Question

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of 18cm during winter. Representing the same experiment during summer, she measures the column length to be xcm for the second resonance. Then,
(A) $18 > x$
 (B)$x > 54$
(C)$54 > x > 36$
(D)$36 > x > 18$

Answer 1

Hint: Firstly, we will see the expression of frequency and analyze how frequency is dependent on velocity of sound and its wavelength. Then, we are going to find velocity of sound in summer as well as in winters. Now, by equating these two expressions of velocities, we will get the length of the column.

Complete answer:
We will use formula of frequency here,
$f=\dfrac{v}{\lambda }$ (Where, v = velocity of sound)
As given in the question first resonance condition during summer occurred at length (l1=18cm),
$\therefore $ Length of first resonance:
 ${{l} _ {1}} =\dfrac {\lambda} {4} $
$18=\dfrac {\lambda} {4} $
$\lambda =4\times {{l} _ {1}} $………………….. (1)
As we know that frequency of first resonance will be –
$f=\dfrac{{{v}_{1}}}{\lambda }=\dfrac{{{v}_{1}}}{4\times {{l}_{1}}}$ (Where, v1 = velocity of 1st resonance at temperature T1)
$\begin {align}
  & \therefore {{l} _ {1}} =\dfrac {{{v} _ {1}}} {4\times f} \\
 & 18=\dfrac {{{v} _ {1}}} {4\times f} \\
 & \\
\end{align}$
$\therefore {{v} _ {1}} =72\times f\, \dfrac{cm}{s}$ ………………………. (2)
A/C to question, length of second resonance occurred at length (l2= x cm) during winter,
$\therefore $ Length of second resonance:
 ${{l} _ {2}} =\dfrac {3\times \lambda} {4} $
  ${{\lambda} _ {2}} =\dfrac {4\times {{l} _ {2}}} {3} $ ………………………… (3)
So, frequency of second resonance will be –
${{f}_{2}}=\dfrac{{{v}_{2}}}{{{\lambda }_{2}}}=\dfrac{3\times {{v}_{2}}}{4\times {{l}_{2}}}$ (Where, v2 = velocity of second resonance at temperature T2)
 $\therefore {{v} _ {2}} =\dfrac {4\times f\times {{l} _ {2}}} {3} $ ………………………… (4)
$\Rightarrow $ On comparing equation (2) and (4):
${{v} _ {2}}> {{v} _ {1}} $
$\begin {align}
 & 72f>\dfrac {4\times f\times {{l} _ {2}}} {3} \\
 & 72>\dfrac {4\times {{l} _ {2}}} {3} \\
\end{align}$
$\therefore {{l} _ {2}}>54$
$\therefore x>54$

So, the correct option will be (B).

Note:
The length of the second resonance (i.e. in winter season) is more than that of length we measure in summer season. Because in summer the velocity of the sound is greater than that of in winter because the air particles are not closely packed due to humidity.