An organ pipe filled with a gas at 27^{circ}C | vedclass

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(B) The fundamental frequency of an organ pipe is given by $n = \frac{v}{4\ell}$ (for a closed pipe) or $n = \frac{v}{2\ell}$ (for an open pipe). In b

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$440$

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(B) The fundamental frequency of an organ pipe is given by $n = \frac{v}{4\ell}$ (for a closed pipe) or $n = \frac{v}{2\ell}$ (for an open pipe). In both cases,$n \propto v$.Since the speed of sound in a gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$,we have $v \propto \sqrt{T}$.Therefore,the frequency $n$ is directly proportional to the square root of the absolute temperature $T$,i.e.,$n \propto \sqrt{T}$.Given $T_1 = 27^{\circ}C = 300\,K$ and $T_2 = 90^{\circ}C = 363\,K$.Using the ratio: $\frac{n_2}{n_1} = \sqrt{\frac{T_2}{T_1}}$.$\frac{n_2}{400} = \sqrt{\frac{363}{300}} = \sqrt{1.21} = 1.1$.$n_2 = 400 	imes 1.1 = 440\,Hz$.

An organ pipe filled with a gas at $27^{\circ}C$ resonates at $400\,Hz$ in its fundamental mode. If it is filled with the same gas at $90^{\circ}C$,the resonance frequency at the same mode will be $...........\,Hz$.

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