First slide

Stationary waves

Question

Two identical flutes produce fundamental notes of frequency 300 Hz at 27⁰C. If the temperature of air in one flute is increased to 31⁰C, the number of the beats heard per second will be

Moderate

A

1
B

2
C

3
D

4

Solution

Velocity of sound increases if the temperature increases. So with $v = nλ$ , if v increases n will increase
at $27^{o} C, v_{1} = nλ$ , at $31^{o} C, v_{2} = (n + x) λ$
Now using $v \propto \sqrt{T}$ $(∵ v = \sqrt{\frac{γRT}{M}})$
$\frac{v_{2}}{v_{1}} = \sqrt{\frac{T_{2}}{T_{1}}} = \frac{n + x}{n}$
$\Rightarrow \frac{300 + x}{300} = \sqrt{\frac{(273 + 31)}{(273 + 27)}} = \sqrt{\frac{304}{300}} = \sqrt{\frac{300 + 4}{300}}$
$\begin{array}{l} \Rightarrow 1 + \frac{x}{300} = {(1 + \frac{4}{300})}^{1 / 2} = (1 + \frac{1}{2} \times \frac{4}{300}) \Rightarrow x = 2 \end{array}$
$[∵ {(1 + x)}^{n} = 1 + nx]$

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