Two identical flutes produce fundamental note | vedclass

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(B) The velocity of sound $v$ is directly proportional to the square root of the absolute temperature $T$,i.e.,$v \propto \sqrt{T}$.Since the length o

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$2$

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(B) The velocity of sound $v$ is directly proportional to the square root of the absolute temperature $T$,i.e.,$v \propto \sqrt{T}$.Since the length of the flute remains constant,the wavelength $\lambda$ of the fundamental note remains constant. Given $v = n\lambda$,we have $n \propto v$,so $n \propto \sqrt{T}$.Let $n_1 = 300 \ Hz$ at $T_1 = 27 + 273 = 300 \ K$.Let $n_2 = 300 + x$ at $T_2 = 31 + 273 = 304 \ K$,where $x$ is the number of beats per second.Using the ratio: $\frac{n_2}{n_1} = \sqrt{\frac{T_2}{T_1}}$$\frac{300 + x}{300} = \sqrt{\frac{304}{300}} = \sqrt{1 + \frac{4}{300}}$Using the binomial approximation $(1 + z)^n \approx 1 + nz$ for small $z$:$1 + \frac{x}{300} \approx 1 + \frac{1}{2} 	imes \frac{4}{300}$$\frac{x}{300} = \frac{2}{300}$$x = 2$.Thus,the number of beats heard per second is $2$.

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

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