Angular Velocity CalculatorRPM to rad/s · ω = Δθ ÷ Δt
Angular velocity (ω) is how fast something rotates: the angle turned per unit of time, in radians per second. One rpm equals 2π ÷ 60 ≈ 0.1047 rad/s. Convert rpm to rad/s, or find ω from an angle and a time, from a rotation period, or from linear speed and radius (ω = v ÷ r). The spinning disc on this page turns at your real rate.
Formula verified against OpenStax — University Physics, Volume 1 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Rotation data
Choose what you know. 1 revolution = 2π rad = 360°.
Step-by-step working
What is angular velocity?
Angular velocity is the rate at which an object rotates or revolves: the angle it sweeps out per second, measured in radians per second.
It’s the rotational counterpart of velocity. A car wheel, a hard drive platter and Earth all have an angular velocity, and every point on a rigid spinning object shares it, while points farther from the axis cover more distance per turn.
Quick answers
How do I convert rpm to rad/s?
Multiply rpm by 2π and divide by 60, or simply multiply by 0.10472. A motor at 1,500 rpm turns at 157.1 rad/s. To go back, multiply rad/s by 9.5493.
How do you find angular velocity from linear velocity?
Divide the tangential speed by the radius: ω = v ÷ r. A point on a 0.25 m pulley rim moving at 5 m/s means ω = 20 rad/s (191 rpm).
What is the angular velocity of Earth?
About 7.292 × 10⁻⁵ rad/s, based on one rotation per sidereal day of 23 h 56 min 4 s. At the equator that carries you east at roughly 465 m/s.
Is angular velocity the same as angular frequency?
For steady rotation they share a value and unit (rad/s): ω = 2πf. Angular frequency is also used for oscillations like a pendulum, where nothing literally spins.
Angular velocity formulas
Angles must be in radians for ω = v ÷ r and v = ωr to work.
| Symbol | Meaning | SI unit |
|---|---|---|
| ω | Angular velocity | rad/s |
| Δθ | Angle turned | rad |
| T | Period of one revolution | s |
| f | Rotational frequency | Hz (rev/s) |
| v | Tangential speed at radius r | m/s |
| r | Radius | m |
How to calculate angular velocity
- Convert the angle to radians: multiply degrees by π ÷ 180.
- Divide by the time taken to turn through it.
- From rpm, multiply by 2π and divide by 60.
- From a rim speed, divide the speed (m/s) by the radius (m).
Worked example: a bicycle wheel at 25 km/h
A road bike rolls at 25 km/h on wheels with a 0.34 m radius. How fast do the wheels spin?
ω = v ÷ r = 6.944 ÷ 0.34 = 20.42 rad/s
rpm = 20.42 × 60 ÷ 2π = 195 rpm
Earth is far slower: one turn per sidereal day (86,164 s) is 7.29 × 10⁻⁵ rad/s. That tiny number still produces the Coriolis effect that steers hurricanes.
RPM to rad/s conversion table
Handy values for motors, engines and spinning equipment.
| RPM | rad/s | deg/s | rev/s (Hz) |
|---|---|---|---|
| 1 | 0.1047 | 6 | 0.0167 |
| 33⅓ (vinyl) | 3.491 | 200 | 0.556 |
| 60 | 6.283 | 360 | 1 |
| 1,000 | 104.72 | 6,000 | 16.67 |
| 3,000 | 314.16 | 18,000 | 50 |
| 7,200 (HDD) | 753.98 | 43,200 | 120 |
Common mistakes to avoid
- Leaving the angle in degrees. ω = v ÷ r only gives rad/s with radians.
- Confusing rpm with rad/s. 3,000 rpm is 314 rad/s, not 3,000.
- Mixing up angular and linear speed. Every point on a wheel has the same ω, but the rim moves faster than the hub.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Angular Velocity Calculator: frequently asked questions
ω = Δθ ÷ Δt, the angle turned divided by time. Equivalent forms are ω = 2π ÷ T, ω = 2πf and ω = v ÷ r.
Radians per second (rad/s) in SI. Revolutions per minute (rpm) and degrees per second are also common.
1 rpm = 2π ÷ 60 ≈ 0.10472 rad/s.
Angular speed is the magnitude only; angular velocity is a vector along the rotation axis, with its direction given by the right-hand rule.
v = ωr. Points farther from the axis move faster in a straight-line sense, even though they share the same ω.
Yes. By convention, counter-clockwise is positive and clockwise is negative when viewed from a chosen side.
References & sources
- OpenStax — University Physics, Volume 1 — Ch. 10.1, Rotational Variables
- HyperPhysics, Georgia State University — angular velocity and rotational kinematics
- NIST — CODATA 2022 Fundamental Physical Constants — SI derived units
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.