# String tension problem

1. Apr 8, 2013

### myrek

1. The problem statement, all variables and given/known data
A guitarist has a problem that the E-string (330Hz) often breaks when tuning it. The string is made of Copper and it has a diameter of 0.3mm. After some quick calculations based on the length of the guitar neck we can determine that the wave velocity is 530m/s

It is known that string on musical instruments often break if the tension exceeds 2% what advice would you give the guitarist to avoid breaking the strings?

2. Relevant equations
I'm not sure what to calculate using youngs modulus.

3. The attempt at a solution
I have been battling this problem for some time and I'm not sure exactly what the "answer" should be. I have approached it the following way.

first I calculated the wave length
λ=v/f, λ=530/330, λ=1.6m

I then know that the length of the string is
L=λ/2, L=0.8m

since I know the length and diameter of the wire i looked up the density and calculated the mass of the string according to the following calculation
m=A*L*ρ

m= (∏*0.00015*0.00015)*0.8*8900
m=5.03E-4 kg

I then calculated the linear mass density according to formula μ= 5.03E-4 /0.8
μ=6.29E-4 kg/m

finally I can calculate the string tension
F=v2
F=530*530*6.29E-4 = 177N

So now I know that the string tension is 177N. I'm not sure exactly what to do with it.
my prof gave me the tip to use youngs modulus.
I looked up youngs modulus for Cu to be 120GPa.

The equation looks like Y = (F/A)/(ΔL/L)

but I'm not sure exactly what answer he is looking for. I think ΔL/L = 1.02 because of the maximum tension allowed is 2%
Should I rearrange the equation to see which stress is required to extend the wire 2%?

Any help appreciated.

2. Apr 8, 2013

### barryj

Well, a guitarist probably knows nothing about tensions, young modulus, velocity, and etc. So, what would be reasonable to tell him?