Angular frequency

In a rotating or orbiting object, there is a relation between distance from the axis, ${\displaystyle r}$, tangential speed, ${\displaystyle v}$, and the angular frequency of the rotation. During one period, ${\displaystyle T}$, a body in circular motion travels a distance ${\displaystyle vT}$. This distance is also equal to the circumference of the path traced out by the body, ${\displaystyle 2\pi r}$. Setting these two quantities equal, and recalling the link between period and angular frequency we obtain:

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${\displaystyle {\textbf {F}}={\frac {\mathrm {d} }{\mathrm {d} t}}(m{\textbf {v}})}$ Second law of motion
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An object attached to a spring can oscillate. If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by[6]

$${\displaystyle \omega ={\sqrt {\frac {k}{m}}},}$$

where

• k is the spring constant,
• m is the mass of the object.

ω is referred to as the natural frequency (which can sometimes be denoted as ω₀).

As the object oscillates, its acceleration can be calculated by

where x is displacement from an equilibrium position.

Using "ordinary" revolutions-per-second frequency, this equation would be

$${\displaystyle a=-4\pi ^{2}f^{2}x.}$$

The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance (C measured in farads) and the inductance of the circuit (L, with SI unit henry):[7]

$${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}.}$$

Adding series resistance (for example, due to the resistance of the wire in a coil) does not change the resonant frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonant frequency does depend on the losses of parallel elements.