Chapter 15, Problem 86A

To determine

To calculate:The tension in the string for the production of a note of frequency $147\text{ Hz}$ .

Answer to Problem 86A

The tension in the string should be $\text{178}\text{.9 N}$ to produce a note of frequency $147\text{ Hz}$ .

Explanation of Solution

Given info:

Equation for the speed of a wave on a string

  $v=\sqrt{\frac{F_T}{\mu }}$

Where $F_T$ is the tension in the string and $\mu$ is the mass per unit length of the string.

Mass of the guitar string = $3.2\text{ g}$

Length of the string = $65\text{ cm}$

Frequency of sound = $147\text{ Hz}$

Formula used:

The length-wavelength relationship for a string

  $\lambda =2L$

Where $\lambda$ represents wavelength of sound and $L$ is the string length.

The frequency-wavelength relationship

  $f=\frac{v}{\lambda }$

$f$ represents the frequency of sound, $v$ represents the speed of wave in string and $\lambda$ represents the wavelength of the resonant sound.

Calculation:

As per problem,

Mass of the guitar string ( $m$ ) = $3.2\text{ g =3}\text{.2}\times {\text{10}}^{-3}\text{ kg}$

Length of the string ( $L$ ) = $65\text{ cm =0}\text{.65 m}$

Calculate the mass per unit length $\mu$

  $\begin{array}{l}\mu \text{=}\frac{m}{L}\\ \mu =\frac{3.2\times {10}^{-3}\text{ kg}}{0.65\text{ m}}\\ \mu =4.9\times {10}^{-3}\text{ kg/m}\end{array}$

Calculate wavelength using formula

  $\lambda =2L$

Substitute $L=0.65\text{ m}$

  $\begin{array}{l}\lambda =2(0.65\text{ m)}\\ \lambda =1.3\text{ m}\end{array}$

Calculate speed of wave in string rearranging formula

  $\begin{array}{l}f=\frac{v}{\lambda }\\ v=f\lambda \end{array}$

Substitute the values $f=147\text{ Hz}$ and $\lambda =1.3\text{ m}$

  $\begin{array}{l}v=(147\text{ Hz)(1}\text{.3 m)}\\ v\text{=191}\text{.1 m/s}\end{array}$

Use the value of speed $v$ and mass per unit length $\mu$ calculated above in the given formula to calculate tension in the string

  $\begin{array}{l}v=\sqrt{\frac{F_T}{\mu }}\\ F_T=v^2\cdot \mu \text{ [square both sides and rearrange to get tension]}\\ F_T={(}^{191.1}(4.9\times {10}^{-3}\text{ kg/m})\\ F_T=178.9\text{ N}\end{array}$

Conclusion:

Thus, to produce a note of frequency $147\text{ Hz}$ , the tension in the string should be $\text{178}\text{.9 N}$ .