Two tuning forks \(X\) and \(Y\) are of frequencies \(280 \mathrm{~Hz}\) and \(284 \mathrm{~Hz}\). A third…
Two tuning forks \(X\) and \(Y\) are of frequencies \(280 \mathrm{~Hz}\) and \(284 \mathrm{~Hz}\). A third tuning fork \(Z\) is of unknown frequency. When \(X\) and \(Z\) are sounded together certain beats are heard per second. When \(Y\) and \(Z\) are sounded together beat frequency is found to be thrice as great. The frequency of \(Z\) is
\(282 \mathrm{~Hz}\)
\(286 \mathrm{~Hz}\)
\(280 \mathrm{~Hz}\)
\(278 \mathrm{~Hz}\)
Solution
Given, frequency of tuning fork \(X\) and \(Y\) are,
\(\begin{aligned}
& n_X=280 \mathrm{~Hz} \\
& n_Y=284 \mathrm{~Hz} \\
& n_Z=?
\end{aligned}\)
When tuning forks \(X\) and \(Z\) are sounded together, then beats produced is \(b\) (assume).
\(\therefore \quad n_Z-n_X=b\) ...(i)
Again, when \(Y\) and \(Z\) are sounded together, then
\(n_Z-n_Y=3 b\) ...(ii)
Dividing Eq. (i) by Eq. (ii), we have
\(\begin{array}{ll}
& \frac{n_Z-n_X}{n_Z-n_Y}=\frac{b}{3 b} \Rightarrow \frac{n_Z-280}{n_Z-284}=\frac{1}{3} \\
\Rightarrow \quad & 3 n_Z-840=n_z-284 \\
\Rightarrow \quad & 3 n_z-n_Z=840-284 \\
\Rightarrow \quad & 2 n_Z=556 \Rightarrow n_Z=278 \mathrm{~Hz}
\end{array}\)