Whether you are diagnosing an electric motor, tuning an audio signal, calibrating a hard drive, or working through a textbook dynamics problem, you often need to switch between revolutions per minute (RPM), hertz (Hz), radians per second (rad/s) and the related period. This calculator handles all of those conversions simultaneously, shows the full working, and provides a reference table covering common real-world speeds.
How it works
All conversions flow through hertz as the canonical base unit — one hertz equals one complete cycle (or revolution) per second. From that shared baseline the tool fans out to every output unit in a single pass:
- RPM = Hz × 60 — multiply by the number of seconds in a minute.
- RPS = Hz — revolutions per second and hertz are numerically identical.
- rad/s = Hz × 2π — each revolution sweeps a full circle (2π radians).
- Period T = 1 ÷ Hz — the time for one complete revolution, in seconds.
Accepted input units are Hz, kHz (divided by 1 000 to reach Hz), RPM (divided by 60), RPS (already in Hz), and rad/s (divided by 2π). Once the base Hz value is known, all seven output values update at the same time.
Worked example
A 3-phase induction motor nameplate says 1 450 RPM.
- Convert to Hz: 1 450 ÷ 60 = 24.167 Hz
- Angular velocity: 24.167 × 2π = 151.8 rad/s
- Period: 1 ÷ 24.167 = 0.04138 s = 41.38 ms per revolution
The synchronous speed for a 4-pole motor on a 50 Hz supply is 120 × 50 ÷ 4 = 1 500 RPM. The difference (1 500 − 1 450 = 50 RPM) is the slip — a normal consequence of the induction principle. This calculator handles all three steps automatically, so you can check nameplate data against supply frequency in seconds.
Formula reference
| Conversion | Formula |
|---|---|
| Hz → RPM | RPM = Hz × 60 |
| RPM → Hz | Hz = RPM ÷ 60 |
| Hz → rad/s | ω = Hz × 2π |
| rad/s → Hz | Hz = ω ÷ 2π |
| RPM → rad/s | ω = RPM × 2π ÷ 60 |
| Hz → period (s) | T = 1 ÷ Hz |
| RPM → period (s) | T = 60 ÷ RPM |
The angular velocity ω in rad/s appears in many physics and engineering equations, including torque-power relationships (P = τ × ω) and rotational kinetic energy (KE = ½Iω²). Converting from the more familiar RPM to rad/s is therefore a constant requirement in mechanical and electrical engineering.
Practical contexts
Electric motors — UK and EU motors run on 50 Hz mains. A 2-pole motor has a synchronous speed of 3 000 RPM; 4-pole gives 1 500 RPM; 6-pole gives 1 000 RPM. Slip in induction motors typically reduces the actual shaft speed by 2–4 %.
Hard drives — consumer drives spin at 5 400 RPM (90 Hz) or 7 200 RPM (120 Hz). That 120 Hz means one platter revolution every 8.33 ms, which determines the maximum rotational latency.
Internal combustion engines — a typical petrol car idles near 750 RPM (12.5 Hz) and redlines near 6 000–8 000 RPM (100–133 Hz). Diesel engines generally have lower redlines, closer to 4 000–5 000 RPM.
Audio — concert pitch A4 is 440 Hz. While “revolutions” is not the usual term for sound, the relationship holds for rotating machinery that generates tonal noise: a fan blade passing a strut at 3 000 RPM with 4 blades produces a tone at 3 000 ÷ 60 × 4 = 200 Hz, which sits in the easily audible mid-frequency range.
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