Two identical violin strings, when i tune and stretched with the same tension, have a fundamental frequency of 440Hz. One of the strings is retuned by adjusting the tension. When both strings are pluckes simultaneously, after one has been retuned, 1.5 beats per second are heard.

A) What are the possible fundamental frequencies of the retuned string?

B) By what fractional amount was the string tension changed if it was I)increased, ii)decreased

the possible frequencies are 440+-1.5 hz

Use the law of tensions,,,
frequency is proportinal to the square root of tension, or tension is prop to freq squared.

To solve this problem, we can use the concept of beats produced by two slightly different frequencies.

Let's start by calculating the frequency difference between the two strings. The beats occur at a rate of 1.5 beats per second, which means the frequency difference between the retuned string (f_retuned) and the original string (f_original) is 1.5 Hz.

A) Now, let's find the possible fundamental frequencies of the retuned string. We know that the original string has a fundamental frequency of 440 Hz. So, we can write the equation:

f_retuned - 440 Hz = 1.5 Hz

Rearranging the equation, we get:

f_retuned = 441.5 Hz

Therefore, the possible fundamental frequencies of the retuned string can be 441.5 Hz.

B) Next, let's determine how the tension of the retuned string was changed. The frequency produced by a string is directly proportional to the square root of the tension applied. Let's consider two scenarios:

i) If the tension was increased: In this case, the frequency of the retuned string increased. Since the fundamental frequency is directly proportional to the square root of the tension, we can express the change in tension as:

tension_change = (f_retuned^2 - f_original^2) / f_original^2

Plugging in the given values, we get:

tension_change = ((441.5 Hz)^2 - (440 Hz)^2) / (440 Hz)^2

Calculating this expression will give you the fractional amount by which the tension was increased.

ii) If the tension was decreased: Similarly, if the tension was decreased, the frequency of the retuned string would be lower. Using the same formula as above, we can calculate the tension change by substituting the appropriate values.

By following these steps, you should be able to find the possible fundamental frequencies of the retuned string and determine how the tension was changed, depending on whether it was increased or decreased.