Problem 2: Note: beats, which we discussed in lecture, are also covered in secti

Question

Problem 2: Note: beats, which we discussed in lecture, are also covered in section 16-6 of the text and do not require specific knowledge of the physics of sound. There is a cool trick in hints.pdf A violinist plays his A string while tuning up, and listens to oboist playing her perfect A at 440Hz. He notices a beat with frequency 1.5 Hz between the two tones. By what percentage must he change the tension in his string? (He will have to experiment to see whether he needs to make it tighter or looser).

Explanation / Answer

The frequency of the violin string is directly prportional to the square root of tension in the string.

so, f = kF

Violinist notices a beat of 1.5 Hz, which means the difference in the frequency of violin and oboe is 1.5 Hz, which means either the violinist is playin a tune of f = 440Hz - 1.5 Hz = 438.5 Hz or a tune of f = 440Hz + 1.5 Hz = 441.5 Hz.

Suppose he is playing a tune of 438.5 Hz, then he will have to increase the frequency to 440 hz for which he will have to tighten the string.

Let tension in the string be F1 for frequency f1 =  438.5 Hz, and let tension in the string be F2 for frequency f2 =  440 Hz

f2/f1 = [F2/F1]

or, F2/F1 = [440 Hz/438.5 Hz]2

or, percentage change in tension is;

%Change = [(F2 - F1)/F1]X100 = [F2/F1 - 1]X100

or, %Change = 0.68%, so the tension should be increased by 0.685%

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Suppose he is playing a tune of 441.5 Hz, then he will have to decrease the frequency to 440 Hz for which he will have to tighten the string.

Let tension in the string be F1 for frequency f1 =  441.5 Hz, and let tension in the string be F2 for frequency f2 =  440 Hz

f2/f1 = [F2/F1]

or, F2/F1 = [440 Hz/441.5 Hz]2

or, percentage change in tension is;

%Change = [(F2 - F1)/F1]X100 = [F2/F1 - 1]X100

or, %Change = -0.678%, -ve sign indicates decrease in tension

So the tension should be decreased by 0.678%.

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This concludes the answers. Check the answer and let me know if it's correct. If you need any more clarification or correction, feel free to ask.....

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