Question

A tuning fork vibrates at a frequency of 800 Hz and produces resonance in a resonance column tube. The upper end is left open and the lower end is closed by a water surface which can be varied. Successive resonances are observed at lengths 9.75 cm, 31.25 cm and 52.75 cm. With the help of above data, obtain the speed of sound in air.

Answer 1

Firstly, we consider the successive resonance lengths as $l_1,l_2,l_3$ and then these are substituted in the formula of resonant frequency, $v = \dfrac{{nu}}{{4l}}$. Three equations $(i),(ii),(iii)$ are obtained. Then by solving them, expressions for $l_3 - l_2,l_2 - l_1$, equation $(iv)$ are deduced, further values of $l_3 - l_2,l_2 - l_1$ from experimental data given in question are obtained which are found to be equal. Now the obtained values are equated with the expression $\left( {iv} \right)$. Thus, by simple calculation value of $u$,i.e., speed of sound in air is calculated.

Complete step by step solution:
Consider successive resonance lengths as $l_1,l_2$ and $l_3$ such that, $l_1 = 9.75cm$, $l_2 = 31.25cm$ and $l_3 = 52.75cm$.
Now, for the resonance column tube open at one end,
the resonant frequency, $v = \dfrac{{nu}}{{4l}}$
where,$n$is an odd-positive integer,
and,$v$ is the resonance frequency
Suppose the frequency of tuning fork is $\nu$ and $l_1,l_2$ and $l_3$ are the successive lengths of tube in resonance with it, then, we have,
$\dfrac{{nu}}{{4l_1}} = \nu$ …$(i)$
$\dfrac{{(n + 2)u}}{{4l_2}} = \nu ...(ii)$
$\dfrac{{(n + 4)u}}{{4l_3}} = \nu ...(iii)$
From $(iii) - (ii)$and $(ii) - (i)$, we get,
$l_3 - l_2 = l_2 - l_1 = \dfrac{u}{{2\nu }}...(iv)$
According to given data, $l_3 - l_2 = (52.75 - 31.25)cm = l_2 - l_1 = (31.25 - 9.75)cm = 21.50 cm$
Now, from eq. $(iv)$,
$\dfrac{u}{{2v}} = 21.50 cm$
Or, $u = 2v \times 21.50 cm = 2 \times 800{s^{ - 1}} \times 21.50 cm = 344 m{s^{ - 1}}$

Thus, speed of air as calculated from the given data is $344 m{s^{ - 1}}$.

Note: The above method used for determination of speed of sound is referred to as ‘Resonance Column Method’. Resonance Column apparatus is used to measure the speed of sound in the air. It is one of the simplest techniques to determine the speed of sound by taking the note of harmonics.