The frequency that is characteristic of a given freely oscillating system, with no applied driving force. If the frequency is is in oscillations per unit time, it is represented by the symbol ν0. The angular natural frequency is a measure of the angle per unit time, assuming that one full cycle is equal to a full rotation around a circle. This frequency is represented by the symbol ω0, and is often measured in radians per second. If that is the case, then the relationship between the two forms is
\begin
[\omega_
= 2 \pi \nu_
]\end
For a mass on a spring, the natural frequency is given by
\begin
[ \omega_
= \sqrt{\frac
{m}} ]\end
while for a simple pendulum of mass m on an arm of length L the natural frequency is
\begin
[ \omega_
= \sqrt{\frac
{L}} ]\end
See Simple Harmonic Motion for fuller details.
The natural frequencies are related to the period T by:
\begin
[ T = \frac
{\nu_{0}} = \frac
{\omega_{0}} ]\end