Vibration and the Tuning Fork
An acoustic resonator, or a tuning fork, is a tool made out of elastic metal that is used as a standard comparison to tune other instruments. The tuning fork resonates at a specified constant pitch (for example, an C Chord). When the fork is hit against a surface it emits a pure musical tone. The pitch of the fork depends on the length and mass of the two prongs. And one can make different pitched forks by altering the length and mass to produce different notes.
f is the frequency the fork vibrates at in Hertz. 1.875 the smallest positive solution of cos(x)cosh(x) = -1.
l is the length of the prongs in meters.
E is the Young's modulus of the material the fork is made from in pascals. Young's Modulus is a measure of how stiff the material is
I is the second moment of area of the cross-section in metres to the fourth power.
ρ is the density of the material the fork is made from in kilogrammes per cubic meters.
A is the cross-sectional area of the prongs (tines) in square meters.
What is important here is the scaling f ~ 1/I^2 sqrt(E I / p A)
Take Away: longer tuning forks oscillate slower (the 1/I^2), stiffer things oscillate faster E, and thicker things oscillate faster I