Question

Two identical flutes produce fundamental notes of frequency $300\, Hz$ 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 $v \propto \sqrt{T}$
$\left(\because v=\sqrt{\frac{\gamma R T}{M}}\right)$
$\frac{v_{2}}{v_{1}}=\sqrt{\frac{T_{2}}{T_{1}}}=\frac{n+ x}{n}$
$\Rightarrow \frac{300+x}{300}=\sqrt{\frac{(273+31)}{(273+27)}}$
$=\sqrt{\frac{304}{300}}=\sqrt{\frac{300+4}{300}}$
$\Rightarrow 1+\frac{x}{300}=\left(1+\frac{4}{300}\right)^{1 / 2}$
$=\left(1+\frac{1}{2} \times \frac{4}{300}\right)$
$\Rightarrow x=2 .$
$\left[\because(1+x)^{n}=1+n x\right]$