Question

A whistle whose air column is open at both ends has a fundamental frequency of $5100\;{\rm{Hz}}$. If the speed of sound in air is $340\;{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}$, the length of the whistle, in cm is:
A) $\dfrac{5}{3}$
B) $\dfrac{10}{3}$
C) $5$
D) $\dfrac{20}{3}$

Answer 1

Hint: The speed of sound and length of the whistle is used to determine the value of the fundamental frequency. If we know two values out of three then the third value can be calculated by use of the relation between the speed of air, fundamental frequency, and the length of the whistle.

Complete step by step answer:
Given, the speed of the sound in air is $v = 340\;{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}$.
The fundamental frequency of a whistle whose air column is open at both ends is $f = 5100\;{\rm{Hz}}$.
Suppose that the length of the whistle that is open at both the end be L.
The fundamental frequency is the ratio of the velocity of sound and two times the length of the pipe. The standard measurement unit for frequency is Hertz.
We know that the fundamental frequency is given by $f = \dfrac{v}{{2L}}$.
For the length of the whistle, we will reshuffle the above equation in terms of the length of the whistle.
$L = \dfrac{v}{{2f}}$
Here, the length of the pipe is $L$, the fundamental frequency is $f$ and the velocity of the sound in the air is $v$.
We will now substitute the known values in the above formula of the fundamental frequency.
$\Rightarrow L = \dfrac{{340\;{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}}}{{2 \times 5100\;{\rm{Hz}}\left( {\dfrac{{1\;{{\rm{s}}^{ - {\rm{1}}}}}}{{1\;{\rm{Hz}}}}} \right)}}$
On simplification,
$\Rightarrow L = 0.0333\;{\rm{m}}\left( {\dfrac{{10\;{\rm{cm}}}}{{1\;{\rm{m}}}}} \right)$
On further simplification,
$\Rightarrow L = 3.33\;{\rm{cm}}$
$\Rightarrow L= \dfrac{{{\rm{10}}}}{3}\;{\rm{cm}}$

$\therefore$ The length of whistle in cm is calculated to be $\dfrac{10}{3}cm$ and thus from the given options, only option (B) is correct.

Note:
Make sure to convert the final answer from meter to centimeter otherwise your answer did not match with the given option. Also do not get confused between Hertz and the inverse of second both representing the fundamental frequency and equal in amount.