Angular Frequency Calculator (ω = 2πf = 2π/T, spring–mass and pendulum)

Understanding the angular frequency calculator — concept and principle

Angular frequency expresses how fast something oscillates or rotates in radians per second. This calculator converts between angular frequency, ordinary frequency, period and rpm, and can also find the natural angular frequency of a mass on a spring or a simple pendulum.

One cycle is 2π radians, so 50 Hz mains sweeps 314.159 rad every second.

How it is calculated

ω = 2πf = 2π ÷ T, because one full cycle is 2π radians.
From rpm: f = rpm ÷ 60.
Mass on an ideal spring: ω = √(k ÷ m).
Simple pendulum at small angles: ω = √(g ÷ L).
Degrees per second = ω × 180 ÷ π.

The principle

Any repeating motion can be pictured as a point going round a circle. One full cycle corresponds to 2π radians, so multiplying cycles per second by 2π gives radians per second. Angular frequency appears naturally in the equations of oscillation: a displacement x = A cos(ωt) repeats every 2π ÷ ω seconds. For a spring, a stiffer spring or lighter mass raises ω; for a pendulum, a shorter string raises it. Note that a pendulum's ω describes the swing, not the bob's angular speed.

Worked example

Default inputs: start from frequency, f = 50 Hz (mains electricity in many countries).

ω = 2π × 50 = 100π ≈ 314.159 rad/s.
Frequency = 50 Hz; period T = 1 ÷ 50 = 0.02 s (20 ms).
rpm = 50 × 60 = 3,000.
Degrees per second = 314.159 × 180 ÷ π = 18,000°/s.

(The spring inputs, k = 200 N/m and m = 0.5 kg, would give ω = 20 rad/s; the 1 m pendulum would give about 3.13 rad/s.)

Each point is the calculator re-run with only “Frequency, period or rpm” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Frequency, period or rpm	Angular frequency ω (rad/s)
12.5	78.5398
25	157.08
37.5	235.619
50	314.159
62.5	392.699
75	471.239
100	628.319

How to read the result

Use ω in formulas for AC circuits (reactance X_L = ωL, X_C = 1 ÷ ωC), vibration and wave motion. Use f when talking about hertz, and T when timing cycles. A machine running at 3,000 rpm turns at 50 revolutions per second, matching a 50 Hz two-pole synchronous motor.

Common mistakes

Confusing f (Hz) with ω (rad/s), which differ by a factor of 2π.
Using the pendulum formula for large swings, where the period grows.
Forgetting to convert rpm to revolutions per second.
Using the spring formula when the spring's own mass is not negligible.

Keep in mind: Spring and pendulum results assume ideal, undamped systems. Real machine and vibration design requires appropriate standards and a qualified engineer.

Frequently asked questions

What is the difference between frequency and angular frequency?

Frequency counts cycles per second (Hz); angular frequency counts radians per second, ω = 2πf.

What is the angular frequency of 60 Hz?

2π × 60 ≈ 376.99 rad/s.

Why do AC formulas use ω?

Because sinusoidal voltages and currents are written as sin(ωt), and reactances depend directly on ω.

Sources: Halliday, Resnick & Walker, Fundamentals of Physics (oscillations) · OpenStax, University Physics Volume 1, Chapter 15 Oscillations

Checked 9 Nov 2026. OpenStax, University Physics Volume 1 (2016), §15.1 “Simple Harmonic Motion” (f = 1/T, Eq. 15.1; ω = 2π/T = 2πf; mass–spring ω = √(k/m)) and Chapter 15 Key Equations (simple pendulum ω = √(g/L)).

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