Frequency from angular frequency: f = ω / (2π)
Period (optional derived): T = 1/f = 2π/ω
- ω is angular frequency in radians per second (rad/s).
- f is frequency in cycles per second (Hz).
- T is the time for one cycle in seconds (s).
- Enter the angular frequency value (ω).
- Select the unit of ω (rad/s, rad/min, rad/hr, deg/s, or deg/min).
- Choose your desired frequency output unit (Hz, kHz, MHz, GHz, or rpm).
- Set rounding and scientific notation if needed.
- Click Calculate Frequency to view results above the form.
- Use Download CSV or Download PDF to export the results.
| Angular Frequency ω (rad/s) | Frequency f (Hz) | Period T (s) | rpm |
|---|---|---|---|
| 6.2832 | 1.0000 | 1.0000 | 60.0000 |
| 31.4159 | 5.0000 | 0.2000 | 300.0000 |
| 62.8319 | 10.0000 | 0.1000 | 600.0000 |
| 314.1593 | 50.0000 | 0.0200 | 3000.0000 |
Values use π ≈ 3.14159 and f = ω/(2π).
1) Understanding Angular Frequency
Angular frequency (ω) measures how fast a repeating motion advances through phase. It is expressed in radians per second and is widely used in oscillators, waves, and vibration models because equations in radians often simplify. In sinusoidal motion, ω links phase, time, and frequency.
2) Converting ω to Frequency in Hertz
Frequency (f) counts complete cycles per second. Because one cycle equals 2π radians, the conversion is f = ω/(2π). With ω = 62.8319 rad/s, the result is f ≈ 10 Hz. The calculator also supports kHz and MHz outputs for faster signals.
3) Period and Cycle Time
The period (T) is the time for one cycle. Since T = 1/f, substitution gives T = 2π/ω. Higher ω means a shorter period. For ω ≈ 314.1593 rad/s, you get f ≈ 50 Hz and T ≈ 0.02 s. Period is helpful for timing and sampling windows.
4) Typical Values Across Fields
Power systems operate at 50–60 Hz, corresponding to ω ≈ 314.16–376.99 rad/s. Mechanical vibration studies can span from a few Hz (low ω) to kilohertz ranges (high ω), depending on the structure and excitation. In controls and signal work, ω is often reported directly in transfer functions and specifications.
5) RPM Equivalence for Rotation
When ω comes from rotating machinery, rpm is often more familiar. The calculator reports rpm = 60·f to compare motor specs with analysis outputs. For example, f = 10 Hz equals 600 rpm, which is a quick sanity check against nameplates and tachometer readings.
6) Units and Common Pitfalls
Most mistakes come from mixing degrees with radians or using per-minute data as per-second. This tool accepts rad/s, rad/min, rad/hr, deg/s, and deg/min, then converts internally to rad/s before computing f and T. Always confirm the instrument’s unit and time base.
7) Reading the Results
Use Hz when matching spectrum plots, sampling rates, or compliance limits. Use ω when working with differential equations, transfer functions, or phase calculations. The period helps estimate cycle timing for experiments, fatigue loading, and data logging intervals.
8) Professional Calculation Workflow
Confirm the unit of ω at the source, select the desired output unit for f, and choose decimal precision appropriate to your measurement resolution. Enable scientific notation for extremely large or small inputs. Export CSV for records or generate a PDF snapshot for reports and reviews.
1) What is the difference between frequency and angular frequency?
Frequency (f) is cycles per second (Hz). Angular frequency (ω) is radians per second and describes phase change rate. They represent the same repetition using different units.
2) Why does 2π appear in the conversion?
One full cycle equals 2π radians. Since ω is in radians per second and f is in cycles per second, dividing by 2π converts radians to cycles.
3) Can I use degrees per second instead of radians per second?
Yes. Degrees are converted to radians using π/180 internally. Select the degree-based ω unit to avoid manual conversion and reduce unit mistakes.
4) How accurate are the results?
Results reflect your input precision and rounding choice. Calculations use floating‑point arithmetic and a high‑precision value of π, then format outputs to your selected decimals or scientific notation.
5) What does the period value mean physically?
The period (T) is the time for one complete cycle. Smaller T means faster repetition; larger T means slower oscillation. It is especially useful when timing measurements or events.
6) How do I convert to RPM and when is it useful?
RPM is revolutions per minute: rpm = 60·f. It is useful for motors, fans, and shafts where specifications and tachometer readings are reported in rpm.
7) What if my ω is provided in rad/min or rad/hr?
Select the matching ω unit in the form. The calculator converts to rad/s first, then computes frequency, period, and rpm consistently for documentation and comparison.