Physics Question on Waves

Question

A tuning fork of frequency $512\, Hz$ makes $4$ beats per second with the vibrating string of a piano. The beat frequency decreases to $2$ beats per sec when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was

Answer 1

Solution

Let the frequencies of tuning fork and piano string be $\upsilon_1$ and $\upsilon_2$ respectively
$\therefore\, \, \upsilon_2=\upsilon_1\pm 4=512 \,Hz \pm 4$
$=516\, Hz$ or $508\, Hz$

Increase in the tension of a piano string increases its frequency
If $\upsilon_2=516 Hz,$ further increase in $\upsilon_2,$ resulted in an increase in the beat frequency. But this is not given in the question.
If $\upsilon_2=508\, Hz$ , further increase in $\upsilon_2$ resulted in decrease in the beat frequency. This is given in the question. When the beat frequency decreases to $2$ beats per second. Therefore, the frequency of the piano string before increasing the tension was $508\, Hz$ .