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{excerpt} The frequency that is characteristic of a given freely oscillating system, with no applied driving force.{excerpt} If the frequency is is in oscillations per unit time, it is represented by the symbol {*}ν{~}0{~}{*}. 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 {*}ω{~}0{~}{*}, and is often measured in radians per second. If that is the case, then the relationship between the two forms is 

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{latex}\begin{large} \[\omega_{0} = 2 \pi \nu_{0} \]\end{large}{latex}


For a [mass on a spring], the natural frequency is given by
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{latex}\begin{large} \[ \omega_{0} = \sqrt{\frac{k}{m}} \]\end{large}{latex}
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while for a simple [pendulum] of [mass] *m* on an arm of length *L* the natural frequency is
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{latex}\begin{large} \[ \omega_{0} = \sqrt{\frac{g}{L}} \]\end{large}{latex}

See [Simple Harmonic Motion] for fuller details.

The *natural frequencies* are related to the [period] *T* by:
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{latex}\begin{large} \[ T = \frac{1}{\nu_{0}} = \frac{2 \pi}{\omega_{0}} \]\end{large}{latex}