Question

An open organ pipe and a closed organ pipe have the frequency of their first overtone identical. The ratio of length of open pipe to that of closed pipe is ______.

Options

• 1 : 2

• 3 : 4

• 4 : 3

• 2 : 1

Answer 1

An open organ pipe and a closed organ pipe have the frequency of their first overtone identical. The ratio of the length of open pipe to that of closed pipe is 4 : 3.

Explanation:

The frequency higher than the fundamental frequency of sound is known as overtone.
For open organ pipe, first overtone is

`("v"_"o") = "2v"/"2L"_"o" = "v"/"L"_"o"`

For closed organ pipe, first overtone is

`"v"_"e" = "3v"/(4"L"_"c")`

It is given that, their first overtone are identical i.e.,

v_o = v_c

`"v"/"L"_"o" = "3v"/(4"L"_"c")`

`"L"_"o"/"L"_"c" = 4/3`

OR

`ℓ_o/ℓ_c` = ?

For closed pipe `n_p = n_o(2p + 1) ⇒ n_{1c} = n_0^3`

For open pipe `n_p = n_o(p + 1) ⇒ n_{1o} = n_o^2`

`n_{1c} = 3n_0 = (3V)/(4ℓ_c)`

`n_{1o} = 2n_o = (2V)/(4ℓ_o)`

∴ `(3V)/(4ℓ_c) = (2V)/(4ℓ_o)`

∴ `ℓ_o/ℓ_c = 4/3`