Question

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

Answer: The initial frequency of the violin string is 398 Hz. To tune the violin, the string should be tightened to match the frequency of the tuning fork, 400 Hz.

Step-by-step solution

Identify the beat frequency

We know that when the violin string is played along with the tuning fork, it produces 2 beats per second. This means that the beat frequency is 2 Hz.

Find the initial frequency of the violin string

Since beat frequency is the absolute difference between the frequencies of the two sound waves (tuning fork and violin string), we can set up the following equation: \(f_{beat} = |f_{violin} - f_{fork}|\) We know that the beat frequency \(f_{beat} = 2\: Hz\) and the frequency of standard fork \(f_{fork} = 400\: Hz\). We want to find the initial frequency of the violin string \(f_{violin}\): \(2 = |f_{violin} - 400|\) The beat frequency increases when the string is tightened, meaning that initially, the frequency of the violin string must be slightly lower than 400 Hz. We can write this as: \(f_{violin} = 400 - 2 = 398\: Hz\) So, the initial frequency of the violin string was 398 Hz. Answer (a): The frequency of the violin at first was 398 Hz.

Explain what should be done to tune the violin

To tune the violin, we want to have the frequency of the violin string to match the frequency of the tuning fork (400 Hz). Since we know that the frequency of the violin's string is 398 Hz and that tightening the string will increase the frequency, we should tighten the string. This will increase its frequency until it matches the frequency of the tuning fork (400 Hz). Answer (b): The violin string should be tightened to match the frequency of the tuning fork, 400 Hz, in order to properly tune the violin.