Angular frequency

Angular frequency
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Angular_frequency".

Do not confuse with angular velocity

Angular frequency is a measure of how fast an object is rotating

In physics (specifically mechanics and electrical engineering), angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector $\vec{\omega}$ is sometimes used as a synonym for the vector quantity angular velocity .

In SI units, angular frequency is measured in radians per second, with dimensions s⁻¹ since radians are dimensionless.

One revolution is equal to 2π radians, hence

$\omega = {{2 \pi} \over T} = {2 \pi f} = \frac {|v|} {|r|}$

where

ω is the angular frequency or angular speed (measured in radians per second),

T is the period (measured in seconds),

f is the frequency (measured in hertz),

v is the tangential velocity of a point about the axis of rotation (measured in metres per second),

r is the radius of rotation (measured in metres).

Angular frequency is therefore a simple multiple of ordinary frequency. However, using angular frequency is often preferable in many applications, as it avoids the excessive appearance of $π$ . In fact, it is used in many fields of physics involving periodic phenomena, such as quantum mechanics and electrodynamics.

For example:

$a = - \omega^2 x \;$

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

$a = - 4 \pi^2 f^2 x\;$

Another often encountered expression when dealing with small oscillations is:

$\omega^{2} = \frac{k}{m}$

where

k

is the spring constant

m

is the mass of the object.

Angular frequency inside an LC circuit can also be defined as the inverse of the square root of the capacitance (measured in farads), times the inductance of the circuit (in henrys).

$\omega = \sqrt{1 \over LC}$

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