What this calculator does
Angular velocity describes how fast something turns, independent of its size. Every point on a spinning disc shares the same angular velocity, even though a point near the rim travels much further each second than one near the centre.
Converting to linear speed simply requires a radius. That conversion is why the outer edge of a large wheel moves so much faster than a small one turning at the same rate.
The formula
Divide the angle turned by the time taken, converting degrees to radians first. Multiplying angular velocity by radius gives the tangential speed, and scaling by 60 over 2π converts to revolutions per minute.
| Term | Meaning |
|---|---|
| Radian | The natural unit of angle: the angle subtending an arc equal to the radius. A full turn is 2π radians. |
| Angular velocity (ω) | Rate of rotation, in radians per second. |
| Tangential speed | The linear speed of a point at a given radius, equal to ω times r. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle turned (°) | The angle turned through, in degrees. |
| Time taken (s) | The time taken, in seconds. |
| Radius (for linear speed) (m) | The radius at which to calculate the linear speed, in metres. |
When to use it
Converting between RPM and radians per second
Machinery is specified in RPM while physics formulas need radians per second, and the conversion is needed constantly.
Finding rim speed
Cutting tools and grinding wheels have surface speed limits, which depend on both rotation rate and diameter.
Analysing rotating machinery
Angular velocity is the starting point for torque, power and angular momentum calculations.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do rotation rate and rim speed relate?
Different angles turned in the same half second.
| Angle turned | Angular velocity (rad/s) | Rotational speed |
|---|---|---|
| 90° | 3.142 rad/s | 30.00 RPM |
| 180° | 6.283 rad/s | 60.00 RPM |
| 360° | 12.566 rad/s | 120.00 RPM |
Questions
Why use radians rather than degrees?
Because radians make the relationship between angular and linear quantities direct: tangential speed is simply ω times r with no conversion factor. In degrees an extra constant appears in every formula.
Do all points on a spinning wheel share the same angular velocity?
Yes, that is what makes it useful. Every point completes a full turn in the same time. What differs is linear speed, which grows with distance from the axis.
How do I convert RPM to radians per second?
Multiply by 2π and divide by 60. One RPM is about 0.105 rad/s, so 1,000 RPM is around 104.7 rad/s.
What is the difference between angular velocity and angular acceleration?
Velocity is the rate of turning; acceleration is the rate at which that turning rate changes. A wheel spinning steadily has angular velocity but no angular acceleration.
For how quickly that rotation rate changes, see the angular acceleration calculator. For the momentum it carries, see the angular momentum calculator.