Angular velocity formulas and conversions
Angular velocity measures how quickly angular position changes. It is commonly expressed in radians per second, revolutions per minute, or degrees per second.
The calculator preserves both the angle-time model and the relationship between tangential speed, radius, and angular velocity.
How to use the angular velocity calculator
- Choose a model: Use angle and time or linear speed and radius.
- Choose the unknown: Select the value to solve for.
- Enter values: Enter measurements and select matching units.
- Calculate: Review the converted result and formula.
Formula and variables
Angular velocity equals angular displacement divided by elapsed time.
ω = Δθ / Δt- ω — Angular velocity
- Rate of rotation (rad/s)
- Δθ — Angular displacement
- Angle swept (rad)
- Δt — Time interval
- Elapsed time (s)
Rotating wheel example
A wheel turns 12 radians in 3 seconds.
- Angle
- 12 rad
- Time
- 3 s
- ω = 12 / 3
- ω = 4 rad/s
Result: The angular velocity is 4 rad/s.
The wheel sweeps four radians each second.
Understanding your results
Reading the result
A larger magnitude means faster rotation; the sign can represent the chosen direction.
Assumptions
- Rotation is measured about a fixed axis.
- Average angular velocity is used over the entered interval.
- Linear velocity is tangential to the circular path.
Limitations
- Changing angular velocity requires an angular-acceleration model.
- Direction must be handled with a consistent sign convention.
Common mistakes
- Using revolutions as radians without conversion.
- Mixing minutes and seconds.
- Using diameter where the formula requires radius.
Practical use cases
Rotating equipment
Convert RPM and relate rim speed to shaft rotation.
Physics study
Check circular-motion homework and lab measurements.
Frequently asked questions
How do I convert RPM to rad/s?
Multiply RPM by 2π and divide by 60.
What is the relationship to linear velocity?
Tangential linear velocity is v = rω.
Sources and review
- SI Brochure, 9th edition — BIPM. Accessed 2026-07-11.
- Special Publication 811 — NIST. Accessed 2026-07-11.
Reviewed 2026-07-11.