Angular frequency

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angular frequency

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(′an·gyə·lər ′frē·kwən·sē)

(physics) For any oscillation, the number of vibrations per unit time, multiplied by 2π. Also known as angular velocity; radian frequency.

Angular frequency

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Not to be confused with angular velocity.

Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π.

Classical mechanics
History of classical mechanics · Timeline of classical mechanics
Branches Statics · Dynamics / Kinetics · Kinematics · Applied mechanics · Celestial mechanics · Continuum mechanics · Statistical mechanics
Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics: Lagrangian mechanics Hamiltonian mechanics
Fundamental concepts Space · Time · Velocity · Speed · Mass · Acceleration · Gravity · Force · Impulse · Torque / Moment / Couple · Momentum · Angular momentum · Inertia · Moment of inertia · Reference frame · Energy · Kinetic energy · Potential energy · Mechanical work · Virtual work · D'Alembert's principle
Core topics Rigid body · Rigid body dynamics · Euler's equations (rigid body dynamics) · Motion · Newton's laws of motion · Newton's law of universal gravitation · Euler's laws of motion · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Linear motion · Mechanics of planar particle motion · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement
Scientists Galileo Galilei · Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon Denis Poisson
v t e

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital 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.^[1]

One revolution is equal to 2π radians, hence^[1]^[2]

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

where

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

T is the period (measured in seconds),

f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν),

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

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

Contents

1 Units
- 1.1 Oscillations of a spring
- 1.2 LC Circuits
2 See also
3 References and notes
4 External links

Units

In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. From the perspective of dimensional analysis, the unit Hertz (Hz) is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention helps avoid confusion.^[3]

In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency.

For example

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

where x is displacement from an equilibrium position.

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

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

Oscillations of a spring

Another often encountered expression when dealing with small oscillations or where damping is negligible is:^[4]

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

where

k is the spring constant

m is the mass of the object.

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

LC Circuits

The resonant angular frequency in an LC circuit equals the square root of the inverse of capacitance (C measured in farads), times the inductance of the circuit (L in henrys).^[5]

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

References and notes

^ ^a ^b Cummings, Karen; Halliday, David (Second Reprint: 2007). Understanding physics. New Delhi: John Wiley & Sons Inc., authorized reprint to Wiley - India. pp. 449, 484, 485, 487. ISBN 978-81-265-0882-2. http://books.google.com/?id=rAfF_X9cE0EC&printsec=copyright. (UP1)
^ Holzner, Steven (2006). Physics for Dummies. Hoboken, New Jersey: Wiley Publishing Inc. pp. 201. ISBN 978-0-7645-5433-9. http://books.google.com/?id=FrRNO6t51DMC&pg=PA200&dq=angular+frequency.
^ Lerner, Lawrence S. (1996-01-01). Physics for scientists and engineers. p. 145. ISBN 978-0-86720-479-7. http://books.google.com/books?id=eJhkD0LKtJEC&pg=PA145.
^ Serway,, Raymond A.; Jewett, John W. (2006). Principles of physics - 4th Edition. Belmont, CA.: Brooks / Cole - Thomson Learning. pp. 375, 376, 385, 397. ISBN 978-0-534-46479-0. http://books.google.com/?id=1DZz341Pp50C&pg=PA376&dq=angular+frequency.
^ Nahvi, Mahmood; Edminister, Joseph (2003). Schaum's outline of theory and problems of electric circuits. McGraw - Hill Companies (McGraw - Hill Professional). pp. 214, 216. ISBN 0-07-139307-2. http://books.google.com/?id=nrxT9Qjguk8C&pg=PA103&dq=angular+frequency. (LC1)

Related Reading:

Olenick ,, Richard P.; Apostol, Tom M.; Goodstein, David L. (2007). The Mechanical Universe. New York City: Cambridge University Press. pp. 383–385, 391–395. ISBN [[Special:BookSources/978-0-521-17592-8|978-0-521-17592-8]]. http://books.google.com/?id=xMWwTpn53KsC&pg=RA1-PA383&dq=angular+frequency.

External links

Frequency/angular frequency/wavelenght/period online calculator

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