Question

Vibrating tuning fork of frequency $n$ is placed near the open end of a long cylindrical tube. The tube has a side opening and is fitted with a movable reflecting piston. As the piston is moved through $8.75cm$, the intensity of sound changes from a maximum to minimum. If the speed of sound is $350m/s$ . Then $n$ is

A. $500\,Hz$
B. $1000\,Hz$
C. $2000\,Hz$
D. $4000\,Hz$

Answer 1

Hint: This problem is based on sound waves where we have to find $n$ i.e., frequency. We know that to find the frequency we must have the value of wavelength. As the intensity changes from maximum to minimum, the path difference will be $2l$. Use mathematical relation $f = \dfrac{v}{\lambda }$ to find the frequency of a sound wave.

Formula used:
$\text{Path Difference} = \dfrac{{(n + 1)\lambda }}{2} - \dfrac{{(n)\lambda }}{2}$
$\Rightarrow \text{Path Difference}= \dfrac{\lambda }{2}$
Here, $\lambda$ is the wavelength of sound.
$f(\text{frequency}) = \dfrac{{v(\text{velocity})}}{{\lambda (\text{wavelength})}}$

Complete step by step solution:
As the piston is moved through $l = 8.75\,cm$ (given), therefore path difference must be$2l = 2 \times 8.75 = 17.5\,cm$ … (1)
Also, the intensity changes from maximum to minimum (given)
$\text{Path Difference} = \dfrac{{(n + 1)\lambda }}{2} - \dfrac{{(n)\lambda }}{2}$
$\Rightarrow \text{Path Difference}= \dfrac{\lambda }{2}$
i.e., From equation (1), we get
$\dfrac{\lambda }{2} = 17.5cm$
$ \Rightarrow \lambda = 35cm = 0.35m$
We know that,
$f(\text{frequency}) = \dfrac{{v(\text{velocity})}}{{\lambda (\text{wavelength})}}$
In this case,
$ \Rightarrow n = \dfrac{v}{\lambda } = \dfrac{{350}}{{0.35}}$
$ \therefore n = 1000\,Hz$
Thus, the frequency of a sound wave produced by a tuning fork due to a change in intensity of sound from a maximum to minimum is $1000\,Hz$.

Hence, the correct option is B.

Note: Since this is a numerical problem on sound waves hence, it is essential that phase difference must be calculated first as it will help us in finding the wavelength then the given parameters must be used very carefully to give an accurate solution. While writing an answer, always remember to put the units after results.