What Is Angular Frequency?
Angular frequency is a scalar physical quantity that measures how quickly an object rotates or oscillates with respect to time. It is usually denoted by the Greek letter ω (omega) — and sometimes Ω — and is measured in radians per second (rad/s), the SI unit, or in rpm (revolutions per minute).
Angular frequency differs from ordinary frequency f by a factor of 2π: while frequency counts how many full cycles occur each second (in hertz), angular frequency expresses the same rate in radians per second.
What Is the Angular Frequency Formula?
Depending on the situation, you can find the angular frequency in different ways. This calculator supports three approaches:
1. Rotating Object (ω = Δθ / Δt)
For a rotating object, the angular frequency equals the amount of rotation Δθ (in radians) divided by the elapsed time Δt (in seconds):
ω = Δθ / Δt
where Δθ is the amount of rotation expressed in radians, and Δt is the time taken in seconds.
Example: if a tire turns 200 radians in 40 seconds, how do you calculate the angular velocity of the wheel?
ω = 200 rad / 40 s = 5 rad/s.
2. Simple Harmonic Motion — From Frequency (ω = 2πf)
For an oscillating object, the angular frequency is 2π times the ordinary frequency f:
ω = 2πf
Example: if a pendulum oscillates with a frequency of 2 Hz, how do you find its angular frequency?
ω = 2 × 3.142 × 2 = 12.566 rad/s.
3. Simple Harmonic Motion — From Period (ω = 2π / T)
If you know the period T — the time for one full oscillation — the angular frequency is 2π divided by T:
ω = 2π / T
Here π (Pi) is a mathematical constant approximately equal to 3.142, defined as the ratio of a circle's circumference to its diameter.
How to Use the Angular Frequency Calculator?
- Pick a calculation mode that matches what you know — rotation (angle & time), frequency, or period.
- Enter the known value(s) and choose the appropriate units (the calculator accepts both metric and American/imperial inputs such as radians, degrees, revolutions, seconds, minutes and hours).
- Click Calculate. The angular frequency is shown simultaneously in rad/s, rpm, °/s and rev/s, along with the derived ordinary frequency (Hz) and period (s).
Angular Frequency Units
- rad/s — radians per second (SI standard unit)
- rpm — revolutions per minute (engineering standard)
- °/s — degrees per second
- rev/s — revolutions per second (numerically equal to Hz)
Conversion: 1 rpm = 2π/60 rad/s ≈ 0.10472 rad/s = 6 °/s = 1/60 rev/s
Angular Frequency vs. Angular Velocity
Angular frequency and angular velocity share the same symbol ω and the same numerical value for circular motion, but they differ conceptually. Angular velocity is a vector quantity with a direction (along the rotation axis) that describes actual rotation, whereas angular frequency is a scalar used in wave and oscillation equations such as x = A·cos(ωt).
FAQs
What is the SI unit of angular frequency?
Radians per second (rad/s). Because the radian is dimensionless (arc length ÷ radius), the SI unit is technically s⁻¹.
How is angular frequency related to ordinary frequency?
They differ by a factor of 2π: ω = 2πf. So a frequency of 1 Hz corresponds to an angular frequency of about 6.283 rad/s.
What is the angular frequency of a pendulum oscillating at 2 Hz?
ω = 2πf = 2 × 3.142 × 2 ≈ 12.566 rad/s.
How do you calculate the angular velocity of a wheel that turns 200 radians in 40 seconds?
ω = Δθ / Δt = 200 / 40 = 5 rad/s.