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Calculate the frequency of the tuning fork which when vibrated with the tuning fork of frequency \[256{\text{ Hz}}\] produces \[6{\text{ beat/s}}\] and when vibrated with a tuning fork having frequency \[253{\text{ Hz}}\] produces \[{\text{3 beat/s}}\].

Last updated date: 24th Jul 2024
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Hint: A tuning fork vibrates when one of its ends is hit with another object. This causes the tunes to vibrate, thereby causing disturbances in the surrounding air giving out sound waves. The fork shape of this instrument produces sound waves with a very pure tone. The frequency of the sound waves produced by the tuning fork depends on the dimensions of the fork and the material from which it is made.

Complete step by step answer:
Let the tuning fork which is hit upon being denoted by \[A\], and the other two tuning forks be denoted by \[B\]and \[C\], respectively.
Given in the question,
Frequency of tuning fork \[B\], \[{f_B} = 256{\text{ Hz}}\]
Frequency of tuning fork \[C\], \[{f_C} = 253{\text{ Hz}}\]
Now, the tuning fork \[A\] produces \[6{\text{ beat/s}}\] when hit by the tuning fork \[B\]. Let, the number of beats be denoted by \[n\]
Calculate the frequency of the tuning fork \[A\], \[{f_A}\] as
\[{f_A} = {f_B} \pm n{\text{ }}......{\text{ (1)}}\]
Substitute \[6{\text{ }}\]for \[n\], and \[256{\text{ Hz}}\] for \[{f_B}\] in equation (1) as
  {f_A} = 256 \pm 6 \\
  {f_A} = 250{\text{ Hz or 262 Hz}} \\
Now, the tuning fork \[A\] produces \[{\text{3 beat/s}}\] when hit by the tuning fork \[C\] whose frequency is \[253{\text{ Hz}}\]. This is possible only when the frequency of the tuning fork \[A\] is \[250{\text{ Hz}}\].

Note: A tuning fork is an acoustic resonator and can be implemented to tune musical instruments. Tuning forks are also used in various clocks and watches, medicinal and scientific instruments, level sensors, etc. The oscillations produced by a tuning fork are not damped as because of the fork shape, it can be held at the base, which leads to symmetric vibrations of the tuning fork.