Question

Two identical flutes produce fundamental notes of frequency $300Hz$ at $27^{\circ }C$. If the temperature of air in one flute is increased to $31^{\circ }C$, the number of the beats heard per second will be

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

Velocity of sound increases if the temperature increases. So with $v=n\lambda$, if $v$ increases $n$ will increase at $27^{\circ }C,v_1=n\lambda$, at $31^{\circ }C,v_2=(n+x)\lambda$
Now using $v\propto \sqrt{T}$
$\left(\because v=\sqrt{\frac{\gamma RT}{M}}\right)$
$\frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}}=\frac{n+x}{n}$
$\Rightarrow \frac{300+x}{300}=\sqrt{\frac{(273+31)}{(273+27)}}$
$=\sqrt{\frac{304}{300}}=\sqrt{\frac{300+4}{300}}$
$\Rightarrow 1+\frac{x}{300}={\left(1+\frac{4}{300}\right)}^{1/2}$
$=\left(1+\frac{1}{2}\times \frac{4}{300}\right)$
$\Rightarrow x=2.$
$\left[\because (1+x{)}^n=1+nx\right]$