Angular frequency

Examples

Circular motion

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: ${\displaystyle \omega =v/r.}$ Circular motion on the unit circle is given by $${\displaystyle \omega =\frac{2\pi }{T}=2\pi f,}$$ where:

• ω is the angular frequency (SI unit: radians per second),
• T is the period (SI unit: seconds),
• f is the ordinary frequency (SI unit: hertz).

Oscillations of a spring

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

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

ω is referred to as the natural angular frequency (sometimes denoted as ω₀).

As the object oscillates, its acceleration can be calculated by $${\displaystyle a=-{\omega }^{2}x,}$$ where x is displacement from an equilibrium position.

Using standard frequency f, this equation would be $${\displaystyle a=-(2\pi f{)}^{2}x.}$$

LC circuits

The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance (C, with SI unit farad) and the inductance of the circuit (L, with SI unit henry): $${\displaystyle \omega =\sqrt{\frac{1}{LC}}.}$$

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.

Terminology

Although angular frequency is often loosely referred to as frequency, it differs from frequency by a factor of 2π, which potentially leads confusion when the distinction is not made clear.