| ω (rad/s) | f (Hz) | RPM | Period (ms) | Notes |
|---|---|---|---|---|
| 2π ≈ 6.283185 | 1 | 60 | 1000 | One cycle per second. |
| 10π ≈ 31.41593 | 5 | 300 | 200 | Common lab shaker speed reference. |
| 100 | 15.91549 | 954.9297 | 62.832 | Non-integer frequency. |
| 0.5 | 0.079577 | 4.77465 | 12566 | Slow oscillations. |
Angular frequency ω (radians per second) and frequency f (cycles per second) are linked by:
- ω = 2πf
- f = ω / (2π)
The period is T = 1/f. For rotation, RPM = 60f.
- Select the conversion mode you need.
- Enter your value and pick the correct unit.
- Choose display precision, then press Calculate.
- Read results above, including period and RPM.
- Use Download CSV or Download PDF for records.
Angular Frequency and Hertz: What You’re Converting
Angular frequency (ω) measures how fast a repeating motion advances in radians each second. Hertz (Hz) measures how many full cycles happen each second. The two describe the same rhythm, just with different “counting units.” In sinusoidal motion, ω appears inside sin(ωt) and cos(ωt).
Why 2π Matters
One full cycle equals 2π radians. That constant links cycle-based frequency to angle-based speed. If something completes 1 cycle every second, it advances 2π radians every second, so ω is 2π rad/s. A common reference is 50 Hz, which maps to ω ≈ 314.159 rad/s.
Core Relationship Used by the Calculator
The calculator applies ω = 2πf and f = ω/(2π). It also reports period T = 1/f, which is the time for one cycle. For rotating machines, it shows RPM using RPM = 60f. In simple harmonic motion, x(t)=A sin(ωt) uses ω directly.
Unit Options and Normalization
You can enter ω in rad/s, rad/min, deg/s, deg/min, or RPM. Internally, every input is normalized to rad/s before conversion. Degree inputs are converted using π/180, while RPM is converted using ω = 2π·(RPM/60). In reverse mode, Hz, kHz, and MHz inputs are normalized to Hz. Results also show degrees per second and period in seconds.
Worked Example with Real Numbers
Suppose ω = 31.4159 rad/s. Dividing by 2π gives f ≈ 5 Hz. The period becomes T = 1/5 = 0.2 s (200 ms). The same motion corresponds to RPM = 60·5 = 300 RPM. In reverse mode, entering 5 Hz returns ω ≈ 31.4159 rad/s.
Interpreting Small and Large Values
Very small ω values create long periods, useful for slow oscillations and timing circuits. Very large ω values produce short periods seen in acoustics and vibration testing. For example, 1 kHz has a 1 ms period and ω ≈ 6283.185 rad/s. Use the precision setting to control displayed rounding.
Use Cases and Reporting
This tool helps when moving between textbook formulas (often written in ω) and instrument readouts (often in Hz or RPM). It also supports quick checks: compare motor speeds to resonant frequencies, or convert lab shaker settings to theoretical models. Export results to CSV for spreadsheets or to a compact PDF report for documentation.
Frequently Asked Questions
1) What is the difference between ω and f?
ω is angular frequency in radians per second, while f is frequency in cycles per second. They describe the same repetition rate, connected by ω = 2πf.
2) Why does the calculator show the period?
The period T is the time for one full cycle. It is the inverse of frequency: T = 1/f. Seeing T helps interpret how fast or slow a vibration or rotation feels.
3) How is RPM related to Hz?
RPM is revolutions per minute, while Hz is cycles per second. Because one revolution is one cycle, RPM = 60 × Hz, and Hz = RPM / 60.
4) Can angular frequency be negative?
Yes. A negative value usually indicates direction or phase convention (for example, clockwise vs counterclockwise). The magnitude |ω| determines the oscillation rate; period is based on positive frequency.
5) What happens if I enter kHz or MHz?
In reverse mode, the calculator first converts kHz and MHz to Hz, then computes ω. This keeps the formulas consistent and avoids unit mistakes when working with audio or RF values.
6) Why do my results differ slightly from another calculator?
Small differences typically come from rounding, displayed precision, or using an approximate π value. Set a higher precision if you need closer agreement for reporting or intermediate design steps.