Question

A guitar string is 95 cm long and has a mass of 3.3 g. The length L from the bridge to the support post is 66 cm, and the string is under a tension of 580 N. What is the frequency of the fundamental tone?

Answer 1

The frequency of the fundamental tone is 309.56 hz

Step-by-step explanation:

The frequency of the fundamental tone is given by the following equation:

`f_(1) =(v)/(2L)`

Where
`f_(1)` is the frequency of the fundamental tone, v is the velocity of the wave on the string and L is the length of the string that is vibrating.

From the question we know that L is 66 cm.

On the other hand, the velocity can be calculate as:

`v=\sqrt{(T)/((m_(t) )/(L_(t) ) ) }`

Where T is the tension,
`m_(t)` is the total mass of the string and
`L_(t)` is the total length of the string

From the question we know that T is 580N,
`m_(t)` is 3.3g and
`L_(t)` is 95 cm.

For not have problems with units, T needs to be in Newtons, L and
`L_(t)` needs to be in meters and m need to be in Kg. So the transform values are

L=0.66m

`L_(t)=0.95 m`

m=0.0033 Kg

T=580N

Replacing on the equation of velocity we get:

`v=\sqrt{(580N)/((0.0033Kg)/(0.95m) ) } \\v=408.619 m/s`

Now with the value of the velocity of the wave. we can calculate the value of the frequency of the fundamental tone as:

`f_(1) =(408.619m/s)/(2*0.66m)`

`f_(1) =309.56 Hz`

So, The frequency of the fundamental tone is 309.56 hz