Question

A certain fork is found to give $3\,beats/s$ when sounded in conjunction with a stretched wire vibrating under a tension of either $4.32kg - wt$ or $3kg - wt$ . The frequency of fork is
A. $39\,Hz$
B. $30\,Hz$
C. $33\,Hz$
D. $36\,Hz$

Answer 1

A tuning fork is said to be an acoustic resonator which consists of a two pronged fork which is in the form of a U-shaped bar of elastic metal which is usually made up of steel. Pitch is responsible for the vibration. The pitch of the tuning fork depends on the tension, length and mass. The pitch is one of the characteristics of the sound which depends on the frequency.

Formula used:
Frequency of tuning fork is given by,
$f = \dfrac{1}{{2l}}\sqrt {\dfrac{T}{m}}$ ……….$\left( 1 \right)$
Where, $l =$ Length, $T =$ Tension and $m =$ Mass.
From equation frequency is related to tension as,
$f = \sqrt T$

Complete step by step answer:
We know that, frequency = $\sqrt {Tension}$
Therefore, ${f_1} = \sqrt {{T_1}}$ and ${f_2} = \sqrt {{T_2}}$
Given, ${T_1} = 4.32$ and ${T_2} = 3$
${f_1} - {f_0} = 3$
Therefore, ${f_1} = {f_0} + 3$ ………. $\left( 1 \right)$
${f_0} - {f_2} = 3$
Therefore, ${f_2} = {f_0} - 3$ ……… $\left( 2 \right)$
Hence, it can be written as
$\dfrac{{{f_1}}}{{{f_2}}} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}}$
Substituting the values of ${T_1}$ and ${T_2}$ in the above equation we get
$\dfrac{{{f_1}}}{{{f_2}}} = \sqrt {\dfrac{{4.32}}{3}}$ ………… $\left( 3 \right)$
Substituting equation $\left( 1 \right)$ and equation $\left( 2 \right)$ in equation $\left( 3 \right)$ we get
$\dfrac{{{f_0} + 3}}{{{f_0} - 3}} = \sqrt {\dfrac{{4.32}}{3}}$
On simplifying, we get
$\left( {\dfrac{{{f_0} + 3}}{{{f_0} - 3}}} \right) = 1.2$
$\Rightarrow {f_0} + 3 = 1.2\left( {{f_0} - 3} \right)$
On further simplification
${f_0} + 3 = 1.2{f_0} - 3.6$
$\Rightarrow {f_0} - 1.2{f_0} = - 3.6 - 3$
$\Rightarrow - 0.2{f_0} = - 6.6$
$\Rightarrow {f_0} = \dfrac{{ - 6.6}}{{ - 0.2}}$
$\therefore {f_0} = 33\,Hz$

Hence, option C is correct.

Note: The frequency of the tuning fork and the frequency of the string will be the same if the string and the tuning fork vibrate at resonance. The tension in the string is defined as the pulling force that is transmitted axially by means of a string. The sound depends on the characteristics that are loudness which depends on the amplitude, quality which depends on the nature of waveform and pitch which depends on the frequency.