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Step-by-Step Solution
Step 1: Identify the Given Data
• Length of sonometer wire, L = 1 \text{ m}
• Initial tension in the wire, T_1 = 6 \text{ N}
• New tension in the wire, T_2 = 54 \text{ N}
• Number of beats per second with new tension = 12 \text{ Hz}
• Frequency of the tuning fork = f_F (to be determined).
Step 2: Use the Formula for Frequency of a Vibrating String
The frequency f of a string of length L , linear mass density \mu , under tension T , is given by:
f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}.
From this relationship, we see that f \propto \sqrt{T} when L and \mu are constant.
Step 3: Express the Initial and New Frequencies
• Initial frequency (under tension T_1 ):
f_1 = \frac{1}{2L}\sqrt{\frac{T_1}{\mu}}.
• New frequency (under tension T_2 ):
f_2 = \frac{1}{2L}\sqrt{\frac{T_2}{\mu}}.
Since T_2 = 9 \times T_1 , we get:
f_2 = \frac{1}{2L}\sqrt{\frac{9T_1}{\mu}} = 3 \times \left(\frac{1}{2L}\sqrt{\frac{T_1}{\mu}}\right) = 3f_1.
Hence, f_2 = 3f_1 .
Step 4: Use the Beat Frequency to Find f_F
When the tension is T_1 , the wire resonates with the tuning fork, so f_F = f_1 .
At the new tension T_2 , the wire’s frequency is 3f_1 , and 12 beats per second are heard. Beat frequency is given by the absolute difference between the two frequencies:
|f_2 - f_F| = 12.
Substituting f_2 = 3f_1 and f_F = f_1 :
|3f_F - f_F| = 12.
|2f_F| = 12 \quad \Rightarrow \quad 2f_F = 12 \quad \Rightarrow \quad f_F = 6 \text{ Hz}.
Step 5: Final Answer
Therefore, the frequency of the tuning fork is:
\boxed{6 \text{ Hz}}
