Article — Angular Velocity Calculator
Angular Velocity Calculator: rad/s, RPM, Hz, and v = ωr
Angular velocity ω is the rotational analogue of linear velocity. SI unit is rad/s. The two conversion identities you need: ω = 2π RPM / 60 = 2π f = v / r.
Angular velocity shows up wherever something rotates — wheels, motors, gears, planets, hard disks, washing machines. The SI unit (radians per second) is the right choice for physics formulas, but engineers and the public almost always quote rotation in RPM. The calculator above handles both directions and adds frequency, period, and linear velocity for good measure.
What angular velocity measures
Angular velocity ω (lowercase Greek omega) is the rate at which an object's angular position changes with time. One full rotation is 2π radians (about 6.283), so an object completing one rotation per second has ω = 2π rad/s = 6.283 rad/s. The SI unit is rad/s; the engineering unit is RPM (revolutions per minute).
Angular velocity is a vector — strictly, a pseudovector — pointing along the axis of rotation with magnitude ω. By the right-hand rule, counterclockwise rotation viewed from one end has ω pointing toward you, while clockwise has it pointing away. For most everyday work the scalar magnitude is sufficient.
Angular velocity formulas
Four equivalent expressions tie together the standard rotational quantities. From RPM: ω = 2π × RPM / 60. From frequency: ω = 2π f. From period: ω = 2π / T. From linear velocity at radius r: ω = v / r. All four describe the same motion in different units.
ω = 2π RPM / 60 from RPMω = 2π f from frequency (Hz)ω = 2π / T from period (s)ω = v / r from linear speedv = ω r tangential velocitya_c = ω² r centripetal accelerationConverting RPM to rad/s
Engineers report rotation in RPM; physicists need rad/s. The conversion is simple but worth memorising. 1 RPM equals 2π/60 ≈ 0.1047 rad/s. So 1000 RPM = 104.7 rad/s. Going the other way, 1 rad/s ≈ 9.55 RPM. A common round number worth remembering: 60 RPM = 2π rad/s = 1 Hz exactly.
The original 78 RPM phonograph records were chosen because 78 RPM at the outer edge of a 10-inch (25 cm) record produces about 104 cm/s outer-track velocity, comfortable for early steel needles. The later 33⅓ RPM long-play format dropped the speed by a factor of 2.34, letting the same disc hold over twice the music.
Angular velocity and linear velocity
For any point on a rotating body at distance r from the axis, the linear (tangential) velocity is v = ω r. Points farther from the axis move faster in linear units even though they share the same ω. The rim of a 20-cm bike wheel at 100 rad/s moves at 0.20 × 100 = 20 m/s — about 72 km/h.
This relationship explains why every point on a rigid rotating body keeps the same angular velocity even though their linear velocities differ. It is also the basis for the speedometer formula: the rotation rate of the drive wheels, multiplied by their effective rolling radius, gives the car's speed.
Angular velocity and centripetal acceleration
A point moving in a circle is constantly accelerating toward the centre, even at constant speed, because its velocity direction changes. The centripetal acceleration is a_c = ω² r = v² / r. For a 0.3 m radius wheel at 100 rad/s, a_c = 100² × 0.3 = 3000 m/s², or about 306 g.
This is why centrifuges hit such large accelerations at modest RPM. A 10 cm radius rotor at 10 000 RPM (ω = 1047 rad/s) generates a_c = 1047² × 0.1 = 110 000 m/s², over 11 000 g. Ultracentrifuges spin to 100 000 RPM, hitting roughly 1 million g.
- Ceiling fan 60 RPM ≈ 6.28 rad/s
- Vinyl LP 33⅓ RPM = 3.49 rad/s
- Car wheel at 100 km/h (r = 0.3 m) ≈ 884 RPM = 92.6 rad/s
- Industrial 4-pole motor (50 Hz) = 1500 RPM = 157 rad/s
- Hard disk 10 000 RPM = 1047 rad/s
- Lab centrifuge 15 000 RPM = 1571 rad/s
- Turbocharger up to 250 000 RPM = 26 180 rad/s
Earth's angular velocity
Earth rotates once relative to the distant stars in 23 hours 56 minutes 4 seconds — the sidereal day of 86 164 seconds. That gives ω = 2π / 86 164 = 7.2921 × 10⁻⁵ rad/s, about 0.0042°/s. At the equator (r ≈ 6378 km), the tangential speed is ω × r ≈ 465 m/s, or 1670 km/h.
The solar day (24 hours, 86 400 s) is slightly longer because Earth also moves forward in its orbit. The 4-minute difference accumulates to one full extra rotation per year — the reason the sidereal year has 366.25 rotations while the solar year has only 365.25.
Angular velocity in motors and machines
Synchronous electric motors lock to the grid frequency. At 50 Hz line frequency, a 2-pole motor runs at exactly 3000 RPM (50 Hz × 60), a 4-pole motor at 1500 RPM. At 60 Hz (North America), 2-pole = 3600 RPM, 4-pole = 1800 RPM. Variable-frequency drives bypass this lock by synthesising arbitrary frequencies.
Common angular velocity mistakes
Always convert RPM to rad/s before computing centripetal acceleration or tangential velocity. The factor of 2π/60 is easy to forget, and a missing factor of 6.28 leads to results 40 times off in centripetal acceleration (which goes as ω squared).
If you use ω = 2π RPM/60 and get a number, that number is in radians per second. Multiplying by r gives metres per second only when ω is in rad/s, not deg/s. The factor between is 180/π ≈ 57.3 — large enough to invalidate any unit-mixed result.
Other recurring slips: confusing angular velocity ω with angular frequency (they are the same thing despite the different names), assuming all points on a rotating body have the same linear velocity (they share ω but not v), and forgetting that ω is signed — clockwise and counterclockwise rotation have opposite-direction angular velocity vectors. For most everyday calculations, magnitudes are enough, but check directions when adding rotations or computing Coriolis effects.
Induction motors run slightly slower than the synchronous speed because they need slip to develop torque — typically 2 to 5 percent slip at full load. A nameplate 1800 RPM motor actually runs at 1750 to 1780 RPM under load. This matters when matching a motor to a fan or pump: at the slip-reduced speed, fan airflow and pump head drop slightly from the catalogue figures.
Computer hard drives once relied on tight angular velocity control to read data reliably. A 7200 RPM drive sustains its speed within 0.1 percent or the read head loses synchronisation with the magnetic patterns on the platter. Solid-state drives have eliminated the mechanical constraint, but optical drives (CD, DVD, Blu-ray) still spin to several thousand RPM under software control. The fast outer-track reads happen at variable angular velocity, holding the linear track speed under the laser constant rather than ω.
One final correction many students miss: when an object slows down or speeds up its rotation, the angular acceleration α (units rad/s²) is what describes the change. The kinematic equations of rotation mirror those of linear motion: ω = ω₀ + αt, θ = ω₀t + ½αt², and ω² = ω₀² + 2αθ. Use these to track spin-up and spin-down problems just as you would track linear acceleration.