Chapter 12, Problem 75GP

(a)

To determine

To Find: The wave speed and tension in the string

Answer to Problem 75GP

Explanation of Solution

Given:

Length of violin

  $\begin{array}{l}l\text{= 32 cm}\hfill \\ \text{ = 0}\text{.32 m}\hfill \end{array}$

Fundamental frequency $f_1=440\text{ Hz}$

Mass per unit length $\mu =6.1\times {10}^{-4}\text{kg/m}$

Formula used:

  $v=f\times 2l\text{ m/s}$

Calculation:

Fundamental frequency has 2 nodes at the end and one anti-node in the middle, vibrating length of the string is 2l, so, the speed becomes

  $v=f\times 2l\text{ m/s}$

  $v=440\times 2\times 0.32\text{ m/s}$

  $v=281.6\text{ m/s}$

The wave speed is 281.6 m/s.

Now, find the tension in the string, T

  $T=v^2\mu \text{ N}$

  $T={281.6}^2\text{ m/s}\times 6.1\times {10}^{-4}\text{ kg/m N}$

  $T=48.37\text{ N}$

The tension in the string is $48.37\text{ N}$

Conclusion:

The wave speed and the tension in the string is $281.6\text{ m/s}$ and $48.37\text{ N}$ respectively.

(b)

To determine

To Find: The length of the tube of a simple wind instrument close at one end

Answer to Problem 75GP

  $L=0.195\text{ m}$

Explanation of Solution

Given:

Fundamental frequency $f_1=440\text{ Hz}$

Mass per unit length $\mu =6.1\times {10}^{-4}\text{kg/m}$

Speed of the sound $v=343\text{ m/s}$

Formula used:

  $L=\frac{v}{4l}\text{ m}$

Calculation:

The fundamental frequency is $440\text{Hz}$ .

The tube is closed, so the frequency calculated as

  $L=\frac{v}{4l}\text{ m}$

  $L=\frac{343\text{ m/s}}{4\times 440\text{Hz}}\text{ m}$

  $\begin{array}{l}L=0.1948\text{ m}\\ L=0.195\text{ m}\end{array}$

The length of the tube is $0.195\text{ m}$

Conclusion:

The length of the tube is $0.195\text{ m}$ of a simple wind instrument close at one end whose fundamental is also $440\text{Hz}$ .

(c)

To determine

To Find: The frequency of first overtone of each instrument.

Answer to Problem 75GP

For Violin, $f_2=880\text{ Hz }$

For string, $f_2=1320\text{ Hz}$

Explanation of Solution

Given:

Fundamental frequency $f_1=440\text{ Hz}$

Speed of the sound $v=343\text{ m/s}$

Formula used:

  $f_2=\frac{v}{2L}\times \text{2 Hz}$

  $f_2=3\times f_1\text{ Hz}$

Calculation:

To determine the frequency of first overtone at each instrument, use the following equations.

It is to be known that,

For Violin,

  $f_2=\frac{v}{2L}\times \text{2 Hz}$

  $f_2=2\times 440\text{ Hz }$

  $f_2=880\text{ Hz }$

For String,

  $f_2=3\times f_1\text{ Hz}$

  $f_2=3\times 440\text{ Hz}$

  $f_2=1320\text{ Hz}$

Conclusion:

The frequency of overtone of a violin and the string are $\text{880 Hz, 1320 Hz}$ respectively.