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

`v_1 = 301 Hz`

`v_2 = 601 \ \ Hz`

`v_3 = 901 \ Hz`

Step-by-step 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 = (n)/(2 * L ) [\sqrt{ (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 = (m_s)/(l)`

=>
`\mu = (0.0035 )/( 0.9 )`

=>
`\mu = 0.003889 \ kg/m`

So for n = 1

`v_1 = (1)/(2 * 0.62 ) [\sqrt{ ( 540)/(0.003889) } ]`

`v_1 = 301 \ Hz`

So for n = 2

`v_2 = (2)/(2 * 0.62 ) [\sqrt{ ( 540)/(0.003889) } ]`

`v_2 = 601 \ Hz`

So for n = 3

`v_3 = (3)/(2 * 0.62 ) [\sqrt{ ( 540)/(0.003889) } ]`

`v =901 \ Hz`