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

Answer:

The frequency of the fundamental tone is 309.56 hz

Explanation:

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

$f_{1} =\frac{v}{2L}$

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:

$v=\sqrt{\frac{T}{\frac{m_{t} }{L_{t} } } }$

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:

$v=\sqrt{\frac{580N}{\frac{0.0033Kg}{0.95m} } } \\v=408.619 m/s$

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

$f_{1} =\frac{408.619m/s}{2*0.66m}$

$f_{1} =309.56 Hz$

So, The frequency of the fundamental tone is 309.56 hz