Convert angular velocity between rad/s, RPM, degrees per second, and Hz — or calculate angular velocity from linear speed and radius. 4 modes covering every rotational motion scenario, NIST verified formulas, instant results.
✓Verified: NIST Physics Reference & ISO 80000-3 Rotational Mechanics
Enter angular velocity in any unit to convert to all others instantly:
Enter value in selected unitEnter valid angular velocity.
Select input unitSelect unit.
Calculate angular velocity from tangential (linear) velocity and radius: ω = v / r
Tangential speed at the rimEnter valid velocity.
Distance from rotation axisEnter valid radius.
Calculate period and frequency from angular velocity, or vice versa:
What you knowSelect option.
Enter valueEnter valid value.
Calculate tangential velocity from angular velocity and radius: v = ω × r
Enter valueEnter valid angular velocity.
Select unitSelect unit.
Distance from rotation centreEnter valid radius.
Angular Velocity
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⚠️ Disclaimer: Results use standard NIST physics formulas. Angular velocity calculations assume rigid-body rotation. For high-speed machinery, always verify with a tachometer and consult engineering specifications.
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📚 Sources & Methodology
All angular velocity formulas verified against authoritative sources:
NIST Physics Reference Data — SI unit definitions for angular velocity (rad/s), angular frequency, and rotational mechanics constants physics.nist.gov
ISO 80000-3:2019 — Quantities and units, Part 3: Space and time — angular velocity, angular frequency, and period definitions iso.org
Angular velocity (ω, omega) is the rate at which an object rotates, expressed as the angle swept per unit time. The SI unit is radians per second (rad/s). One full revolution equals 2π radians (approximately 6.2832 rad), so a wheel making one complete turn per second has ω = 2π rad/s = 1 Hz = 60 RPM. Angular velocity is a vector quantity — its direction follows the right-hand rule along the rotation axis.
In electrical engineering, angular frequency ω = 2πf appears everywhere: AC circuit impedance (X_L = ωL, X_C = 1/ωC), phasor analysis, signal processing, and motor synchronous speed calculations. A 50 Hz power system has ω = 314.16 rad/s; a 60 Hz system has ω = 376.99 rad/s.
Angular Velocity Formulas (NIST / ISO 80000-3)
omega = delta-theta / delta-t [angle / time, rad/s]omega = 2 * pi * f [from frequency in Hz]omega = 2 * pi * N / 60 [from RPM]omega = v / r [from linear velocity v (m/s) and radius r (m)]omega = 2 * pi / T [from period T in seconds]Conversion: rpm to rad/s = rpm * 2*pi/60 = rpm * 0.10472Conversion: rad/s to rpm = rad/s * 60/(2*pi) = rad/s * 9.5493Conversion: rad/s to deg/s = rad/s * 180/pi = rad/s * 57.2958Tangential velocity: v = omega * r (m/s)Centripetal acceleration: a = omega^2 * r = v^2 / r (m/s^2)
Angular Velocity Unit Conversion Reference
rad/s
RPM
deg/s
Hz
Common Application
0.1047
1 RPM
6
0.01667
Slow rotation, clock hands
1.0
9.549 RPM
57.296
0.1592
Reference: 1 rad/s
2π = 6.2832
60 RPM
360
1
1 revolution per second
52.36
500 RPM
3,000
8.333
Slow motor
104.72
1,000 RPM
6,000
16.67
Typical motor
157.08
1,500 RPM
9,000
25
Synchronous (50 Hz, 4-pole)
188.50
1,800 RPM
10,800
30
Synchronous (60 Hz, 4-pole)
314.16
3,000 RPM
18,000
50
Synchronous (50 Hz, 2-pole)
376.99
3,600 RPM
21,600
60
Synchronous (60 Hz, 2-pole)
Angular Velocity in Electric Motors
Electric motor synchronous speed: N_s = 120f/P, where f is supply frequency (Hz) and P is the number of poles. For a 2-pole 50 Hz motor: N_s = 120 × 50 / 2 = 3000 RPM = 314.16 rad/s. For a 4-pole 60 Hz motor: N_s = 120 × 60 / 4 = 1800 RPM = 188.5 rad/s. Induction motors run slightly below synchronous speed (slip = 2–8%). The angular velocity appears in torque calculation: P(watts) = τ × ω, so τ = P/ω (Nm).
Angular Velocity vs Angular Acceleration
Angular velocity (ω) describes instantaneous rotational speed. Angular acceleration (α) describes how quickly ω changes: α = dω/dt (rad/s²). The rotational analogue of Newton’s second law: τ = I × α, where I is moment of inertia (kg·m²) and τ is torque (Nm). Kinematic equations apply: ω = ω_0 + αt; θ = ω_0 t + ½αt².
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Motor power formula: Power = Torque × Angular velocity: P = τ × ω. A motor producing 50 Nm at 1500 RPM (157.1 rad/s) outputs P = 50 × 157.1 = 7,854 W = 7.85 kW = 10.53 HP. This is the fundamental relationship linking mechanical torque, speed, and power output for any rotating machine.
❓ Frequently Asked Questions
Angular velocity (omega) is the rate of rotation — how many radians (or degrees, revolutions) an object sweeps per second. SI unit is rad/s. Formula: omega = 2pi x f = 2pi x N/60 (where N is RPM). A motor at 3000 RPM has omega = 3000 x 2pi/60 = 314.16 rad/s. Use the Unit Converter tab above for instant conversion between rad/s, RPM, deg/s, and Hz.
rad/s = RPM x 2pi/60 = RPM x 0.10472. Example: 1800 RPM = 1800 x 0.10472 = 188.5 rad/s. Reverse: RPM = rad/s x 9.5493. A 60 Hz 2-pole motor runs at 3600 RPM = 376.99 rad/s. Use the Unit Converter tab above for any value.
omega = v / r, where v is linear (tangential) velocity in m/s and r is radius in meters. Example: a point on a wheel rim moving at 20 m/s at 0.4 m radius: omega = 20/0.4 = 50 rad/s = 477.5 RPM. Use the From Linear Velocity tab above.
Period T = 2pi/omega = 1/f = 60/RPM. A motor at 1500 RPM: T = 60/1500 = 0.04 seconds (40 ms per revolution). Frequency f = 1/T = RPM/60 = omega/(2pi). Use the Period and Frequency tab for instant calculation from any starting value.
Angular frequency omega = 2pi x f. For a 50 Hz system: omega = 314.16 rad/s. For 60 Hz: omega = 376.99 rad/s. Capacitor reactance Xc = 1/(omega x C). Inductor reactance XL = omega x L. Angular frequency appears in all AC circuit analysis and phasor calculations.
v = omega x r. Example: grinding wheel at 2000 RPM (209.4 rad/s) with 100 mm radius: v = 209.4 x 0.1 = 20.94 m/s (75.4 km/h at the rim). Use the Tangential Velocity tab above with any angular velocity unit.
rad/s = deg/s x pi/180 = deg/s x 0.017453. Example: 360 deg/s = 360 x pi/180 = 2pi rad/s = 6.283 rad/s = 1 Hz = 60 RPM. Reverse: deg/s = rad/s x 180/pi = rad/s x 57.2958. The Unit Converter tab handles this automatically.
Synchronous speed N_s = 120 x f / P (RPM), where f = supply frequency (Hz) and P = number of poles. 2-pole 50 Hz: 3000 RPM = 314.2 rad/s. 4-pole 60 Hz: 1800 RPM = 188.5 rad/s. 6-pole 50 Hz: 1000 RPM = 104.7 rad/s. Induction motors run at slightly less due to slip (typically 2-5%).
Centripetal acceleration a = omega^2 x r = v^2/r. A drum at 1200 RPM (125.7 rad/s) with 0.25 m radius: a = 125.7^2 x 0.25 = 3,952 m/s^2 = 403 g. Centripetal force F = m x omega^2 x r. This force is directed toward the rotation centre and keeps the object in circular motion.
Electric motors: 1500-3600 RPM (157-377 rad/s). Hard drives: 5400-7200 RPM (565-754 rad/s). Car engine: 700-7000 RPM. Turbines: up to 15000 RPM. Centrifuges: 10,000-100,000+ RPM. Washing machine spin: 800-1600 RPM. PC CPU fan: 1000-3000 RPM. Household ceiling fan: 200-400 RPM.
Earth rotates once per sidereal day (86,164 seconds). omega = 2pi/86164 = 7.292 x 10^-5 rad/s = 0.00417 deg/s = 0.00007292 RPM. At the equator, linear velocity = 7.292e-5 x 6,371,000 m = 465 m/s = 1,674 km/h. Earth orbits the Sun at omega = 2pi/(365.25 x 86400) = 1.991 x 10^-7 rad/s.
Torque (Nm) = Power (W) / omega (rad/s) = Power x 60 / (2pi x RPM). A 5 kW motor at 1500 RPM: omega = 1500 x 2pi/60 = 157.1 rad/s, Torque = 5000/157.1 = 31.8 Nm. Also: Torque (Nm) = HP x 7121 / RPM. A 10 HP motor at 1800 RPM: Torque = 10 x 7121/1800 = 39.6 Nm = 29.2 ft-lb.