In a resonance tube experiment, three successive resonances are produced by adjusting the air column of the tube in the ratio 1 : 5 : x . If the frequency of tuning fork is 380 Hz and velocity of soun

Question

In a resonance tube experiment, three successive resonances are produced by adjusting the air column of the tube in the ratio 1 : 5 : x . If the frequency of tuning fork is 380 Hz and velocity of sound at 2 7 ∘ C is 340 m/s, then the value x is

TestHub · Accepted Answer

In a resonance tube experiment, the difference between successive resonant lengths of the air column is constant and equal to $\frac{λ}{2}$ , where $λ$ is the wavelength of the sound wave.

Let the three successive resonant lengths be $L_{1}$ , $L_{2}$ , and $L_{3}$ .

We have:
$L_{2} - L_{1} = \frac{λ}{2}$
$L_{3} - L_{2} = \frac{λ}{2}$

This implies that the resonant lengths $L_{1}$ , $L_{2}$ , and $L_{3}$ are in an arithmetic progression (A.P.).

Given that the lengths are in the ratio $1 : 5 : x$ , we can assume:
$L_{1} = k$
$L_{2} = 5 k$
$L_{3} = x k$

Since they are in A.P., the common difference must be the same:
$L_{2} - L_{1} = L_{3} - L_{2}$
$5 k - k = x k - 5 k$
$4 k = (x - 5) k$

Dividing by $k$ (since $k \neq = 0$ ):
$4 = x - 5$
$x = 9$

The given values of the tuning fork's frequency and the velocity of sound are extraneous information and are not required to find the value of $x$ .

Answer: $9$

Physics - Sound Wave Question with Solution | TestHub

Options:

Doubts & Discussion