Question

A column of air at ${51^ \circ }\,C$ and a tuning fork produces $4$ beats per second when sounded together. As the temperature of the air column is decreased, the number of beats per sec tends to decrease and when the temperature is ${16^ \circ }\,C$ the two produce $1$ beats per second. If the frequency of the tuning fork is $10\,k$, then find the value of $k$.
(A) $8$
(B) $5$
(C) $6$
(D) $7$

Answer 1

By using the relation that the frequency is directly proportional to the velocity of the sound and the velocity of the sound is directly proportional to the square of the temperature in kelvin. By using this relation, the solution can be determined.

Useful formula
The relation that the frequency is directly proportional to the velocity of the sound and the velocity of the sound is directly proportional to the square of the temperature in kelvin, then
$f \propto V \propto \sqrt T$
Where, $f$ is the frequency, $V$ is the velocity and $T$ is the temperature.

Complete step by step solution
Given that,
The air at ${51^ \circ }\,C$ produces $4$ beats per second.
The air at ${16^ \circ }\,C$ produces $1$ beats per second.
By the above-mentioned formula,
$\Rightarrow f \propto \sqrt T$
By the given information,
$\Rightarrow f + 4 \propto \sqrt {273 + 51} \,....................\left( 1 \right)$
The temperature is converted from Celsius to kelvin,
$\Rightarrow f + 1 \propto \sqrt {273 + 16} \,....................\left( 2 \right)$
Now the equation (1) is divided by the equation (2), then
$\Rightarrow \dfrac{{f + 4}}{{f + 1}} = \dfrac{{\sqrt {273 + 51} }}{{\sqrt {273 + 16} }}$
By adding the above equation, then
$\Rightarrow \dfrac{{f + 4}}{{f + 1}} = \dfrac{{\sqrt {324} }}{{\sqrt {289} }}$
By taking the square root, then the above equation is written as,
$\Rightarrow \dfrac{{f + 4}}{{f + 1}} = \dfrac{{18}}{{17}}$
By rearranging the terms, then the above equation is written as,
$\Rightarrow 17\left( {f + 4} \right) = 18\left( {f + 1} \right)$
By multiplying the terms, then the above equation is written as,
$\Rightarrow 17f + 68 = 18f + 18$
By rearranging the terms, then the above equation is written as,
$\Rightarrow 18f - 17f = 68 - 18$
On further simplification, then
$\Rightarrow f = 50\,Hz$.
By comparing the above equation with tuning fork given in the question, then
$\Rightarrow 50 = 10k$
By keeping $k$ in one side, then
$\Rightarrow k = \dfrac{{50}}{{10}}$
On dividing the terms, then
$\Rightarrow k = 5$

Hence, the option (C) is the correct answer.

Note: From this relation the frequency is directly proportional to the velocity of the sound and the velocity of the sound is directly proportional to the square of the temperature in kelvin. If any one of the parameters increases, then the other parameters also increases.