Question

The end correction of a resonance column is $1cm.$If the shortest length resonating with the tuning fork is $10cm,$the next resonating length should be
A. $32cm$
B. $40cm$
C. $28cm$
D. $36cm$

Answer 1

The shortest length corresponds to the first mode of vibration which is one fourth of wavelength. Then the next resonating length will be 3/ 4th of wavelength.
Formula used:
Use the equations $R{L_1} + e = \lambda /4$ and $R{L_2} + e = 3\lambda /4$

Complete step by step answer:
Given that end correction$= e = 1cm$
Let the shortest resonating length be $R{L_1} = {\text{ }}10{\text{ }}cm$
And the next resonating length be $R{L_2}$
Then, $R{L_1} + e = {\text{ }}\lambda /4$
So, putting $R{L_1} = 10$ cm and $e = 1$ cm, we will get
$\lambda = \left( {10 + 1} \right)4$
$\lambda = 44$
Now equation for $R{L_2}$ will be,
$R{L_2} + e = \;3\lambda /4$
Putting $\lambda = 44$
$R{L_2} + 1\; = \;44 \times 3/4$
$R{L_2} = 33-1 = 32$
$R{L_2} = 32{\text{ }}cm$
Hence, the correct option is (A).
Additional Information:
(i) The phenomenon of setting a body into oscillations of large amplitude by the influence of another vibrating body having the same natural frequency is called resonance.
(ii) Sometimes, Resonance is harmful.

Note: Resonance occurs in an air column when the length of resonance columns (closed organ pipe) is odd multiple of one fourth of the wavelength of sound.