Question

There are two open organ pipes of exactly the same length and material but different radii. The frequencies of their fundamental notes will be
A) Wider pipe has lower frequency
B) Narrower pipe has lower frequency
C) Both the pipes have same frequency
D) B and C are correct

Answer 1

Hint
We are given 2 open ended organ pipes, we just need to find a relation between the radius and the frequency of the pipe as the other quantities are constant for both of them. We know that the frequencies are independent of the dimension of the pipe (like the thickness or the circular cross-sectional area of the pipe) which is perpendicular to the axis of the standing wave.

Complete step-by-step answer
As we know that the frequency of an open-ended organ pipe follows the formula as:
${\text{f = }}\dfrac{{{\text{( 2n}}\,{\text{ - }}\,{\text{1)}}\,{\text{v}}}}{{{\text{4( L + D )}}}}$
And the frequency of a closed organ pipe is given by:
${\text{f = }}\dfrac{{{\text{nv}}}}{{{\text{2}}\,{\text{(}}\,{\text{L + 1}}{\text{.4D}}\,{\text{)}}}}$
Where, $n$ is the number of nodes formed
$V$ is the velocity of the wave inside the organ pipe
$L$ is the length of the organ pipe
$D$ is the diameter of the organ pipe
From these 2 equations, we can see a similarity that $f$ is inversely proportional to the diameter of the organ pipe, irrespective of whether it is a closed ended pipe or an open ended. Therefore, the narrower pipe will have a higher fundamental note frequency.
So, the option with the correct answer is option B.

Note
One might forget the formulae of frequencies in open and closed ended pipes. The organ pipes, irrespective of their openings are proportional to the same set of quantities. The only thing that differs is the factors associated with them. The width of the pipe is perpendicular to the direction of the axis of the waves in the pipe.
Wider pipe means more air particles. This means the intensity will be higher. But the frequencies will still remain the same for wide or narrower pipes.