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Chapter 16: Problem 34

A string of a violin produces 2 beats per second when sounded along with a standard fork of frequency \(400 . \mathrm{Hz}\). The beat frequency increases when the string is tightened. a) What was the frequency of the violin at first? b) What should be done to tune the violin?

Short Answer

Expert verified
Answer: The initial frequency of the violin string was 398 Hz. To tune the violin, the string should be tightened until its frequency reaches 400 Hz to match the frequency of the tuning fork.

Step by step solution

01

Understand the beat frequency

Beat frequency is the difference between the frequencies of two waves. In this case, the beat frequency is created when the violin string is played along with a standard tuning fork. The beat frequency is given as 2 beats per second (Hz).
02

Use the beat frequency formula to find the initial frequency

The formula for beat frequency is: Beat frequency = |f1 - f2| where f1 and f2 are the frequencies of the two waves. In this case, we are given the beat frequency (2 Hz) and the frequency of the tuning fork (400 Hz). Let the initial frequency of the violin string be f_violin. Then we have: 2 = |f_violin - 400|
03

Solve for the initial frequency of the violin string

We have two possibilities for the initial frequency: 1. f_violin - 400 = 2, which implies f_violin = 402 Hz 2. 400 - f_violin = 2, which implies f_violin = 398 Hz Since we are given that the beat frequency increases when the string is tightened, we can infer that the initial frequency of the violin string must be 398 Hz (lower than the tuning fork). Therefore, the initial frequency of the violin string is 398 Hz. a) The frequency of the violin at first was 398 Hz.
04

Determine the action needed to tune the violin

We are given that the beat frequency increases when the string is tightened, which means the frequency of the violin string also increases as it approaches the frequency of the tuning fork (400 Hz). To tune the violin, we need to adjust the string to match the frequency of the tuning fork. b) To tune the violin, we should tighten the string until the frequency of the violin string reaches 400 Hz and matches the frequency of the tuning fork. At this point, there will be no beats as both the sources have the same frequency.

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