This angular velocity converter changes a rotation rate between revolutions per minute (rpm), radians per second, degrees per second and revolutions per second. All values go through rad/s, the SI unit. One revolution is 2π rad, so 1 rpm = 2π/60 rad/s, about 0.10472 rad/s, and 1 rad/s is about 9.5493 rpm.
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How to use the angular velocity converter
- Enter the rotation rate, for example 3,000 from a motor nameplate.
- In the angular velocity converter, choose its unit and the target unit, such as rpm to rad/s.
- Read the angular velocity converter result: 3,000 rpm = 314.16 rad/s.
For rad/s to rpm, multiply by 60/(2π), about 9.5493. Degrees per second are rpm × 6.
What is angular velocity?
Angular velocity is the rate of rotation. The SI unit is the radian per second; engines and motors are rated in revolutions per minute (RPM), and 1 RPM = 2π/60 ≈ 0.1047 rad/s.
Conversion formula
Every unit is stored as a multiple of the base unit, the radian per second (rad/s). To convert:
value in target unit = value × (factor of source unit ÷ factor of target unit)
where each factor is the size of the unit in radians per second.
Common angular velocity conversions
| From | To | 1 unit equals |
|---|---|---|
| revolution per minute (rpm) | radian per second (rad/s) | 0.1047198 rad/s |
| radian per second (rad/s) | revolution per minute (rpm) | 9.549297 rpm |
| degree per second (°/s) | radian per second (rad/s) | 0.01745329 rad/s |
| revolution per minute (rpm) | degree per second (°/s) | 6 °/s |
Angular velocity converter worked example
Convert 25 revolutions per minute to degrees per second:
- Factor of rpm = 0.10471976 rad/s
- Factor of °/s = 0.017453293 rad/s
- 25 × 0.10471976 ÷ 0.017453293 = 150 °/s
Angular velocity unit reference
| Unit | Symbol | Group | Equals (rad/s) |
|---|---|---|---|
| Radian per Second | rad/s | SI | 1 |
| Degree per Second | °/s | Common | 0.01745329252 |
| Revolution per Minute | rpm | Common | 0.1047197551 |
| Revolution per Second | rps | Common | 6.283185307 |
| Radian per Minute | rad/min | SI | 0.01666666667 |
| Degree per Minute | °/min | Common | 0.0002908882087 |
| Degree per Hour | °/h | Common | 4.84814 × 10−6 |
| Milliradian per Second | mrad/s | SI | 0.001 |
Reverse factors for each angular velocity unit
The reference table gives each unit in radian per seconds. To go the other way, divide by that factor; the list below does the division for you.
- Degree per Second (°/s), Common: 1 °/s = 0.01745329 rad/s, so 1 rad/s = 57.29578 °/s.
- Revolution per Minute (rpm), Common: 1 rpm = 0.1047198 rad/s, so 1 rad/s = 9.549297 rpm.
- Revolution per Second (rps), Common: 1 rps = 6.283185 rad/s, so 1 rad/s = 0.1591549 rps.
- Radian per Minute (rad/min), SI: 1 rad/min = 0.01666667 rad/s, so 1 rad/s = 60 rad/min.
- Degree per Minute (°/min), Common: 1 °/min = 0.0002908882 rad/s, so 1 rad/s = 3,437.747 °/min.
- Degree per Hour (°/h), Common: 1 °/h = 4.84814 × 10−6 rad/s, so 1 rad/s = 206,264.7 °/h.
- Milliradian per Second (mrad/s), SI: 1 mrad/s = 0.001 rad/s, so 1 rad/s = 1,000 mrad/s.
Motor and disc speeds in rad/s
Typical rotation speeds converted with the exact factor 2π/60. AC induction motors on a 60 Hz supply run just under 1,800 rpm (4-pole) or 3,600 rpm (2-pole) synchronous speed; 33 rpm is close to a record turntable:
| rpm | rad/s | deg/s | Hz (rev/s) |
|---|---|---|---|
| 33 | 3.46 | 198 | 0.55 |
| 60 | 6.28 | 360 | 1 |
| 1,000 | 104.72 | 6,000 | 16.67 |
| 1,800 | 188.5 | 10,800 | 30 |
| 3,000 | 314.16 | 18,000 | 50 |
| 3,600 | 376.99 | 21,600 | 60 |
| 7,200 | 753.98 | 43,200 | 120 |
Synchronous speed in rpm is 120 × supply frequency ÷ number of poles.
From angular velocity to linear speed
A point at radius r on a rotating part moves at v = ω × r, with ω in rad/s. A 0.3 m radius wheel turning at 10 rad/s has a rim speed of 3 m/s (10.8 km/h). This is why rad/s, not rpm, appears in physics formulas. Deg/s to rad/s is multiply by π/180. For rotational acceleration, see the angular acceleration converter.
Everyday examples
Some reference values help you check a result from the angular velocity converter. The Earth turns once per sidereal day of about 86,164 seconds, an angular velocity of about 7.29 × 10−5 rad/s. A 7,200 rpm hard disk spins at about 754 rad/s, and a car engine idling at 800 rpm turns at about 84 rad/s. If the angular velocity converter gives an answer far from these orders of magnitude for a similar object, check the input unit.
Related converters and calculators
Related rotation and motion tools:
- Angular acceleration converter: rad/s² and rpm/s
- Angle converter: degrees and radians
- Torque converter: N·m and lbf·ft
- Moment of inertia converter: kg·m²
- Speed converter: linear speed
- Frequency to wavelength: Hz
Sources
BIPM: SI Brochure; NIST Special Publication 811, Appendix B.
Questions about this calculator
What is the SI unit of angular velocity?
Radians per second (rad/s). One full revolution is 2π radians.
How do I convert rpm to rad/s?
Multiply rpm by 2π/60 (about 0.10472). For example, 1,500 rpm = 157.08 rad/s.
How do I convert rad/s to rpm?
Multiply by 60/(2π), about 9.5493. For example, 50 rad/s = 477.46 rpm.
How do I convert deg/s to rad/s?
Multiply by π/180 (about 0.017453). For example, 90 deg/s = 1.5708 rad/s.
Is rpm the same as Hz?
No. Hz here means revolutions per second, so divide rpm by 60: 3,600 rpm = 60 rev/s.
What is the difference between angular velocity and angular frequency?
Both are measured in rad/s. Angular velocity describes rotation; angular frequency (ω = 2πf) describes an oscillation.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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