Answer :
The beat frequency heard when the two strings are played together will be 2940 Hz.
The beat frequency \( f_b \) between two vibrating strings with frequencies f_1 and f_2 is given by:
[tex]\[ f_b = | f_1 - f_2 | \][/tex]
In this case, both violin strings are initially tuned to 294 Hz, so f_1 = f_2 = 294 Hz. When the tension in one string is decreased by 4.0, its frequency changes. Let's denote the new frequency of this string as \( f_2' \).
The change in frequency due to tension is given by the formula:
[tex]\[ \Delta f = \frac{f}{2L} \cdot \Delta T \][/tex]
Where:
[tex]- \( \Delta f \) is the change in frequency.\\- \( f \) is the initial frequency (294 Hz in this case).\\- \( L \) is the length of the string.\\- \( \Delta T \) is the change in tension (4.0 in this case).[/tex]
Given that \[tex]( f = 294 \) Hz and \( \Delta T = -4.0 \) (decreased tension), we can calculate \( \Delta f \):[/tex]
[tex]\[ \Delta f = \frac{294}{2L} \cdot (-4.0) \][/tex]
Now, we need to find \( f_2' \), the new frequency of the string with decreased tension. It is given by:
[tex]\[ f_2' = f_2 + \Delta f \][/tex]
Since both strings were initially tuned to the same frequency, f_2 = 294 Hz. Substitute the values to find f_2':
[tex]\[ f_2' = 294 + \frac{294}{2L} \cdot (-4.0) \][/tex]
Let's assume a typical length for a violin string, such as 32 cm (0.32 m):
[tex]\[ f_2' = 294 + \frac{294}{2(0.32)} \cdot (-4.0) \]\[ f_2' = 294 + \frac{294}{0.64} \cdot (-4.0) \]\[ f_2' = 294 - 735 \cdot (-4.0) \]\[ f_2' = 294 - 2940 \]\[ f_2' = -2646 \text{ Hz} \][/tex]
Now, calculate the beat frequency:
[tex]\[ f_b = | f_1 - f_2' | \]\[ f_b = | 294 - (-2646) | \]\[ f_b = | 294 + 2646 | \]\[ f_b = 2940 \text{ Hz} \][/tex]
So, the beat frequency heard when the two strings are played together will be 2940 Hz.
