Question

A tuning fork vibrates with frequency 256Hz and gives one beat per second with the third normal mode of vibration of an open pipe. What is the length of the pipe? (Speed of sound of air is $340m{s^{ - 1}}$).
A. $190cm$
B. $180cm$
C. $220cm$
D. $200cm$

Answer 1

Here, we are given the frequency of the fork and the thing which we have to determine is the length of the pipe. Therefore, we have to find the frequency of the pipe first which can be obtained by the given frequency of the fork. After that, by using the equation of frequency we can get our answer.

Formulas used:
$f = \dfrac{{nv}}{{2l}}$, where, $f$ is the frequency of open pipe, $n$ is the number of normal mode of vibration, $v$ is the speed of sound of air and $l$ is the length of open pipe.

Complete step by step solution:
Our first step is to find the frequency of the open pipe from the frequency of vibrating fork which is given to us in the question.We know that,
Frequency of open pipe= Frequency of fork $\pm 1$
$f = 256 \pm 1Hz$
$\Rightarrow f = 257Hz$ or $255Hz$
We have to calculate the length of the pipe for both these frequencies.
So let us first calculate for $f = 257Hz$
We know that $f = \dfrac{{nv}}{{2l}}$
Here, it is given that the fork gives one beat per second with the third normal mode of vibration of an open pipe. Therefore we will take $n = 3$ and Speed of sound of air is $340m{s^{ - 1}}$
$\Rightarrow 255 = \dfrac{{3 \times 340}}{{2 \times l}} \\\ \Rightarrow l = 2m = 200 cm$
Similarly for $f = 255Hz$, we get
$\Rightarrow 255 = \dfrac{{3 \times 340}}{{2 \times l}} \\\ \therefore l = 2m = 200 cm$
We can see that for both the cases, the value of the length is approximate $200cm$.

Hence, option D is the right answer.

Note: Here, we have used the formula $f = \dfrac{{nv}}{{2l}}$ for finding the length of the pipe. Similarly when the length of the pipe is given, we can find frequency of both open pipe and vibrating fork by using the same equation.