Question

A piano tuner uses a tuning fork that emits sound with a frequency of$440{\text{s}}^{-1}$ . Calculate the wavelength of the sound from this tuning fork and the time the sound takes to travel$10.0\text{m}$ across a large room. Take the speed of sound in air to be$343{\text{ms}}^{-1}$ .

Short answer

The wavelength of sound and time taken by sound to travel $10.0\text{m}$is given as

$\lambda =7.8·{10}^8\text{nm}$and $t=0.029\text{s}$.

Step-by-step solution

Given data

A frequency of the sound of the tuning fork is$440{\text{s}}^{-1}$ .

The length of the room is 10m and the speed of sound in air is $343{\text{ms}}^{-1}$.

Concept of wavelength

To calculate the wavelength of the sound, use the formula for frequency $\nu =\frac{c}{\lambda }$and rearrange it to obtain $\lambda =\frac{c}{\nu }$.

Calculate the wavelength

The wavelength is calculated below:

$$\begin{gathered} \lambda =\frac{c}{\nu } \\ \lambda =\frac{343{\text{ms}}^{-1}}{440{\text{s}}^{-1}} \\ \lambda =0.78\text{m} \end{gathered}$$

Convert units of wavelength

Convert a meter into a nanometre $\left(mtonm\right)$

Since, $1\text{m}={10}^9\text{nm}$

$0.78\text{m}·\frac{{10}^9\text{nm}}{1\text{m}}=7.8·{10}^8\text{nm}$

Calculation of time taken

Now calculate the time needed for sound to travel across the room.

For calculation, divide the distance that sound travels by its speed.

$$\begin{gathered} t=\frac{s}{v} \\ t=\frac{10\text{m}}{343{\text{ms}}^{-1}} \\ t=0.029\text{s} \end{gathered}$$