First slide

Beats

Question

Two identical flutes produce fundamental notes of frequency 300 Hz at $27^{0} C$ . If the temp of the air in one flute is increased to $31^{0} C$ . The number of beats heard per second will be

Moderate

A

1
B

2
C

3
D

5

Solution

$\frac{V_{2}}{V_{1}} = {(\frac{273 + 31}{273 + 27})}^{1 / 2} = {(\frac{304}{300})}^{1 / 2} = {(\frac{152}{150})}^{1 / 2} W e k n o w, f r e q u e n c y i s p r o p o r t i o n a l t o s p e e d o f s o u n d \begin{array}{l} ∴ \frac{n_{2}}{n_{1}} = \frac{V_{2}}{V_{1}} = {(1 + \frac{2}{150})}^{1 / 2} ≃ (1 + \frac{1}{150}) u s e b i n o m i a l a p p r o x \\ \Rightarrow n_{2} = 300 \times (1 + \frac{1}{150}) = 300 + 2 = 302 H z \\ ∴ b e a t f r e q u e n c y = 302 - 300 = 2 H z \end{array}$

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