Question

The frequency of a fork is $200\;{\rm{Hz}}$. The distance through which sound travels by the time the fork make $16$ vibrations is (velocity of sound in air is $340\;{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}}$):
a). ${\rm{34}}\;{\rm{m}}$
b). $21.25\;{\rm{m}}$
c). $425\;{\rm{m}}$
(D). ${\rm{27}}{\rm{.2}}\;{\rm{m}}$

Answer 1

We know that the total Distance covered in one time period or in one vibration is known as wavelength. So, we first calculate the wavelength for first vibration then multiply it by $16$ to find the total distance covered in $16$ vibrations.

Complete step by step answer: Given:
The velocity of sound in air is $v = 340\;{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}}$.
The frequency of a fork is $f = 200\;{\rm{Hz}}$.

The vibration or oscillation is a type of motion that reoccurs in a regular interval of time. The swinging of a pendulum and the tines of a tuning fork are the examples of vibrations.
The study of oscillatory motions of any body and the forces acting on them are studied under the theory of vibrations.

In the other words, the wavelength is the measure of length between two similar crest of a waveform created by the propagation of wave along a wire or in space. The unit of wavelength can be meters.

The formula to calculate the wavelength is given by,
$\lambda = \dfrac{v}{f}$
Here, $v$ is the velocity of the sound in air, $f$ is the frequency of a fork, and $\lambda$ is the wavelength.

Substituting the values of $v$and $f$ in the equation $\lambda = \dfrac{v}{f}$, we get,
$\begin{array}{c} \lambda = \dfrac{{340}}{{200}}\\\ = 1.7\;{\rm{m}} \end{array}$

Find the distance covered in $16$ vibrations as follows.
$s = 16\lambda$

Substituting the value of $\lambda$ in the equation $s = 16\lambda$, we get,
$\begin{array}{c} s = 16\left( {1.7\;{\rm{m}}} \right)\\\ = 27.2\;{\rm{m}} \end{array}$

Hence, the option (D) is the correct answer.

Note: We can use an alternate method to solve this problem. First calculate the time period by using formula ${\rm{Time}}\;{\rm{period}} = \dfrac{1}{{{\rm{Frequency}}\left( f \right)}}$, then calculate the total time by using formula ${\rm{Total}}\;{\rm{time}} = {\rm{Time}}\;{\rm{period}} \times {\rm{number}}\;{\rm{of}}\;{\rm{vibrations}}$, and finally required distance is calculated by using formula ${\rm{Distance}}\;{\rm{travelled}} = {\rm{Speed}} \times {\rm{Total}}\;{\rm{time}}$.