When the air column of a resonance tube is vibrated together with a tuning fork, 3 beats are heard per…
When the air column of a resonance tube is vibrated together with a tuning fork, 3 beats are heard per second, either the temperature of the air column is \(51^{\circ} \mathrm{C}\) or \(16^{\circ} \mathrm{C}\). The frequency of the tuning fork is
\(128 \mathrm{~Hz}\)
\(98 \mathrm{~Hz}\)
\(105 \mathrm{~Hz}\)
\(256 \mathrm{~Hz}\)
Solution
Number of beats per second when the air column of resonance tube, \(n=3\)
If \(n\) be the frequency of tuning fork, Then, \(n \propto \sqrt{T}[T \rightarrow\) temperature \(]\)
\(\frac{n_1}{n_2}=\sqrt{\frac{T_1}{T_2}}\)
At \(51^{\circ} \mathrm{C}, T_1=273+51=324 \mathrm{~K}\)
\(n_1=n+3\{\) frequency of tuning fork increase at the higher temperature \}
At \(16^{\circ} \mathrm{C}, T_2=273+16=289 \mathrm{~K}\)
\(n_2=n-3\) [At lower temperature, frequency of the tuning fork decreases.]
\(\therefore\) From Eq. (i),
\(\begin{array}{ll}
& \frac{n_1}{n_2}=\sqrt{\frac{324}{289}} \Rightarrow \frac{n+3}{n-3}=\frac{18}{17} \\
& 18 n-54=17 n+51 \\
\Rightarrow \quad 18 n-17 n=54+51 \Rightarrow n=105 \mathrm{~Hz}
\end{array}\)