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Description
A tuning fork is a simple device having two equal-length tines that meet in the center as a ̶0;U̶1; shape. The tines are usually round or square in cross-section and are thin relative to their length, which makes up the majority of the tuning fork̵7;s height. Typical musical tuning forks measure from about five to eight inches in length and have a handle at the base. Tuning forks may be passive, requiring a small strike to start the vibrations, or they may have a driver device which makes them vibrate continuously. Most tuning forks are standard references for musical pitches, though some are frequency standards for science or timekeeping.
Young's Modulus
Engineers looking to build structures use Young̵7;s Modulus to determine if a material will hold up under an expected amount of strain. It measures how much a material deforms under strain; harder and stronger substances such as steel deform less than soft ones such as wood and plastic. Measurements of Young̵7;s Modulus for materials have units of pressure, such as psi or gigapascals. In a tuning fork, the tines must bend slightly at the bottom of the U-shape in order for them to vibrate; it takes pressure to produce the bend. Young̵7;s Modulus determines how much pressure the tines take to make them bend, and by how much they bend.
Calculation
The following formula determines a tuning fork̵7;s frequency:
f = (1/2*pi*L^2) * sqrt(A*E/rho)
F is frequency in cycles per second, pi = 3.14159, L is tine length, sqrt() is the square root function, A is the cross-section area of the tines, E is Young̵7;s Modulus for the tine material, and rho is the density of the material. Remember that as length increases, frequency decreases, and as the cross-section area and Young̵7;s Modulus increase, f increases.
Temperature
Implied but not expressed in the above frequency formula is the relationship between Young̵7;s Modulus and temperature. Many substances soften as temperature increases and stiffen as it decreases. This changes a material̵7;s response to stress and its Young̵7;s Modulus. If you heat a tuning fork, its resonant frequency decreases because its Young̵7;s Modulus decreases. This decrease is a slow one, however. The formula shows that frequency varies as the square root of the Young̵7;s Modulus. The frequency remains accurate for most practical purposes from 50 to 100 degrees Fahrenheit.