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:

Part of a series on
Classical mechanics
${\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}$ Second law of motion
History Timeline Textbooks
Branches Applied Celestial Continuum Dynamics Kinematics Kinetics Statics Statistical mechanics
Fundamentals Acceleration Angular momentum Couple D'Alembert's principle Energy kinetic potential Force Frame of reference Inertial frame of reference Impulse Inertia / Moment of inertia Mass Mechanical power Mechanical work Moment Momentum Space Speed Time Torque Velocity Virtual work
Formulations Newton's laws of motion Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton–Jacobi equation Appell's equation of motion Koopman–von Neumann mechanics
Core topics Damping Displacement Equations of motion Euler's laws of motion Fictitious force Friction Harmonic oscillator Inertial / Non-inertial reference frame Mechanics of planar particle motion Motion (linear) Newton's law of universal gravitation Newton's laws of motion Relative velocity Rigid body dynamics Euler's equations Simple harmonic motion Vibration
Rotation Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational frequency Angular acceleration / displacement / frequency / velocity
Scientists Kepler Galileo Huygens Newton Horrocks Halley Maupertuis Daniel Bernoulli Johann Bernoulli Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Cauchy Routh Liouville Appell Gibbs Koopman von Neumann
Physics portal  Category

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[7]

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

where

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

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

As the object oscillates, its acceleration can be calculated by

where x is displacement from an equilibrium position.

Using standard frequency f, this equation would be

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

The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance (C, with SI unit farad) and the inductance of the circuit (L, with SI unit henry):[8]

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

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.