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### Tuning Fork Formulae and Calculator

Strength of Materials and Applications

Tuning fork (cylindrical prongs) Equation and Calculator

A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs (tines) formed from a U-shaped bar of elastic metal (usually steel). It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a moment to allow some high overtones to die out. The pitch that a particular tuning fork generates depends on the length and mass of the two prongs. It is frequently used as a standard of pitch to tune musical instruments.The frequency of a tuning fork depends on its dimensions and the material from which it is made. If the prongs are cylindrical the frequency of the tuning fork is related to the length of the radius of the cylinder section.

N = 1.875^{2 }

Note:1.875 is the smallest positive solution of
cos(x)cosh(x) = -1

Eq. 1

$f=\frac{N}{2\cdot \pi \cdot {L}^{2}}\sqrt{\frac{E\cdot I}{\rho \cdot A}}$

where

f = frequency the fork vibrates at, (SI units: 1/s) (hertz)

N ≈ 3.516015 is the square of the smallest positive solution to cos(x)cosh(x) = −1, which arises from the boundary conditions of the prong’s cantilevered structure.

L = length of the prongs, (m)

E = Young's modulus (elastic modulus or stiffness) of the material the fork is made from, (Pa or N/m* ^{2}* or kg/(ms

*))*

^{2}I = second moment of area of the cross-section, (m

^{4})

ρ = density of the fork's material (kg/m

^{3}), and

A = cross-sectional area of the prongs (tines) (m

*).*

^{2} The ratio I/A in the equation above can be rewritten as r* ^{2}*/4 if the prongs are cylindrical with radius r, and a2/12 if the prongs have rectangular cross-section of width a along the direction of motion.

Reference:

- Feldmann, H. (1997). "History of the tuning fork. I: Invention of the tuning fork, its course in music and natural sciences. Pictures from the history of otorhinolaryngology, presented by instruments from the collection of the Ingolstadt German Medical History Museum".
*Laryngo-rhino-otologie*.

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