How is angular velocity different from regular frequency?

The normal frequency is given in Hertz and indicates the number of cycles per second while angular velocity gives the number of radians per second. Both describe the same motion, with the difference being only a factor of 2π. So omega equals 2π multiplied by f.

Are angular velocity and circular frequency the same?

Both have the unit radians per second and for objects rotating at a constant speed they are numerically identical. The difference in name is due to context; angular velocity describes how fast something rotates while circular frequency describes oscillations or waves.

What is the circular frequency of a pendulum or spring?

In a simple pendulum, omega is determined by gravity and length. In a mass attached to a spring, it's determined by the spring constant and the mass. In both cases, the angular frequency is not determined by amplitude but by the system itself.

What unit is used for angular frequency?

The SI unit is radians per second. As a radian has no dimension it is sometimes expressed as the reciprocal of seconds. This tool can also display results in revolutions per minute or revolutions per second.

Angular Frequency Calculator

What you know

Frequency (f)

Add wave speed (wavelength and wavenumber)

Enter a wave speed to also get the wavelength and wavenumber.

Add an amplitude (peak speed and acceleration)

Enter the amplitude to get the peak speed and acceleration of the motion.

Your answer

Angular frequency (ω)

Frequency (f)

Period (T)

Angular frequency (ω)

rad/s

Ordinary frequency (f): Hz
Period (T): sec

That is 2π times the ordinary frequency of 50 Hz, or one full cycle every 0.02 sec.

Waveform

Angular frequency is a measure of how fast an oscillation occurs and is measured in radians per second. This calculator allows you to calculate the angular frequency from any available information, such as regular frequency, period, revolutions per minute (RPM), or physical sources of oscillations like springs, pendulums, or LC circuits. It outputs the angular frequency as well as the corresponding frequency and period, and can also calculate wavelengths and maximum velocities.

What is the circular frequency?

The angular frequency is denoted by the Greek letter Omega (ω) and measures how fast the phase of a wave changes. Since one full period encompasses 2π radians, the angular frequency is equal to the regular frequency multiplied by 2π.

Imagine a point moving along the circumference of a circle. The regular frequency calculates how many rotations per second are taking place. The circular frequency tracks the angle traversed per second. Both describe the same motion, but one is expressed in cycles and the other in radians.

\omega = 2\pi f = \frac{2\pi}{T}

Here f is the frequency in hertz (Hz) and T is the period in seconds (s). The unit of ω is radians per second (rad/s). Since a radian is a pure ratio, angular frequency can also be expressed as the reciprocal of time.

Angular frequency, frequency, angular velocity.

The three terms are easily confused with each other. The normal frequency f is the number of cycles per second in Hertz. The angular frequency ω is the number of radians per second and is the result of 2π multiplied by f. The unit for angular velocity is also radians per second. For an object rotating at a constant rate, this value numerically equals the circular frequency for one full revolution.

The actual difference is the context; angular velocity is used to describe how fast a wheel is spinning and circular frequency is used when describing an oscillation as a sine wave. This page focuses on the meaning of oscillations, which are used in waves and simple harmonic motion.

How to use this calculator:

Choose the known quantity from the top. If you know the frequency, enter it in hertz. If you know the period, enter the time for one complete cycle in seconds. If you already have the value of ω, you can directly input that to check the frequency and period.

If you select RPM, then you can convert the motor or turntable speed in revolutions per minute to frequency. This is because the number of rotations per minute is divided by 60 to give a frequency in hertz. If you select oscillation system, then you can calculate ω directly from a spring, pendulum or LC circuit. The two radio buttons allow you either add wave velocity to determine wavelength and wavenumber, or add amplitude to find the maximum speed and acceleration of the motion.

Examples of calculations:

Suppose a particular oscillation repeats itself with a frequency of 50 hertz. The angular frequency is the result of multiplying 2 pi by 50.

\omega = 2\pi \times 50 \approx 314.16\ \text{rad/s}

Since the period is the inverse of frequency, T = 0.02 seconds. This means that this motion completes one cycle every twenty milliseconds. Conversely, a pendulum swinging with an angular velocity of 43.98 radians per second has a frequency of seven hertz. This is because 43.98 divided by 2 pi is approximately equal to seven.

Angular frequencies of typical systems:

In simple harmonic motion, the angular frequency is determined solely by the system and does not depend on amplitude. So a pendulum can measure time precisely no matter whether the amplitude is large or small. Three typical cases are presented here.

System	Angular frequency	Inputs
Mass on a spring	omega = sqrt(k / m)	stiffness k, mass m
Simple pendulum	omega = sqrt(g / L)	gravity g, length L
LC circuit	omega = 1 / sqrt(L C)	inductance L, capacitance C

\omega = \sqrt{\frac{k}{m}} \qquad \omega = \sqrt{\frac{g}{L}} \qquad \omega = \frac{1}{\sqrt{LC}}

Angular frequency in waves:

A propagation wave links the circular frequency with space through the wave velocity. When a wave propagates at speed v, its wavelength is the result of dividing the speed by the frequency while the wavenumber represents this same quantity as the number of radians per meter.

\lambda = \frac{v}{f} \qquad k = \frac{2\pi}{\lambda} = \frac{\omega}{v}

For an oscillator with amplitude A the maximum speed during its motion is proportional to omega and the maximum acceleration increases even faster, as shown in the following formula. For a given amplitude, larger angular frequency leads to faster and steeper oscillations.

v_{\max} = \omega A \qquad a_{\max} = \omega^2 A

Applications of angular frequency

Angular frequency is a natural language for describing all periodic phenomena. It determines the argument of the sine function in waves, resonance points in alternating current circuits and the intrinsic rhythm of springs or pendulums. In signal processing it is measured as radians per second and used in rotating machinery to convert revolutions per minute into a smooth phase position.

Frequently asked questions

What is the circular frequency?: Angular frequency is denoted by ω and measures the rate at which the phase of a wave changes, in radians per second. It is the result of multiplying the normal frequency by 2π, since one complete cycle covers an arc length of 2π radians.
How is angular velocity different from regular frequency?: The normal frequency is given in Hertz and indicates the number of cycles per second while angular velocity gives the number of radians per second. Both describe the same motion, with the difference being only a factor of 2π. So omega equals 2π multiplied by f.
Are angular velocity and circular frequency the same?: Both have the unit radians per second and for objects rotating at a constant speed they are numerically identical. The difference in name is due to context; angular velocity describes how fast something rotates while circular frequency describes oscillations or waves.
What is the circular frequency of a pendulum or spring?: In a simple pendulum, omega is determined by gravity and length. In a mass attached to a spring, it's determined by the spring constant and the mass. In both cases, the angular frequency is not determined by amplitude but by the system itself.
What unit is used for angular frequency?: The SI unit is radians per second. As a radian has no dimension it is sometimes expressed as the reciprocal of seconds. This tool can also display results in revolutions per minute or revolutions per second.

Related calculators

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Wavelength CalculatorCalculate wavelength from frequency, from wave speed and frequency, or from photon energy, and work backwards from a wavelength to frequency and energy. Handles the refractive index of a medium, in any unit.

Angular Velocity CalculatorCalculate angular velocity (ω) from an angle and time, a linear speed and radius, a period or a frequency, and read it in rad/s, RPM, rev/s and deg/s. Solves every method backwards too.

Beat Frequency CalculatorCalculate the beat frequency of two overlapping waves from their frequencies or wavelengths, find the beat period and carrier frequency, or tune against a reference.

Angular Displacement CalculatorFind angular displacement from a constant angular velocity (theta = omega t), from an initial angular velocity, angular acceleration and time (theta = omega-zero t + half alpha t squared), from an arc length and a radius (theta = s/r), or from a start and end angle. Solve for any variable, in radians, degrees or revolutions.

Angular Momentum CalculatorFind the angular momentum of a spinning body from its moment of inertia and angular velocity (L = I omega), or of a point mass from its mass, speed and radius (L = m v r). Solve for any variable, work the moment of inertia out from a shape, apply the conservation of angular momentum, and read the rotational kinetic energy.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

OpenStax, University Physics Volume 1: Simple Harmonic Motion
Free textbook treatment of oscillations, period, and angular frequency.
Wikipedia: Angular frequency
Definition, relationship to frequency and period, and common examples.
HyperPhysics (Georgia State University): Simple Harmonic Motion
Reference on the angular frequency of springs, pendulums, and resonant circuits.