Two piano strings are supposed to be vibrating at 220 Hz , but a piano tuner hea

Question

Two piano strings are supposed to be vibrating at 220 Hz , but a piano tuner hears three beats every 3.4 s when they are played together.

Part A

If one is vibrating at 220 Hz , what must be the frequency of the other (is there only one answer)?

Express your answer using four significant figures. If there is more than one answer, enter them in ascending order separated by commas.

Part B

By how much (in percent) must the tension be increased or decreased to bring them in tune?

Express your answer using two significant figures. If there is more than one answer, enter them in ascending order separated by commas.

Explanation / Answer

(a)
Beat frequency is:
f_beat = 3/3.4s = 0.882Hz

It is equal to the magnitude of the difference in frequency of the tones
0.882Hz = |f - 220Hz|

Hence,
f = 220Hz - 0.882Hz = 219.118Hz (lowest possible answer)
or
f = 220Hz + 0.882Hz = 220.882Hz (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 = (220 / 219.118)² = 1.00807
That means you need to change the tension by
(1.00806 - 1)100% = +0.806%

When frequency is to high
T'/T = (220 / 220.882)² = 0.99203
That means you need to change the tension by
(0.99202 - 1)100% = -0.798%

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