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String Gauges - Hooke's Law

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Mathematically, Hooke's law states that



x is the displacement of the spring's end from its equilibrium position (a distance, in SI units: meters);
F is the restoring force exerted by the spring on that end (in SI units: N or kg·m·s-2); and
k is a constant called the rate or spring constant (in SI units: N·m-1 or kg·s-2).

Now, lets use this to prove that heavy gauge strings have a smaller amplitude than a light gauge string when plucked with the same force. For this example, we are going to compare Nickel strings.

A string under tension behaves as a simple spring (stretching it
along its length should be equivalent to stretching it by plucking).
Given the diameter, initial length and Young's Modulus of nickel, the
spring constant of each guage of string can be calculated (k = EA/L).
Young's Modulus (E) of nickel is 200GPa. A (the cross-section area) is
equal to pi*r2.

Converted from inches to millimeters, the A value for each string is:
A1 = (3.14159)(0.254mm/2)2 = 0.051mm2
A2 = (3.14159)(0.2794mm/2)2 = 0.061mm2

The spring constants of each guage of string work out as follows:

k1 (of the 0.010" string) = ((200GPa)(0.051mm2))/647.7mm = 0.015748031
k2 (of the 0.011" string) = ((200GPa)(0.061mm2))/647.7mm = 0.018835880

Hooke's Law states that the force of a spring is equal to the distance
it is stretched times its spring constant (F = kx).

Since we tune the string, and then apply a constant force to pluck it,
we can say that the total force on the string is equal to the spring
constant times the total distance stretched (the string stretches some
when tuned, as you know). In math, that looks like:

(F0 + Fp) = k(x0 + xp)
*F0 is the force applied by tuning to E, Fp is the force of the pluck,
x0 is the stretch caused by tuning to E and xp is the stretch caused
by the pluck. x0 is, out of necessity, directly correlated to the
amplitude of the standing wave in the string (I think this makes

Multiply thtough to get: F0 + Fp = kx0 + kxp
but we know that F0 = kx0 so: F0 + Fp = F0 + kxp
F0 cancels and it's simply: Fp = kxp

Isolate xp: xp = Fp/k