Two forks \(\mathrm{A}\) and \(\mathrm{B}\) when sounded together produce 4 beats \(/ \mathrm{s}\). The fork A is in unison with \(30 \mathrm{~cm}\) length of a sonometer wire and \(B\) is in unison with \(25 \mathrm{~cm}\) length of the same wire at the same tension. The frequencies of the fork are (A) \(24 \mathrm{~Hz}, 28 \mathrm{~Hz}\) (B) \(20 \mathrm{~Hz}, 24 \mathrm{~Hz}\) (C) \(16 \mathrm{~Hz}, 20 \mathrm{~Hz}\) (D) \(26 \mathrm{~Hz}, 30 \mathrm{~Hz}\)

The beat frequency is the absolute difference between the frequencies of the two tuning forks. If the two forks produce 4 beats per second, we can write the equation: \[ |f_A - f_B| = 4 \]

When a tuning fork is in unison with a sonometer wire, it means that their frequencies are related. Depending on the length, tension, and mass per unit length of the sonometer wire, its fundamental frequency is proportional to the inverse of its length. We can write this proportionality relation as: \[ f_A \propto \frac{1}{l_A} \] and \[ f_B \propto \frac{1}{l_B} \] Where \(l_A = 30 cm\) and \(l_B = 25 cm\).

From the proportionality relations in step 2, we can derive the ratio of frequencies as: \[ \frac{f_A}{f_B} = \frac{l_B}{l_A} \] Substituting the given lengths into the equation, we get: \[ \frac{f_A}{f_B} = \frac{25}{30} = \frac{5}{6} \]

Using the beat frequency equation and the ratio of frequencies equation, we can set up a system of equations to find the frequencies of the forks. Let \( x = f_A \) and \( y = f_B \), then our equations are: \[ |x - y| = 4 \] \[ \frac{x}{y} = \frac{5}{6} \] We can use the second equation to express one variable in terms of the other and then substitute it into the first equation. For example, express x in terms of y: \[ x = \frac{5}{6}y \] Substituting into the first equation, we get two cases: \[ (\frac{5}{6}y) - y = 4 \quad or \quad y - (\frac{5}{6}y) = 4 \] Solving both cases, we find that the frequencies are: Case 1: \( f_A = 20 Hz \) and \( f_B = 24 Hz \) Case 2: \( f_A = 24 Hz \) and \( f_B = 20 Hz \) In both cases, we have the same pair of frequencies, so the correct option is: (B) \(20 \mathrm{~Hz}\), \(24 \mathrm{~Hz}\)