This angular velocity calculator converts rotation between RPM, rad/s and revolutions per second — and finds the linear speed of a point at any radius: v = ωr.
| Angular velocity ω | – |
| Revolutions | – |
| Period T (per revolution) | – |
| Linear speed at radius | – |
| Centripetal acceleration ω²r | – |
ω Connects the Spinning World to the Straight-Line One
Every point on a rigid rotator shares ω but owns its linear speed v = ωr — the rim outruns the hub. What that speed demands in force is our Centripetal Force Calculator. Part of our Physics Calculators collection.
Worked Example (Verified)
- 300 RPM (defaults): ω = 300 × 2π/60 = 31.42 rad/s; at r = 0.5 m the rim moves 15.71 m/s (56.5 km/h) with centripetal acceleration ≈ 493 m/s² ≈ 50 g
- Vinyl check: 33⅓ RPM = 3.49 rad/s; the groove at r = 0.146 m travels 0.51 m/s under the needle — why inner grooves (same ω, smaller r, slower v) squeeze high frequencies
- Earth: ω = 2π/86,164 s (sidereal) = 7.292×10⁻⁵ rad/s — tiny ω, but r = 6,371 km makes the equator’s v = 465 m/s
- Period is the reciprocal view: T = 2π/ω = 1/f; 300 RPM means T = 0.2 s per turn
Units Cheat Sheet
- 1 rev = 2π rad exactly — all conversions here are that identity wearing units.
- RPM ÷ 60 = rev/s (Hz); × 2π = rad/s. Degrees/s ÷ 57.30 = rad/s.
- Motor nameplates speak RPM; physics speaks rad/s; datasheets make you bilingual or sorry.
- Angular velocity is a vector (right-hand rule, along the axle) — sign conventions matter in coupled systems.
Related Tools
- Centripetal Force Calculator — F = mv²/r
- Simple Harmonic Motion Calculator — oscillation’s ω
- Physics Calculators — the full collection
Frequently Asked Questions
How do you convert RPM to rad/s?
Multiply by 2π and divide by 60: 300 RPM × 2π/60 = 31.42 rad/s. Reverse: rad/s × 60/(2π) = RPM. This calculator also reports rev/s, degrees/s, period and the linear speed at your radius.
What is the difference between angular and linear velocity?
Angular velocity ω (rad/s) describes the rotation rate shared by the whole rigid body; linear velocity v = ωr describes a specific point and grows with its radius. Two points on one wheel have identical ω and different v — the source of rim-vs-hub speed differences everywhere from vinyl to turbines.
How do you find centripetal acceleration from ω?
a = ω²r (equivalently v²/r). At 300 RPM and 0.5 m: 31.42² × 0.5 ≈ 493 m/s² ≈ 50 g — the reason high-RPM machinery lives or dies by balance and containment.
Is frequency the same as angular velocity?
No — frequency f counts cycles per second (Hz); angular velocity ω = 2πf counts radians per second. They differ by exactly 2π, and confusing them is the classic factor-of-six error in rotation and oscillation problems.
Why is Earth’s angular velocity based on 86,164 seconds?
Earth rotates once relative to the stars in a sidereal day (23 h 56 m 4 s = 86,164 s), while the solar day (86,400 s) includes ~1° of orbital travel. Physics ω uses the sidereal figure: 7.292×10⁻⁵ rad/s.
Is this angular velocity calculator free?
Yes — free, instant, no sign-up.