High School

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Two violin strings are tuned to the same frequency of 294 Hz. The tension in one string is then decreased by 4.0 units. What will be the beat frequency heard when the two strings are played together?

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.

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Rewritten by : Barada