Question

In a resonance column, first and second resonance are obtained at depths 22.7 cm and 70.2 cm. The third resonance will be obtained at a depth:
A) 117.7 cm
B) 92.9 cm
C) 115.5 cm
D) 113.5 cm

Answer 1

Hint
Here, we will use the condition of resonance that is when the frequency of waves in the air column becomes equal to the natural frequency of tuning fork, a loud sound is produced in the air column.

Complete step-by-step answer:
Here the depth of first and second are given so we will use the resonance condition. When the frequency of waves in the air column becomes equal to the natural frequency of the tuning fork, a loud sound is produced in the air column. It is the condition for resonance. It occurs only when the length of the air column is proportional to one-fourth of the wavelength of sound waves having frequency equal to frequency of tuning fork.
In resonance column, first resonance occurs at
⇒ ${l_1} + x = \dfrac{\lambda }{4}$…………………. (1)
Where, l1 is the first resonance depth
Second resonance occurs at
⇒ ${l_2} + x = \dfrac{{3\lambda }}{4}$………………. (2)
Where, l2 is the second resonance depth.
From equation (1) and (2) we get
⇒ ${l_2} + x = 3\left( {{l_1} + x} \right)$
⇒ $x = \dfrac{{3{l_1} - {l_2}}}{2}$
As it is given that $\begin{gathered}
  {l_1} = 22.7cm \\
  {l_2} = 70.2cm \\
\end{gathered} $
⇒ $x = \dfrac{{3 \times 22.7 - 70.2}}{2} = 1.05cm$
Now, for third resonance depth
⇒ ${l_3} + x = \dfrac{{5\lambda }}{4}$……………. (3)
⇒ ${l_3} = \dfrac{{5\lambda }}{4} - x$
On putting the value of λ from equation (1) to equation (3), we get
⇒ ${l_3} = 5\left( {{l_1} + x} \right) - x = 5{l_1} + 4x$
On putting the values of x and l1 in above equation, we get
⇒ ${l_3} = 5 \times 22.7 + 4 \times 1.05 = 117.7cm$
Hence, option A is correct.

Note
Here the conditions of first, second and third resonance depth conditions, which are obtained due to the production of standing waves.