Question

A guitar string 0.65 m long has a tension of 61 N and a mass per unit length of 3.0 g/m. (i) What is the speed of waves on the string when it is plucked? (ii) What is the string's fundamental frequency of vibration when plucked? (iii) At what other frequencies will this string vibrate?

Answer 1

Answer:

i

  $v = 142.595 \ m/s$

ii

  $f = 109.69 \ Hz$

iii1 )

  $f_2 =219.4 Hz$

iii2)

   $f_3 =329.1 Hz$

iii3)

    $f_4 =438.8 Hz$

Explanation:

From the question we are told that

    The length of the string is  $l = 0.65 \ m$

     The tension on the string is  $T = 61 \ N$

     The mass per unit length is  $m = 3.0 \ g/m = 3.0 * \frac{1}{1000} = 3 *10^{-3 } \ kg /m$

     

The speed of wave on the string is mathematically represented as

       $v = \sqrt{\frac{T}{m} }$

substituting values

      $v = \sqrt{\frac{61}{3*10^{-3}} }$

     $v = 142.595 \ m/s$

generally the  string's  frequency is mathematically represented as

         $f = \frac{nv}{2l}$

n = 1  given that the frequency we are to find is the fundamental frequency

So

      substituting values

       $f = \frac{142.595 * 1 }{2 * 0.65}$

       $f = 109.69 \ Hz$

The  frequencies at which the string would vibrate include

1       $f_2 = 2 * f$

Here $f_2$ is  know as the second harmonic and the value is  

      $f_2 = 2 * 109.69$

      $f_2 =219.4 Hz$

2

$f_3 = 3 * f$

Here $f_3$ is  know as the third harmonic and the value is  

      $f_3 = 3 * 109.69$

     $f_3 =329.1 Hz$

3

     $f_3 = 4 * f$

Here $f_4$ is  know as the fourth harmonic and the value is  

      $f_3 = 4 * 109.69$

     $f_4 =438.8 Hz$