An open pipe resonates to a frequency `n_1` and closed pipe to a frequency `n_2`. If they are joined to form a longer closed pipes , then its fundamental frequency of resonance will be

Options

A. `(n_1n_2)/(2n_2+n_1)`
B. `(2n_2n_1)/(2n_2+n_1)`
C. `(2n_2n_1)/(n_1+n_2)`
D. `(n_2+2n_1)/(n_1n_2)`

Answer

A

Explanation

For a open pipe
`n_1 = (v)/(2l_1) therefore l_1 = (v)/(2n_1)`
For a closed pipe
`n_2=(v)/(4l_2) therefore l_2=(4)/(4n_2)`

For a single pipe when both are joined
`L=l_1+l_2`
`L=(v)/(2n_1) +(v)/(4n_2)`
`n=(v)/(4L) = (v)/(4[(v)/(2n_1) +(v)/(4n_2)])`
`n=(v)/((4v)/(2)[1/n_1+ 1/(2n_2)])`
`=(n_1n_2)/(2n_2+n_1)`