About

Miscalculating angular velocity in rotating machinery leads to bearing failure, resonance-induced fractures, or centrifuge sample destruction. This calculator derives ω (angular velocity in rad/s) from four independent input methods: frequency f, period T, the linear-velocity/radius pair v / r, or direct angular displacement Δθ over elapsed time Δt. Results are cross-converted to RPM, deg/s, and rev/s in a single pass. Derived quantities - tangential velocity and centripetal acceleration - are computed when a radius is provided.

The underlying relation ω = 2πf assumes rigid-body rotation about a fixed axis. For flexible shafts or eccentric orbits the effective radius varies with angular position; treat outputs as first-order approximations in those cases. All conversions use the exact factor π = 3.14159265358979 rather than truncated constants, so round-trip unit errors stay below 10⁻¹².

Formulas

The fundamental relation links angular velocity to rotational frequency:

ω = 2πf

When the period T is known instead:

ω = 2πT

From linear (tangential) velocity and radius:

ω = vr

From angular displacement over time:

ω = ΔθΔt

Derived quantities when radius r is available:

v = ω ⋅ r

a_c = ω² ⋅ r

Where: ω = angular velocity (rad/s), f = frequency (Hz), T = period (s), v = tangential velocity (m/s), r = radius (m), Δθ = angular displacement (rad), Δt = time interval (s), a_c = centripetal acceleration (m/s²).

Reference Data

Unit	Symbol	Factor to rad/s	Typical Use
Radians per second	rad/s	1	SI standard, physics equations
Revolutions per minute	RPM	0.10472 (2π÷60)	Motors, engines, turntables
Revolutions per second	rev/s	6.28318 (2π)	Centrifuges, turbines
Degrees per second	deg/s	0.01745 (π÷180)	Gyroscopes, robotics joints
Radians per minute	rad/min	0.01667 (1÷60)	Slow-rotation equipment
Hertz (cycles/s)	Hz	6.28318 (2π)	Frequency analysis
Milliradians per second	mrad/s	0.001	Satellite attitude control
Degrees per minute	deg/min	0.000291	Astronomical tracking
Degrees per hour	deg/h	4.848×10⁻⁶	Earth rotation rate
Turns per hour	TPH	1.745×10⁻³	Industrial mixers
Common Reference Speeds
Earth rotation	-	7.292×10⁻⁵ rad/s	Geophysics, Coriolis calcs
Audio CD	-	200-500 RPM	CLV playback
Hard-drive platter	-	7200 RPM	Data storage
Jet engine LP spool	-	2500-4000 RPM	Aviation propulsion
Jet engine HP spool	-	9000-15000 RPM	Aviation propulsion
Dental drill	-	400000 RPM	Medical equipment
Ultracentrifuge	-	150000 RPM	Biochemistry separation
Car wheel at 100 km/h (r = 0.3 m)	-	≈92.6 rad/s	Automotive engineering
ISS solar array joint	-	4 deg/min	Orbital tracking
Helicopter main rotor	-	300-500 RPM	Rotorcraft design

Frequently Asked Questions

Angular speed is the scalar magnitude of angular velocity. Angular velocity ω is a pseudovector directed along the rotation axis (right-hand rule). This calculator outputs the magnitude. For three-dimensional problems you must track the axis direction separately.

One revolution equals 2π radians. One minute equals 60 seconds. Therefore ω (rad/s) = RPM × 2π ÷ 60. The inverse: RPM = ω × 60 ÷ (2π).

Centripetal acceleration a_c = ω²r arises from differentiating the velocity vector (which itself already contains ω) with respect to time. Doubling rotational speed quadruples the centripetal load, which is why centrifuge safety margins are critical.

The calculator assumes constant angular velocity over the measurement interval. For accelerating shafts, the output represents the average ω during Δt. Instantaneous angular velocity requires the limit as Δt → 0, which demands continuous sensor data.

When the tangential velocity v = ωr exceeds roughly 0.1c (3 × 10⁷ m/s), special-relativistic effects become non-negligible (errors above 0.5%). For nearly all engineering applications this threshold is never reached.

Select the "degrees" option in the angle unit dropdown. The calculator converts internally: θ_rad = θ_deg × π ÷ 180. All results remain in rad/s canonical form before final unit conversion.