Question

(a) Find the speed of waves on a violin string of mass $\mathbf{\text{800mg}}$and length$\mathbf{\text{22.0cm}}$if the fundamental frequency is$\mathbf{\text{920Hz}}$.

(b) What is the tension in the string? For the fundamental, what is the wavelength of (c) the waves on the string and (d) the wavelength of sound waves emitted by the string?

Short answer

1. Thespeed of waves on a violin string of mass$800\mathrm{mg}$ and length$22.0\mathrm{cm}$ if the fundamental frequency is$920\mathrm{Hz}$ is $405\mathrm{m}/\mathrm{s}$.
2. Thetension in the string is $596\mathrm{N}$.
3. Thewave on the string is $44.0\mathrm{cm}$.
4. The wavelength of sound waves emitted by the string is $37.3\mathrm{cm}$.

Step-by-step solution

Given

• Mass of string ,

$\begin{array}{c}\mathrm{m}=800\text{\hspace{0.17em}mg}\\ =8\times {10}^{\text{-4}}\text{kg}\end{array}$

• Length of string,

$\begin{array}{c}\mathrm{L}=22.0\mathrm{cm}\\ =0.2\text{2m}\end{array}$

• Fundamental frequency,

$\mathrm{f}=920\text{Hz}$

Determining the concept

Apply the formulas in terms of wavelength and frequency to find the velocity. Tension can be found from the formula for velocity in terms of tension and linear density.

The expression for the frequency is given by,

${\mathrm{f}}_0=\frac{1}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$

Where,${\mathrm{f}}_0$is frequency, $L$is length, $\mathrm{T}$ is tension and $\mathrm{\mu }$is mass per unit length.

(a) Determine the speed of waves on a violin string of mass 800 mg and length role="math" localid="1661417326558" 22.0 cm  if the fundamental frequency is  920 Hz

For fundamental frequency, length of string will be half of wavelength,

$\mathrm{L}=\frac{\mathrm{\lambda }}{2}$

$\begin{array}{c}\mathrm{\lambda }=2\mathrm{L}\\ =2\left(0.22\text{m}\right)\\ =0.44\text{m}\end{array}$

$\begin{array}{c}\text{speed}=\mathrm{v}\\ ={\mathrm{f}}_0\mathrm{\lambda }\\ =\text{920Hz}\left(\text{0.44m}\right)\\ =404.8\mathrm{m}/\mathrm{s}\end{array}$

Hence, the speed of waves on a violin string of mass$\text{800mg}$ and length $\text{22.0cm}$if the fundamental frequency is$\text{920Hz}$ is$404.8\mathrm{m}/\mathrm{s}$ .

(b) Determine the tension in the string

Mass per unit length,

$\begin{array}{c}\mathrm{\mu }=\frac{\mathrm{m}}{\mathrm{L}}\\ =\frac{\left({\text{8×10}}^{\text{-4}}\text{kg}\right)}{\text{0.22m}}\\ =3.64\times {10}^{-3}\text{kg/m}\end{array}$

Fundamental frequency is defined as,

${\mathrm{f}}_0=\frac{1}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$

$\begin{array}{c}\mathrm{T}={\mathrm{f}}_0^2\left(4{\mathrm{L}}^2\right)\mathrm{\mu }\\ ={\left(920\text{Hz}\right)}^2(4\times {\left(0.22\text{m}\right)}^2)\left(3.64\times \frac{{\text{10}}^{\text{-3}}\text{kg}}{\text{m}}\right)\\ =596\mathrm{N}\end{array}$

Hence, the tension in the string is $596\mathrm{N}$.

(c) Determine the waves on the string

Wavelength,

$\begin{array}{c}\mathrm{\lambda }=2\mathrm{L}\\ =2\left(0.\text{22m}\right)\\ =0.44\text{\hspace{0.17em}m}\\ =44.\text{0\hspace{0.17em}cm}\end{array}$

Hence, the wave on the string is$44.0\mathrm{cm}$ .

(d) Determine the wavelength of sound waves emitted by the string

Since, speed of sound,

$\mathrm{v}=343\mathrm{m}/\mathrm{s}$ (at, room temperature),

$\begin{array}{c}\mathrm{\lambda }=\frac{\mathrm{v}}{{\mathrm{f}}_0}\\ =\frac{343\mathrm{m}/\mathrm{s}}{\text{920Hz}}\\ =0.373\mathrm{m}\end{array}$

Hence, the wavelength of sound waves emitted by the string is $0.373m$.