Question

A guitar string is 90 cm long and has a mass of 3.5g . The distance from the bridge to the support post is L=62cm, and the string is under a tension of 540N . What are the frequencies of the fundamental and first two overtones? Express your answers using two significant figures. Enter your answers in ascending order separated by commas.

Answer 1

Answer:

$v_1 = 301 Hz$

$v_2 = 601 \ \ Hz$

$v_3 = 901 \ Hz$

Explanation:

From the question we are told that

     The  length of the string is  $l = 90 \ cm = 0.9 \ m$

     The mass of the string is  $m_s = 3.5 \ g =0.0035 \ kg$

     The  distance  from the bridge to the support post $L = 62 \ c m = 0.62 \ m$

    The tension is $T = 540 \ N$

Generally the frequency is mathematically represented as

        $v = \frac{n}{2 * L } [\sqrt{ \frac{T}{\mu} } ]$

Where n is and integer that defines that overtones

i.e  n =   1 is for fundamental frequency

      n =  2   first overtone

       n =3   second overtone

Also  $\mu$ is the linear density of the string which is mathematically represented as

           $\mu = \frac{m_s}{l}$

=>        $\mu = \frac{0.0035 }{ 0.9 }$

=>       $\mu = 0.003889 \ kg/m$

So for   n = 1

     $v_1 = \frac{1}{2 * 0.62 } [\sqrt{ \frac{ 540}{0.003889} } ]$

     $v_1 = 301 \ Hz$

So for  n =  2

     $v_2 = \frac{2}{2 * 0.62 } [\sqrt{ \frac{ 540}{0.003889} } ]$

     $v_2 = 601 \ Hz$

So for  n =  3

     $v_3 = \frac{3}{2 * 0.62 } [\sqrt{ \frac{ 540}{0.003889} } ]$

     $v =901 \ Hz$