Finally, the lowest resonant frequency (in Hz) of the spring is found from the simple equation,
where k is the spring constant from above and M is the spring mass (see derivation). The spring mass M can be found by weighing the spring, or by finding the spring volume and multiplying by its material density,
We can express the spring's lowest resonance in terms of basic spring geometry if we substitute for k and M in the equation for f_{res} (and then eliminate L_{wire}). Doing so gives,
For springs with small rise angles and several active coils we can make the approximation,
If we also allow the approximation,
we can then simplify the resonant frequency formula to a form that can be found in several reference books,
