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I was playing with a tuning fork and got to wondering how to find it's spring constant (assuming damped oscillation). I can find plenty of resources about materials for springs, but not a whole lot about tuning fork-style oscillators. How do people determine something like that in the real world? Bonus points if you can help describe the role played by material density, since that was what I was curious about initially.

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Each prong of the tuning fork is a cantilever with a rectangular cross section. The spring constant k of the cantilever is $ k = \frac{F}{\delta} = \frac{Ewt^3}{4L^3} $. Its resonance frequency is $ f = \frac{1}{2\pi} \sqrt{\frac{k}{m_\text{equivalent}}} $.
The frequency of a tuning fork is $ f = \frac{1.875^2}{2\pi L^2} \sqrt\frac{EI}{\rho A} $ with $\frac{I}{A}= \frac{t^2}{12} $.
As the mass of the cantilever is $ M = \rho w t L $, the frequency is $ f = \frac{1.875^2}{2\pi \sqrt{3}} \sqrt{\frac{k}{M}} $

See 1 and 2.

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