The Emperor has asked Darth Maul to tune the piano that is used in the Galactic

Question

The Emperor has asked Darth Maul to tune the piano that is used in the Galactic Senate.

Darth Maul plays a note from a frequency generator at 277 Hz and then hits a key on the piano that should have the same frequency. Darth Maul hears 12 beats every 4.0 s when they are played together.

(a) If the tuner is indeed vibrating at 277 Hz, what must be the frequency of piano string be?

1- lowest possible answer
Hz?
2- highest possible answer
Hz?

(b) By how much (in percent) must the tension of the string be increased for them to be perfectly in tune, assuming the piano is actually playing the lower of your two answers you found from part (a)?

Explanation / Answer

(a)

Beat frequency is:
f_beat = 12/4.0s = 3Hz
It is equal to the magnitude of the difference in frequency of the tones
3Hz = |f - 277Hz|
Hence,
f = 277Hz - 3Hz = 274Hz (lowest possible answer)
or
f = 277Hz + 3Hz = 280Hz (highest possible answer)

(b)
The frequency in a vibrating string is proportional to the square root of the tension:
f ~ T  
f'/f = (T'/T)
T'/T = (f'/f)²

If frequency is to low

T'/T = (277 / 274)²

T'/T = 1.022

That means you need to change the tension by

(1.022 - 1)100% = +2.20%

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