Calculator
Example data
| Given | Input | Key formula | ω (rad/s) |
|---|---|---|---|
| Frequency | f = 60 Hz | ω = 2πf | ≈ 376.991 |
| Period | T = 0.50 s | ω = 2π/T | ≈ 12.566 |
| RPM | rpm = 1800 | ω = 2π(rpm/60) | ≈ 188.496 |
| Speed & radius | v = 10 m/s, r = 0.50 m | ω = v/r | 20.000 |
| aₙ & radius | aₙ = 9.81 m/s², r = 1.0 m | ω = √(aₙ/r) | ≈ 3.132 |
Formulas used
- From frequency: ω = 2πf
- From period: ω = 2π/T, where T = 1/f
- From rpm: f = rpm/60, then ω = 2πf
- From linear speed: v = ωr ⇒ ω = v/r
- From centripetal acceleration: aₙ = ω²r ⇒ ω = √(aₙ/r)
- Conversions: deg/s = ω·180/π, rpm = 60·ω/(2π)
How to use this calculator
- Select the input method that matches your data.
- Enter values, then choose the correct units.
- Pick decimal places and enable steps if needed.
- Click Calculate to see ω plus conversions.
- Use the download buttons to export your results.
Tip: Keep units consistent when comparing multiple cases.
Article
What this tool calculates
Angular frequency (ω) describes how fast something rotates or oscillates in radians per second. It connects the angle of rotation to time, which is why it appears in sine and cosine models. Radians measure turns naturally in calculus, making derivatives and integrals cleaner. Use it for motors, rotors, fans, gears, pendulums, and any periodic signal.
Inputs you can start from
You can compute ω from frequency f, period T, revolutions per minute, linear speed v with radius r, or centripetal acceleration aₙ with radius. This flexibility matters because real projects report different measurements. A tachometer gives rpm, a stopwatch gives T, and a data logger often outputs Hz.
Core relationships behind the math
The main identity is ω = 2πf, where 2π radians equals one full turn. Because f = 1/T, you can also use ω = 2π/T. For circular motion, v = ωr rearranges to ω = v/r. With centripetal acceleration, aₙ = ω²r gives ω = √(aₙ/r), useful when acceleration comes from an IMU.
Unit conversions you get instantly
The calculator converts ω between rad/s and deg/s, then back-calculates frequency, period, and rpm. These conversions make it easy to compare a specification sheet in rpm with a physics equation in radians. It also helps when mixing disciplines, like mechanical rotation rates and electrical signal frequencies.
Example with common power frequency
If your supply is 60 Hz, ω = 2π × 60 ≈ 376.99 rad/s. That equals about 21,600 deg/s. In rotating machinery, it corresponds to 3,600 rpm for a two‑pole synchronous speed, while a four‑pole machine halves that speed. Comparing ω and rpm side by side is a fast sanity check.
Accuracy and rounding tips
Small unit mistakes create large errors. Always confirm whether a value is rpm, Hz, deg/s, or rad/s before you calculate. Convert centimeters or millimeters into meters when using v/r or √(aₙ/r), and keep gravity units consistent if you enter aₙ in g. Choose more decimal places for simulations and fewer for reports.
Where engineers use ω daily
Angular frequency appears in vibration analysis, bearing diagnostics, control loops, flywheel sizing, and gear trains. It defines reactance in AC circuits and the phase rate in sinusoidal models, so it is shared language between mechanical and electrical teams. Consistent ω values improve troubleshooting, documentation, and repeatable test results. It also helps interpret resonance peaks on frequency plots.
FAQs
What is angular frequency measured in?
Angular frequency is measured in radians per second (rad/s). It can also be expressed in degrees per second, but rad/s is standard for physics and engineering formulas.
How do I convert Hz to angular frequency?
Use ω = 2πf. Multiply the frequency in hertz by 2π to get rad/s. For example, 10 Hz becomes about 62.832 rad/s.
How do I convert rpm to angular frequency?
First convert rpm to hertz: f = rpm/60. Then apply ω = 2πf. The calculator performs both steps and also returns rpm from ω.
Why does ω use 2π?
One full revolution equals 2π radians. Frequency counts cycles per second, so multiplying by 2π converts cycles into radians per second.
Can I compute ω from linear speed and radius?
Yes. For circular motion, v = ωr, so ω = v/r. Make sure v is in m/s and r is in meters to avoid scaling errors.
When should I use √(aₙ/r) for ω?
Use it when you know centripetal acceleration aₙ and radius r. Since aₙ = ω²r, taking the square root gives ω. Ensure aₙ is positive and r is greater than zero.