Angular Velocity Converter
rad/s · °/s · RPM · RPS · rad/min
Convert angular velocity units instantly — rad/s, RPM, degrees/s, RPS and rad/min. Click any row in the conversion table to set it as the target unit.
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1 RPS = 2π rad/s ≈ 6.283 rad/s
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1 RPM = π/30 rad/s ≈ 0.1047 rad/s
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360°/s = 1 RPS = 60 RPM
About Angular Velocity Units
Angular velocity measures how fast an object rotates — the rate of change of its angular position with respect to time. The SI unit is radians per second (rad/s), but RPM (revolutions per minute) is the most common practical unit in engineering. Angular velocity is fundamental in rotating machinery, motors, gears, astronomy and signal processing.
📐 Radians & Degrees per Second
The SI unit rad/s is preferred in physics and calculus because 1 rad/s × radius (m) = linear velocity (m/s) directly. One full revolution = 2π rad ≈ 6.283 rad. Degrees/second (°/s) is more intuitive — 360°/s = one full rotation per second. Conversion: 1°/s = π/180 rad/s ≈ 0.01745 rad/s.
⚙️ RPM, RPS & rad/min
RPM (revolutions per minute) is standard in motors, engines and turbines. RPS (revolutions per second) is used in high-speed applications. 1 RPM = 2π/60 rad/s ≈ 0.1047 rad/s. 1 RPS = 2π rad/s = 60 RPM. rad/min is used for slow-moving machinery like telescope mounts and clocks — Earth rotates at about 0.00437 rad/min.
Worked Examples
Car Engine at 3,000 RPM
3,000 RPM = 314.16 rad/s
× π/30 = 3000 × 0.10472
= 50 RPS
= 18,000 °/s
= 3,141.6 rad/min
Typical highway cruising speed
= 50 RPS
= 18,000 °/s
= 3,141.6 rad/min
Typical highway cruising speed
Earth's Rotation
7.292 × 10⁻⁵ rad/s
= 0.00417 °/s
= 0.000698 RPM
= 4.178 × 10⁻³ rad/min
1 full revolution per sidereal day
= 0.000698 RPM
= 4.178 × 10⁻³ rad/min
1 full revolution per sidereal day
Hard Drive at 7,200 RPM
7,200 RPM = 753.98 rad/s
= 120 RPS
= 43,200 °/s
= 7,539.8 rad/min
Standard desktop HDD speed
= 43,200 °/s
= 7,539.8 rad/min
Standard desktop HDD speed
50 Hz AC Motor
314.16 rad/s = 3,000 RPM
ω = 2π × 50 = 314.16 rad/s
= 50 RPS
= 18,000 °/s
Synchronous speed at 50 Hz, 2-pole
= 50 RPS
= 18,000 °/s
Synchronous speed at 50 Hz, 2-pole
Who Uses Angular Velocity Conversion?
Mechanical Engineers
Convert between RPM (motor specs) and rad/s (torque/power calculations). Power = torque × angular velocity in rad/s — so conversions are essential for drivetrain design.
Electrical Engineers
AC motors and generators use angular frequency ω = 2πf rad/s. A 50 Hz motor has ω = 314.16 rad/s. Convert between electrical frequency (Hz) and mechanical speed (RPM).
Aerospace Engineers
Calculate gyroscope precession, reaction wheel angular momentum for satellite attitude control, and turbine blade tip velocities in jet engines.
Astronomers
Express planetary rotation rates, pulsar spin frequencies and orbital angular velocities. Pulsars can spin at up to 716 Hz = 4,500 rad/s, among the fastest rotating objects known.
Automotive Engineers
Convert engine RPM to wheel angular velocity for drivetrain ratio design. Calculate tyre angular velocity from road speed: v = ω × r, where r is tyre radius in metres.
Physics Students
Convert between rpm and rad/s for rotational kinetics problems. Angular momentum L = Iω, kinetic energy KE = ½Iω² — ω must be in rad/s for these equations to work in SI units.
Mental Math Tips
RPM to rad/s: divide by ~9.55 (or × 0.1047)
1 RPM = π/30 ≈ 0.10472 rad/s. Quick mental shortcut: RPM ÷ 10 gives a good approximation. Example: 3,000 RPM ÷ 10 ≈ 300 rad/s (actual: 314.16). For precision: 3,000 × 0.1047 = 314.1 rad/s.
1 RPS = 2π rad/s ≈ 6.28 rad/s = 60 RPM
One revolution per second is the key bridge unit. 360°/s = 1 RPS = 2π rad/s = 60 RPM. So 7,200 RPM ÷ 60 = 120 RPS = 120 × 2π = 753.98 rad/s. Memorise this chain and you can convert any angular velocity mentally.
ω (rad/s) × radius (m) = linear speed (m/s)
This is why rad/s is the natural unit. A wheel at 100 rad/s with radius 0.3 m has rim speed = 100 × 0.3 = 30 m/s ≈ 108 km/h. No conversion needed when ω is in rad/s and r in metres — making rad/s the preferred engineering unit.
Frequently Asked Questions — Angular Velocity Converter
Angular velocity measures how fast an object rotates — the rate of change of its angular position. The SI unit is rad/s. One full rotation = 2π radians ≈ 6.283 rad. It differs from linear velocity: a point on the edge of a spinning wheel moves faster than one near the centre, even at the same angular velocity.
RPM (revolutions per minute) is the most common practical unit for rotational speed. Convert RPM to rad/s: multiply by 2π/60 ≈ 0.10472. Example: 3,000 RPM × 0.10472 = 314.16 rad/s. Convert rad/s to RPM: multiply by 60/(2π) ≈ 9.5493. Example: 100 rad/s × 9.5493 = 954.93 RPM.
Earth rotates once per sidereal day (86,164 s). Angular velocity = 2π/86164 ≈ 7.292 × 10⁻⁵ rad/s ≈ 0.00417 deg/s ≈ 0.000698 RPM. Earth also orbits the Sun at about 1.99 × 10⁻⁷ rad/s (one revolution per year).
HDD: 5,400–7,200 RPM · Car engine idle: 700–900 RPM · Car redline: 6,000–8,000 RPM · Washing machine spin: 800–1,600 RPM · Ceiling fan: 200–350 RPM · Helicopter rotor: 200–500 RPM · Jet turbine: 10,000–25,000 RPM · Dental drill: up to 400,000 RPM.
A radian is the SI unit of angle — the angle subtended at the centre of a circle by an arc equal to the radius. There are 2π ≈ 6.2832 radians in a full circle (360°). So 1 rad = 180°/π ≈ 57.296°. Rad/s simplifies physics equations: angular velocity × radius = linear velocity, with no conversion factor needed.
Multiply by π/180 ≈ 0.017453. Example: 360°/s × 0.017453 = 6.2832 rad/s (one full revolution per second). To convert rad/s to deg/s: multiply by 180/π ≈ 57.296. Example: 1 rad/s × 57.296 = 57.296°/s.
Angular velocity (ω) and angular frequency (ω) use the same symbol and unit (rad/s) and are numerically equal for circular motion. Angular velocity is a vector (has direction); angular frequency is a scalar. For a 50 Hz AC supply: ω = 2π × 50 = 314.16 rad/s.