Question

An air column at temperature of ${51}^{\circ }\text{C}$ produces $4$ beats in one second with a tuning fork.
When the temperature of the air column decreases to ${16}^{\circ }\text{C}$ then one beat per second is produced with same tuning fork. The frequency of the tuning fork is

• A. $80\text{Hz}$
• B. $70\text{Hz}$
• C. $60\text{Hz}$
• D. $50\text{Hz}$

Answer 1

The correct option is D $50\text{Hz}$

Let us suppose, the frequency of the tuning fork is $f_o$ and the frequency of the sound source in the air column is $f$.
So, beat frequency,
$f_b=|f_o-f|$
At $51{}^{\circ }\text{C}$, $f_b=4\text{Hz}$
At $16{}^{\circ }\text{C}$, $f_b=1\text{Hz}$
As temperature reduces, beat frequency reduces, hence
we can write $f_b=f-f_o$
Let $f_1$ be frequency at $51{}^{\circ }\text{C}$, $f_2$ be frequency at $16{}^{\circ }\text{C}$,
$\Rightarrow f_1=4+f_o$ and
$\Rightarrow f_2=1+f_o$
We know that $f\propto \sqrt{T}$
$\frac{f_1}{f_2}=\sqrt{\frac{T_1}{T_2}}=\sqrt{\frac{324}{289}}=\frac{18}{17}=\frac{4+f_o}{1+f_o}$
$18+18f_o=68+17f_o$
$f_o=50\text{Hz}$