Angular Frequency Calculator – rad/s from Hz or Period
Convert frequency or period to angular frequency ω in rad/s using ω = 2πf or ω = 2π/T with multiple unit options.
Choose the 'From Frequency' tab to convert Hz, RPM, kHz, or MHz, or the 'From Period' tab to convert seconds, milliseconds, minutes, or hours.
Angular Frequency Calculator – rad/s from Hz or Period
Convert frequency or period to angular frequency ω in rad/s using ω = 2πf or ω = 2π/T with multiple unit options.
About the Angular Frequency Calculator
Angular frequency, denoted ω (omega), measures how fast an object or signal completes its rotational or oscillatory cycle, expressed in radians per second (rad/s). It is directly related to ordinary frequency f (cycles per second, Hz) by the formula ω = 2π × f, and to the period T (time per cycle) by ω = 2π / T. Because one complete cycle spans 2π radians, multiplying frequency in Hz by 2π converts it to the angular measure.
Angular frequency appears extensively in physics and engineering. In mechanics, it characterises the rotation rate of shafts, rotors, wheels, and planetary bodies. In wave physics and signal processing, it describes the oscillation rate of sinusoidal signals: a sine wave of frequency f Hz has angular frequency ω = 2πf, and the wave can be written as sin(ωt + φ). In AC electrical circuits, the mains supply at 60 Hz (North America) has ω ≈ 377 rad/s, while the European 50 Hz mains gives ω ≈ 314 rad/s. Impedance of capacitors and inductors in AC analysis depends on ω rather than f.
This calculator supports multiple input units for convenience. Frequency can be entered in Hz (standard unit for audio, electrical, and mechanical oscillations), RPM (revolutions per minute, common for motors and engines), kHz (kilohertz, common for radio-frequency and audio applications), or MHz (megahertz, common for processors, RF signals, and wireless communications). Period can be entered in seconds (standard), milliseconds (short-duration oscillations), minutes (slow rotations), or hours (very slow rotations such as planetary or antenna scanning motions).
The angular frequency is distinct from the angular velocity of a spinning object, although they share the same symbol ω and the same unit rad/s. In practice, they are numerically equal when the oscillation or rotation is steady, but the conceptual context differs: angular velocity describes how fast a body is rotating in space, while angular frequency characterises the rate of a periodic phenomenon. For simple harmonic oscillators such as springs and pendulums, the angular frequency determines the oscillation period via T = 2π / ω.
Angular Frequency Examples
Four real-world examples illustrating frequency and period conversions.
| Input | Result | Notes |
|---|---|---|
| Vinyl record: 33.33 RPM | ω ≈ 3.4907 rad/s | ω = 2π × 33.33 / 60. Standard LP record speed. |
| Computer fan: 3000 RPM | ω ≈ 314.159 rad/s | ω = 2π × 3000 / 60. Typical CPU cooling fan. |
| AC mains: 60 Hz | ω ≈ 376.991 rad/s | ω = 2π × 60. North American electrical grid frequency. |
| Earth's rotation: T = 24 h | ω ≈ 7.272×10⁻⁵ rad/s | ω = 2π / (24 × 3600). Earth completes one rotation per sidereal day. |
How to use the Angular Frequency Calculator
- Click 'From Frequency' if you know the frequency, or 'From Period' if you know the time per cycle.
- For frequency, enter the numerical value and select the unit: Hz, RPM, kHz, or MHz.
- For period, enter the numerical value and select the unit: s, ms, min, or h.
- Click Calculate to see the angular frequency ω in rad/s. Click Reset to clear the fields.
Angular Frequency FAQ
What is the difference between frequency and angular frequency?
Ordinary frequency f (Hz) counts complete cycles per second. Angular frequency ω (rad/s) measures the same oscillation in radians per second. Since each cycle spans 2π radians, ω = 2π × f. They describe the same physical oscillation but in different units.
Why is angular frequency used instead of frequency?
Angular frequency simplifies the mathematics of wave equations and oscillator models. For example, the displacement of a simple harmonic oscillator is x = A cos(ωt), which is cleaner than using 2πf explicitly. In electrical engineering, impedance formulas such as Z_C = 1/(jωC) use ω naturally.
How do I convert RPM to rad/s?
Multiply RPM by 2π/60 ≈ 0.10472. For example, 3000 RPM × (2π/60) ≈ 314.16 rad/s. This calculator does that conversion automatically when you select the RPM unit.
What is the angular frequency of a pendulum?
For a simple pendulum of length L (m) in gravitational field g (m/s²), the angular frequency is ω = √(g/L). On Earth with g ≈ 9.81 m/s², a 1 m pendulum has ω ≈ 3.13 rad/s and period T = 2π/ω ≈ 2 s.
What is the angular frequency of AC mains electricity?
In North America (60 Hz): ω = 2π × 60 ≈ 376.99 rad/s. In Europe and most of the world (50 Hz): ω = 2π × 50 ≈ 314.16 rad/s. This affects the reactance of capacitors and inductors in AC circuits.
Can angular frequency be negative?
In mathematics, negative angular frequency is used to describe waves propagating in the opposite direction or to represent the conjugate component in complex Fourier analysis. In physical terms, angular frequency is always reported as a positive magnitude.