Question

A sonometer wire under tension \[64\,{\text{N}}\] vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of \[10\,{\text{cm}}\] and mass \[1\,{\text{g}}\]. The vibrating tuning fork is now moved away from the vibrating wire at a constant speed and an observer standing near the sonometer hears one beat per second. Then the speed with which the tuning fork is moved, if the speed of sound in air is \[330\,{\text{m/s}}\]
A. \[0.82\,{\text{m/s}}\]
B. \[1.5\,{\text{m/s}}\]
C. \[3\,{\text{m/s}}\]
D. \[6\,{\text{m/s}}\]

Answer 1

Hint: We can use the formula for the fundamental frequency of the vibrating string. First calculate the mass per unit length of the sonometer wire. Then calculate the fundamental frequency of the sonometer wire and the frequency of the sound heard by the person. Then we have to use the formula for the frequency heard by the stationary observer when the source of sound is moving away from the source.

Formulae used:
The expression for the fundamental frequency of vibration in a vibrating string is given by
\[f = \dfrac{1}{{2L}}\sqrt {\dfrac{T}{\mu }} \] …… (1)
Here, \[L\] is length of the wire, \[T\] is tension in the string and \[\mu \] is mass per unit length of the string.
The frequency \[f'\] of the sound heard by the observer moving away from a stationary source is given by
\[f' = \left( {\dfrac{v}{{v + {v_s}}}} \right)f\] …… (2)
Here, \[v\] is the velocity of sound, \[{v_s}\] is velocity of source of sound and \[f\] is frequency of the sound from the source.

Complete step by step answer:
We have given that the tension in the sonometer wire is \[64\,{\text{N}}\].
\[T = 64\,{\text{N}}\]
The length and mass of the sonometer wire are \[10\,{\text{cm}}\] and \[1\,{\text{g}}\] respectively.
\[L = 10\,{\text{cm}}\]
\[ \Rightarrow m = 1\,{\text{g}}\]
The speed of the sound is \[330\,{\text{m/s}}\].
\[v = 330\,{\text{m/s}}\]
Let us first calculate mass per unit length of the given sonometer wire.
\[\mu = \dfrac{{1\,{\text{g}}}}{{10\,{\text{cm}}}}\]
\[ \Rightarrow \mu = \dfrac{{{{10}^{ - 3}}\,{\text{kg}}}}{{0.1\,{\text{m}}}}\]
\[ \Rightarrow \mu = 0.01\,{\text{kg/m}}\]
Hence, mass per unit length of the given sonometer wire is \[0.01\,{\text{kg/m}}\].

Let us now calculate the frequency of the sonometer wire.We have given that the sonometer wire is in resonance with the tuning fork. Hence, the fundamental frequency of the tuning fork and the sonometer wire is the same.Substitute \[10\,{\text{cm}}\] for \[L\], \[64\,{\text{N}}\] for \[T\] and \[0.01\,{\text{kg/m}}\] for \[\mu \] in equation (1).
\[f = \dfrac{1}{{2\left( {10\,{\text{cm}}} \right)}}\sqrt {\dfrac{{64\,{\text{N}}}}{{0.01\,{\text{kg/m}}}}} \]
\[ \Rightarrow f = \dfrac{1}{{2\left( {0.1\,{\text{m}}} \right)}}\sqrt {6400} \]
\[ \Rightarrow f = \dfrac{{80}}{{2\left( {0.1\,{\text{m}}} \right)}}\]
\[ \Rightarrow f = 400\,{\text{Hz}}\]
Hence, the fundamental frequency of the sonometer wire is \[400\,{\text{Hz}}\].We have given that the observer near the sonometer hears the sound with beat frequency of one beat per second.Hence, the frequency of the sound heard by the person decreases by one hertz.

Thus, the frequency of the sound heard by the person is \[399\,{\text{Hz}}\].
\[f' = 399\,{\text{Hz}}\]
Let us now calculate the velocity with which the tuning fork is moved.Substitute \[399\,{\text{Hz}}\] for \[f'\], for \[v\] and \[400\,{\text{Hz}}\] for \[f\] in equation (2).
\[399\,{\text{Hz}} = \left( {\dfrac{{330\,{\text{m/s}}}}{{\left( {330\,{\text{m/s}}} \right) + {v_s}}}} \right)\left( {400\,{\text{Hz}}} \right)\]
\[ \Rightarrow \dfrac{{399}}{{400}} = \dfrac{{330}}{{330 + {v_s}}}\]
\[ \Rightarrow 330 + {v_s} = \dfrac{{330 \times 400}}{{399}}\]
\[ \Rightarrow {v_s} = \dfrac{{330 \times 400}}{{399}} - 330\]
\[ \therefore {v_s} = 0.82\,{\text{m/s}}\]
Therefore, the velocity with which the tuning fork is moved is \[0.82\,{\text{m/s}}\].

Hence, the correct option is A.

Note: The students should keep in mind that we have given that the beat frequency for the sound heard by the person is one beat per second. This means that the frequency of the sound heard by the person near the sonometer wire is less by one hertz than the fundamental frequency of the vibrating sonometer wire and not greater than one hertz.