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A piano tuner sounds two strings simultaneously. One has been previously tuned to vibrate at 293.0 Hz. The tuner hears 3.0 beats per second. The tuner increases the tension on the as-yet untuned string, and now when they are played together the beat frequency is
1.0s−1.
(a) What was the original frequency of the untuned string?
(b) By what percentage did the tuner increase the tension on that string?

Answer :

Answer:

Part a)

[tex]f_B = 290 Hz[/tex]

Part B)

percentage increase is

[tex]percentage = 1.38[/tex]%

Explanation:

Part a)

As we know that the beat frequency is

[tex]f_A - f_B = 3[/tex]

after increasing the tension the beat frequency is decreased and hence the tension in string B will increase

So we have

[tex]293 - f_B = 3[/tex]

[tex]f_B = 290 Hz[/tex]

Part B)

percentage increase in the tension of the string will be given as

[tex]f_A - f_B' = 1[/tex]

[tex]f_B' = 292 Hz[/tex]

now we have

[tex]f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex]

so we have

[tex]T_1 = C (290)^2[/tex]

[tex]T_2 = C(292)^2[/tex]

so we have

[tex]\frac{\Delta T}{T} = \frac{292^2 - 290^2}{290^2}[/tex]

percentage increase is

[tex]percentage = 1.38[/tex]

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