Angular Velocity Calculator
Turn RPM and radius into angular speed and rim speed.
What it does
Spinning parts are described two ways. Angular velocity tells how fast the angle turns, in radians per second. Rim speed tells how fast a point on the edge moves, in m/s or ft/s. A small and large wheel can share angular speed while rim speeds differ.
The tool above takes RPM and radius and returns angular velocity, rim speed, and period. The defaults 60 RPM with 0.5 m match the worked example, so the page opens showing 6.28 rad/s. Switch to imperial for feet.
The links are direct. One revolution is 2 pi radians, and one minute is 60 seconds, so omega = 2 pi N / 60. Rim speed v = omega r, so doubling radius doubles rim speed at the same RPM. Period T = 60 / N is seconds per turn.
Car tires show the split. At fixed road speed, a larger tire turns fewer RPM because v = omega r. Bike gears, drills, and centrifuges use the same trade between spin rate and edge speed.
This page shows the formulas, one full hand checked example at 60 RPM, a reference table, and practice checks that run through the tool in both unit systems.
How to use
- Pick Metric (m) or Imperial (ft) for radius and rim speed.
- Type RPM, for example 60.
- Type radius from center to rim in the active unit.
- Read Angular velocity in rad/s.
- Read Rim speed in m/s or ft/s at your radius.
- Read Period as seconds for one full turn.
Formula
- RPM
- Revolutions per minute.
- omega
- Angular velocity in radians per second.
- r
- Radius in m or ft. Rim speed follows its unit.
Radians are unit free as arc over radius, so omega carries rad/s by convention. For degrees per second, multiply rad/s by 57.2958.
Worked example
Example: 60 RPM at 0.5 m radius
A wheel turns 60 RPM with 0.5 m radius. Find spin and rim speed.
- Write the inputs: N = 60 RPM, r = 0.5 m.
- Angular: omega = 2 x 3.14159 x 60 / 60 = 6.28319 rad/s.
- Round: 6.28 rad/s.
- Rim: v = 6.28319 x 0.5 = 3.14159 m/s.
- Round: 3.14 m/s.
- Period: T = 60 / 60 = 1 s. Imperial mirror at 1.64 ft gives 10.3 ft/s.
Reference table
| RPM | Angular vel | Period |
|---|---|---|
| 30 | 3.14 rad/s | 2 s |
| 60 | 6.28 rad/s | 1 s |
| 120 | 12.57 rad/s | 0.5 s |
| 600 | 62.83 rad/s | 0.1 s |
Limits
- Steady spin assumed. Speeding up needs angular acceleration terms.
- Rigid radius assumed. Belts that slip break v = omega r.
- RPM cap is 100000. Turbine speeds above that need special tools.
- No load effects. Air drag and bearing loss are ignored.
Common mistakes
- Using diameter instead of radius, which doubles rim speed.
- Forgetting to divide RPM by 60, mixing minutes with seconds.
- Reporting rad/s as rev/s. One rev is 2 pi rad.
- Mixing meters with feet in v = omega r.
- Reading period as frequency. Period is seconds per turn, frequency is turns per second.
Try it yourself
Quick checks. Answers sit tucked under each question.
120 RPM at 0.25 m?
12.57 rad/s, rim 3.14 m/s, period 0.5 s.
30 RPM at 1 m?
3.14 rad/s, rim 3.14 m/s, period 2 s.
Questions
RPM to rad/s?
Multiply RPM by 2 pi / 60, about 0.10472.
What is rim speed?
omega times radius. Same spin, larger radius means faster edge.
What is period?
60 / RPM in seconds. 60 RPM gives 1 s per turn.
Does radius change omega?
No. Omega follows RPM only. Radius changes rim speed.
How fast is a car tire at 60 RPM?
At 0.5 m radius, rim is 3.14 m/s. Road speed matches rim without slip.
Degrees or radians?
This tool uses rad/s. Multiply by 57.2958 for deg/s.