Question

In a resonance tube, the first resonance with a tuning fork occurs at $ 16{\text{ }}cm $ and second at $ 49{\text{ }}cm $ . If the velocity of sound is $ 330{\text{ }}m/s, $ the frequency of tuning fork is:
(A) $ 500Hz $
(B) $ 300Hz $
(C) $ 330Hz $
(D) $ 165Hz $

Answer 1

Hint: A resonance tube is like a closed organ pipe with a tube partially filled with water and the rest of the column. It consists of a resonance tube that is connected through a rubber tube to the water reservoir. This rubber tube is fitted with a clamp that regulates the height of the liquid in the tube. If we hit the tuning fork just above the mouth the air above the liquid vibrates. The condition of the resonance is achieved when the natural frequency of the air and the fork matches.

Complete answer:
Given that the first resonance occurs at $ 16{\text{ }}cm $ and the second resonance occurs at $ 49{\text{ }}cm $
Also given that the velocity of the sound is $ 330{\text{ }}m/s, $ we need to find the frequency of the tuning fork. The formula used for this is,
 $ \lambda = \dfrac{v}{n} $ ………. (1)
Here, $ \lambda $ is the wavelength
 $ v $ is the velocity of vibration
 $ n $ is the frequency of vibration.
We have given that resonance occurs two times at two lengths. The formula for finding the wavelength using the given information is given as
 $ \dfrac{\lambda }{2} = ({l_2} - {l_1}) $
Here $ {l_2} - {l_1} $ is the difference in the two lengths when the resonance occurred
Now rewriting the wavelength in terms of equation (1), we get
 $ \dfrac{v}{n} = 2({l_2} - {l_1}) $
To find the frequency of the tuning fork, we can rearrange the above formula.
 $ n = \dfrac{v}{{2({l_2} - {l_1})}} $
Substituting all the known values in the above formula we get
 $ n = \dfrac{{330}}{{2(0.49 - 0.16)}} $
 $ n = 500Hz $
Correct Answer: Therefore the correct option is A.

Note:
We have seen that wavelength is directly proportional to the velocity. We can see some examples regarding this statement. If we consider a situation where the frequency is constant if the wavelength of a wave gets doubled then the velocity of the wave gets doubled. Similarly, if the wavelength gets decreased then the velocity of the wave also decreases.