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

Hint: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.