All Equivalents Table
| Angular Velocity Unit | Equivalent Value |
|---|
Angular velocity (symbolized by ω) measures the vector rate of angular rotation per unit of time (ω = dθ ÷ dt). Across mechanical engineering, automotive powertrain design, robotics control, industrial electric motors, wind turbine energy generation, aerospace gyroscopes, and orbital mechanics, rotational speed is expressed in three primary unit families: International System of Units (SI metric fundamental: Radians per Second / rad/s, Radians per Minute / rad/min), engineering machinery rotation metrics (Revolutions per Minute / RPM, Revolutions per Second / RPS), and angular degree sweep rates (Degrees per Second / °/s, Degrees per Hour / °/h).
Our free online Velocity – Angular Converter provides instant, high-precision conversions across all SI metric, industrial engineering, and astronomical rotational speed units:
- Revolutions per Minute to Radians per Second (RPM to rad/s): Multiply RPM by
π ÷ 30(or0.104719755). Example: 1,000 RPM = 1,000 × 0.10472 = 104.72 rad/s. - Radians per Second to RPM (rad/s to RPM): Multiply rad/s by
30 ÷ π(or9.54929658). Example: 100 rad/s = 100 × 9.5493 = 954.93 RPM. - Revolutions per Second to rad/s (RPS to rad/s): Multiply RPS by
2π(1 RPS = 2π rad/s ≈ 6.2831853 rad/s = 360 °/s = 60 RPM). - Degrees per Second to rad/s (°/s to rad/s): Multiply °/s by
π ÷ 180(1 °/s ≈ 0.01745329 rad/s = 0.166667 RPM). - Radians per Second to °/s: Multiply rad/s by
180 ÷ π(1 rad/s ≈ 57.2957795 °/s = 9.5493 RPM). - Earth Planetary Angular Velocity (ωearth):
1 rev/sidereal day = 7.292115 × 10-5 rad/s = 15.041 °/hour.
Master Angular Velocity & Rotational Speed Conversion Table
The table below displays exact mathematical relationships, SI rad/s multipliers, and degree/RPM equivalents relative to 1 Radian per Second (1 rad/s):
| Angular Velocity Unit Name | Symbol | Exact Value in rad/s | RPM, RPS & Degree/sec Equivalent | Domain & System Application |
|---|---|---|---|---|
| 1 Radian per Second | rad/s |
1.0 rad/s (Base SI Unit) |
9.549297 RPM (0.159155 RPS / 57.29578 °/s) |
SI Metric Fundamental Angular Velocity Unit |
| 1 Revolution per Minute | RPM, r/min |
0.104719755 rad/s (π/30 rad/s) |
1.0 RPM (0.016667 RPS / 6.0 °/s) |
Global Engine, Motor & Shaft Speed Standard |
| 1 Revolution per Second | RPS, r/s |
6.283185307 rad/s (2π rad/s) |
60.0 RPM (1.0 RPS / 360.0 °/s) |
Rotational Frequency (Hertz / AC Power 60 Hz) |
| 1 Degree per Second | °/s |
0.017453293 rad/s (π/180 rad/s) |
0.166667 RPM (0.002778 RPS / 1.0 °/s) | Robotic Arm Joint Tracking & Gyroscopes |
| 1 Degree per Minute | °/min |
0.000290888 rad/s | 0.002778 RPM (0.016667 °/s) | Precision Astronomical Telescope Tracking |
| 1 Degree per Hour | °/h |
4.848137 × 10-6 rad/s | 0.0000463 RPM (0.000278 °/s) | Slow Celestial Ecliptic Drift Rates |
| 1 Radian per Minute | rad/min |
0.016666667 rad/s (1/60 rad/s) | 0.159155 RPM (0.95493 °/s) | Industrial Mixer & Agitator Drive Shafts |
| 1 Radian per Hour | rad/h |
0.000277778 rad/s (1/3,600 rad/s) | 0.002653 RPM (0.015915 °/s) | Geophysical Tectonic & Ocean Current Rotation |
| 1 Revolution per Day | r/d |
7.272205 × 10-5 rad/s | 0.0006944 RPM (0.004167 °/s / 15.0 °/h) | Diurnal Solar Planetary Rotation Rate |
Step-by-Step Rotational Speed Calculation Example
To convert an automotive engine speed of 3,500 Revolutions per Minute (3,500 RPM) into SI Radians per Second (rad/s) and Degrees per Second (°/s):
rad/s = 3,500 × (π ÷ 30) = 3,500 × 0.104719755 = 366.5191 rad/s (366.52 rad/s)
°/s = 3,500 × 6 = 21,000.0 Degrees per Second (21,000 °/s)
RPS = 3,500 ÷ 60 = 58.3333 Revolutions per Second (58.33 RPS)
Thus, 3,500 RPM equals 366.52 rad/s (or 58.33 RPS / 21,000 °/s).
Real-World Industrial Machinery & Celestial Benchmarks
Below is a comparative reference chart showing rotational speeds across wind turbines, automotive engines, electric vehicle motors, dental drills, and planetary rotation:
| Rotational Equipment / Celestial Body | Speed in Engineering Units (RPM / RPS) | Speed in SI Metric Units (rad/s / °/s) | Engineering & Dynamic Context |
|---|---|---|---|
| Earth Diurnal Sidereal Rotation | 0.000696 RPM (1 rev / sidereal day) | 7.2921 × 10-5 rad/s (15.041 °/h) | Planetary 360° rotation in 23h 56m 4.09s relative to stars |
| Commercial Wind Turbine Large Rotor | 15.0 RPM (0.25 RPS) | 1.5708 rad/s (90.0 °/s) | Low-speed direct-drive generator rotor speed |
| Automotive Gasoline Engine Idle Speed | 800.0 RPM (13.33 RPS) | 83.7758 rad/s (4,800.0 °/s) | Standard 4-cylinder engine stationary idle rotation |
| Automotive Engine Peak Horsepower Speed | 6,000.0 RPM (100.0 RPS) | 628.3185 rad/s (36,000.0 °/s) | High-performance engine redline shaft speed |
| Tesla Model 3 EV Electric Traction Motor | 18,000.0 RPM (300.0 RPS) | 1,884.9556 rad/s (108,000.0 °/s) | High-speed AC permanent magnet electric motor rotor |
| Medical High-Speed Dental Air Turbine Drill | 400,000.0 RPM (6,666.67 RPS) | 41,887.902 rad/s (2,400,000.0 °/s) | Precision miniature air-bearing turbine rotation |
History & Dynamics: James Watt’s 1788 Flyball Governor vs. Harmonic Frequency ω = 2πf
James Watt’s 1788 Centrifugal Flyball Governor
In 1788, Scottish engineer James Watt adapted the centrifugal flyball governor to regulate the rotational speed (RPM) of his rotary steam engines. As engine speed increased, centrifugal force swung two weighted metal balls outward on lever arms, closing the steam throttle valve to maintain constant RPM. Watt’s governor was the first major industrial application of negative feedback control loops and established Revolutions per Minute (RPM) as the universal benchmark for mechanical engine speed.
Harmonic Oscillations & AC Electrical Power (ω = 2πf)
In classical mechanics and electrical engineering, angular velocity (ω) is intimately linked to rotational frequency (f in Hertz or RPS) via the fundamental harmonic identity: ω = 2πf. In AC electrical power grids, a 60 Hz electrical generator rotates at an angular frequency of ω = 2 × π × 60 = 376.991 rad/s (3,600 RPM for a 2-pole generator), while a 50 Hz European generator rotates at ω = 314.159 rad/s (3,000 RPM).
Popular direct tools:
Frequently Asked Questions (FAQ)
How do you convert RPM to Radians per Second (rad/s)?
To convert Revolutions per Minute (RPM) to Radians per Second (rad/s), multiply RPM by π ÷ 30 (approximately 0.10472). For example, 1,000 RPM × 0.10472 = 104.72 rad/s.
How do you convert rad/s to RPM?
To convert Radians per Second (rad/s) to RPM, multiply rad/s by 30 ÷ π (approximately 9.5493). For example, 100 rad/s × 9.5493 = 954.93 RPM.
What is 1 Revolution per Second (RPS) in rad/s and RPM?
1 Revolution per Second (RPS) equals exactly 2π rad/s (approximately 6.28319 rad/s), 360 °/s, or 60 RPM.
Why do physicists use rad/s instead of RPM?
Physicists and control engineers use Radians per Second (rad/s) because radians are a natural dimensionless metric derived directly from circle geometry (arc length s = r · θ). Using rad/s simplifies rotational kinetic energy (Ek = 1/2 I ω2) and power equations (P = τ · ω) without requiring extra conversion factors.