Angular Velocity Calculator
Solve for the missing variable — ω, Δθ, Δt, f, T, v, or r — with the three core relations ω = Δθ/Δt, ω = 2πf, and v = ωr. Results in rad/s, rpm, deg/s, rev/s, and Hz.
ω = Δθ / Δt — solver
ω here is the average angular velocity over Δt. A negative Δθ gives a negative ω — the sign is the rotation direction.
ω = 2πf — solver
Frequency f is a magnitude (≥ 0). Period T = 1/f is always derived alongside ω.
v = ωr — tangential speed solver
v = ωr holds for rigid circular motion (constant r) — a point on a spinning wheel, pulley, or shaft.
Angular velocity unit converter
Exact factors: 1 rev = 2π rad, 1 min = 60 s — so 1 rpm = 2π/60 rad/s ≈ 0.10471975512 rad/s.
Verified examples
Δθ = 10 rev × 2π = 62.831853 rad; ω = Δθ/Δt = 62.831853/5 = 12.566371 rad/s = 120 rpm.
ω = 2πf = 2π × 60 = 376.99112 rad/s; T = 1/60 = 0.016666667 s = 16.666667 ms.
v = 27.777778 m/s; ω = v/r = 27.777778/0.35 = 79.365079 rad/s = 757.88068 rpm.
3000 × 2π/60 = 314.15927 rad/s.
Methodology
All computation runs in your browser in IEEE-754 double precision (~15–16 significant digits). Every solver function was checked against an independent Python oracle (50-digit Decimal) on 22 frozen test vectors — including the 60 Hz → 376.99111843 rad/s anchor, signed (negative) rotation, the r = 0 edge case, and round-trip unit conversions — with zero mismatches before release.
Unit factors are exact definitions: 1 revolution = 2π radians, 1 minute = 60 seconds, 1 inch = 0.0254 m. Displayed values are rounded to 8 significant digits; the underlying math keeps full precision.
Scope boundary — which tool do you need?
- This page is the physics solver: given any two of {Δθ, Δt} find ω, given f find ω and T, given any two of {v, ω, r} find the third.
- hz-to-rads (Hz to rad/s converter) is the pure unit-conversion specialist — Hz ↔ rad/s ↔ RPM ↔ rev/s ↔ deg/s, motor synchronous speed, and one-direction tip speed. If you only need "60 Hz in rad/s", go there.
- frequency-converter (Frequency Converter hub) covers SI prefix scaling (Hz ↔ kHz ↔ MHz ↔ GHz ↔ THz), wavelength, and Nyquist analysis; its ω↔f tab is a static converter, not a solver.
- angular-acceleration-calculator (Angular Acceleration) computes α (rad/s²) — how fast ω itself changes. A different quantity.
- Angular displacement (θ, the position angle) is a companion tool; here Δθ appears only as an input to ω = Δθ/Δt.
Honest limitations
- Average, not instantaneous. ω = Δθ/Δt is the average over the interval. Instantaneous angular velocity is dθ/dt (calculus); if the spin speeds up or slows down, the average differs from any single instant.
- v = ωr needs rigid circular motion. Constant radius r — a point on a wheel, disc, or shaft. It does not apply to elliptical orbits, spirals, or flexible bodies where r changes.
- Frequency is a magnitude. f ≥ 0 and T = 1/f; direction lives in the sign of ω (or Δθ), not in f.
- Results are for calculation, learning, and design estimates — verify independently before safety-critical engineering.
Frequently asked questions
What is the angular velocity formula?
How do you calculate angular velocity from RPM?
What is the relationship between angular velocity and frequency?
How do you find tangential velocity from angular velocity?
What are the units of angular velocity?
Is angular velocity the same as angular frequency?
Can angular velocity be negative?
Average vs instantaneous angular velocity — what is the difference?
When does v = ωr NOT apply?
I only need Hz to rad/s — which tool should I use?
Related tools
Free tool by NRTechStudio