz.

 

[ortegavs]

So there is no way of knowing with the information given if the starting frequency is 438 or 442? I thought about that also and reason that the final f should be found with the information given but the starting frequency, I just couldn't make a connection with anything.

 

[rl.bhat]

So there is no way of knowing with the information given if the starting frequency is 438 or 442? I thought about that also and reason that the final f should be found with the information given but the starting frequency, I just couldn't make a connection with anything.

The additional information should be either the beats decrease first and then increase to 4 Hz or increase to 4 Hz.

 

[rwisz]

Your thinking process is correct. The relationship between change in v and change in beat frequency is that as the tension (T) in the string decreases, the velocity (v) of the wave traveling through the string also decreases. This results in a decrease in the frequency of the note produced by the string. As the frequency of the note produced by the string decreases, the beat frequency also decreases. This is because the beat frequency is the difference between the frequencies of the two notes being played (in this case, the out of tune A note and the tuning fork). So as the frequency of the out of tune A note decreases, the beat frequency also decreases. Therefore, when the tension in the string is decreased, the beat frequency decreases from 2 Hz to 4 Hz, indicating that the frequency of the out of tune A note has also decreased.