[robj144]

This is what I wrote in an archived post:

You seem to have a fascination of string tension. Why not learn the science behind it? To find the tension of a string in Newtons use:

Tension = 4* (frequency)^2 (length)^2 (mass/unit length)

frequency = frequency of string in Hz (can be googled)
length = length of the string (scale length) in meters (google the conversion to meters)
mass/unit length = mass of a string in kg divided by it's length in meters (this can also be googled)

To find the pressure exerted on the string use:

Pressure = Tension/(3.1415 radius^2)

radius = half the string diameter in meters (it would be 0.004 inches for 0.008 gauge string, but convert to meters)

Compare this pressure to the ultimate strength of steel which can also be found by a google search. If the pressure you calculate is greater or close to this value the string will most likely break.

What is also fun is examining the equation for tension again:

Tension = 4* (frequency)^2 (length)^2 (mass/unit length)

It's proportional to the frequency squared. That means doubling the frequency without changing any other variable will quadruple the tension. Increasing the frequency by 41% or 1.41 times the original frequency, will double the tension. It's also proportional to the length squared, but the length does not change drastically compared to the frequency.

Also note, each half step corresponds to a 2^(1/12) increase in frequency. So from hi e to hi a, which is 5 semitones, the increase is 2^(5/12) = 1.33. So (1.33)^2 = 1.78, and the tension will increase by 78% compared to high e.

And then this:

I'm bumping this up, because I just realized something when it comes to the stress (what I called "pressure" before, but it should really be termed "stress") on a string. The tension on the string is:

Tension = 4*frequency^2*length^2*(mass/unit length)

The mass of the string is:

mass = density*volume = density*(cross sectional area)*Length,

So, the mass unit length is:

mass/length = density*(cross sectional area)*Length/Length = density*(cross sectional area)

So, the tension is:

Tension = 4*frequency^2*length^2* density*(cross sectional area)

In other words, if two strings are the same length, tuned to the same frequency, and are made out of the same material, the one with the greatest cross sectional area will have the greater tension. Since the high e and b strings both have the same densities, but different cross sectional areas, lengths, and are tuned to different frequencies, they will have different tensions.

However, the stress is:

stress = Tension/(cross sectional area) = 4*frequency^2*length^2* density*(cross sectional area)/(cross sectional area)

So,

stress = 4*frequency^2*length^2* density

It does not depend on the string gauge at all... just the density of the string, the frequency, and length of the string. In other words, if the high e and b strings were tuned to the same frequency and were the same length, the tension, would be different because the cross sectional area's are different, but the stress on each would be the same. If they were both the same length, the would both break at the same frequency.

Sorry for the math, but I never realized that before.