Waves

JEE Main level, numerical answer, medium

85 of 142

Question 85 of 142

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is (a−1a)\left(\frac{a-1}{a}\right) then the value of a\mathrm{a} is ________

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Answer: 16

For closed organ pipe
fc=(2n+1)v4ℓ=15v4ℓ\mathrm{f}_{\mathrm{c}}=(2 \mathrm{n}+1) \frac{\mathrm{v}}{4 \ell}=\frac{15 \mathrm{v}}{4 \ell}
For open organ pipe
fo=(n+1)v2ℓ=8v2ℓfcfo=1516=a−1a⇒a=16\begin{aligned} & \mathrm{f}_{\mathrm{o}}=(\mathrm{n}+1) \frac{\mathrm{v}}{2 \ell}=\frac{8 \mathrm{v}}{2 \ell} \\ & \frac{\mathrm{f}_{\mathrm{c}}}{\mathrm{f}_{\mathrm{o}}}=\frac{15}{16}=\frac{\mathrm{a}-1}{\mathrm{a}} \\ & \Rightarrow \mathrm{a}=16\end{aligned}

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