Case 1
- Frequency
- 50Hz
314.1593rad/s
Open with these valuesAngular Frequency Calculator
314.1593rad/s
Result: 314.1593 rad/sAngular frequency is the ordinary frequency times 2π, because one full cycle covers 2π radians. Mains at 50 Hz runs at 314.159265 rad/s. It is the ω that appears in every sine wave, in reactance formulas and in simple harmonic motion.
The numbers at a glance
| Frequency (Hz) | Result (rad/s) |
|---|---|
| 0.00 | 0.0000 |
| 20.00 | 125.6637 |
| 40.00 | 251.3274 |
| 50.00Your value | 314.1593 |
| 60.00 | 376.9911 |
| 80.00 | 502.6548 |
| 100.00 | 628.3185 |
Worked examples
Case 2
- Frequency
- 60Hz
376.9911rad/s
Open with these valuesCase 3
- Frequency
- 440Hz
2,764.6015rad/s
Open with these valuesHow it's calculated
ω = 2 × π × f
- StepEnter the frequency in hertz — cycles per second.
- StepMultiply by 2π, the radians in one full turn.
- ResultRead ω in radians per second.
Reference table
| Frequency (Hz) | Where it turns up | ω (rad/s) |
|---|---|---|
| 0.5 | One half-cycle per second | 3.1416 |
| 1 | One cycle per second | 6.2832 |
| 50 | Mains in Europe | 314.1593 |
| 60 | Mains in North America | 376.9911 |
| 440 | Concert pitch A | 2764.6015 |
Questions
What is angular frequency?
It is how fast the phase of an oscillation advances, measured in radians per second. One full cycle is 2π radians, so ω is simply 2π times the frequency in hertz.
How do I convert hertz to rad/s?
Multiply by 2π, roughly 6.283185. Fifty hertz becomes 314.159265 rad/s.
How do I go back from rad/s to hertz?
Divide by 2π. An ω of 314.159265 rad/s is 50 Hz again.
Why do formulas use ω instead of f?
Sine and cosine take an angle, not a count of cycles, so the 2π has to appear somewhere. Writing ω once keeps it out of every later line.
Sources and last check
Information, not professional advice.