Question

283.Two closed organ pipes A and B have the same length and A is wider than B.They resonate in the fundamental mode at frequencies nA and nB respectively

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{Let}\mathrm{l}\mathrm{be}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{pipe}\mathrm{and}\mathrm{d}\mathrm{be}\mathrm{the}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{pipe}. \\ \mathrm{For}\mathrm{fundamental}\mathrm{mode}\mathrm{of}\mathrm{vibration}, \\ \frac{\mathrm{\lambda }}{4}=\mathrm{l}+0.3\mathrm{d} \\ \mathrm{\lambda }=4\left(\mathrm{l}+0.3\mathrm{d}\right) \\ \mathrm{Frequency},\mathrm{n}=\frac{\mathrm{V}}{\mathrm{\lambda }} \\ \Rightarrow \mathrm{n}=\frac{\mathrm{V}}{4\left(\mathrm{l}+0.3\mathrm{d}\right)} \\ \mathrm{As}\mathrm{l}\mathrm{is}\mathrm{same},\mathrm{the}\mathrm{wider}\mathrm{the}\mathrm{tube}\mathrm{will}\mathrm{resonate}\mathrm{at}\mathrm{lower}\mathrm{frequency}. \\ \mathrm{Therefore},{\mathrm{n}}_{\mathrm{A}}<{\mathrm{n}}_{\mathrm{B}} \end{gathered}$$